SPRINGER BRIEFS IN ECONOMICS
Gourishankar S. Hiremath
Indian Stock
Market
An Empirical Analysis
of Informational
Efficiency
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Gourishankar S. Hiremath
Indian Stock Market
An Empirical Analysis of Informational
Efficiency
123
Gourishankar S. Hiremath
Assistant Professor of Economics and Finance
Department of Humanities and Social Sciences
IIT Kharagpur
Kharagpur, West Bengal
India
ISSN 2191-5504
ISBN 978-81-322-1589-9
DOI 10.1007/978-81-322-1590-5
ISSN 2191-5512 (electronic)
ISBN 978-81-322-1590-5 (eBook)
Springer New Delhi Heidelberg New York Dordrecht London
Library of Congress Control Number: 2013946889
The Author(s) 2014
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To
My mother, Shakuntala
and
My father, Sharanayya Swami
Who exemplified through their life that love
of humanity is above an ideological talk
Foreword
There is a paucity of rigorous academic work on securities markets in India. This is
surprising, especially given the long tradition of statistical research pioneered by
Professors P.C. Mahalanobis, C.R. Rao, and their colleagues at the Indian Statistical Institute in Calcutta, and the availability of detailed data on Indian stock
and commodities markets, at least since the early 1990s. Indeed, despite the large
numbers of Indian Institutes of Technology and Indian Institutes of Management
that have been established over the years, India continues to be under-represented
in research published in leading international academic journals, particularly in
finance and economics.
In this context, it is refreshing to read this monograph by Professor Gourishankar S. Hiremath, which covers a broad array of topics on the major Indian
stock market indices, using daily data from the period following liberalization,
mostly since 1997, soon after the National Stock Exchange was established. The
topics covered range from assessing the efficiency of the Indian stock markets to
studying the hypotheses of mean-reversion and structural breaks in the data, to
analyzing the characteristics of the volatility process underlying the returns from
these indices.
Professor Hiremath’s analysis is thorough and comprehensive, and tries to
connect the conclusions of his research with the major economic events that
occurred during the past two decades in India, including major structural and
regulatory changes in the economy at large, and the stock market in particular. In
particular, he places his statistical results firmly in the context of the most
important macro-economic and market developments of the period. He investigates structural breaks in the data and relates them to global and domestic shocks
to conclude that there are indications that the efficient market hypothesis may not
strictly hold in the Indian markets, which appear to be strongly mean-reverting.
This monograph provides much food for thought for future research on Indian
capital markets with questions relating to asset pricing, market microstructure, and
the effect of capital flows on stock returns, and other, more tangential issues
relating to the disclosure of private information. I am sure that it will provide
vii
viii
Foreword
useful background information for academics, institutions, and regulators that have
an interest in Indian security markets. Any student of the Indian markets will find
the book to be rewarding reading.
October 2, 2013
Marti G. Subrahmanyam
Leonard N Stern School of Business
New York University
New York
Charles E. Merill
Professor of Economics and Finance
Preface
Efficient market hypothesis is the well-known yet highly controversial theory of
the Neoclassical School of Finance. In an informationally efficient market, price
fully and instantly reflects available information in such a way that there are no
opportunities for the agents to predict prices and make excess profits. An inefficient market distorts efficient allocation of capital in the economy. This book
presents an empirical analysis of the informational efficiency of the Indian Stock
Market.
India began the process of economic reforms in 1991 in the wake of the balance
of payments crisis. The reforms were intended to achieve higher growth, efficiency, and macro economic stability. A number of financial sector reforms were
initiated and microstructure and trading practices in the Indian Stock Market have
undergone drastic changes. The policy reforms in the financial sector have given
rise to a need for re-looking the behavior of stock returns in India. The past two
decades also witnessed the burst of the tech boom bubble, volatile exchange
markets, sub-prime crisis, and global financial crisis. The present work is motivated from these changes and situates the objectives of the study in these contexts.
This volume examines random walk hypothesis and focuses on issues like nonlinear dynamics, structural breaks, and long memory properties of stock returns,
which are of special interest in recent times.
This book caters to the needs of a wider audience. Apart from serving the needs
of students of Economics and Finance, the empirical work will be of special
interest to people in academia and in decision-making organizations. Instructors in
universities, who teach topics like market efficiency, will find the present volume
useful in relating the theory to the empirical evidence. The book also provides
good coverage on latest sophisticated time series techniques which are useful to
analyze time series data. The general reader, who is interested in knowing the
Indian Stock Market, will also find this book informative.
This volume would not have been possible without the help of several people.
The first encouragement I received was from Late Prof. Basavaraj Nimbur. I will
never forget the advice of Late Prof. Mallappa Amravati and his confidence in me.
I greatly miss these Teachers, who left the classroom before the lecture was over.
I particularly wish to thank Prof. B. Satyanarayana and late M. Upender, Osmania
University, Hyderabad for their continuous support, and more importantly for
introducing me to Prof. Kamaiah.
ix
x
Preface
Words fail me while expressing my sincere gratitude to my affectionate Teacher, Prof. Bandi Kamaiah, University of Hyderabad, India for introducing me to
the area of Finance and Time Series Analysis, supervising my dissertation, and for
his personal care. I am indebted to Prof. Marti Subrahmanyam, Charles E. Merrill
Professor of Finance at the Leonord N Stern School of Business at New York
University, for writing the foreword to this volume and I express my profound
gratitude to him for the encouragement.
Over the years, I had opportunities to draw freely from the expertise of Drs.
Allen Roy, Debasish Acharya, Amaresh Samantarya, Jitendra Mahakud, Phanidra
Goyari, and late Amanulla. I would like to thank the Editors of International
Journal of Economics and Finance, Artha Vijnana, Journal of Quantitative Economics, Banking and Finance Letters, Economics, Management and Financial
Markets and, Journal of Business and Economics Studies. I thank NSE India,
Wiley and Princeton University Press for the permissions.
The discussions with Ikshwaku, Ramesh, Santosh Dhani, Niranajan, and
Krishna ‘spoiled me’ and I thank them for the same. I got many insights by
probing questions of my students at ICFAI Business School, Hyderabad, Gokhale
Institute of Politics and Economics, Pune and Indian Institute of Technology,
Jodhpur.
I would like to acknowledge Indian Institute of Technology, Kharagpur for
providing the necessary resources and infrastructure to carry out this work. I thank
my Head and colleagues in the Department of Humanities and Social Sciences, IIT
Kharagpur for their encouragement and wishes. I especially appreciate the help of
Rajesh Acharya, Ansu Louis and Vidya Sarvesaran, highly valued friends and
colleagues. It is indeed K. J. George, my colleague and philosopher-friend, who
advised me to write this book. I thank him for his suggestion and support during
the project.
I owe my thanks to Springer, particularly to Sagarika Ghosh, Publishing Editor,
Noopur Singh for their guidance and keen interest from the first stage of proposal
to the last. It was pleasure to have the cheerful company of my friend, Shree
Prakash Tiwari. I thank him for his support particularly at difficult and frustrating
times.
Back at home, I appreciate the help by Malappa Dandagunda, Ambaraya
Hagargi, Mallinath Rasure and Santosh Madki to my parents in my absence. I take
this opportunity to acknowledge the continued source of inspiration and support of
my father, Sharayanayya Swamy and mother Shakuntala. I thank my sisters,
Channamma and Nagaveni, whose very existence is a gift of life, for showering
their unconditional love and affection on me and thanks to Vinayak Hiremath,
Yuvaraj, and Shreya for cheering me always. To end with, I thank Jyoti Kumari for
her patience.
Gourishankar S. Hiremath
Contents
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1
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7
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2
Random Walk Characteristics of Stock Returns
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
2.2 Review of Previous Work. . . . . . . . . . . . . .
2.3 Weak Form Efficiency: Empirical Tests . . . .
2.3.1 Parametric Tests . . . . . . . . . . . . . .
2.3.2 Non-Parametric Tests . . . . . . . . . . .
2.4 Discussion on Empirical Results . . . . . . . . .
2.5 Concluding Remarks . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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19
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37
37
3
Nonlinear Dependence in
3.1 Introduction . . . . . .
3.2 Methodology. . . . . .
3.3 Empirical Results . .
3.3.1 1997–1998 .
3.3.2 1998–1999 .
3.3.3 1999–2000 .
3.3.4 2000–2001 .
3.3.5 2001–2002 .
3.3.6 2002–2003 .
3.3.7 2003–2004 .
3.3.8 2004–2005 .
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41
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54
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Background . . . . . . . . . . . . . . . . . . . . .
1.2 Theoretical Foundations . . . . . . . . . . . . .
1.2.1 Forms of Efficiency . . . . . . . . . .
1.3 Random Walk Model . . . . . . . . . . . . . . .
1.4 Policy Reforms, Growth, and Emergence
of Stock Market in India . . . . . . . . . . . .
1.5 Issues and Scope of the Study. . . . . . . . .
1.6 Nature and Sources of Data . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . .
Stock Returns .
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xi
xii
Contents
3.3.9 2005–2006 .
3.3.10 2006–2007 .
3.3.11 2007–2008 .
3.4 Concluding Remarks
References . . . . . . . . . . .
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54
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Mean-Reverting Tendency in Stock Returns . . . . . . . . . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Review of Previous Works . . . . . . . . . . . . . . . . . . . . . .
4.3 Data and Methodology . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Zivot and Andrews (1992) Sequential Break Test
4.3.2 Lee-Strazicich (2003) LM Unit Root Multiple
Breaks Test. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Empirical Findings . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Variance Ratios, Structural Breaks and Nonrandom
Walk Behavior in the Indian Stock Returns . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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59
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5
Long Memory in Stock Returns: Theory and Evidence . . .
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Theory of Long Memory . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Meaning and Definitions . . . . . . . . . . . . . . . .
5.2.2 ARFIMA Model . . . . . . . . . . . . . . . . . . . . . .
5.3 Review of Previous Work. . . . . . . . . . . . . . . . . . . . . .
5.4 Testing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Geweke and Porter-Hudak Semiparametric Test
5.4.2 Robinson’s Gaussian Semiparametric Test . . . .
5.4.3 Andrews and Guggenberger Bias-Reduced Test.
5.5 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6
Long Memory in Stock Market Volatility .
6.1 Introduction . . . . . . . . . . . . . . . . . . .
6.2 Review of Previous Work. . . . . . . . . .
6.3 Data and Methodology . . . . . . . . . . . .
6.4 Empirical Results . . . . . . . . . . . . . . .
6.5 Concluding Remarks . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . .
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Contents
xiii
Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111
About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
Index Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
121
Abbreviations
ACF
AGBR
AR
ARCH
ARFIMA
ARIMA
ARMA
AMH
BDS
BSE
CRR
ECBs
EMH
FDI
FEMA
FIGARCH
FIIs
GARCH
GPH
IGARCH
IRDA
LMVR
MENA
MLE
NSE
NYSE
QMLE
R/S
RBI
RGSE
RWH
RMW
SEBI
Autocorrelation function
Andrews and Guggenberger bias reduced test
Autoregressive
Autoregressive conditional heteroscedasticity
Autoregressive fractionally integrated moving average
Autoregressive integrated moving average
Autoregressive moving average
Adaptive Market Hypothesis
Broack, Dechert, Sheinkman, LeBaron (1996)
Bombay Stock Exchange
Cash reserve ratio
External commercial borrowings
Efficient market hypothesis
Foreign direct investment
Foreign Exchange Management Act
Fractionally integrated generalized autoregressive conditional
heteroskedasticity
Foreign institutional investors
Generalized autoregressive conditional heteroskedasticity
Geweke Porter-Hudak semiparametric test
Integrated generalized autoregressive conditional heteroskedasticity
Insurance Regulatory and Development Authority
Lo and MacKinlay variance ratio test
Middle East and North Africa
Maximum likelihood estimation
National Stock Exchange
New York Stock Exchange
Quasi maximum likelihood estimator
Rescaled range statistics
Reserve Bank of India
Robinson’s Gaussian semiparametric estimation
Random walk hypothesis
Random walk model
Securities and Exchange Board of India
xv
xvi
SSS
UTI
WRSVR
Abbreviations
Small-shuffle surrogate
Unit Trust of India
Wright’s ranks and sighs variance ratio test
Figures
Fig. 1.1
Fig. 1.2
Fig. 1.3
Fig.
Fig.
Fig.
Fig.
2.1
4.1
6.1
6.2
Daily price movement of major indices. . . . . . . . . . . . . . .
Market capitalization as a per cent of GDP: select markets.
Data Source: World Bank (2012) . . . . . . . . . . . . . . . . . . .
Turnover ratio: select markets. Data Source:
World Bank (2012). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Autocorrelation function of index returns . . . . . . . . . . . . .
Plot of index stock returns with structural break. . . . . . . . .
Daily closing index values. . . . . . . . . . . . . . . . . . . . . . . .
Daily log index returns . . . . . . . . . . . . . . . . . . . . . . . . . .
...
11
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12
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104
106
xvii
Tables
Table 1.1
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
1.2
2.1
2.2
2.3
2.4
2.5
2.6
3.1
3.2
3.3
3.4
4.1
4.2
4.3
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
Table
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
5.1
5.2
5.3
6.1
Table 6.2
Market capitalizations of listed companies—a global
comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Data sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Descriptive statistics . . . . . . . . . . . . . . . . . . . . . . . .
Autocorrelations of index returns . . . . . . . . . . . . . . .
Variance ratio tests statistic for index returns. . . . . . .
Multiple variance ratio test statistics for index returns
Runs test statistics for index returns . . . . . . . . . . . . .
BDS test statistics for index returns . . . . . . . . . . . . .
McLeod-Li, Tsay, and bi spectrum test statistics . . . .
BDS test statistics . . . . . . . . . . . . . . . . . . . . . . . . .
Hinich bicorrelation (H) statistics for full sample . . . .
Windowed test results of Hinich H statistic . . . . . . . .
Unit root test results . . . . . . . . . . . . . . . . . . . . . . . .
Zivot-Andrew sequential trend break test statistics . . .
Lee-Strazicich LM unit root two structural breaks
test statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
WRSVR test results—NSE: full sample . . . . . . . . . .
WRSVR test results—BSE: full sample . . . . . . . . . .
WRSVR test results—NSE: period-I. . . . . . . . . . . . .
WRSVR test results—BSE: period-I . . . . . . . . . . . . .
WRSVR test results—NSE: period-II . . . . . . . . . . . .
WRSVR test results—BSE: period-II . . . . . . . . . . . .
WRSVR test results—NSE: period-III . . . . . . . . . . .
WRSVR test results—BSE: period-III. . . . . . . . . . . .
GPH estimates of ‘d’ . . . . . . . . . . . . . . . . . . . . . . .
RGSE estimates of ‘d’ . . . . . . . . . . . . . . . . . . . . . .
AGBR estimates of ‘d’ . . . . . . . . . . . . . . . . . . . . . .
Estimates of GARCH model for NSE
and BSE index returns . . . . . . . . . . . . . . . . . . . . . .
FIGARCH estimates for NSE and BSE index returns .
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10
15
29
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33
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45
46
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66
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95
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96
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107
108
xix
Chapter 1
Introduction
Abstract Theory of efficient market is one of most debated yet controversial
theory of Neoclassical School of Finance. The efficient market hypothesis states
that in an efficient market, current prices instantly and correctly reflects all the
available and relevant information and such market does not provide consistent
abnormal returns. Despite a voluminous research, there is no consensus among
economists whether financial markets are efficient. This chapter briefly provides
theoretical foundations and empirical perspectives of theory of efficient market.
Further, the financial sector reforms and changes in market microstructure and
trading practices in India, emergence of Indian stock market in recent past,
motivation, and need to relook the issue of efficient market in changed environment are explained. Lastly, present chapter presents the issues such as episodic
nonlinear dependence, structural breaks and long memory in stock returns
addressed in the present volume, nature of data, and scope of the study.
Keywords Efficient market hypothesis
Random walk Identical and independent distribution Emerging market Market microstructure Financial sector
reforms Nonlinear dependence Long memory Mean reversion NSE BSE
1.1 Background
India is one of the major emerging economies of the world that has witnessed
tremendous economic growth over the last 15 years. The reforms in the financial
sector were introduced to infuse energy and vibrancy to the process of economic
growth. In addition, the drastic changes in the market microstructure since the
mid-1990s sought a transparent, fair, and efficient market. As a result, India’s
financial system grew by leaps and bounds. In other words, the Indian stock market
has witnessed tremendous growth after financial liberalization in terms of size,
liquidity, volume, and total turnover. As per the S & P Fact book (2012), Indian
G. S. Hiremath, Indian Stock Market, SpringerBriefs in Economics,
DOI: 10.1007/978-81-322-1590-5_1, The Author(s) 2014
1
2
1 Introduction
stock market now has the largest number of listed companies on its exchanges. The
growing percentage of market capitalization to the GDP and the increasing integration of the Indian market with the global economy indicate the phenomenal
growth of the Indian equity market and its growing importance in the economy.
The capital market of India emerged as one of the important destinations for
investment, and the Indian stock market has received ample attention from the
media and academia. Notwithstanding the recent notable growth, investors, traders
and policy-makers have their own misgivings regarding the efficiency of the Indian
stock market.
Among several channels promoting sustained economic growth, financial
market development in general, and stock market in particular is the most relevant
channel. An efficient equity market plays a vital role in the economy. In its
absence, the allocation of capital will not meet the demands of the economy and,
consequently, economic growth will be retarded. Hence, market efficiency has
been a focal point of research in finance literature. Efficient Market Theory is the
most important theory of neoclassical school of finance. Unlike Pareto efficiency,
a stock market is said to be efficient if it is informationally efficient. In an informationally efficient market, current prices instantly and correctly reflect all the
available and relevant information (Fama 1970). Such markets do not provide
consistent abnormal returns. The early empirical evidence on EMH demonstrated
that stock returns follow a random walk process. The implication of the random
walk process is that it is not possible to predict future returns based on the past
information of stock returns. In short, it is not possible to ‘beat the market’.
1.2 Theoretical Foundations
Theory of efficient market originated during the beginning of twentieth century.
The seminal work of Bachelier (1900) laid theoretical foundation for the theory of
market efficiency. Bachelier (1900) in his investigation of French Government
Bonds concludes that prices fluctuate randomly as they are independent and
identically distributed (i i d). Bachelier (1900) observes that ‘past, present, and
even discounted future events are reflected in market prices, but often show no
apparent relation’. In other words, past movement of prices would not guide future
movement of prices.1 Further, Kendall (1953) finds no predictable components in
stock prices and therefore stock prices appeared to evolve randomly.
The pioneering work of Samuelson (1965) added rigor to the theory of stock
market efficiency. He argues that ‘‘in competitive market, there is a buyer for every
seller. If one is sure that a price would rise, it would have risen’’ and hence
changes in prices follow a random walk. Utilizing a framework of general
1
The work of Bachelier (1900) did not come to light for a long time. Its English translation
appeared in Cootner (1964). Osborne (1959) reports similar results.
1.2 Theoretical Foundations
3
stochastic model of price, Samuelson (1965) deduces his theorem in which future
changes in prices are uncorrelated with past changes in prices. In other words, as
the current prices properly anticipate information, prices fluctuate randomly in
response to new information.
In a survey of efficient capital markets, Fama (1970) explicitly formalized the
efficient market hypothesis (EMH). Fama (1970) states that a market in which,
prices always ‘‘fully reflect’’ available information is called ‘‘efficient’’. In such a
market, when new information (news) arrives, security prices quickly and correctly respond to that information and incorporate all information at any point of
time and reach a new equilibrium. The theory of efficient market argument is
grounded in rational expectation theory. It assumes that investors arrive at rational
expectation forecast about future security returns. According to Fama (1970),
expected returns represent the conditions of market equilibrium and such
‘expected returns equilibrium is function of its risk’. Following Fama (1970), let
returns of an asset given as
Rt ¼ wt1 ftm þ pt
ð1:1Þ
m
where Rt is stock returns, wt1 ft represents equilibrium return expected at t-1
period, pt is abnormal or excess component. Market uses information to arrive at
equilibrium return. Let St represent information set. Then
ð1:2Þ
wt1 ftm ¼ w ftm jSt1
where St is the information set available to the market at time t-1. Equation (1.2)
implies that stock market would be efficient when it uses all the relevant information correctly and quickly in determining market price (Fama 1970). In such a
case, it is not possible to use information set St1 or any other information set to
make excess profits. The observed return is then equal to random market return. In
other words, there will be no excess returns over and above the random market
return. In symbols
wðpt jSt1 Þ ¼ 0
ð1:3Þ
In an informationally efficient market, with a given set of information, current
equilibrium returns reflect all available information, and the expected returns can
reach new equilibrium returns only due to the arrival of new information, which
comes randomly. Hence, future returns are not predictable based on the past
history of stock returns and such market mechanism rules out profit in excess of
expected profits. Ross (2005) explains that the idea behind EMH is that in a
competitive market, security prices are resultants of decisions made by individual
agents and prices, therefore, depend on information underlying those decisions. An
investor, whose information is inferior or already possessed by the market, cannot
outguess the market.
4
1 Introduction
1.2.1 Forms of Efficiency
Different forms of efficiency stems from the interpretation of ‘fully’ and ‘available
information’ found in the definition of market efficiency. Roberts (1959) notes
three forms of efficiency.
1.2.1.1 Weak Form of Efficiency
Weak form efficiency is one where the information set includes only past
sequences of returns. That is, St-1 contains only past history of returns, then
Eq. (1.3) is written as
ð1:4Þ
w pt . . .; Rt2; Rt1; ¼ 0
When information instantaneously absorbs in current returns, then such
mechanism will not ensure consistent abnormal returns.
1.2.1.2 Semi-Strong Form Efficiency
When information set includes all publicly available information like information
on macroeconomic variables, company’s performance, etc., including past
sequences of returns, it is termed as semi-strong efficiency,2 Let k be publicly
available information set. Rewriting Eq. (1.3), definition of semi-strong efficiency
can be represented as
ð1:5Þ
w pt . . .; Rt2; Rt1; ktn ¼ 0
1.2.1.3 Strong Form Efficiency
The information set includes private or monopolistic information. If private
information denotes f, then Eq. (1.3) becomes
w pt . . .; Rt2; Rt1; ktn ; 1 ¼ 0
ð1:6Þ
2
According to Professor Bandi Kamaiah, theoretically there is no difference between weak and
semi-strong forms of efficiency. For empirical testing, it is convenient to test weak and semistrong separately. However, the conventional classification is followed here.
1.2 Theoretical Foundations
5
The above equation asserts that even with monopolistic access to certain
information, it is not possible to outguess the market.3 The efficient market
hypothesis has been based on the following assumptions/conditions:
•
•
•
•
•
No transaction costs in trading4
Information is freely available to market participants
All participants are rational profit seeking maximizing investors
New information arrives into the market randomly
All participants are aware of implications of current information.
Thus, in a market comprizing of rational profit seeking investors, prices completely incorporate information and perfect arbitrage is possible. In such efficient
market, collection of information is costly and there will be no returns on such
actions. Even if informed traders observe bullish market based on their information, they bid up the prices and bearish, their trading put pressure on price
downward and these trading strategies get reflected in prices and thus information
hidden by informed traders become public. Hence, in an informationally efficient
market, it would not be possible to earn excess returns. Under such conditions, a
simple buying and holding diversified security strategy cannot be outperformed by
fundamental or technical analysts.5
A counter theoretical argument to EMH was provided by many scholars.
Grossman (1975, 1977) shows that information collection activity is costly due to
the presence of noise in the prices. The prices aggregate diverse information perfectly in an efficient market and this mechanism eliminates private incentive to
collect information. As a result, no equilibrium exists when no one collects
information or when one makes positive return on collected information. In other
words, an ‘‘over information’’ market, where there is no noise and information
collection is costly, eventually breaks down. Alternatively, price aggregation is not
perfect when there is noise and results into break down of allocation efficiency of
competitive market (Grossman 1976). According to Grossman and Stiglitz (1980),
costless information is a necessary condition and not a sufficient condition of
efficient market. The competitive market where information is costless is necessarily a thinly traded market. Extending the noise rational expectation model of
Lucas (1972), Grossman and Stiglitz (1980) argue that informed traders could earn
return on their efforts in gathering information because that information enables
informed traders to take better positions than others. In an informationally efficient
market, information is costly and there would be no incentive or reward to collect
any sort of information. Therefore, in a competitive market, informed traders could
3
No-trade theorem argument is that even one knows what other does not know, also then it is not
possible to make profit from such information. A brilliant description of No trade argument and
efficient market can be found in Ross (2005).
4
In empirical testing of EMH, transaction costs are ignored. Now, with screen-based trading,
transaction costs are considerably minimal.
5
Malkiel (1973) put it that ‘a blindfolded chimpanzee throwing darts at the Wall Street could
select a portfolio that would do as well as the experts’.
6
1 Introduction
stop the endeavor of collecting information which is a costly affair. It eventually
leads to break down of market (Grossman and Stiglitz 1980).
The efficient market hypothesis demands homogeneity in beliefs of traders.
However, in reality, market consists of people of different beliefs, and not
homogenous participants. Without heterogeneous investors, trading is not possible
and consequently market does not work. A market works only when people with
different beliefs trade. According to Black (1986), financial market is characterized
by noise. Investors trade on noise, thinking that they are trading on information. It
is noise which makes trading possible in financial markets, but it also makes these
markets inefficient. Furthermore, price would not reveal all information in the
presence of noise and difference in beliefs would not be arbitraged completely
(Grossman 1977).
There are other important schools of thought which describe the behavior of
stock returns. The Fundamental School believes that certain fundamental factors
determine behavior of stock returns. The Fundamentalists seek to analyze stock
prices on the basis of earnings and dividend prospects of the firm, corporate
governance, macroeconomic variables, and other key-decision variables. They also
analyze the quality of firm’s management, status of industry, business cycles,
financial statements of firms, etc. The efficient market theory rules out the usefulness of such fundamental analysis since such past information is already known
to the rational agents and it is correctly reflected in the stock prices.
Technical school or analysis, one of the popular methods among traders, asserts
that information possessed in stock prices is of great use to predict future returns.
Technical analysts, popularly known as chartists, use a variety of approaches such
as Dow Theory, filter rules, trading range breaks, wave principles, moving averages, relative strength, candle sticks, distribution line, direction index, stochastic
oscillator, etc. The basic contention of technical analysis is that forces of demand
and supply reflect in the pattern of trade volume and prices and these patterns get
repeated. By a careful analysis of past sequences of prices, future prices could be
predicted, thus making it possible to ‘beat the market’. According to advocates of
EMH, the technical analysis based on the past history of prices is a futile exercise.
An emerging school known as Behavioral School criticizes the EMH on the
ground that neo classical finance ignores the behavioral aspects of decision making of
investors. According to the advocates of behavioral school, market not only consists
of rational agents but also irrational agents. Due to fear, greed, over confidence,
heuristic attitude, there are behavioral biases in information processing by agents and
this limits the arbitrage and hence prices are not perfect. The mere absence of excess
profit opportunities does not necessarily imply that markets are efficient.
A large volume of empirical research examined various aspects of stock market
efficiency.6 The next section discusses the random walk model, which is used to
test the EMH.
6
A collection of the seminal works on theory of efficient market and its anomalies, and random
walk hypothesis can be found in Lo (1997).
1.3 Random Walk Model
7
1.3 Random Walk Model
Random walk model (RWM) or random walk hypothesis (RWH) has been one of
the important and effective models employed to examine the EMH in empirical
research. There are various definitions of random walk, but the main contention of
random walk is that asset prices move in a random manner.
Let us consider the following equation:
Rt ¼ d þ Rt1 þ et
ð1:7Þ
where Rt is stock returns at time t, Rt-1 is stock returns at time t-1, d is the drift
parameter (or expected returns), et is error term. The stochastic variable of stock
returns Rt is said to be random walk, with a drift parameter d, if
et ð0; r2 Þ
ð1:8Þ
where white noise term, et is independent and identically distributed with mean
zero and constant variance r2 . Thus, the value of Rt at time t is equal to its value at
time t-1 plus a random shock. An important feature of RWM is the persistence of
random shocks. A particular shock does not die away. If the process {et} in
addition to conditions mentioned in Eq. (1.7) is normally distributed, then it is
equivalent to arithmetic Brownian motion (Cambell et al. 1997). The independence of increments fet g implies that the process is strictly white noise process. It
is a stricter definition of RWH. Cambell et al. (1997), in addition to it, define less
restrictive definitions of RWM, which are as following.
From time to time, changes in technology, institutions, regulation, and market
microstructure have been in order. Hence, it is difficult to find identical distribution
of increments. Independent increments version of RWM is one which requires
increments to be independent but not identically distributed. It allows for
unconditional heteroscedasticity in e0t . By relaxing independence assumption, the
uncorrelated increments version of RWM refers to a process with dependent but
uncorrelated increments.
1.4 Policy Reforms, Growth, and Emergence of Stock
Market in India
In the early 1980s, protagonists of globalization advocated free market economy
for a sustained economic growth and also made a sustained attack on state
intervention, and public sector dominance.7 The global institutions like World
7
For e.g., see Krueger (1974), Joshi and Little (1994), Bhagwati (1982), Bhagwati and Desai
(1970), Bhagvati and Srinivasan (1993), Lal (1993), Shroff (1993), Srinivasan (2001).
8
1 Introduction
Bank prescribed the financial sector reforms for under-developed economies.8
India started liberalizing the economy from 1991 and has been transforming from a
closed economy to an open economy and emerged as one of the leading economies. The pre-reform period (1947–1991) was essentially characterized by the
dominance of public sector, industrial licensing, excessive restrictions on capacity
creation, high tax rates, restriction on foreign trade and finance, and administered
prices. The financial sector in particular was led by public sector banks. The
administered interest rates, capital controls, and direct credit program were features of controlled financial sector regime. The Industries (Development and
Regulation) Act 1951, Monopolies Restrictive Trade Practices Act 1969, and
Foreign Exchange Regulation Act (FERA) 1973 provided the legal framework for
the controls. India experienced low productivity in manufacturing, a slow economic growth rate, stagnant employment rate, high inflation, mounting fiscal
deficit, and growing debt. According to Srinivasan (2001), ‘‘the grossly regulated
system was responsible for chaotic, incentive structure, and political corruption.
Indeed, it became a cancer in the body politic’’.
The financial sector identified with poorly developed money and capital markets, weak banking sector, inadequate prudential regulations, and lack of financial
innovation. The administered interest rates resulted in cross subsidization and
regulation of deposit rates which severally affected profitability of banks. The
Capital Issues (Control) Act of 1947 imposed severe restrictions which discouraged the firms to go public for resource mobilization. The imposition of dividend
restriction ordinance led to the significant erosion of market capitalization. The
mid 1980s reforms under Rajiv Gandhi’s Government revived the capital market
but gains remained limited. The cumulative effect of these was the balance of
payments crisis in 1991 which called for overarching economic reforms.
The new industrial policy of 1991 was the first step in the direction of economic
reforms aimed at the liberalization and privatization of the Indian economy.9 The
abolition of industrial policy, dismantling of control on private sector, fiscal
reforms, opening of economy for foreign trade and investment were the major
policy changes initiated by the Government. A series of financial sector reforms
were introduced since 1991. The importance of efficient and stable financial system was increasingly felt. The financial sector reforms aimed at an efficient,
vibrant, and stable financial system. The first generation reforms reduced the
statutory liquidity ratio (SLR) and cash reserve ratio (CRR), and banks were given
operational flexibility. The capital market witnessed sea saw changes in the form
of repealing of the Capital Issues (Control) Act 1947. The pricing of financial
assets was set free for market, new stock exchanges were established, private
8
World Bank (1989) reports recommended structural reforms in financial sector and prescribed
opening of sector for global capital to attain desired macro-economic stability and efficient
resource use.
9
The reforms were initiated in 1991 under the compulsion of crisis rather than the realization of
their significance. Hence these reforms are sometime termed as crisis driven reforms. For e.g., see
Basu (1993).
1.4 Policy Reforms, Growth, and Emergence of Stock Market in India
9
mutual funds were permitted, and Securities Exchange Board of India (SEBI) was
set up in April 1992 as regulator of the Indian capital market. As a part of financial
sector reforms, National Stock Exchange (NSE) was established in April 1993.
The products, trading, clearing, settlement, and regulations are the major
constituents of the microstructure of capital markets. Several changes in market
microstructure and trading practices had been taking place in Indian equity market
to bring transparency. The NSE became a market leader and forerunner of many
changes in market microstructure and trading practices and set the international
standards, which were later followed by many other exchanges. The 19 stock
exchanges in India are now corporatized and demutualized. Fully automated
screen-based trading system is in place. As per the NSE (2012) review, Indian
equity market now has nationwide network of trading, and over 4,827 corporate
brokers and about 10,165 trading members are registered with the SEBI. An
important landmark is the establishment of the NSE that started its operation in
November, 1994. The NSE is the first stock exchange in the world to use satellite
communication technology for trading. During the same period, web-based
Internet trading was allowed both at the NSE and Bombay Stock Exchange (BSE).
In order to ensure counterparty guarantee, Clearing Corporation of India Limited
(CCIL) was set up which acts as a counter party to both buyers and sellers through
novation. The NSE and BSE introduced several new financial products. Derivative
instruments such as Index Options, Index Futures, Single Stock Futures, and
individual stock options were introduced between 2000 and 2001 in order to
improve risk management and efficiency. The NSE and BSE launched many new
indices of different cap and floated sectoral indices.
SEBI, the regulatory authority also undertook several regulatory and procedural
changes to improve efficiency and protect the interest of investors.10 The regulatory authority permitted the foreign institutional investors (FIIs) into Indian capital
market and approved short selling for all kinds of investors including the FIIs.
Further, the SEBI allowed direct market access (DMA)11 facility for institutional
investors. These market microstructure changes and regulatory measures had the
objective to improve efficiency and transparency in security market of India.
How liberalization can affect financial sector did not leave the researchers and
policy makers free from botheration. The monetary and fiscal policy instruments
need to be modified to tackle the adverse consequences of liberalized economy
which is exposed to world economy. A robust policy framework is required to save
the economy from the possible ill-effects of deregulated markets. In late 1980s and
early 1990s, doubts were raised that the policy makers have not given sufficient
thought to the potential ill-effects of opening the economy for global capital (e.g.,
see Patnaik 1986, 1994a; Sau 1995). After a careful analysis and comparing the
10
For detailed information on various regulatory measures initiated by the SEBI, see annual
reports of the SEBI.
11
DMA allows brokers to offer clients direct access to the exchange trading system through the
broker’s infrastructure without manual intervention by the broker. This facility is available from
April, 2008. See, NSE (2008, 2009).
10
1 Introduction
Table 1.1 Market capitalizations of listed companies—a global comparison
Markets
Market capitalization Market capitalization
ratio (in per cent)
US $ (in million)
Developed market
USA
UK
France
Germany
Emerging markets
China
India
Brazil
Russia
Indonesia
World total
India as percent of World
33,169,049
15,640,707
1,202,031
1,568,730
1,184,459
11,913,772
3,389,098
1,015,370
1,228,969
796,376
390,107
45,082,821
2.3
–
103.62
49.43
56.57
33.17
68.42
63.81
47.13
46.37
21.04
Number of
listed companies
27,497
4,171
2,001
893
670
22,056
2,342
5,112
366
327
440
49,553
10.32
Source NSE (2012)
India’s reforms with the experience of other economies that embarked on reform
process, Nayyar (1993) concludes that economic policy reforms need careful
‘consideration and substantial correctives’. The policy of allowing FIIs was criticized on the ground that the entry leads to destructive impact on capital market
activities in India because FIIs soon act like owners than passive investors
(e.g., Ghosh 1993). The reforms were also criticized as an ideology of imperialism
(Diwan 1995; Clairmont 2002).12 Patnaik (1994b) terms the perspective of
financial sector reforms as ‘‘transcendental marketism’’. The voices of protests
were largely ignored by the policy makers and ruling class.
In post liberalization era, Indian equity market has made substantial growth on
many fronts.13 India has the largest number of listed companies (5,112) on its
exchanges and it has secured 11th and 17th position in terms of market capitalization of listed companies and total value traded on exchange, respectively
(Table 1.1). Despite tremendous growth, India’s share in world market capitalization stood 2.3 % in 2011 while China constituted 7.5 %. As per the World
Development Indicators 2012, the share of emerging markets in general declined
in the last 3 years. However, there was a marginal decline in turnover ratio during
the same period (Table 1.1). The benchmark indices, Nifty and Sensex have shown
increasing trends since 1994 (see Fig. 1.1). Before the sub-prime crisis in 2007,
both indices reached their highest points. The period 2007–2009, is period of high
volatility.
12
Patnaik (1994b) presents a thoughtful insights and an insightful critique of India’s economic
reforms.
13
NSE (2012) in its review of capital market provided a detailed discussion on recent
developments and achievements of Indian equity market.
1.4 Policy Reforms, Growth, and Emergence of Stock Market in India
11
Fig. 1.1 Daily price movement of major indices
Figure 1.2 presents the stock market capitalization as percentage of GDP for
select markets. The stock market capitalization as percentage of GDP in India
increased from 17 % in 1991 to 89 % in 2006 and stood 93.6 % in 2010. The
increased percentage of market capitalization to GDP indicates the growing
importance of equity market in the Indian economy. Since 2007, there has been
decline in the stock market capitalization of both developed and emerging markets,
nevertheless, the US and the UK markets continued to top the list. The stock
market capitalization to India’s GDP declined to 54 % in 2011, but it is still higher
than other emerging markets like China, Brazil, Russia, and Indonesia and
developed markets like France and Germany. Figure 1.3 shows significant increase
in turnover ratio of India from 53 % in 1991 to 192 % 2001, next to the turnover
ratio of the US, indicates that India is becoming a liquid market. However, after
the dot com bubble bust, the liquidity indicated by turnover ratio is draining and it
stood 56 % while other emerging markets like China and Brazil, are experiencing
relatively high liquidity. According to NSE (2012) estimates, trading volume was
$203 billion in 2002–2003 and it increased continuously since then. The rate of
1 Introduction
160
90
140
80
120
70
100
60
80
50
60
40
40
30
20
20
0
Percent
Percent
12
10
1991
1992
1993
1994
1995
Period
US
FRANCE
CHINA
RUSSIA
UK
GERMANY
BRAZIL
INDONESIA
INDIA
Fig. 1.2 Market capitalization as a per cent of GDP: select markets. Data Source: World Bank
(2012)
350
200
300
175
150
200
125
150
100
100
Turnover Ratio
Turnover Ratio
250
75
50
0
50
1991
1992
1993
1994
1995
Period
US
FRANCE
CHINA
RUSSIA
UK
GERMANY
BRAZIL
INDONESIA
INDIA
Fig. 1.3 Turnover ratio: select markets. Data Source: World Bank (2012)
increase was 76.8 % in 2007–2008, but it plunged continuously and stood $34,843
billion in the post global economic crisis. Nevertheless, the phenomenal growth
achieved in the past concentrated only in the NSE and BSE. These two exchanges
account for 99.98 % of total turnover as on March 2012 (NSE 2012). Even the
BSE, the oldest stock exchange in Asia accounts 19.2 % of turnover ($130,482).
It is the NSE which is the market leader accounting 80.7 % of total turnover
($549,469). In the past decade, with its commencement of business in 1994, the
NSE emerged as largest stock exchange in India.
In the backdrop of recent policy reforms in the financial sector and market
microstructure, and a phenomenal growth of stock market, need is felt for relooking the behavior of stock returns and for examining the informational efficiency of the Indian stock market.
1.5 Issues and Scope of the Study
13
1.5 Issues and Scope of the Study
The behavior of stock returns has been extensively debated over the years. The
empirical studies have examined the EMH and random walk characterization of
returns and also alternatives to random walk. The presence of linear dependence in
stock returns provides opportunities for potential excess profit in the market to the
agents. In recent years, studies show that nonlinear dependence in stock returns
indicates the possibility of predictability and thus violates the EMH. Further, there
exists a tendency for the stock returns to return to its trend path which is termed as
mean-reversion. It is one of the competing alternatives to the random walk
character of stock returns.
Another aspect of stock market returns which departs from random walk
hypothesis is long memory or long-range dependence. Long memory or long-range
dependence is a process in which its autocovariances are not absolutely summable
and underlying time series realizations are temporally dependent at distant lags.
The autocorrelation function of such stationary series decays hyperbolically. The
persistent temporal dependence between distant observations indicates possibilities of predictability and hence provides opportunity to speculators to forecast
future returns based on past information and make abnormal profits. Presence of
long memory has important theoretical and practical implications.
The stock market volatility indicates the future growth prospect of stock market
and influences the economic growth and stability of the economy. The conventional models of volatility view the variance of the disturbance terms as constant
over time. Later, the modern financial analysis observed that volatility cannot be
constant as it evolves over time and shocks persist for a longer time and thus
exhibits periods of unusually high or low volatility periods. However, there is a
possibility that the mean-reverting hyperbolic rate of decay in the variance may be
slow, thus indicating a long memory in volatility. The conventional models of
volatility could not capture such persistence in volatility. In the presence of long
memory in volatility, those models which use short memory such as derivative
pricing, value at risk models would not be reliable.
Against this background and in the context of drastic changes due to policy
reforms in the economy in general, and market microstructure changes in equity
market in India in particular, the present study seeks to examine the issues of
informational efficiency of Indian stock market. Specifically, the present study
formulates following objectives. First, empirically investigate Indian stock returns
behavior by testing validity of random walk hypothesis. Second, understand
nonlinear dependent structure in underlying stock returns and explain how such a
phenomenon contradicts EMH. Third, check whether stock returns exhibit a meanreverting tendency and also address the issue of accounting for structural breaks.
Fourth, detect long memory or long-range dependence in mean returns and
volatility
The past three decades and a half have produced a large volume of research on
stock market efficiency mainly focused on developed markets. The quest for
14
1 Introduction
analyzing stock market efficiency in India began with the early work of Rao and
Mukherjee (1971). The studies by Amanulla (1997), Amanulla and Kamaiah
(1998), and Poshakwale (2002), are important additions to the recent literature. A
quick review of the previous work reveals that consensus on this issue has been
elusive. Nevertheless, the on-going scientific debate, as Lo and MacKinlay (2001)
observe, has provided new insights into the economic structure of financial
markets.
The main purpose of the present study is to examine the returns behavior in
the Indian equity market in the changed market environment. Departing from the
previous studies on Indian stock market efficiency, the present study has made the
following improvements. First, the available studies refer to the 1980s and early
1990s and hence could not capture the changes in nature of stock market efficiency
in the post financial sector reforms and drastic transformation in market microstructure of Indian stock market. This study covers the period (1997–2010) of such
structural changes is in order. Second, earlier studies in India focused on the BSE
and (mostly confined to the BSE Sensex index) with the belief that every other
stock exchange and investors in India follow the BSE. However, in the last few
years, the NSE has emerged as the largest stock exchange in India besides being
the current leader of the market. In this light, to obtain a comprehensive picture of
the growth and efficiency of the Indian stock market, this study uses new and
disaggregate data from both the NSE and the BSE. Third, the dataset of different
indices has another advantage as it helps to measure relative efficiency represented
by different indices on the same exchange. It also helps to understand sensitivity of
stock returns to market capitalization and liquidity. Fourth, majority of the studies
in India used conventional tests to examine the issue of market efficiency. The
present study has employed certain state-of-the-art methods and techniques, which
are first of their kind in the Indian context. Finally, the issue of nonlinearity, longrange dependence and long memory in volatility have been addressed in the
present volume.
1.6 Nature and Sources of Data
Data of daily values14 of 8 indices from the NSE and 6 indices from the BSE for
the period June 2, 1997 to March 31, 2010 are considered for the study. Table 1.2
provides the details of sample data. This large and varied data sample is expected
to reflect drastic changes taken place in Indian equity market. The data range is
different for different indices, as shown in Table 1.2. The launching of different
14
Taylor (2005) suggests that time interval between prices should be sufficient enough so that
trade takes place in most intervals. Selecting daily values will be both appropriate and
convenient.
1.6 Nature and Sources of Data
15
Table 1.2 Data sample
Sl. No
Index
Time period
01
02
03
04
05
06
07
08
09
10
11
12
13
14
S & P CNX Nifty
CNX Nifty Junior
S & P CNX Defty
CNX 100
CNX 500
BSE Sensex
BSE 100
BSE 200
BSE 500
BSE Midcap
BSE Smallcap
CNX IT
CNX Bank Nifty
CNX Infrastructure
02/06/1997–31/03/2010
02/06/1997–31/03/2010
02/06/1997–31/03/2010
01/01/2003–31/03/2010
07/06/1999–31/03/2010
01/01/1998–31/03/2010
01/01/1998–31/03/2010
01/01/1998–31/03/2010
03/01/2000–31/03/2010
01/01/2004–31/03/2010
01/01/2004–31/03/2010
02/06/1997–31/03/2010
01/01/2000–31/03/2010
01/01/2004–31/03/2010
Source NSE and BSE
indices at different points of time by the exchanges dictated the different sample
range (Table 1.2).
The selected indices have at least 6 years of daily values, thus providing enough
number of observations to perform advanced time series econometric models and
for accurate estimation.15 Indices namely, S & P CNX Nifty, CNX Nifty Junior, S
& P CNX Defty CNX 100, and CNX 500 are selected from the NSE and BSE
Sensex, BSE 100, BSE 200, BSE 500, BSE Midcap, and BSE Smallcap are from
BSE. Considering the growing importance of information technology, banking and
infrastructure sectors in the economy, respective indices of these sectors from the
NSE namely, CNX IT, CNX Bank Nifty, and CNX Infrastructure are also added.
This comprehensive and updated disaggregated data sample reflects sensitiveness
of results to the composition of indices and relative performance of the indices.
The daily index values are collected from official websites of the NSE and BSE.16
The volume is organized into six chapters. The first chapter recapitulated the
basic tenets and brief history of theory of stock market efficiency. It also includes
problem identification, nature, and sources of sample data, and scope of the study.
The second chapter provides evidence from parametric and nonparametric tests of
random walk hypothesis. In the third chapter, the nonlinear dependence structure
in stock returns is discussed. The fourth chapter treats the issue of mean-reversion
and structural breaks with empirical evidences. The issue of long memory in
15
Taylor (2005) opines that at least 4 years of daily values (more than 1,000) observation are
required to obtain interesting results.
16
Appendix presents a brief description of the selected indices.
16
1 Introduction
returns is examined in the fifth chapter. The sixth and final chapter explains the
long memory in volatility followed by presentation of summary and conclusion.
References
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University of Hyderabad, Hyderabad
Amanulla S, Kamaiah B (1998) Indian stock market: is it informationally efficient? Prajnan
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Bachelier L (1900) Theory of speculation. Faculty of the Academy of Paris, Paris
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Malkiel BG (1973) A random walk down Wall Street. W. W. Norton & Co, New York
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Shorff M (1993) Indian economy at the crossroads. Econ Polit Week 28(19):934–944
Srinivasan TN (2001) Indian economic reforms: background, rationale, achievements and future
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Chapter 2
Random Walk Characteristics of Stock
Returns
Abstract This chapter studies the behavior of stock returns in India. For this
purpose, data from 1997 to 2010 of 14 indices traded on the National stock
exchange (NSE) and Bombay stock exchange (BSE) are used and several parametric and non-parametric methods are employed to empirically test the random
walk characteristics of stock returns and examine the weak form efficiency of the
Indian stock market. The results from parametric tests are mixed and validity of
random walk hypothesis (RWH) is suggested only for large cap and high liquid
indices traded on the BSE. However, the same is not true in the case of NSE index
returns. The non-parametric tests resoundingly reject the null of random walk for
the chosen indices. The results broadly suggest non-random walk behavior of stock
returns and invalidate the weak form efficiency in case of India. The evidence of
dependence in stock returns call for appropriate regulatory and policy changes to
ensure further dissemination of information and quick and correct price aggregation in the market.
Keywords Random walk Market efficiency Weak form of efficiency Stochastic process Abnormal returns Variance ratio Autocorrelation Serial
dependence
2.1 Introduction
The behavior of stock returns has been extensively debated over the years.
Researchers have examined the efficient market hypothesis (EMH) and random
walk characterization of returns and alternatives to random walk. In an informationally efficient market, current prices quickly absorbs information and hence
such a mechanism does not provide scope for an investor to make abnormal returns
(Fama 1970). In respect of empirical evidence, earlier studies have found evidence
in favor of random walk hypothesis (RWH) (Working 1960; Fama 1965;
G. S. Hiremath, Indian Stock Market, SpringerBriefs in Economics,
DOI: 10.1007/978-81-322-1590-5_2, The Author(s) 2014
19
20
2 Random Walk Characteristics of Stock Returns
Niederhoffer and Osborne 1966). Later studies however, documented mean
reversion tendency in stock returns (Jennergren and Korsvold 1974; Solnik 1973;
Keim and Stambaugh 1986; Jagadeesh 1990). Further, anomalies to EMH were
also observed in the empirical research (Fama 1998). Fama’s informationally
efficient market model is criticized for its assumption that market participants
arrive at a rational expectation forecast. It is argued that trade implies heterogeneity (bull and bear traders) and therefore returns can be predicted. Further,
psychological and behavioral elements in stock price determination help predict
future prices. In contrast to Fama’s model, Campbell et al. (1997) states that asset
returns are predictable to some degree. The consensus on this issue, thus, continues
to be elusive. In this context, an attempt is made to empirically check whether
stock returns in India, one of the emerging markets, follow random walk or not.
The specific focus of the present chapter is to test linear dependence or lack of it in
stock returns at the two premier exchanges in India namely, the National stock
exchange (NSE) and Bombay stock exchange (BSE). The remainder of this
chapter is structured as follows. In Sect. 2.2, a brief review of literature is offered.
Section 2.3 describes the time series techniques carried out for the purpose.
Sect 2.4 presents a discussion on empirical evidence and Sect. 2.5 concludes with
a summary of the main findings.
2.2 Review of Previous Work
Literature on random walk characters of stock returns and EMH is truly abundant.
Here an attempt is made to present a selective review of recent work.1 Bachelier
(1900) is perhaps the first who theorized the concept of market efficiency. In his work,
he shows that the successive price changes are independent and identically distributed (i.i.d) because of randomness of information and possible unsystematic patterns
in noise trading. In other words, the mathematical expectation of the speculation is
zero. Osborne (1959) also provides a similar argument. The seminal works of
Samuelson (1965) and Fama (1965, 1970) triggered much interest in this area.
Fama (1965) carries out empirical testing, shows the independence of price
changes and concludes that the chartists exercise has no value. The studies of
Working (1960), Niederhoffer and Osborne (1966) suggest that stock price
movements are not serially correlated and, therefore, it is impossible to make
abnormal profits from investment strategies. The independence of price changes
remained unchallenged however. Jennergren and Korsvold (1974) in their study of
45 stocks on Norwegian and Swedish markets reject RWH and conclude that these
markets may be ‘weakly inefficient’. Solnik (1973) observes more apparent
1
Fama (1970, 1998) present an excellent review of work on theory of efficient market and its
genesis and history. The review of previous work carried out in the present study mainly focused
on evidences from emerging markets.
2.2 Review of Previous Work
21
deviations from random walk in European markets and cites inadequate disclosure
norms, thin trading and insider trading as possible reasons for the inefficiency.
French and Roll (1986) document a statistically significant negative serial correlation in daily returns but they are sceptical about the economic significance of
such returns. In a similar vein, Keim and Stambaugh (1986) find statistically
significant consistent predictability in stock prices by using forecasts of predetermined variables. Jagadeesh (1990) also reports predictability of stock returns.
Frennberg and Hansson (1993) find serial dependence in stock returns of Sweden.
However, Fama and French (1988) who documented negative autocorrelation in
long horizon returns, suggest that such evidence does not necessarily imply
inefficient market but may be the result of time-varying equilibrium expected
returns generated by rational investors’ behavior.
The early studies on market efficiency used serial correlation, runs, and spectral
tests to check whether stock returns are characterized by random walk. The conventional techniques such as serial correlation seem to suffer from restrictive
assumptions. They tend to be less efficient to capture the patterns in the returns.
A new test, which is robust to heteroscedasticity, was proposed by Lo and
MacKinlay (1988). In their study of weekly stock returns in the US, Lo and
MacKinay (1988) reject the RWH for the weekly returns. They conclude that the
mean reverting models of Poterba and Summers (1988), and Fama and French
(1988) cannot give a satisfactory description of behavior of stock returns in the
backdrop of strong evidence of positive correlation in the returns.
The most popular test carried out in the empirical testing of random walk since
the publication of Lo and MacKinlay (1988) is the variance ratio test (henceforth,
LMVR) proposed by them. Emerging and developing markets are expected to
strongly reject random walk process of underlying returns because of underdevelopment of markets, thin trading and several frictions. However, similar to
developed markets, studies from the emerging markets also have thrown inconsistent evidence. Butler and Malaikah (1992) empirically conclude that returns in
Kuwait followed a random walk while rejecting RWH for Saudi Arabia. Abraham
et al. (2002), who applied LMVR on emerging markets, observed dependence in
index returns of Saudi Arabia, Kuwait, and Bahrain. However, the corrected
returns support a weak form of market efficiency. The rejection of random walk in
Middle Eastern markets has been identified to be the result of thin and infrequent
trading (Butler and Malaikah 1992; Abraham et al. 2002).
The non-random walk behavior of stock returns is not just confined to the
emerging Middle Eastern markets. Such behavior has been found in other
emerging markets too. Urrutia (1995) finds positive autocorrelation in monthly
returns of some Latin American countries. The studies by Ojah and Karemera
(1999) and Greib and Reyes (1999) from Latin America empirically report mixed
results. While the former finds evidence in support of random walk for Latin
America, the latter finds significant autocorrelation in the Mexican market and
random walk behavior in the Brazilian market.
The empirical results reported from Asian emerging markets are also mixed.
Huang (1995), Alam et al. (1999), and Chaing et al. (2000) find that emerging
22
2 Random Walk Characteristics of Stock Returns
Asian markets, are not weak form efficient. In support of these findings, Husain
(1997) concludes that RWH is not valid in Pakistan’s equity markets because of
strong dependence of stock returns. Thin trading, as in case of Middle Eastern
markets, is one of the important sources of significant correlation in returns
(Mustafa and Nishat 2007). Empirical findings on China, the leading emerging
market, are quite inconsistent. Liu et al. (1997) upholds weak form efficiency for
Chinese markets. Darant and Zhong (2000) and Lee et al. (2001) report independence of returns series for Chinese markets. Nevertheless, conflicting results in
the same market were observed by Lima and Tabak (2004). While the Chinese-A2
shares and Singapore stock market are weak form efficient, the Chinese-B shares
and Hong Kong market revealed autocorrelation in the returns. The authors note
that market capitalization and liquidity explain such conflicting results in the same
market. The empirical findings of Lock (2007),Charles and Darne (2008) and,
Fifield and Jetty (2008) support the earlier evidence on China that Share-A was
weak form efficient while Share-B evidenced against it.
The LMVR tests individual variance ratios for a specific aggregation investment horizon and thus may result in size distortions. In order to overcome such
deficiency in LMVR, later studies employed multiple variance ratio tests along
with other tests. Ayadi and Pyun (1994) observed linear autocorrelation in Korean
stock returns. Smith (2007) who investigated whether Middle East stock markets
follow a random walk or not found that largely Israeli, Jordanian, Lebanese
markets are weak form efficient while Kuwait and Oman markets reject the RWH.
Smith et al. (2002) reports autocorrelation in return in Botswana, Egypt, Kenya,
Mauritius, Morocco, Nigeria, and Zimbabwe. The study finds empirical evidence
in support of random walk only in South Africa.
The empirical analysis for Australia for a longer period, 1875–2004, carried out
by Worthington and Higgs (2009) rejected RWH and thus revealing strong serial
dependence in the stock returns. Hoque et al. (2007) also observes autocorrelation
in the majority of eight emerging markets researched. Using the multiple variance
tests, an attempt was made by Benjelloun and Squalli (2008) to unmask sectoral
efficiency in markets of Jordan, Qatar, Saudi Arabia, and United Arab Emirates.
The study obtained inconsistent results among different sectors and different
economies. The EMH in the European stock market was investigated by Borges
(2011). The study employed tests namely, autocorrelation, runs, ADF unit root,
and multiple variance ratio to test RWH. The study found that while the markets in
France, Germany, the UK, and Spain followed a random walk, there was positive
serial correlation in returns of Greece and Portugal. Nakamura and Small (2007)
by using small-shuffle surrogate method found random walk characters in the US
and Japanese stock returns.
2
The ownership of Share A, denominated in local currency of China are restricted to domestic
investors, while Share B denominated in US $ are exclusively for foreign investors. However,
Chinese government from 2001 allowed domestic investors to trade Share B.
2.2 Review of Previous Work
23
The early study on Indian stock market efficiency was perhaps carried out by
Rao and Mukherjee (1971). Later, in a comparative study between BSE and
NYSE, Sharma and Kennedy (1977) using runs test and spectral technique found
that monthly returns on BSE were characterized by random walk. Similar evidence
of random walk behavior was noted by Barua (1981), Gupta (1985).3 Furthermore,
Amanulla (1997), Amanulla and Kamaiah (1998), examined the behavior of stock
returns on BSE Sensex, BSE National Index,4 and 53 individual stocks. In addition
to serial correlation and rank correlation tests, these two studies used the ARIMA
(0, 1, 0) model to examine the distribution pattern of increments that received less
focus on stock market efficiency studies in India. They concluded that the equity
market in India was of weak form efficieny. However, Poshakwale (2002) found
evidence against RWH. Thus, as in case of other markets, the results for India too
remain inconclusive.
To sum up, although the literature on random walk and market efficiency is
vast, there is no consensus among the researchers regarding efficiency of the
market. The different tests implemented in the empirical investigation yielded
different results. The empirical results of various studies appear to be sensitive to
the tests employed for the analysis. However, conventional tests provide evidence
in support of the RWH. Thin trading or non-synchronous trading, disclosure
norms, various restrictions, and incomplete reforms are cited as important factors
for the rejection of the random walk characterization of returns particularly in
emerging markets. The review of literature shows mixed empirical evidence
regarding the behavior of stock returns. In this context, the present chapter
investigates the validity of the RWH in the Indian context by using the empirical
tests described in the next section.
2.3 Weak Form Efficiency: Empirical Tests
This section presents description of time series techniques used to test the RWH.
2.3.1 Parametric Tests
2.3.1.1 Autocorrelation Test
Autocorrelation estimates may be used to test the hypothesis that the process
generating the observed return is a series of i.i.d random variables. It helps to
3
Amanulla and Kamaiah (1996) presented an excellent and comprehensive review of early
Indian evidence on market efficiency. Also see, Barua et al. (1994). Repetition is avoided here.
4
This is now known as BSE 100 Index traded on BSE.
24
2 Random Walk Characteristics of Stock Returns
evaluate whether successive values of serial correlation are significantly different
from zero. To test the joint hypothesis that all autocorrelation coefficients qk are
simultaneously equal to zero, Ljung and Box’s (1978) portmanteau Q-statistic is
used in the study. The test statistic is defined as
Xm q
^2k
LB ¼ nðn þ 2Þ
ð2:1Þ
k¼1 n k
where n is number of observation, m lag length. The test follows Chi square (v2 )
distribution.
2.3.1.2 Lo and MacKinlay (1988) Variance Ratio Test
Lo and MacKinaly (1988) proposed the variance ratio test, which is capable of
distinguishing between several interesting alternative stochastic processes5 For
example, if the stock prices are generated by a random walk process, then the
variance of monthly sampled log-price relatives must be four times as large as the
variance of weekly return.
Let a stochastic process represented by
rt ¼ l þ ln Pt Pt1 þ et
ð2:2Þ
where rt is stock returns, l is drift parameter, ln Pt and Pt1 is log price at t time
and Pt1 is price at t – 1. Under random walk, increments of et are i.i.d. and
disturbances are uncorrelated. Under RWH for stock returns rt ,the variance of
rt ? rt1 are required to be twice the variance of rt . Following Campbell et al.
(1997), let the ratio of the variance of two period returns, rt ð2Þ rt rt1 , to
twice the variance of a one-period return rt . Then variance ratio VR(2) is
VRð2Þ ¼
¼
Var ½rt ð2Þ Var ½rt þ rt1
¼
2Var½rt
2Var½rt
2 Var½rt þ 2 Cov½rt ; rt 1
2 var½rt
VRð2Þ ¼ 1 þ qð1Þ
ð2:3Þ
where q(1) is the first order autocorrelation coefficient of returns frt g. RWH which
requires zero autocorrelations holds true when VR(2) = 1. The VR(2) can be
extended to any number of period returns, q. Lo and MacKinaly (1988) showed
that q period variance ratio satisfies the following relation:
5
A detailed discussion on the test and its empirical application can be seen in Campbell et al
.(1997).
2.3 Weak Form Efficiency: Empirical Tests
25
Xq1
Var½rt ðqÞ
k k
¼1þ2
VRðqÞ ¼
1 q
k¼1
q:Var½rt
q
ð2:4Þ
where rt ðkÞ rt þ rt1 þ . . . þ rtkþ1 and q(k) is the kth order autocorrelation
coefficient of frt g: Equation (2.4) shows that at all q, VR(q) = 1. For all definition
of random walk6 (as defined in Chap. 1) to hold, variance ratio is expected to be
equal to unity (Campbell et al. 1997). The test is based on standard asymptotic
approximations. Lo-MacKinlay proposed Z(q) standard normal test statistic7 under
the null hypothesis of homoscedastic increments and VR(q) = 1, test statistic
Z(q) is given by
Z ð qÞ ¼
VRðqÞ 1
UðqÞ1=2
which is asymptotically distributed as N (0,1).
In the Eq. (2.5), asymptotic variance UðqÞ is defined as
2ð2q 1Þðq 1 2
UðqÞ ¼
3q
ð2:5Þ
ð2:6Þ
To ensure rejection of RWH that is not because of heteroscedasticity, a common feature of financial returns, Lo-MacKinlay constructed a heteroscedastic
robust test statistic, Z*(q)
Z ð qÞ ¼
VRðqÞ 1
U ðqÞ1n2
ð2:7Þ
which follows standard normal distribution asymptotically. The asymptotic variance U ðqÞ is
Xq1 2ð2q 1Þ2
dð j Þ
ð2:8Þ
U ðqÞ ¼
j¼1
q
where
dð j Þ ¼
Pnq
2
ðrt b
l Þ2 rtj b
l
hP
i2
nq
l Þ2
t¼1 ðrt b
t¼jþ1
ð2:9Þ
Thus, according to variance ratio test, the returns process is a random walk
when variance ratio at a holding period q is expected to be unity. If it is less than
6
These definitions are independence and identical distributions, independent increments, and
uncorrelated elements. Also see Campbell et al. (1997).
7
A detailed discussion on sampling distribution, size and power of the test can also be found in
Lo and MacKinlay (1999).
26
2 Random Walk Characteristics of Stock Returns
unity, it implies negative autocorrelation and if it is great than one, indicates
positive autocorrelation.
2.3.1.3 Chow and Denning (1993) Multiple Variance Ratio Test
The variance ratios test of Lo and MacKinlay (1988) estimates individual variance
ratios where one variance ratio is considered at a time, for a particular holding
period (q). Empirical works examine the variance ratio statistics for several
q values. The null of random walk is rejected if it is rejected for some q value.
Therefore, it is essentially an individual hypothesis test. The variance ratio of Lo
and MacKinlay (1988) tests whether variance ratio is equal to one for a particular
holding period, whereas the RWH requires that variance ratios for all holding
periods should be equal to one and the test should be conducted jointly over a
number of holding periods. The sequential procedure of this test leads to size
distortions and the test ignores joint nature of random walk. To overcome this
problem, Chow and Denning (1993) proposed multiple variance ratio test wherein
a set of multiple variance ratios over a number of holding periods can be tested to
determine whether the multiple variance ratios (over a number of holing periods)
are jointly equal to one. In Lo-MacKinlay test, under null VRðqÞ ¼ 1, but in
multiple variance ratio test, Mr ¼ ðqi Þ ¼ VRðqÞ 1 ¼ 0. This can be generalized to a set of m variance ratio tests as
fMr ðqi Þji ¼ 1; 2. . .; mg
ð2:10Þ
Under RWH, multiple and alternative hypotheses are as follows
H0i ¼ Mr ¼ 0 for i ¼ 1; 2; . . .; m
ð2:11aÞ
H1i ¼ Mr ðqi Þ 6¼ 0 for any i ¼ 1; 2; . . .; m
ð2:11bÞ
The null of random walk is rejected when any one or more of H0i is rejected.
The homoscedastic test statistic in Chow-Denning is as
pffiffiffiffi
CD1 ¼ T maxj1 i Z ðqi Þj
ð2:12Þ
In Eq. (2.12), Z ðqi Þ is defined as in Eq. (2.5). Chow-Denning test follows
studentized maximum modulus, SMMða; m; TÞ, distribution with m parameters
and T degrees of freedom. Similarly, heteroscedastic robust statistic of
Chow-Denning is given as
pffiffiffiffi
CD2 ¼ T maxj1 i Z ðqi Þj
ð2:13Þ
where Z ðqi Þ is defined as in Eq. (2.7). The RWH is rejected if values of standardized test statistic, CD1 or CD2 is greater than the SMM critical values at
chosen significance level.
2.3 Weak Form Efficiency: Empirical Tests
27
2.3.2 Non-parametric Tests
2.3.2.1 Runs Test
Runs test is one of the important non-parametric tests of RWH. A run is defined as
the sequence of consecutive changes in the return series. If the sequence is positive
(negative), it is called positive (negative) run and if there are no changes in the
series, a run is zero. The expected runs are the change in returns required, if the
data is generated by a random process. If the actual runs are close to expected
number of runs, it indicates that the returns are generated by random process. The
expected number of runs, ER, is computed as
P
X ðX 1Þ 3i¼1 c2i
ER ¼
ð2:14Þ
X
where X is total number of runs, ci is number of returns changes of each category
of sign (i = 1, 2, 3). The ER in Eq. (2.14) has an approximate normal distribution
for large X. Hence, to test null hypothesis, standard Z-statistic can be used.8
2.3.2.2 BDS Test
Brock et al. (1996) developed a portmanteau test for time-based dependence in a
series, which is popularly known as BDS (named after its authors). The test can be
used for testing against a variety of possible deviations from independence
including linear dependence, nonlinear dependence, or chaos. The BDS test uses
correlation dimension of Grassberger and Procaccia (1983). To perform the test9
for a sample of n observations {x1,……..,xn}, an embedding dimension m, and a
distance e, the correlation
integral Cm (n, e) is estimated by
1 if jxs xt j\e;
I (xs, xt, e) =
0 otherwise,
Qm1
Im (xs, xt, e) = k¼0 I ðxsþk ; xtþk ; eÞ;
Cm ðn; eÞ ¼
Xnm Xnmþ1
2
I ðx ; x ; eÞ
t¼Sþ1 m s t
s¼1
ðn mÞðn m þ 1Þ
ð2:15Þ
The function I () indicates whether the observations at times s and t are near
each other or not, as determined by the distance e. The product Im () is only one
when the two m-period histories (xs, xs ? 1, ……, xs ? m - 1) and (xt, xt ? 1,
…….., xt ? m - 1) are near each other in the sense that each term xs + k is near
xt + k. The estimate of the correlation integral is the proportion of pairs of m-period
8
9
For further discussion on runs test, see Siegel (1956).
The BDS test discussion is based on Taylor (2005).
28
2 Random Walk Characteristics of Stock Returns
histories that are near each other. For observations from many processes, limit is
defined as
lim Cm ðn; eÞ
n!1
When the observations are from an i.i.d processes, the probability of m consecutive near pairs of observations is simply the product of m equal probabilities
and hence
Cm ðeÞ ¼ C1 ðeÞm
When the observations are from a chaotic process, the conditional probability of
xs + k being near xt + k, given that xs + j is near xt + j for 0 B j \ k, is higher than
the conditional probability and hence
Cm ðeÞ [ C1 ðeÞm
The BDS considers the random variable Hn(Cm(n, e)-C1(n, e)m which, for an
i.i.d process, converges to a normal distribution as n increases. The test statistic is
given below.
rffiffiffiffiffiffi
n
W m ðeÞ ¼
ð2:16Þ
ðCm ðn; eÞ C1 ðn; eÞm Þ
^
Vm
^m is given by
where the consistent estimator of Vm namely, V
Xm1
^m ¼ 4ðkm þ ðm 1Þ2 C 2m m2 kC 2m2 þ 2
V
kmj C2j
j¼1
ð2:17Þ
with C = C1 (n, e) and
K¼
i
Xnm hXs1
ihXnmþ1
6
I
ð
x
;x
Þ
I
x
x
m
s
t
m
r;
s
t¼Sþ1
r¼1
S¼2
ðn m 1Þðn mÞðn m þ 1Þ
ð2:18Þ
It has power against a variety of possible alternative speciations like nonlinear
dependence and chaos. The BDS statistics is commonly estimated at different m,
and e.
2.4 Discussion on Empirical Results
This section discusses the empirical results of parametric and non-parametric tests
that are carried out in this study. The descriptive statistics for the 14 indices are
given in Table 2.1. The highest average returns are obtained in CNX 100. The
CNX Infrastructure and CNX Bank Nifty are the other indices, which show higher
mean returns. This reflects the performance of these indices owing to the considerable growth of infrastructure and banking sector in India because of the
Min
Max
Standard deviation
Skeweness
Kurtosis
J-B test statistics
P value of JB test
S & P CNX Nifty
CNX Nifty Junior
S & P CNX Defty
CNX 100
CNX 500
BSE Sensex
BSE 100
BSE 200
BSE 500
BSE Midcap
BSE Smallcap
CNX IT
CNX Bank Nifty
CNX Infrastructure
-0.130538
-0.131333
-0.141130
-0.130493
-1.288471
-0.118091
-0.599342
-2.299381
-0.249827
-0.120764
-0.108357
-2.365839
-0.151380
-0.150214
0.079690
0.082922
0.089858
0.080065
0.076944
0.079310
0.552933
2.297634
0.075327
0.104317
0.132050
0.145567
0.114014
0.102127
0.017485
0.020528
0.018532
0.018059
0.017744
0.017810
0.023934
0.063972
0.018659
0.018377
0.019092
0.051938
0.021785
0.021826
-0.512508
-0.668462
-0.472054
-0.835206
-0.761208
-0.399402
-1.459145
-0.068990
-1.690044
-1.266593
-0.874436
-32.15014
-0.423283
-0.758949
4.366738
3.746319
4.548736
5.683283
4.460254
3.339056
241.725
1188.688
17.02682
7.827763
5.399936
1462.399
4.036178
5.930724
2479.67
1950.09
2659.12
2282.31
2272.06
1377.15
6827.37
1650.41
2901.32
3689.15
1755.87
2631.86
1638.38
2042.52
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000352
0.000458
0.000234
0.000667
0.000436
0.000345
0.000400
0.000412
0.000273
0.000144
0.000171
0.000187
0.000614
0.000659
2.4 Discussion on Empirical Results
Table 2.1 Descriptive statistics
Index
Mean
Note Basic statistics for 14 indices are given in the table. The null of skewness and kurtosis = 0, is significantly rejected for all the chosen index
29
30
2 Random Walk Characteristics of Stock Returns
Autocorrelation Function of S & P CNX DEFTY
Autocorrelation Function of S & P NIFTY
1.00
1.00
0.75
0.75
0.50
0.50
0.25
0.25
0.00
0.00
-0.25
-0.25
-0.50
-0.50
-0.75
-1.00
-0.75
Q= 41.81 P-value 0.00024
1
2
3
-1.00
4
5
6
7
8
9
10
11
12
13
14
15
Q= 40.35 P-value 0.00040
1
Autocorrelation Function of CNX NIFTY JUNIOR
1.00
1.00
0.75
0.75
0.50
0.50
0.25
0.00
-0.25
-0.50
-0.50
-0.75
2
3
4
-1.00
5
6
7
8
9
10
11
12
13
14
15
5
6
7
8
9
10
11
12
13
14
15
13
14
15
Q= 38.62 P -value 0.00073
1
Autocorrelation Function of CNX 500
2
3
4
5
6
7
8
9
10
11
12
Autocorrelation Function of BSE100
1.00
1.00
0.75
0.75
0.50
0.50
0.25
0.25
0.00
0.00
-0.25
-0.25
-0.50
-0.50
-0.75
-0.75
Q= 19.15 P -value 0.20697
Q= 91.84 P-value 0.00000
-1.00
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
Autocorrelation Function of BSE SENSEX
2
3
4
5
6
7
8
9
10
11
12
13
14
15
13
14
15
Autocorrelation Function of BSE 200
1.00
1.00
0.75
0.75
0.50
0.50
0.25
0.25
0.00
0.00
-0.25
-0.25
-0.50
-0.50
-0.75
-1.00
4
-0.75
Q= 123.55 P- value 0.00000
1
-1.00
3
0.25
0.00
-0.25
-1.00
2
Autocorrelation Function of CNX100
-0.75
Q= 37.53 P -value 0.00106
1
2
3
4
-1.00
5
6
7
8
9
10
11
12
13
14
15
Q= 572.55 P-value 0.00000
1
2
3
4
5
6
7
8
9
10
11
12
Fig. 2.1 Autocorrelation function of index returns
significant increase in the government outlay along with encouraging participation
of private players. Further, the BSE 200 has the highest standard deviation
(0.0639) which represents higher volatility and lowest is of CNX Nifty (0.0174)
and the CNX 500 (0.0177). The CNX IT registered the higher volatility among the
selected sectoral indices due to fluctuation in the international market. The returns
of the selected index series are negatively skewed implying that the returns are
flatter to the left compared to the normal distribution. The significant kurtosis
indicates that return distribution has sharp peaks compared to a normal distribution. Further, the significant Jarque and Bera (1980) statistic confirmed that index
returns are non-normally distributed. This confirms the stylized facts of stock
returns. This study employs Ljung-Box test to check whether all autocorrelations
are simultaneously equal to zero. The plots of autocorrelation function of indices are given in Fig. 2.1which clearly display that autocorrelations even up to 15
lags are significant.
Ljung-Box test statistics are provided in Table 2.2. It is evident from test statistics that the null hypothesis of no serial correlation cannot be rejected at any
conventional significance level for CNX IT and CNX 500 index returns and thus
indicate random walk behavior. The rest of indices show strong autocorrelation in
2.4 Discussion on Empirical Results
31
Autocorrelation Function of BSE 500
Autocorrelation Function of BSE SMALLCAP
1.00
1.00
0.75
0.75
0.50
0.50
0.25
0.25
0.00
0.00
-0.25
-0.25
-0.50
-0.50
-0.75
-1.00
-0.75
Q= 60.66 P- value 0.00000
1
2
3
4
-1.00
5
6
7
8
9
10
11
12
13
14
15
Q= 133.00 P -value 0.00000
1
2
Autocorrelation Function of BSE MIDCAP
1.00
1.00
0.75
0.75
0.50
0.50
0.25
4
5
6
7
8
9
10
11
12
13
14
15
13
14
15
0.25
0.00
0.00
-0.25
-0.25
-0.50
-0.50
-0.75
-1.00
3
Autocorrelation Function of CNX IT
-0.75
Q= 70.33 P -value 0.00000
1
2
3
4
-1.00
5
6
7
8
9
10
11
12
13
14
15
Q= 17.19 P - value 0.30787
1
2
3
4
5
6
7
8
9
10
11
12
Autocorrelation Function of CNX INFRASTRUCTURE
Autocorrelation Function of CNX BANK NIFTY
1.00
1.00
0.75
0.75
0.50
0.50
0.25
0.25
0.00
0.00
0
- .25
-0.25
0
- .50
-0.50
0
- .75
-0.75
Q= 71.06 P-value 0.00000
Q= 42.34 P - value 0.00020
1
- .00
-1.00
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Fig. 2.1 continued
Table 2.2 Autocorrelations of index returns
Index returns
Lags
LB Q statistic
Q significance
S & P CNX Nifty
CNX Nifty Junior
S & P CNX Defty
CNX 100
CNX 500
BSE Sensex
BSE 100
BSE 200
BSE 500
BSE Midcap
BSE Smallcap
CNX IT
CNX Bank Nifty
CNX Infrastructure
41.81
123.55
40.35
38.62
19.15
37.53
91.84
572.55
60.66
70.33
133.00
17.19
71.06
42.34
0.0002*
0.0000*
0.0004*
0.0007*
0.2070
0.0011*
0.0000*
0.0000*
0.0000*
0.0000*
0.0000*
0.3079
0.0000*
0.0002*
15
15
15
15
15
15
15
15
15
15
15
15
15
15
Note The Ljung-Box (LB) Q statistic is given in the table up to 15th order autocorrelation for all
series. Asterisked value rejects the null hypothesis at 1 % level of significance. The critical values
of the test statistics reject null hypothesis of no serial correlation at all conventional significance
level except for CNX IT Junior and CNX 500
32
2 Random Walk Characteristics of Stock Returns
the returns series as the null is rejected at the 1 % significance level (see
Table 2.2).
Furthermore, Lo and MacKinlay (1988) test is carried out and variance ratios
and corresponding homoscedastic increments and heteroscedasticity robust tests
statistic for each index returns at various investment horizons like 2, 4, 8, and 16
are presented in second and third rows, respectively, in Table 2.3. The test results
presented in table show that with the sole exception of BSE 100, variance ratios for
all other indices at all investment horizons are greater than unity. The significant
homoscedastic and heteroscedastic statistics reject RWH for the index returns
namely, Nifty Junior, BSE 500, BSE Midcap, and BSE Smallcap including CNX
500 (with exception at lag 2 and 4) at all investment horizons or holding periods.
The variance ratios for these indices are greater than unity and thus indicate the
presence of significant positive autocorrelations in the returns. However, the test
statistic for CNX IT and the BSE 200, BSE Sensex supports the presence of
random walk, as value of test statistic is lower than the critical value.
The volatility changes over time and therefore rejection of null of variance ratio
equal to unity due to conditional heteroscedasticity is not of much interest and less
relevant for the practical applications. The homoscedastic statistic given in second
row in Table 2.3 for CNX Nifty, CNX Infrastructure at lag 2, CNX Defty, and
CNX 100 at lag 2 and 4 and for BSE 100 at all the investment horizons rejects
RWH. However, heteroscedastic robust statistic is insignificant for these indices at
all lags (investment horizons). This shows that rejection of random walk for these
indices is because of conditional heteroscedasticity. Otherwise, the results conform
to RWH for these index returns and hence rejection of null of random walk is not
meaningful. In short, the LMVR results suggest autocorrelation only in Nifty
Junior, CNX 500, BSE500, BSE Midcap, BSE Smallcap and sectoral index, CNX
Bank Nifty returns.
The mixed results from the LMVR test reveals the fact that the individual
variance ratio test of LMVR do not give consistent evidence at different holding
periods, since the null of random walk requires variance ratios for all holding
periods to be equal to one. In this context, the Chow and Denning (1993) multiple
variance ratio test assumes relevance. The maximum homoscedastic and heteroscedastic robust test statistics are reported in Table 2.4. The maximum
homoscedastic values of BSE Sensex and CNX 500 are less than critical value
(2.49) and hence cannot reject the null of random walk. The statistics for other
index returns reject the null of random walk at 5 % level of significance. However,
the homoscedastic statistics are less relevant for meaningful inferences because of
rejection may be due to heteroscedasticity. The heteroscedastic robust ChowDenning test statistics significantly reject the null of random walk for the CNX
Nifty Junior, CNX 500, BSE 500, BSE Midcap and BSE Smallcap, and the CNX
Bank Nifty returns suggesting serial dependence (see Table 2.4). It is to be noted
that LMVR (both homoscedastic and heteroscedastic) test also rejects null of
RWH for these indices. On the other hand, return indices such as CNX Nifty, CNX
Defty, CNX100, BSE Sensex, BSE 100, BSE 200 and sectoral index CNX IT, and
CNX Infrastructure validate RWH since Chow-Denning statistic values are less
2.4 Discussion on Empirical Results
33
Table 2.3 Variance ratio tests statistic for index returns
Index returns
S & P CNX Nifty
CNX Nifty Junior
S & P CNX Defty
CNX 100
CNX 500
BSE Sensex
BSE 100
BSE 200
BSE 500
BSE Midcap
BSE Smallcap
CNX IT
CNX Bank Nifty
CNX Infrastructure
Lo-MacKinlay variance ratios for different investment horizons
2
4
8
16
1.062
(3.35)*
(1.93)
1.143
(7.75)*
(4.26)*
1.072
(3.89)*
(2.22)*
1.093
(3.65)*
(1.85)
1.138
(6.81)*
(3.63)*
1.070
(3.66)*
(2.34)*
0.840
(-8.37)*
(-0.75)
1.011
(0.61)
(0.81)
1.123
(5.91)*
(3.39)*
1.220
(7.85)*
(3.43)*
1.279
(9.96)*
(5.28)*
1.008
(0.43)
(0.33)
1.123
(5.90)*
(3.21)*
1.091
(3.26)*
(1.63)
1.053
(1.53)
(0.92)
1.209
(6.04)*
(3.46)*
1.094
(2.72)*
(1.62)
1.096
(2.01)*
(1.06)
1.189
(4.98)*
(2.78)*
1.069
(1.94)
(1.26)
0.769
(-6.46)*
(-0.72)
1.014
(0.40)
(0.54)
1.173
(4.42)*
(2.66)*
1.350
(7.85)*
(3.10)*
1.504
(9.60)*
(5.42)*
1.016
(0.47)
(0.43)
1.146
(3.73)*
(2.16)*
1.078
(1.49)
(0.78)
1.036
(0.65)
(0.41)
1.231
(4.22)*
(2.59)*
1.091
(1.66)
(1.04)
1.054
(0.71)
(0.40)
1.221
(3.68)*
(2.21)*
1.034
(0.61)
(0.40)
0.719
(-4.98)*
(-0.75)
1.023
(0.42)
(0.57)
1.217
(3.50)*
(2.24)*
1.464
(5.59)*
(2.90)*
1.733
(8.84)*
(5.45)*
1.026
(0.48)
(0.44)
1.049
(0.80)
(0.50)
1.031
(0.38)
(0.21)
1.087
(1.07)
(0.72)
1.072
(4.96)*
(3.29)*
1.163
(2.00)*
(1.32)
1.126
(1.12)
(0.68)
1.380
(4.25)*
(2.75)*
1.093
(1.10)
(0.76)
0.770
(-2.74)*
(-0.56)
1.058
(0.69)
(0.95)
1.396
(4.29)*
(2.96)*
1.688
(5.57)*
(3.29)*
2.069
(8.65)*
(5.86)*
1.113
(1.39)
(1.20)
1.047
(0.51)
(0.34)
1.061
(0.49)
(0.30)
Note Table report Lo-MacKinalay test results. The variance ratios VR (q) are reported in the main rows
and variance test statistic Z(q) for homoscedastic increments and, for heteroscedastic—robust test
statistics z*(q) are given in the second and third row parentheses respectively. Under the null of random
walk, the variance ratio value is expected to be equal to one. Asterisked values indicate rejection of the
null of random walk hypothesis at 5 % level significance
34
Table 2.4 Multiple variance
ratio test statistics for index
returns
2 Random Walk Characteristics of Stock Returns
Index returns
Homoscedastic statistic
Heteroscedastic
statistic
S & P CNX Nifty
CNX Nifty Junior
S & P CNX Defty
CNX 100
CNX 500
BSE Sensex
BSE 100
BSE 200
BSE 500
BSE Midcap
BSE Smallcap
CNX IT
CNX Bank Nifty
CNX Infrastructure
3.31803*
7.74220*
3.88990*
3.60146*
1.69682
3.30603
8.32605*
8.84280*
5.98647*
6.73254*
9.93798*
1.38788*
5.80567*
3.22557*
1.93554
4.26921*
2.22679
1.85454
3.62379*
2.34265
0.76055
0.82037
3.37238
3.42666*
5.27285*
0.53818
3.23927*
1.60863
Note The multiple variance ratio homoscedastic (CD1)
eroscedastic (CD2) statistic of Chow-Denning test are
here. The critical value is 2.49. Asterisked values
rejection of null of random walk hypothesis at 5 %
significance
and hetreported
indicate
level of
than the critical values for these index returns. Furthermore, Chow-Denning results
are not significantly different from those of LMVR. However, diverse results and
statistical size distortion problem can be mitigated by Chow-Denning test and
therefore results of this test are preferable. It may be noted that the parametric tests
provided diverse results where five out of eight indices traded on NSE and three
out of six indices traded on BSE validate RWH while rest of the indices reject the
RWH. This indicates intra-market and intra-exchange variations in the behavior of
stock returns.
This study also employed two non-parametric tests namely the runs test and the,
BDS test which are robust to distribution of the returns. The choice of these tests is
appropriate especially in the light of the observation made in the present study that
returns series are non-normally distributed (see Table 2.1). The runs test is a
popular non-parametric test of RWH. Table 2.5 provides runs test results.
Actual runs (see, second column of Table 2.1) are number of change in returns,
positive or negative, observed in the returns series. The expected runs given in
third column are the change in returns required, if the data is generated by random
process. If the actual runs are close to expected number of runs, it indicates that the
returns are generated by random process. It can be seen from the table that the
actual runs of index returns namely CNX Nifty, CNX Nifty Junior, CNX Defty,
BSE Sensex, BSE 100, BSE 200, CNX 500, CNX Bank Nifty, and BSE 500 are
less than the expected runs. The significant negative Z values indicate the positive
correlation in these returns series. The number of runs for CNX IT, CNX
Infrastructure, BSE Midcap and BSE Smallcap returns exceeds the expected
2.4 Discussion on Empirical Results
Table 2.5 Runs test statistics
for index returns
35
Index returns
Actual runs
Expected runs
Z-statistic
S & P CNX Nifty
CNX Nifty Junior
S & P CNX Defty
CNX 100
CNX 500
BSE Sensex
BSE 100
BSE 200
BSE 500
BSE Midcap
BSE Smallcap
CNX IT
CNX Bank Nifty
CNX Infrastructure
1,144
1,081
1,193
533
872
1,126
1,104
1,079
851
557
471
1,183
1,114
670
1,258
1,183
1,253
546
993
1,231
1,231
1,228
982
472
219
939
1,259
423
-4.59*
-4.35*
-2.42*
-0.85
-5.5*
-4.29*
-6.41*
-6.10*
-5.10*
4.36*
5.28*
11.32*
-5.83*
17.31*
Note Under null of random walk, actual runs should be equal to
expected runs. Asterisked values indicate rejection of null of
random walk at 1 % level of significance
number of runs. For these indices, the positive sign of the significant Z value
suggest a negative correlation. With the sole exception of CNX 100, the hypothesis
of random walk has been rejected for all the indices. In other words, behaviour of
Indian stock returns is not explained by the random walk theory.
The BDS test is performed at various embedded dimensions (m) like 2, 4, 6, 8,
and 10 and also at various distances like 0.5, 0.75, 1, 1.25, and 1.5 s where s
denotes the standard deviation of the return. The BDS test statistic followed by
p-values in parentheses is furnished in Table 2.6. In the BDS test, the null
hypothesis is that return series are i.i.d and rejection of the null implies that RWH
does not pass the test. It is very clear from the results that BDS test rejects the null
hypothesis of independence and thereby RWH too for all the 14 indices. It shows
that stock returns are linearly dependent. The dependence may be linear or nonlinear in the returns series which is not specified here.10 The BDS test has been
known to be having better statistical power properties than the runs test. Besides,
the latter test suffers from a reduction in test power due to loss of information in
the transformation from returns to their signs. Overall, the results of the runs and
BDS test rejected the null of i.i.d (the stricter definition of random walk) at the
conventional significant levels.
This empirical analysis shows that behavior of stock index returns in India both
on the NSE and BSE, largely, do not support RWH. The parametric multiple
variance ratio test results support the view that stocks returns follow random walk
for indices namely, S& P CNX Nifty, S &P CNX Defty, CNX 100, BSE Sensex,
BSE 100 and sectoral CNX IT, and CNX Infrastructure. The LMVR test results for
10
The issue of alternative specifications like non-linear dependence are detailed in Chap. 4.
36
m = 4, e = 0.75S
m = 6, e = S
m = 8, e = 1.25S
m = 10, e = 1.5S
S & P CNX Nifty
CNX Nifty Junior
S & P CNX Defty
CNX 100
CNX 500
BSE Sensex
BSE 100
BSE 200
BSE 500
BSE Midcap
BSE Smallcap
CNX IT
CNX Bank Nifty
CNX Infrastructure
20.45
24.28
20.31
18.82
23.20
21.42
24.20
23.03
23.35
18.92
17.90
25.36
17.86
17.67
26.80
30.74
26.23
25.53
31.01
28.39
29.30
24.58
30.64
23.91
21.53
25.64
21.90
23.69
31.25
35.60
30.56
29.40
35.92
33.38
30.35
23.51
34.37
25.53
23.32
25.07
24.63
26.67
32.25 (0.000)
36.99 (0.0000)
31.77 (0.0000)
29.72 (0.0000)
36.36 (0.0000)
34.70 (0.0000)
28.83 (0.0000)
21.48 (0.0000)
34.05(0.0000)
24.32 (0.0000)
22.87 (0.0000)
24.16 (0.0000)
25.71 (0.0000)
26.89 (0.0000)
12.38
16.00
12.69
11.48
14.77
12.83
15.55
16.07
14.58
13.39
13.91
19.35
12.18
10.37
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
Note The table reports the BDS test results. Here, ‘m’ and ‘e’ denote the dimension and distance, respectively and ‘e’ equal to various multiples (0.5, 0.75, 1,
1.25 and 1.5) of standard deviation (s) of the data. The value in the each cell is the BDS test statistic followed by the corresponding p value in parentheses.
The asymptotic null distribution of test statistics is N (0.1). The BDS statistic tests the null hypothesis that the increments are independent and identically
distributed, where the alternative hypothesis assumes a variety of possible deviations from independence including nonlinear dependence and chaos. All the
BDS values are statistically significant at 1 % level
2 Random Walk Characteristics of Stock Returns
Table 2.6 BDS test statistics for index returns
Index returns
m = 2, e = 0.5S
2.4 Discussion on Empirical Results
37
these indices are found significant only for short horizons like 2–4 days and following random walk afterwards (longer horizons). The possible explanation for the
this behaviour of stock returns is that the information in short-horizon is not
instantly reflected in returns and thus provide opportunity for excess returns to
those who have access to this information. Later, as the time horizon increases,
information is reflected in stock returns leading to improvement in informational
efficiency. The parametric results for other indices show strong autocorrelation.
The non-parametric tests, which are robust to non-normality, reject random walk
characteristics in Indian stock returns on NSE and BSE. The view that the likelihood of rejection of RWH in case of larger indices having higher market capitalization and higher liquidity is less than their lower counterparts is supported in
case of BSE, as rejection of null is stronger in case of BSE Midcap and BSE
Smallcap. However, this is not fully observed in indices traded on NSE.
2.5 Concluding Remarks
This chapter has investigated the behavior of stock returns by testing RWH, in
emerging Indian equity market. The specific objective of the chapter was to test
weak form of market efficiency in Indian equity market. Toward this end, parametric and non-parametric tests are used to analyze the daily data on 14 market
indices from two major stock exchanges namely, the NSE and BSE. The results
from parametric tests offered mixed results and suggest random walk characteristics in returns of highly liquid and considerable market capitalized indices on
BSE. However, this has not found empirical support from evidence on NSE
indices. The sector-wise results largely indicate random walk behavior for the
selected indices. The empirical results from the non-parametric runs and BDS tests
resoundingly reject the RWH in Indian stock markets. However, it is to be noted
that these two tests examine the stricter definition of random walk. In the light of
the present evidence, it is necessary that the regulatory authority and policy
making body ensure dissemination of information so that price can reflect the
information quickly.
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Chapter 3
Nonlinear Dependence in Stock Returns
Abstract A body of literature focused on testing for linear dependence in stock
returns. The rejection of linear dependence does not necessarily imply independence because of the possibility of a nonlinear structure in the time series realizations. This chapter empirically investigates nonlinear dependence in Indian
stock returns using a set of nonlinearity tests. The daily data between 1997 and
2010 of eight indices from the National Stock Exchange (NSE) and six indices
from the Bombay Stock Exchange (BSE) are used. The results suggest strong
evidence of nonlinear structure in stock returns. The nonlinear dependence,
however, is not consistent throughout the sample period but confined to a few brief
periods. The periods of nonlinear dependence are majorly associated with events
such as uncertainties in international oil prices, volatile exchange markets, solvency issues of cooperatives, US 64 scam, subprime crisis followed by the global
economic meltdown, and political uncertainties among others.
Keywords Nonlinear dependence
Informational efficiency
Pure noise
Episodic dependence Oil shocks Subprime crisis Subprime crisis FIIs
3.1 Introduction
Nonlinear dependence in stock returns has gained importance in recent times
because it indicates the possibility of predictability. The earlier studies that examined the Efficient Market Hypothesis (EMH) largely used conventional tests such as
autocorrelation, variance ratio, and runs tests that are not capable of capturing
nonlinear patterns in the returns series. The earlier evidence of rejection of linear
dependence is not sufficient to prove independence in view of the nonnormality of
the series (Hsieh 1989). The rejection of linear dependence does not necessarily
imply independence (Granger and Anderson 1978). The presence of nonlinearity
provides opportunities for market participants to make excess profits. The use of
linear models in such conditions may give the wrong inference of unpredictability.
Moreover, the presence of nonlinearity in stock returns contradicts EMH.
G. S. Hiremath, Indian Stock Market, SpringerBriefs in Economics,
DOI: 10.1007/978-81-322-1590-5_3, The Author(s) 2014
41
42
3 Nonlinear Dependence in Stock Returns
Hinich and Patterson (1985) were among the first to provide evidence of
nonlinear dependence in NYSE stock returns. Nevertheless, the market crash of
October 1987 shifted the paradigm. The crash was a major event that influenced
the role of nonlinearities in the dynamics of stock returns (Lima 1998). The later
studies challenged the previously considered ‘‘stylized fact’’ that the stock return
series follows a random walk (see e.g., Fama and French 1988; Poterba and
Summers 1988; Lo and MacKinlay 1988) and, instead, nonlinear behavior in the
US exchange rate and stock market were reported (Hsieh 1989; Scheinkman and
LeBaron 1989). Further, Willey (1992), Lee et al. (1993), Pagan (1996), Blasco
et al. (1997), Lima (1998), Yadav et al. (1999) viewed the nonlinear behavior of
stock returns as an alternative to random walk and found nonlinearity in the
underlying returns. Similar results were also reported in the UK market
(Abhyankar et al. 1995; Opong et al. 1999). Solibakke (2005) distinguished
between models that are nonlinear in mean and, hence, they depart from the
martingale hypothesis and models that are nonlinear in variance and, hence, they
depart from the assumption of independence too, but not from the martingale
hypothesis. In his empirical work, Solibakke (2005) found strong nonlinearity in
variance and weak dependence in mean in the case of Norwegian stock returns.
It is important to note that most of the studies cited above are confined to welldeveloped markets. Given this fact, it would be interesting to see whether stock
returns exhibit the same patterns in emerging markets as well. Sewell et al. (1993),
for instance, provided evidence of nonlinearity in the emerging markets. Similarly,
Scheicher (1996) for Vienna, Seddighi and Nian (2004) for China and Panagiotidis
(2005) for Greece found evidence of nonlinearity in stock returns.
The empirical evidence of a nonlinear structure in stock returns since the late
1980s, both from developed and emerging markets, indicates the possible predictability of future returns. However, nonlinear serial dependence present
throughout the sample period or confined to a certain period within a sample
period is significant enough to explore it in detail. Such possibilities cannot be
denied given the changes in institutional arrangements and regulatory norms.
Further, events occurred during a particular period might induce nonlinearity in
stock returns during that period and nonlinear dependence might disappear later. In
case underlying returns are nonlinear for a few episodes, then it would be difficult
to make any forecast of future returns. To examine such possibilities, Hinich and
Patterson (1995) recommend the windowed test procedure.
Ammermann and Patterson (2003) reported brief periods of linear and nonlinear
dependence and disappearance of such dependencies before investors could
exploit them. Bonilla et al. (2006) for Latin America and Lim et al. (2008)
reported similar episodic transient nonlinear dependencies for selected Asian
markets. Conditional heteroscedasticity has been cited in the studies as one of the
factors responsible for observed nonlinear dependence in returns. In previous
studies, one often finds extensive application of BDS test to examine the issue of
nonlinearity. In India, while Amanulla and Kamaiah (1998) reported independence
of returns, Chaudhuri and Wu (2003), Ahmad et al. (2006) concluded that stock
3.1 Introduction
43
returns in India do not follow random walk.1 These studies have employed conventional tests that are not capable of detecting the nonlinear structure in the data.
The issue of nonlinear dependence in stock returns though significant, has not
received due attention in the Indian context, with the exception of the study by
Poshakwale (2002). Given the fact that the stock market in India has witnessed
several changes since the mid 1990s, this chapter assumes relevance, and seeks to
examine nonlinear behavior of stock returns in two premier stock exchanges,
namely, the National Stock Exchange (NSE) and the Bombay Stock Exchange
(BSE). The study considers data of daily stock returns of eight indices from the
NSE and six indices from the BSE from June 1997 to March 20102 (Chap. 1,
Table 1.2). This study has the advantage of updated and disaggregated data,
covering a period during which several major market microstructure changes took
place. To investigate the issue, a set of nonlinearity tests are applied to ensure that
the results are not sensitive to the test carried out. In addition, to examine the
persistence of dependence, the windowed test procedure is followed. Further, an
attempt is made to identify events associated with the periods of presence of
nonlinear dependence.
The remaining part of this chapter is organized in the following sections. Section 3.2 explains the methodology in brief. Section 3.3 discusses
empirical results, and the last section comprises a few concluding remarks.
3.2 Methodology
The study employed a set of nonlinear tests namely, Hinich and Patterson (1989)
bispectrum, McLeod and Li (1983), Tsay (1986), Brock et al. (1996) and Hinich
(1996) bicorrelation tests to examine the nonlinear structure in stock index returns
of the NSE and the BSE. Further, to examine whether the presence of nonlinear
dependence is pertinent during the entire sample period or a few sub-periods,
Hinich and Patterson’s (1995) windowed test procedure is followed. The tests are
implemented after removing linear dependence in daily returns by fitting an
appropriate autoregressive (AR) (q) model. A brief description of these tests is
given below.
McLeod and Li’s (1983) portmanteau test of nonlinearity seeks to discover
whether the squared autocorrelation function of returns is non-zero. Tsay (1986)
test of nonlinearity aims to detect the quadratic serial dependence in the data. It
tests the null that all coefficients are zero. The bispectrum test is a test of linearity
and Gaussianity, as described by Hinich and Patterson (1989). The Hinich bi
spectrum test is a frequency domain test. It estimates bispectrum of stationary time
1
For a survey of literature on Indian stock returns, please see Chap. 2, Sect. 2.2. Also see,
Amanulla and Kamaiah (1996).
2
For details on sample data, see Chap 1, Table 1.2.
44
3 Nonlinear Dependence in Stock Returns
series and provides a direct test of nonlinearity in returns series. The flatness of the
skewness function in this frequency domain test indicates a third order nonlinear
dependence.
Brock et al. (1996) proposed a portmanteau test that is popularly known as the
BDS test, named after its authors, for time-based dependence in a series. It has
power against a variety of possible deviations from independence including linear
dependence, nonlinear dependence or chaos.3 In this test, m denotes the embedded
dimension (period histories), and e is the distance that is used to decide if returns
are near each other. The estimate of the correlation integral value is the proportion
of pairs of m period histories that are near to each other. The BDS statistic is
estimated at different m, and e values.
The portmanteau bicorrelation test of Hinich (1996) is a third order extension of
the standard correlation tests for white noise. The null hypothesis for each window
is that the transformed data are realizations of a stationary pure white noise process
that has zero correlation (C) and bicorrelation (H). Thus, under the null hypothesis,
the correlation (C) and bicorrelation (H) are expected to be equal to zero. The
alternative hypothesis is that the process has some non-zero correlation (second
order linear) or bicorrelations (third order nonlinear dependence). The linear
dependence in returns is removed using an AR (q) model. An appropriate lag is
selected so that there are no significant (C) statistics. Hence, the rejection of null of
pure noise implies nonlinear dependence. Further, Hinich and Patterson (1995) test
procedure involves dividing the full sample period into equal-length non-overlapped windows to capture episodic dependencies in stock returns. The study
divides the whole sample into a set of non-overlapped window of 50 observations
of equal length.4 Then, the Hinich (1996) bicorrelation test is applied to detect
episodic nonlinear dependencies in returns.
3.3 Empirical Results
This section presents nonlinearity test results. The nonlinear dependence in stock
returns is examined by applying the set of nonlinear tests mentioned in the previous section. Before performing these tests, linear dependence is removed by
fitting the AR (q) model so that any remaining dependence would be rendered
nonlinear. The results of McLeod-Li and Tsay tests are reported in Table 3.1. The
former tests the null of i.i.d while the latter tests whether all coefficients are zero.
The rejection of null suggests that the underlying returns series are nonlinearly
dependent.
3
See Chap. 2, Sect. 2.3.2.2 for detailed description of the test.
Hinich and Patterson (1995) suggest that the window length should be sufficiently large to
validly apply bicorrelation test and yet short enough for the data generating process to have
remained roughly constant.
4
3.3 Empirical Results
45
Table 3.1 McLeod-Li, Tsay, and bi spectrum test statistics
Index returns
McLeod-Li
Tsay test statistic
test statistics
Lag 4
Lag 6
(probability)
Bi spectrum
test statistic
S and P CNX Nifty
CNX Nifty Junior
S and P CNX defty
CNX 100
CNX 500
BSE Sensex
BSE 500
BSE100
BSE 200
BSE Midcap
BSE Smallcap
CNX IT
CNX Bank Nifty
CNX Infrastructure
3.75 (0.000)
13.03 (0.000)*
16.64 (0.000)*
17.88 (0.000)
–
7.00 (0.000)*
18.08 (0.000)*
31.26 (0.000)*
–
30.26 (0.000)*
10.19 (0.000)*
–
13.36 (0.000)*
20.3 (0.000)*
0.000*
0.000*
0.000*
0.000*
1.000
0.000*
0.000*
0.000*
0.000*
0.000*
0.000*
1.000
0.000*
0.000*
6.25 (0.000)*
6.97 (0.000)*
6.97 (0.000)*
6.53 (0.000)*
2.42 (0.007)*
5.76 (0.000)*
5.72 (0.000)*
75.11 (0.000)*
91.83 (0.000)*
8.17 (0.000)*
6.37 (0.000)*
1.12 (0.341)
4.05 (0.000)*
5.89 (0.000)*
4.41 (0.000)*
4.16 (0.000)*
4.81 (0.000)*
4.58 (0.000)*
1.71 (0.021)**
3.73 (0.000)*
3.80 (0.000)*
36.66 (0.00)*
44.04 (0.00)*
4.59 (0.000)*
3.70 (0.000)*
13.60 (0.000)*
2.99 (0.000)*
4.56 (0.000)*
Note The McLeod-Li tests the null hypothesis that the increments are independent and identically
distributed (iid). The corresponding p values of test statistics are given in the second column.
Tsay method tests that all coefficients are zero. The alternative hypothesis is that returns series are
characterized by nonlinear dependence. Tsay statistics are calculated at lag 4 and 6 and respective
statistic followed by p values in parentheses is given. The bispectrum method tests the null of
absence of third order nonlinear dependence. The bispectrum statistic is given in the last column
along with p values in parentheses. All the test values are significant at 1 % level. The bispectrum
test could not be calculated for CNX IT, BSE 200 and CNX 500. Asterisked*,** values indicates
rejection of null at statistical significance level of 1% and 5%. The rejection of null indicates
nonlinear dependence in returns
The McLeod-Li test strongly rejects the null of i.i.d as probability values for all
index returns are zero. CNX IT and CNX 500 are, however, exceptions to this
(Table 3.1). The Tsay test results support the presence of nonlinear dependence as
evidenced by the McLeod-Li test. In other words, the Tsay test results suggest that
with sole exception of CNX IT, all other index returns are characterized by
nonlinear dependence (Table 3.1).
Further, the Hinich bispectrum tests the null of absence of third order nonlinear
dependence (flat skewness function). Rejection of null suggests nonlinearity in
stock returns. Unlike other nonlinear tests, the bispectrum directly tests for linearity. Hence, filtering of data is not necessary before performing the test. In other
words, the test is invariant to linear filtering.5 The test statistics present in the last
column of Table 3.1 rejects the null of absence of third order nonlinear dependence for entire index returns.6
5
In the present study, though the bispectrum is performed both on raw data and on residuals, the
results are reported only for raw returns because results for both the series are the same.
6
The bispectrum test could not be calculated for CNX IT, BSE 200 and CNX 500.
46
3 Nonlinear Dependence in Stock Returns
Table 3.2 BDS test statistics
Index returns
m = 2,
e = 0.75 s
S and P CNX Nifty
CNX Nifty Junior
S and P CNX
Defty
CNX 100
CNX 500
BSE Sensex
BSE100
BSE 200
BSE 500
BSE Midcap
BSE Smallcap
CNX IT
CNX Bank Nifty
CNX Infrastructure
m = 4,
e = 1.0 s
m = 8,
e = 1.25 s
m = 10,
e = 1.50 s
12.94 (0.0000)
15.81 (0.0000)
13.15 (0.0000)
20.53 (0.0000)
23.77 (0.0000)
20.56 (0.0000)
31.25 (0.0000)
35.49 (0.0000)
31.04 (0.0000)
32.07 (0.0000)
37.08 (0.0000)
32.18 (0.0000)
11.98
16.89
13.71
18.99
28.16
15.03
11.96
10.20
19.32
12.37
10.27
18.26
21.78
22.00
25.78
27.04
23.10
16.63
13.68
23.39
17.75
16.93
28.44
23.97
34.67
32.72
21.87
34.02
22.94
18.63
25.53
24.94
26.21
28.63
22.08
35.94
31.41
18.91
33.57
22.16
19.12
24.60
25.81
26.13
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
Note The table reports the BDS test results. Here, ‘m’ and ‘e’ denote the dimension and distance,
respectively and ‘e’ equal to various multiples (0.75, 1, 1.25 and 1.5) of standard deviation (s) of
the data. The value in the first row of each cell is a BDS test statistic followed by the corresponding p value in parentheses. The asymptotic null distribution of test statistics is N (0.1).The
BDS statistic tests the null hypothesis that the increments are independent and identically distributed (iid), where the alternative hypothesis is nonlinear dependence. All the results are statistically significant at 1 % level
The BDS test is performed at various embedded dimensions (m) like 2, 4, 8 and
10 at various distances (e) like 0.75, 1.0, 1.25, and 1.50 s where s denotes standard
deviations of the return. The BDS test statistics are furnished in Table 3.2. In the
Table 3.2, the value in each cell represents a BDS test statistic followed by a
probability value in parenthesis. The BDS tests the null hypothesis that return
series are i.i.d. The rejection of the null implies that the random walk hypothesis
does not hold good. It is clear from the statistics in Table 3.2 that null of i.i.d is
rejected for all indices. The rejection of i.i.d for residuals from AR (q) models
indicates presence of nonlinear structure in the returns series. This implies the
possible predictability of future returns based on past information.
The Hinich (1996) bicorrelation (H) test statistics covering the full sample
period are presented in Table 3.3. The null of pure noise is tested. The total
number of bicorrelations and corresponding probability values are provided in the
third and fourth columns of Table 3.3. It is evident from the probability values that
with the exception of CNX IT and CNX 500, as in case of the McLeod-Li and
Tsay tests, the null of pure noise is clearly rejected for all other index returns from
the NSE and BSE. It may be inferred that the returns series are characterized by
nonlinear dependence as the bicorrelation test applied to residuals extracted after
fitting AR (q) model. The null of pure noise could not be rejected for CNX IT and
CNX 500, as the probability value is almost close to 1 (Table 3.3).
3.3 Empirical Results
47
Table 3.3 Hinich bicorrelation (H) statistics for full sample
Index returns
Number of
Number of
bicorrelations
lags
Probability (p)
value for (H) statistic
S and P CNX Nifty
CNX Nifty Junior
S and P CNX Defty
CNX 100
CNX 500
BSE Sensex
BSE 100
BSE 200
BSE 500
BSE Midcap
BSE Smallcap
CNX IT
CNX Bank Nifty
CNX Infrastructure
0.0000*
0.0000*
0.0000*
0.0000*
0.9999
0.0000*
0.0000*
0.00008
0.0000*
0.0000*
0.00008
1.0000
0.0000*
0.0000*
24
24
24
18
23
23
23
23
22
17
17
24
22
17
276
276
276
153
231
253
253
253
231
136
136
276
231
136
Note The table reports Hinich bicorrelation test statistics. Under the null of pure noise, the
bicorrelations are expected to be zero. Rejection of null hypothesis suggests the presence of
nonlinear dependence. Asterisked values indicate rejection of null hypothesis of zero bicorrelation
at the 1 % level of significance
Whether the presence of nonlinear dependence exists throughout the sample
period or confined to a certain subperiod within the sample is an interesting issue
to explore. This helps us understand the nature of market efficiency over a period
of time. To examine the episodic dependence in returns series, the residuals are
divided into a set of non-overlapped windows of 50 observations of equal length
and then H statistics of Hinich (1996) are computed to detect nonlinear dependencies in each window. The lag is selected so that there are no significant
(C) windows at 5 % probability value. Table 3.4 presents the total number of
significant (H) windows in column three, and the percentage of significant windows to the total number of windows are given in column four of Table 3.4.
The results show that the number of significant (H) windows on an average is
low. These significant windows reject the null of pure noise, indicating the presence of nonlinearity confined to these windows. The BSE Midcap and BSE
Smallcap have the highest percentage of significant nonlinear dependence
(38.4 %) followed by CNX Nifty Junior (32.2 %) and CNX 500 (26.5 %). While
the BDS test rejects the null of i.i.d for CNX IT and CNX 500, the other nonlinear
tests including Hinich (1996) test results suggest that these two index returns
validate weak form efficiency. However, it is not un surprising that CNX IT and
CNX 500 possess pockets of nonlinear dependencies that are evident in the results
presented in Table 3.4. The events that occurred during these windows do not
seem to influence the overall performance of CNX IT and CNX 500 index returns.
The evidence from nonlinear tests, namely McLeod-Li, Tsay, Hinich bispectrum, BDS, and Hinich bicorrelation tests employed in the study provide strong
48
3 Nonlinear Dependence in Stock Returns
Table 3.4 Windowed test results of Hinich H statistic
Index returns
Total number
Total number
of significant
of windows
H windows
Percentage
of significant
windows
Windows period
S and P CNX Nifty
59
10
16.9
CNX Nifty Junior
59
19
32.2
S and P CNX Defty
59
10
16.9
CNX 100
31
7
22.5
01/12/98–03/26/98
06/10/98–08/18/98
01/04/01–03/19/01
08/09/01–10/22/01
10/24/02–01/06/03
03/16/04–05/26/04
12/28/04–03/10/05
03/09/06–05/23/06
12/22/06–03/08/07
12/26/07–03/04/08
08/16/99–10/25/99
01/01/00–03/16/00
03/21/00–06/01/00
10/25/00–01/03/01
08/09/01–10/22/01
10/23/01–01/07/02
03/19/02–05/30/02
05/31/02–08/08/02
06/03/03–08/11/03
01/01/04–03/15/04
03/16/04–05/26/04
12/28/04–03/09/05
05/23/05–08/01/05
03/09/06–05/23/06
05/24/06–07/31/06
10/12/06–12/21/06
12/26/07–03/04/08
08/01/08–10/15/08
10/16/08–01/01/09
06/02/97–08/11/97
08/10/00–10/19/01
10/23/02–01/03/03
03/17/04–05/27/04
12/29/04–03/10/05
03/10/06–05/24/06
05/25/06–08/01/06
10/13/06–12/22/06
12/26/06–03/09/07
12/27/07–03/05/08
05/28/03–08/05/03
03/10/04–05/20/04
12/22/04–03/03/05
10/06/05–12/20/05
(continued)
3.3 Empirical Results
49
Table 3.4 (continued)
Index returns
Total number
of windows
Total number
of significant
H windows
Percentage
of significant
windows
CNX 500
49
13
26.5
BSE Sensex
56
8
14.2
BSE 100
55
13
23.6
BSE 200
55
12
21.8
Windows period
03/06/06–05/18/06
12/19/06–03/06/07
02/29/08–05/16/08
10/26/99–01/05/00
01/04/01–03/16/01
08/09/01–10/22/01
10/24/02–01/06/03
06/03/03–08/11/03
03/19/04–05/31/04
12/31/04–03/14/05
10/18/05–12/28/05
03/14/06–05/26/06
05/29/06–08/03/06
05/29/07–08/06/07
12/31/07–03/10/03
08/06/08–10/20/08
10/29/98–01/08/99
10/30/02–01/10/03
10/28/03–01/06/04
03/22/04–06/01/04
12/30/05–03/14/06
03/16/06–05/29/06
05/30/07–08/07/07
10/19/07–12/31/07
06/04/98–08/12/98
03/26/99–06/08/99
01/10/01–03/22/01
08/16/01–10/29/01
10/30/02–01/10/03
03/22/04–06/02/04
01/03/05–03/15/05
10/19/05–12/29/05
03/16/06–05/29/06
03/15/07–05/29/07
01/01/08–03/11/08
03/12/08–05/28/08
08/07/08–10/22/08
10/23/08–01/06/09
01/01/98–03/18/98
03/19/98–06/04/98
06/05/98–08/12/98
01/10/01–03/22/01
08/16/01–10/29/01
(continued)
50
3 Nonlinear Dependence in Stock Returns
Table 3.4 (continued)
Index returns
Total number
of windows
Total number
of significant
H windows
Percentage
of significant
windows
BSE 500
47
8
17.0
BSE Midcap
26
10
38.4
BSE Smallcap
26
10
CNX IT
59
9
15.2
Windows period
10/30/02–01/10/03
03/22/04–06/01/04
01/03/05–03/15/05
03/16/06–05/29/06
05/30/06–08/04/06
10/18/06–12/28/06
03/15/07–05/29/07
01/01/08–03/11/08
03/12/08–05/28/08
08/07/0/–10/21/08
10/22/08–01/06/09
03/14/00–05/29/00
01/01/01–03/13/01
08/06/01–10/17/01
10/21/02–01/01/03
03/11/04–05/21/04
12/23/04–03/04/05
10/09/06–12/18/06
12/19/07–02/28/08
12/28/04–03/09/05
05/23/05–07/29/05
08/01/05–10/11/05
10/13/05–12/23/05
03/09/06–05/23/06
05/24/06–07/31/06
12/22/06–03/08/07
12/24/07–03/04/08
08/01/08–10/15/08
10/16/08–12/31/08
01/01/04–03/15/04
03/16/04–05/26/04
12/28/04–03/09/05
08/01/05–10/11/05
03/09/06–05/23/06
05/24/06–07/31/06
10/12/06–12/21/06
12/26/07–03/04/08
08/01/08–10/15/08
10/16/08–12/31/08
10/24/97–01/07/98
01/08/98–03/24/98
11/05/99–01/17/00
03/31/00–06/13/00
(continued)
3.3 Empirical Results
51
Table 3.4 (continued)
Index returns
Total number
of windows
Total number
of significant
H windows
Percentage
of significant
windows
CNX Bank Nifty
45
7
15.5
CNX Infrastructure
31
7
22.5
Windows period
01/16/01–03/28/01
03/29/01–06/11/01
08/23/01–11/02/01
01/10/07–03/23/07
06/09/0/–08/18/08
08/19/08–10/31/08
10/19/00–12/28/00
08/03/01–10/16/01
05/27/02–08/02/02
03/10/04–05/20/04
12/22/04–03/03/05
10/09/07–12/17/07
12/18/07–02/27/08
05/27/04–08/05/04
03/10/05–05/23/05
12/26/05–03/09/06
10/12/06–12/21/06
03/09/07–05/23/07
12/26/07–03/04/08
01/01/09–03/18/09
Note Total number of significant H windows and the percentage to total number of windows are
furnished in the table. A window is significant if the H statistic rejects the null hypothesis at a 5 %
probability value. Last column of the table presents significant window dates
evidence of nonlinear dependence in both NSE and BSE across all index returns
considered. The windowed Hinich test results document that the reported dependence is confined to a few brief episodes. This implies that the events during the
small number of significant window periods are responsible for the rejection of
null of pure noise for the whole sample period. Given this fact, the events that
occurred during these periods of significant windows provide further insight into
the issue of nonlinearity in returns.
Theoretically, the nonlinear structure in data is explained by different factors.
The characteristics of market microstructure, restrictions on short sales, noise
trading, market imperfections, conditional heteroscedasticity and heterogeneous
beliefs are cited in literature as factors responsible for the nonlinear dependence
structure in stock returns. Following the framework of Lim and Hinich (2005), an
attempt is made here to identify the events that induced nonlinear dependence in
window periods during which significant nonlinear dependence is found. The
period of significant windows of respective indices are given in the last column of
Table 3.4. The major political and economic events that occurred between June
1997 and March 2010 are identified. These events are associated with those
periods of significant windows reported in Table 3.4 based on the Hinich (1996)
52
3 Nonlinear Dependence in Stock Returns
test with windowed procedure. The major events are identified through news
reports and the events cited as important in the various issues of the annual reports
of the Reserve Bank of India (RBI), Securities and Exchange Board of India
(SEBI), Economic Surveys, newspapers, reports of credit rating agencies, etc. For
convenience, the discussion on events is grouped year wise.
3.3.1 1997–1998
The financial year 1997–1998 witnessed a higher level of volatility. The marketfriendly budget of 1997–1998 had a favorable impact as there was a spurt in stock
returns up to the middle of August. The significant window period for CNX IT
falls between October 1997 and January 1998 (Table 3.4). This period is associated with events such as the currency crisis in South East Asia, which generated
panic in the market and resulted in negative net inflows from Foreign Institutional
Investors (FIIs).
3.3.2 1998–1999
The performance of the market in general was gloomy during this year. The
significant windows period during this financial year is associated with events such
as impending sanctions following a nuclear test, instability in exchange rate,
turmoil in the international market, and the bad news of the US-64 scheme of the
UTI scam.
3.3.3 1999–2000
The massive inflow of FIIs and mutual funds in both the NSE and the BSE created
upward pressure on stock returns from August 1999 to October 1999 and late
October 1999 to February 2000. A new government was formed at the Center and
it has passed several reform bills.7 The RBI in its annual report, pointed out that
the market positively responded to the news of rating India as a stable market by
international credit rating agencies. However, with the uncertainty about international oil prices and hike in interest rate by the US, the dot.com bubble bust on
March 10, 2000, and on the political front, the hijack of Air India flight by
‘‘terrorists’’ followed by the war hysteria between India and Pakistan during
7
The bills passed during the year were Insurance Regulatory Authority (IRA) Bill, Foreign
Exchange Management Act (FEMA) Securities Laws (Amendment) Bill.
3.3 Empirical Results
53
January 2000 to March 2000 generated nervousness in the market. The annual
report of the SEBI noted that the behavior of stock returns was not linear during
the year.
3.3.4 2000–2001
The significant windows indicating nonlinearity in the financial year 2000–2001
were for March–June, October–December 2000, and January 2001 to March 2001
(see Table 3.4). The increase in international oil prices and panic in international
equity market were associated with these periods. In general, the Indian equity
market witnessed a sharp decline in all indices during 2000–2001. The last quarter
of January 2000–2001 witnessed high volatility. The RBI noted in its annual report
that Union budget, the expectations of strong earnings and the growth of the new
economy were responsible for the sharp rise. Besides, the fall was due to liquidity/
solvency of some co-operative banks.
3.3.5 2001–2002
During the year, especially from August to October 2001, a bearish sentiment
prevailed in the market. The US stock market crashed following the terrorist attack
on the World Trade Center on September 11, 2001. The slowdown in major
international stock markets aggravated depression and resulted in heavy selling by
FIIs in the Indian stock market.
3.3.6 2002–2003
The events associated with the year 2002–2003 and identified as one of significant
windows (Table 3.4) were the India–Pakistan border tensions, slip in consumer
spending, bad monsoon, tension in the Middle East, and rise in international oil
prices. The Bank Nifty Index’s significant windows during 2002–2003 are associated with information of profitability of banks and the relaxation of Foreign
Direct Investment (FDI) norms for private sector banks.
3.3.7 2003–2004
The Indian equity market witnessed 83 % returns, the highest among any of the
emerging markets. The RBI’s annual report of the year pointed out that the
54
3 Nonlinear Dependence in Stock Returns
improved fundamentals, strong corporate results and initiatives on disinvestment,
and active derivative trading were responsible for the spurt in returns. The SEBI
allowed brokers to extend margin-trading facility. The period from January to
March 2004 was a period of political uncertainties leading to depression in the
market.
3.3.8 2004–2005
The turbulent political conditions of March 2004 continued up to May 2004 and
resulted in lacklustre returns. The BSE Sensex, the major index, reached the lowest
point on May 17, 2004 due to political uncertainties. These uncertainties made the
market nervous. During the subperiods May–July, August–October, November–
December 2004, due to strong economic outlook, and high and sustained portfolio
investment, the market responded quickly and the rally of returns continued.
3.3.9 2005–2006
The first quarter of the financial year March/April 2004 to May 2005 was marked
by the prevalence of a bearish sentiment in the market and events associated with it
during the period are uncertainty relating to the global crude oil prices, rise in
interest rates and turmoil in international stock markets. The corrections during
October 2005 to December 2005 were caused by the response of the market to the
news of the rise in domestic inflation rate and uncertainty regarding crude oil
prices. The proposals of the Union Budget of 2006–2007, including the raising of
FIIs investment limit and the improvement of fundamentals and sound business
outlook were met by a rally in stock returns during the last quarter of 2005–2006.
3.3.10 2006–2007
The period of significant windows during the financial year March 2005–May
2006 was associated with the sharp fall in metal prices, uncertainty in global
interest rate and inflationary pressure on the economy. The hike in the Cash
Reserve Ratio (CRR) and bank rate by the RBI were associated with the significant
window period, October 2006 to December 2006. The impending recession in the
US and deterioration in subprime mortgage banking in the US adversely affected
the Indian stock market.
3.3 Empirical Results
55
3.3.11 2007–2008
The financial year 2007–2008 was highly volatile as the BSE crossed the 20,000
mark and in the same year reached the lowest ever in the Indian equity market. The
first and second quarter (continued with corrections) witnessed a buoyant trend
(May–August 2007). The disarray resulting from the US subprime crisis, surge in
international oil prices, political uncertainties and policy cap on external commercial borrowings (ECBs) generated panic during October–December 2007,
though sharp increases were also observed (This period was highly volatile). The
period of December 2007 to March 2008 was associated with the decline in the
developed equity markets because of subprime crisis, global recession, fear of
credit squeeze, hike in short-term capital gains tax, increase in domestic inflation
rate, etc.
The year 2008 was a year of financial crises and global economic meltdown.
The periods of significant windows during this financial year fell between March
2007 and May 2008, June–August–October 2008 and October 2008 to January
2009. RBI noted in its annual report, that the turbulence in global financial market
began deepening in July 2008. Fannie Mac and Freddie Mac reported drop in fair
value assets. On September 15, 2008, the major US investment bank Lehman
Brothers declared bankruptcy while a merger with the Bank of America saved
Merrill Lynch, another major investment bank in the US. In January 2008, the
Northern Rock Bank crisis aggravated and profits of JP Morgan and Citibank
dived deep. The situation was further aggravated by the Satyam scam.
In summary, the different indices reacted to different events differently. One
possible reason may be due to different market capitalization and liquidity. For
instance, the BSE Midcap and BSE Small cap immediately responded to subprime
crisis and they are found to be more vulnerable than the other high cap indices.
Both positive and negative events are associated with the presence of nonlinearity
in returns. However, negative events have a greater and persistent impact. The
subprime crisis, uncertainties in international oil prices and global financial crisis
have an impact for a longer period and it was so for almost all the indices. The
presence of nonlinearity confounds EMH in the Indian equity market.
3.4 Concluding Remarks
Although the issue of nonlinear dependence has gained importance in recent times,
it is seldom discussed in India. Motivated by this concern, this chapter attempted
to test nonlinear dependence in the stock returns of indices at two top Indian stock
exchanges, namely, the NSE and the BSE. A set of nonlinear tests were applied to
examine the behavior of stock returns. Strong evidence of nonlinear dependence
was found in almost all index returns of NSE and BSE in the study. The results
from the windowed Hinich test show that the reported nonlinear dependencies
56
3 Nonlinear Dependence in Stock Returns
were not consistent during the whole period. It suggests the presence of episodic
nonlinear dependencies in returns series surrounded by long periods of pure noise.
The positive and negative events that occurred during these episodes of presence
of nonlinearity are identified, but the negative events had a greater impact. The
major events identified are South Asian financial crisis, UTI scam, uncertainties in
international oil prices, turbulent world markets, subprime crisis, global economic
meltdown, and political uncertainties, especially border tensions. The nonlinear
dependence found in stock returns indicates predictability of stock returns and
speculative abnormal profits. Nevertheless, in the context of present empirical
evidence of episodic nonlinear dependence in Indian stock returns, the speculators
cannot easily forecast future returns, that can lead to profits. The findings of the
study suggest that episodic nonlinear dependence was due to certain events during
the period particularly those that were negative in nature. Hence, there is a need for
policy reforms to make the market immune to the effects of such events in future.
References
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indices: evidence from the United Kingdom. Econ J 105(431):864–880
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markets. Econ Polit Weekly 41(1):46–56
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24(3):257–280
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Reserve Bank of India Publications. Various Issues
Securities and Exchange Board of India. Annual Reports. Various Issues
Chapter 4
Mean-Reverting Tendency in Stock
Returns
Abstract This chapter re-examines the issue of mean-reversion in Indian stock
market. Unlike earlier studies, the present one carries out multiple structural
breaks tests and uses new and disaggregated data from June 1997 to March 2010.
The study finds significant structural breaks in the returns series of all selected
indices and thus provides evidence of trend stationary process in the Indian stock
returns. The significant structural breaks that are endogenously searched occurred
in the years 2000, 2003, 2006, 2007, and 2008 for most of the indices indicating,
respectively, rise in international oil prices, global recession, erratic fluctuations in
exchange rates, sub-prime crisis and global meltdown. The evidence of structural
breaks and mean-reverting tendency indicates the possibility of prediction of
returns and thus implies that efficient market hypothesis (EMH) does not hold in
Indian context. The study finds that small indices with less liquidity and lower
market capitalization are more vulnerable to shocks particularly external events
rather than the high liquid and Large cap indices. Further, the sub-sample analysis
shows that there was increasing nonrandom walk behavior in stock returns during
the structural breaks periods. The results call for appropriate policies and regulatory measures particularly related to external events to improve the efficiency of
the market.
Keywords Mean reversion Random walk Market efficiency Unit root
Structural breaks Lagrange Multiplier Trend stationary External shocks FIIs
4.1 Introduction
Two extreme views are popular in the literature about behavior of stock returns.
One view is that the financial time series are characterized by nonstationary
processes, and hence do not have the tendency to return to the trend path. In other
words, the shocks have a permanent effect on long-term series. Therefore, it is not
possible to predict their future movements based on past information. The other
G. S. Hiremath, Indian Stock Market, SpringerBriefs in Economics,
DOI: 10.1007/978-81-322-1590-5_4, The Author(s) 2014
59
60
4 Mean-Reverting Tendency in Stock Returns
view is the mean-reversion view, according to which there is a tendency for the
stock returns to return to its trend path. Hence, it is possible to predict future price
movements based on history of prices. Earlier studies supported the stylized fact
that stock returns series follows a random walk (Kendall 1953; Working 1960;
Fama 1965; Niederhoffer and Osborne (1966); Fama 1970). This was challenged
by many later studies which documented mean-reverting tendency in stock returns
(e.g., Fama and French 1988; Poterba and Summers 1988; Balvers et al. 2000).
However, Richardson and Stock (1989), Kim et al. (1991), McQueen (1992),
Richardson (1993) reported evidence against mean-reversion.1
The mean-reverting tendency in stock returns points out possibility of prediction of future returns and consequent abnormal profits. This violates the efficient
market hypothesis (EMH) which states that current prices fully and instantly
reflects information and therefore future returns are unpredictable. The objective
of present chapter is to re-examine the issue of mean-reversion as an alternative to
the random walk hypothesis (RWH) in the Indian stock market. The rest of the
chapter divided into following sections for the convenience. Section 4.2 provides
review of previous works. Section 4.3 describes the data and methodology of the
study. Empirical results are discussed in Sects. 4.4 and 4.5 present the concluding
remarks. A supplement analysis on variance ratio, structural breaks and nonrandom walk behavior is presented at the end of the chapter.
4.2 Review of Previous Works
The conventional view of rejection of unit root (random walk) in the returns series
is that current shocks have only temporary effect and the long-term series remain
unaffected by such shocks. However, Nelson and Plosser (1982) point out that the
random shocks have a permanent effect on the underlying series. Empirical studies
have employed largely conventional unit root tests to examine the issue. However,
in the presence of a structural break, the power of a unit root test decreases when
the stationary alternative is true (Perron 1989). Thus, the inference concerning the
effect of shocks on long-term series, employing conventional unit root tests is
likely to go wrong when the structural break is ignored. An appropriate way would
be to test for the presence of structural break while employing such tests. In this
context, Perron (1989) proposes an alternative test where the break point is known
beforehand. Perron (1989) includes dummy variables to account for one known or
exogenous structural break in the framework of Dickey and Fuller’s (1979, DF)
unit root test. The test allows for a break under the null and alternative hypothesis.
Perron (1989) proposes three models, Model A allows for break in mean, Model B
for break in slope, and Model C for break in both mean and slope. He treated the
Great Depression and Oil shock (1973) solely as exogenous events, which altered
1
Bjorn (2010) discusses methods of derivative pricing of mean-reverting assets.
4.2 Review of Previous Works
61
the long run movement of stock prices. Perron (1989) provided evidence of trend
stationary for 10 of the 13 series used by Nelson and Plosser (1982). A limitation
of this test is that it requires knowledge of break point beforehand, which is more
often than not difficult to ascertain, and also involves subjectivity in the determination of the break point.
To overcome this limitation, Christiano (1992), Banerjee et al. (1992), Zivot
and Andrews (1992), among others, propose test procedures based on different
methods. Zivot-Andrews developed a sequential test procedure, which endogenously searches for a break point and tests for the presence of unit root when the
process has a broken trend. The test selects the break date where t-statistics testing
the null of unit root is minimum (most negative). They provide the evidence in
support of findings of Nelson and Plosser (1982) as they reject null of unit root
only for three out 13 selected data series.
Further, Wu (1997), Chaudhuri and Wu (2003) and Narayan and Smyth (2005)
among others tested for the presence of a structural break by using the ZivotAndrews test. Wu (1997) employed the test on a sample of 11 OECD markets
during 1979–1994. While the conventional unit root test, the augmented DickeyFuller (ADF)2 supported the null of unit root (except for Finland and the UK), the
Zivot-Andrews test showed that stock returns in eight out of 11 markets were
characterized by trend-stationarity. Further, in the analysis of monthly data from
1985 to 1997 of 17 emerging stock markets including India, Chaudhuri and Wu
(2003) found evidence of mean-reversion in 10 emerging markets. In contrast to
the evidence from the emerging markets, the OECD markets documented evidence
against mean-reversion and supported the unit root process of underlying stock
prices (Narayan and Smyth 2005).
In a time series data, due to several structural and regulatory changes, there may be
more than one break in the data series. As mentioned earlier, Perron (1989) points out
that ignoring a structural break may lead to loss of power of a unit root test. Similarly,
ignoring breaks more than one may also lead to loss of power of a test. Motivated by
this concern, Lumsdaine and Papell (1997) proposed two breaks unit root test. They
extended the endogenous break test methodology of Zivot and Andrews (1992), to
allow for two breaks under the alternative hypothesis of unit root test.3
The endogenous break tests of Zivot-Andrews (single break) and LumsdainePapell (two breaks) do not assume break(s) under unit root null and derive their
critical values. This may potentially render the tests biased and lead to size distortions and incorrect inferences (Nunes et al. 1997; Lee and Strazicich 2003).
Lee and Strazicich (2003) propose a Lagrange Multiplier (LM) unit root multiple
breaks test, which incorporates breaks under both the null and alternative. Therefore, rejection of null clearly indicates trend stationarity. Empirically, Lee and
Strazicich (2003) show potential for over rejection of the null in Lumsdaine-Papell
2
The augmented version of Dickey and Fuller (1979) is proposed in Said and Dickey (1984).
Clemente et al. (1998), Ohara (1999), Papell and Prodan (2004) also introduced multiple
breaks tests.
3
62
4 Mean-Reverting Tendency in Stock Returns
test. In the empirics, Cook (2005), Payne et al. (2005), among others, employed the
Lee-Strazicich test to examine the presence of breaks in exchange rates of different
countries. Narayan and Smyth (2007) using the Lee-Strazicich test found evidence
against mean-reversion in six out of seven stock returns of G7 countries.
The works of Rao and Mukherjee (1971), Sharma and Kennedy (1977), Barua
(1981), Gupta (1985), Amanulla (1997), and Amanulla and Kamaiah (1998) on the
behavior of Indian stock returns conclude that Indian stock returns follow a random walk. The study by Poshakwale (2002) rejects random walk for BSE stocks.4
In their study of 17 emerging markets, Chaudhuri and Wu (2003) reject the null of
unit root for India. The authors pointed that rupee devaluation, and the economic
reforms of 1991 were responsible for the structural break during 1991. With the
exception of the study by Chaudhuri and Wu (2003), the previous works on India
examined the issue of mean-reversion of stock returns by using conventional unit
root tests only. These tests are known to be less powerful in the presence of
structural breaks. The study has made major departures from the previous studies
on the following counts. In this study, we carried out the multiple structural breaks
test, which not only has better power properties but also assumes breaks under null
and alternative hypotheses. This provides the advantage of unambiguous results.
The study uses compressive data of 14 indices traded on the NSE and BSE, the top
exchanges in India. The data period from 1997 to 2010 gives the present study the
added advantage of covering the period of major financial reforms and market
microstructure changes.5 In these contexts, the present study is more appropriate
and assumes significance.
4.3 Data and Methodology
The present study employs the Zivot and Andrews (1992) sequential trend break
and Lee-Strazicich (2003) LM Unit root tests. A brief description of these two tests
is presented below.
4.3.1 Zivot and Andrews (1992) Sequential Break Test
Zivot-Andrews developed three models, namely Model A that allows for a break
in intercept only, Model B that allows for a break in trend only, and Model C that
allows for a break each in intercept and trend. Since model C allows single break
each in mean and intercept, it accommodates both model A and B and hence the
present study considers model C as appropriate for examining the issue. Besides,
4
Please refer the Chap. 2, Sect. 2.2 for the detailed discussion. Also see Amanulla and Kamaiah
(1996).
5
For details on sample and indices, see Chap. 1, Table 1.2.
4.3 Data and Methodology
63
Sen (2003) through the Monte Carlo simulation demonstrated that model C yields
more reliable breakpoints than model A when the break is unknown. Model C is
given in the following equation:
DPt ¼ l þ h DUt ðkÞ þ bt þ y DTt ð^
kÞ þ apt1 þ
k
X
j¼1
Uj Dptj þ et
ð4:1Þ
In Eq. (4.1), DPt is the first difference of the process Pt , DUt is a dummy
variable that captures shift in the intercept, and DTt another dummy that represents
a shift in the trend occurring at time TB. l; h; b; c; / and Us are constants, ‘k’
represents location of the break point and et, the shock. These dummy variables are
defined as follows:
1 if t [ TB
DUt ðkÞ ¼
ð4:2aÞ
0 otherwise;
1 if t [ TB
DTt ðkÞ ¼
ð4:2bÞ
0 otherwise;
Zivot-Andrews tests the null that trend (return) variable contains a unit root
with drift that excludes any structural break against the alternative hypothesis of
trend-stationary process with a one-time break in the trend variable. The model
allows for a one-time break in both intercept and trend. The test further allows
testing for a unit root against the alternative of stationary with a structural change
at some unknown point. To determine the break point and compute the test statistics for a unit root, an ordinary least square regression is run with a break at TB,
where TB ranges from 1 to T-2. For each value of TB, the number of extra
regressors k, is chosen following a sequential downward t test on all lags as
suggested by Campbell and Perron (1991).
Furthermore, Ng and Perron (1995) showed that general-to-specific approach
provides test statistics, which have better properties than information, based criteria.6
4.3.2 Lee-Strazicich (2003) LM Unit Root Multiple Breaks
Test
Let the data generating process (yt) be given by
0
fyt g ¼ d Zt þ e;
6
et ¼ bet1 þ et
ð4:3Þ
The sequential procedure suggests first to start with kmax and then estimate the model with kmax
lags. If the coefficient of the last included lag is significant at the 10 % level, select k = kmax.
Otherwise, reduce the lag order by one until the coefficient of the last included lag becomes
significant. For details, see Campbell and Perron (1991).
64
4 Mean-Reverting Tendency in Stock Returns
where Zt a vector of exogenous variables, et a vector of (first order auto-correlated)
errors, d0 a vector of parameters, b a constant, and et an error term with zero mean
and constant variance. Lee-Strazicich by extending the LM unit root test of
Schmidt and Phillips (1992), developed two models namely, model AA and model
CC. In the present study, model CC is employed because model AA allows for two
shifts in intercept only and model CC allows for two shifts each in intercept and
trend. In other words, model CC includes model AA too. The model CC is as
follows:
Let
Zt ¼ ½1; t; D1t ; D2t ; DT1t ; DT2t ;
ð4:4Þ
where Zt a vector of variables, t is time trend, Djt and DTjt (j = 1, 2) are dummy
variables defined as follows:
1 for t [ TBj þ 1
Djt ¼
ð4:5aÞ
0 otherwise;
1 for t [ TBj: þ 1
ð4:5bÞ
DTjt ¼
0 otherwise;
In the above Eqs. (4.5a and b), TBj is the time period when a break occurs. For
model CC, the following null (b = 1) and alternative (b \ 1) hypothesis in which
the process yt includes two trend breaks each in intercept and slope may be
formulated as follows:
ðNÞ
ðNÞ
ðNÞ
ðNÞ
Null : yt ¼ lðNÞ þ d11 B1t þ d12 B2t þ d21 BTit þ d22 BT2t þ yt1 þ m1t
ðAÞ
ðAÞ
ðAÞ
ð4:6aÞ
ðAÞ
Alternative: yt ¼ lðAÞ þ d11 þ D1t þ d12 D2t þ d21 DT1t þ d22 DT2t þ m2t ð4:6bÞ
In (4.6a and b), the superscripts ‘N’ and ‘A’ denote null and alternative,
respectively, m1t and m2t are stationary error terms, and Bjt and BTjt are defined as
follows:
1 for t [ TBj: þ 1
Bjt ¼
ð4:7aÞ
0 otherwise;
1 for t [ TBj: þ 1
ð4:7bÞ
BTjt ¼
0 otherwise;
Under the null hypothesis, it is assumed that
ðNÞ
ðNÞ
ðNÞ
ðNÞ
d11 ¼ d12 ¼ 0
d21 ¼ d22 ¼ 0
4.3 Data and Methodology
65
The two breaks LM unit root test statistics is obtained from the following
regression:
Dyt ¼ d0 DZt þ USt1 þ lt
ð4:8Þ
^ðkÞ ¼ inf ^sðkÞ
inf q
ð4:9Þ
b Zt b
where St ¼ yt w
d t ; t ¼ 2; . . .; T; b
d are coefficients in the regression of
x
b
b
Dyt on DZt ; w x is given by y Z d; and y and Z represent the first observations of yt
and Zt, respectively. The unit root null is described by a = 0, and the LM test
^ and s ¼ t-statistic for testing the unit root null
^ ¼ TU
statistics are given by q
hypothesis that U ¼ 0. The location of the structural break (TB) is determined by
selecting all possible break points for the minimum t-statistics given by:
k
k
The search is carried out over the trimming region (0.15T, 085T), where T is the
sample size. As in the case of Zivot-Andrews test, the numbers of lagged augmentation terms in this test are determined by the general-to-specific procedure
suggested in Ng and Perron (1995). Starting from a maximum of k = 8, lagged
terms, the procedure looks for significance of the last augmented term.
4.4 Empirical Findings
To examine the issue of mean-reversion and for comparison purpose, the conventional ADF of Said and Dickey (1984) and Phillips and Perron (1988) unit root
tests are applied. These two tests have the null hypothesis of unit root (random
walk) against an alternative hypothesis of stationary process. These two tests are
applied with and without deterministic trend variable to test null of unit root
against stationary alternative. The maximum lag length is set to 12 for ADF and
four for PP following the sequential procedure. To take care of possible serial
correlation in error terms, the ADF test adds lagged difference terms of the
regressand in error terms, while the PP test uses nonparametric methods. Table 4.1
presents the ADF and PP test statistics.
The rejection of null in these two tests without trend implies that the returns
series is stationary with a nonzero mean, whereas rejection of null with trend
indicates that the returns series is stationary around a deterministic trend. It is
evident that the ADF test cannot reject the null of unit root for all index returns at
any conventional level of statistical significance both without and with trend.
Similarly, Table 4.1 presents the PP unit root test statistic that strongly supports
the evidence of ADF test. The estimated values of the PP test are less than the
critical values and thus the null of unit root cannot be rejected for any of the
returns series. It implies that the selected index series are nonstationary. As noted
earlier, the conventional unit root tests results may be spurious if there is a
structural break in the series and ignorance of such break in the series leads to
66
4 Mean-Reverting Tendency in Stock Returns
Table 4.1 Unit root test results
Index returns
ADF
S & P CNX Nifty
CNX Nifty Junior
S & P CNX Defty
CNX 100
CNX 500
BSE Sensex
BSE 100
BSE 200
BSE 500
BSE Midcap
BSE Smallcap
CNX IT
CNX Bank Nifty
CNX Infrastructure
PP
Without trend
With trend
Without trend
With trend
-0.812
-1.38445
-0.79145
-1.87274
-1.07358
-0.79724
-0.95140
-1.61487
-0.65320
-1.21907
-1.27030
-1.77224
-0.91384
-1.45136
-1.67048
-1.31551
-1.69247
0.14329
-1.13632
-1.46939
-1.63950
-1.96257
-1.96262
0.13113
0.26128
-2.67936
-1.27534
-0.08777
-0.86015
-1.38200
-0.84017
-1.85944
-1.03165
-0.80977
-0.97153
-1.97679
-0.65687
-1.15561
-1.15293
-1.80138
-1.01344
-1.55606
-1.72273
-1.23069
-1.68636
-0.10249
-1.14111
-1.61725
-1.68309
-2.56982
-1.94189
0.27697
0.57394
-2.73258
-1.68351
-0.18893
Note The table reports augmented Dickey-Fuller (ADF) and Phillips-Perron test statistics for
model with trend and without trend. In the case of both ADF and PP tests, the critical values are
1 % = -3.43, 5 % = -2.86, and 10 % = -2.56 for model without trend, and 1 % = -3.97,
5 % = -3.41, 10 % = -3.13 for model with trend. ADF and PP tests examine the null
hypothesis of a unit root against the stationary alternative. The rejection of null in these two tests
without trend implies returns series is stationary with a nonzero mean, whereas rejection of null
with trend indicates returns series is stationary around a deterministic trend. The values presented
in the table are less than the critical values at all the significance level indicates a unit root in the
returns series
incorrect inference of nonstationarity. Hence, it is vital to see whether there is a
unit root in the returns process while simultaneously taking into account the
possible structural break/s. The trend break tests possess advantage over unit root
tests, and therefore are statistically powerful than the latter tests. Considering this,
Zivot-Andrews sequential break (model C) test, which searches for a break
endogenously, is employed and test results are reported in Table 4.2. The general
to specific procedure is followed to choose extra k regressors. Further, fraction of
data range to skip at either end when examining possible break is fixed as
0.15T (trimming region). Table 4.2 shows that Zivot-Andrews test statistics for all
index series are significant at 1 % level. Thus, this test provides evidence of meanreversion in Indian stock returns.
Figure 4.1 displays the plots of stock returns of 14 indices. The structural break
points for the selected series identified by the Zivot-Andrews test are significant as
it is evident from the minimum t-statistics on vertical axis corresponding to the
break point of each index in the figure. It is significant to investigate the possible
events associated with these structural breaks of different series. For the purpose,
the present study referred Annual Reports of RBI, SEBI, and Economic Surveys
and business newspapers to identify the associated events.
4.4 Empirical Findings
67
Table 4.2 Zivot-Andrew sequential trend break test statistics
Index
Trend
K
break
S & P CNX Nifty
CNX Nifty Junior
S & P CNX Defty
CNX 100
CNX 500
BSE Sensex
BSE 100
BSE 200
BSE 500
BSE Mid Cap
BSE Small Cap
CNX IT
CNX Bank Nifty
CNX
Infrastructure
2003:04:25
2003:03:31
2006:06:14
2008:01:09
2007:08:23
2003:05:12
2000:02:21
2003:04:29
2003:04:01
2008:01:08
2008:01:08
2000:02:21
2006:07:19
2008:01:09
7
2
7
5
0
7
0
6
1
2
2
5
5
5
Minimum T
statistic
-19.286*
-29.698*
-19.038*
-17.652*
-48.288*
-18.674*
-62.221*
-27.186*
-33.143*
-19.528*
-18.169*
-22.037*
-22.125*
-16.647*
Note The table reports Zivot-Andrews test statistics for Model C, which allows for break in both
intercept and trend. The test further allows testing for a unit root against the alternative of
stationary with structural change at some unknown point. The critical values are -5.57 and -508
for 1 and 5 %, respectively. Asterisked values indicate rejection of the null hypothesis at 1 %
level of significance
The analysis show that the structural break for CNX IT and BSE 100 in the year
2000 was associated with the global economic slowdown and the dot-com internet
bubble bust. The structural break for CNX Nifty, BSE Sensex, BSE 200, and BSE
500 occurred in 2003, when there was a rise in international oil prices. The break
points for CNX Bank Nifty and CNX Defty coincides with an unprecedented slide
of the rupee in 2006. The break point for CNX 100, CNX 500, and less liquid and
having lower market capitalization indices namely, CNX Infrastructure, BSE
Midcap, and BSE Smallcap, the structural break point was associated with the
global economic meltdown of 2008 and precipitated by the sub-prime crisis in the
US.
It may be pertinent to note that ignoring a structural break may leads to bias and
loss of the power of unit root test. In the same fashion, ignoring more than one
break reduces the power of the test and results in incorrect inferences. Motivated
by this concern, the present study applied Lee-Strazicich’s two structural breaks
test. The model CC of the Lee-Strazicich test allows for two shifts each in
intercept and trend. The test has an advantage over Zivot-Andrews and Lumsdaine-Papell’s multiple breaks test since it includes breaks both under null and
alternative hypotheses. The rejection of null in this test, unlike Zivot-Andrews, and
Lumsdaine and Papell without any ambiguity implies trend stationary and not
difference stationary. Table 4.3 provides the Lee-Strazicich test statistics along
with break dates. It is evident from the table that LM statistic is statistically
68
4 Mean-Reverting Tendency in Stock Returns
Zivot -Andrews Break Test for S & P NIFTY
-19.00
-19.05
-19.10
-19.15
-19.20
-19.25
-19.30
1999
2000
2001
2002
Jan’ 03 Oct‘03 2004
2005
2006
Dec’06
Zivot -Andrews Break Test for CNX NIFTY JUNIOR
- 29.45
- 29.50
- 29.55
- 29.60
- 29.65
- 29.70
1999
2000
2001
2002
Jan’ 03 Oct’03 2004
2005
Mar’06 Dec’06
Zivot -Andrews Break Test for S & P CNX DEFTY
18.70
18.75
18.80
18.85
18.90
18.95
19.00
19.05
1999
2000
2001
2002
Jan’03 Oct’03 2004
Zivot -Andrews Break Test for CNX
2005
Mar’06 Dec’06
100
-17.25
-17.30
-17.35
-17.40
-17.45
-17.50
-17.55
-17.60
-17.65
-17.70
2005
2004
Oct’05
2006
Zivot -Andrews Break Test for CNX
2007
2008
500
- 47.90
- 47.95
- 48.00
- 48.05
- 48.10
- 48.15
- 48.20
- 48.25
- 48.30
Jan’01
Oct’01
2002
2003
Mar’04
Dec’04
Fig. 4.1 Plot of index stock returns with structural break
2005
2006
2007
4.4 Empirical Findings
69
Zivot -Andrews Break Test for BSE SENSEX
-18.35
-18.40
-18.45
-18.50
-18.55
-18.60
-18.65
-18.70
2000
2001
2002
2002
2003
2004
Mar’05
Dec’05
2006
Dec’05
2006
Dec’05
2006
Zivot -Andrews Break Test for BSE 100
- 62.000
- 62.025
- 62.050
- 62.075
- 62.100
- 62.125
- 62.150
- 62.175
- 62.200
- 62.225
Jun’02
2001
2002
2002
2003
2004
Mar’05
Zivot -Andrews Break Test for BSE 200
- 26.975
- 27.000
- 27.025
- 27.050
- 27.075
- 27.100
- 27.125
- 27.150
- 27.175
- 27.200
2000
Jan’02
2001
Oct’02
2003
2004
Mar’05
Zivot -Andrews Break Test for BSE 500
- 32.90
- 32.95
- 33.00
- 33.05
- 33.10
- 33.15
Jan’02
-
Dec’02
2003
2004
2005
2006
Zivot -Andrews Break Test for BSE MIDCAP
19.10
19.15
19.20
19.25
19.30
19.35
19.40
19.45
19.50
19.55
2004
Fig. 4.1 (continued)
2005
2006
Mar’07
Dec’07
May’08
70
-
4 Mean-Reverting Tendency in Stock Returns
Zivot-AndrewsBreak Test for BSESMALLCAP
17.75
17.80
17.85
17.90
17.95
18.00
18.05
18.10
18.15
18.20
2005
2004
2006
Mar’07
Dec’07
Mayy’08
Zivot -Andrews Break Test for CNX IT
- 21.70
- 21.75
- 21.80
- 21.85
- 21.90
- 21.95
- 22.00
- 22.05
1999
2000
2001
2002
2000
2003
2004
2005
2006
2007
Zivot -Andrews Break Tests for B ANK NIFTY
- 21.90
- 21.95
- 22.00
- 22.05
- 22.10
- 22.15
2002
2003
2004
2005
2006
2007
Zivot -Andrews Break Test for CNX INFRA STRUCTURE
-16.1
-16.2
-16.3
-16.4
-16.5
-16.6
-16.7
2005
2005
2006
2007
2008
Fig. 4.1 (continued)
significant at 1 per cent, thus rejecting the null of unit root. This unambiguously
implies trend stationarity in returns series.
The break dates identified by Zivot-Andrews and Lee-Strazicich, (though
strictly not comparable) suggest different break points. The possible reason may be
the different methods of specification and identification of the break point.7 While
structural break points identified by Zivot-Andrews for BSE 100, CNX Bank Nifty
and BSE 500 are identical to first break point of Lee-Strazicich, and for indices
7
Zivot-Andrews is ADF type test while Lee-Strazicich is a Lagrange multiplier test.
4.4 Empirical Findings
71
Table 4.3 Lee-Strazicich LM unit root two structural breaks test statistics
Index
LM statistic
Trend
Break date
Trend
CNX Nifty
CNX Nifty
Junior
CNX Defty
CNX 100
CNX 500
BSE Sensex
BSE 100
BSE 200
BSE 500
BSE Midcap
BSE Smallcap
CNX IT
CNX Bank Nifty
CNX
Infrastructure
Break date
-23.81*
-18.65*
19.66*
16.83*
1999:01:15
2000:03:13
-19.41*
-18.07*
2008:01:09
2002:02:27
-19.79*
-18.17*
-27.91*
-19.53*
-18.67*
-36.47*
-27.91*
-30.19*
-18.44*
-25.97*
-22.58*
-17.56*
17.86*
14.72*
1.98
-17.68*
17.62*
-27.82*
18.29*
-4.50
15.48*
-19.51*
21.83*
17.16*
2003:10:24
2003:10:24
2001:07:09
2000:03:14
2000:02:25
1999:12:29
2003:05:08
2007:08:27
2007:08:21
2004:05:20
2006:07:19
2006:06:08
-18.23*
-16.17*
-10.50*
18.40*
18.21*
27.93*
3.40
-0.73
-13.37*
11.69*
16.99*
-15.56*
2006:07:06
2008:01:09
2007:09:03
2006:07:06
2003:09:12
2008:01:22
2004:06:18
2008:07:18
2008:03:17
2006:06:14
2008:01:02
2008:01:15
Note The model CC of the Lee-Strazicich test is employed which allows for two shifts each in
intercept and trend. The table reports (Lee and Strazicich 2003) two breaks test statistics. The null
is unit root with breaks and alternative hypothesis is trend stationary with breaks. Asterisked
values indicate rejection of the null at 1 % level of significance. The critical values of the test are
given in Lee and Strazicich (2003)
namely, CNX Defty, CNX 500, CNX 100, CNX Infrastructure, BSE Midcap, and
BSE Smallcap are identical to second break point identified by Lee-Strazicich test.
The break points for the rest of the indices (CNX Nifty, CNX Nifty Junior, BSE
Sensex, and BSE 200) are entirely different. This indicates the importance of
considering two structural breaks against single break test as latter ignores the
other structural breaks, which are important and such ignorance leads to incorrect
inferences. Besides, it also points out the significance of incorporating breaks both
under null and alternative hypotheses. In other words, multiple breaks test is
preferable to Zivot-Andrews’s single break test. Further, Lee-Strazicich test results
are preferable to other multiple structural breaks tests such as Lumsdaine-Papell,
which do not assume breaks under null and thus lead to incorrect inferences.
The Lee-Strazicich test results show that the break points identified around
break dates for various indices are different. Most of the break dates seem to have
occurred during 2000–2003 and 2006–2008. The first break point for CNX Nifty,
CNX Nifty Junior, BSE Sensex, BSE 100, BSE, 200, CNX 500 falls between 1999
and 2001. This was a period when the dot.com bubble was busted and led
recession in the US, and hijack of the Air India flight caused war hysteria between
India and Pakistan. It may also be noted that in March 2000, the government
notified the withdrawal of the ban on futures trading to pave way for derivative
trading in India.
72
4 Mean-Reverting Tendency in Stock Returns
The sluggishness of foreign institutional investors (FIIs), slip in consumer
spending and bad monsoon during 2003 made the market to move within a narrow
range. It was the year when the first break point for CNX Defty, BSE 500, and
CNX 100 was detected. The rise in international oil prices during March–May
2003 is one of the possible factors for the break in these indices. The first structural
break for BSE Midcap and BSE Smallcap occurred in 2007 is associated with
notorious sub-prime mortgage crisis and collapse of many giant investments banks
in a short span of time and there was sustained pull out of investment by FIIs from
Indian markets.
The second structural break point identified by the Lee-Strazicich test for BSE
100 and BSE 500 falls between 2003 and 2004, which coincides with bad monsoons, and international oil price shock. There was sustained pull out of FIIs from
the market and unprecedented slide of rupee in 2006. The second break points for
CNX Defty, CNX IT, and BSE Sensex occurred during this year. The second break
occurred in case of most indices such as CNX Nifty, BSE 200, CNX Bank Nifty,
CNX 100, CNX Infrastructure BSE Midcap, and BSE Smallcap during 2008. This
was the period of global meltdown triggered by sub-prime crisis, which spread to
the financial sector and resulted in an economic crisis. It is important note here that
in late 2007, the Securities and Exchange Board of India (SEBI) banned P-notes
meant for FIIs. The BSE Midcap, BSE Smallcap, CNX Infrastructure, CNX Bank
Nifty were more vulnerable to the financial crisis and market meltdown as they
have lower capitalization and compress less liquid stocks rather than other indices.
Since the trend break test has better power properties than conventional unit
root tests, the former test is better than latter. However, the multiple structural
breaks test is preferable to single structural break test because ignorance of multiple breaks leads to spurious results. Further, the Lee-Strazicich test is preferable
to other multiple breaks tests such as Lumsdaine-Papell because the former
includes breaks under null and alternative hypotheses, and therefore the rejection
of null unambiguously indicates trend-stationarity. In other words, the results of
Lee-Strazicich are unambiguous and reliable. The results of the present study
indicated trend stationary process in stock returns of the NSE and the BSE. The
different index series have different structural breaks. The difference in liquidity
and market capitalization of indices is one of the explanations for such differences.
The study observes that indices composing stocks having relatively less liquidity
and lower capitalization were quick in responding to the shocks particularly
external such as financial crisis, oil prices fluctuations, and global economic
meltdown than their high liquid and large cap counterparts.
4.5 Conclusion
The present chapter re-examines the issue of mean-reversion and structural break
in NSE and BSE. Zivot and Andrews (1992) sequential break test, and Lee and
Strazicich (2003) are employed on a sample of 14 indices of NSE and BSE
4.5 Conclusion
73
between 1997 and 2010. The conventional unit root tests results of the present
study indicate unit root process in stock returns. The Zivot and Andrews (1992)
test provides the mean-reverting tendency as the test strongly rejects null of unit
root for all the index returns. However, since the test assumes breaks only in
alternative hypothesis, the rejection of the null does not necessarily imply trend
stationarity. It may be only difference stationary and inferences would not be
reliable. Therefore, the Lee and Strazicich (2003) LM unit root test is performed,
which assumes breaks both under null and alternative hypothesis. The test results
clearly provide evidence of trend stationarity in Indian stock returns. This suggests
that the shocks trigged by structural or policy changes may have only a temporary
impact on stock returns and there is tendency for the returns to return to trend path.
The breaks that occurred in 2000, 2003, 2007, and 2008 are associated with
structural reforms, global economic recession, and ban on P-notes, sub-prime crisis
and economic meltdown. The study also suggests that the less liquid indices are
more vulnerable to the external shocks.
The substantial evidence of mean-reversion in Indian stock returns across
indices has important theoretical, practical, and policy implications. The observed
mean-reverting tendency indicate possibility of prediction of stock returns based
on past history of returns and thus clearly rejects EMH in the context of Indian
equity market. The financial sector reforms and changes in market microstructure,
which aimed at improving efficiency of market, have not brought desired results.
The external events have always created panic in the Indian equity market. The
events identified around trend breaks in the present study were mostly external
events. Whenever there were some shocks, it was found that there was net outflow
of FIIs. This calls for an appropriate regulation of external sector and FIIs and
further disclosure from them. It is found in the study that smaller indices were
more vulnerable to shocks than large indices. To improve the performance of small
indices having lesser liquidity, it is important to improve liquidity of such indices.
This can be achieved by encouraging retail trading in the market. Because, presently NSE and BSE together constitute 99.9 % of Indian market while trade is not
taking place in other 17 exchanges in India.
4.6 Variance Ratios, Structural Breaks and Nonrandom
Walk Behavior in the Indian Stock Returns
In view of the present evidence of structural breaks in stock index returns
presented in this chapter, it is important to analyze the behavior of stock returns
before and after the breaks, occurring due to financial and economic events.
In other words, it is essential to understand whether nonrandom walk behavior of
stock returns is consistent throughout the period or there are periods characterized
by random walk and periods by nonrandom walk behavior. In other words, the
sample may have periods of predictability and periods of unpredictability of stock
returns. To examine such possibilities, the whole sample is divided into three
74
4 Mean-Reverting Tendency in Stock Returns
sub-sample periods based on the break dates. Then, Wright (2000)’s ranks and
signs variance (WRSVR) ratio test is carried out on full and sub-sample periods.8
The description of the test is given here followed by discussion on test results.
Wright (2000) proposes ranks (R1 and R2) and signs (S1 and S2) based variance
ratio test. He demonstrates that the test has better power properties than conventional variance ratio test. Let r (yt) be the rank of yt among y1 … yT.
ðrðyt Þ Tþ1
2 Þ
ffi ...
r1t ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðT1ÞðTþ1Þ
12
ð4:10Þ
Under the null hypothesis that yt is generated from independent and identical
distribution (iid) sequence, r (yt) is random permutation of the numbers 1,…,
T with equal probability. Wright (2000) proposes the statistics
!
PT
ffi
2
1
2ð2k 1Þðk 1Þ 1=2
t¼kþ1 ðr1t þ r1t1 þ r1tk Þ
Tk
R1 ¼
1
P
T
1
2
3kT
t¼1 r1t
T
ð4:11Þ
which follows an exact sampling distribution. Further, he proposes use of an
alternative standardization
ffi
1 rðyt Þ
r2t ¼ U
ð4:12Þ
T þ1
where U is the standard normal cumulative distribution function. This gives rise to
the R2 statistics:
!
PT
ffi
2
1
2ð2k 1Þðk 1Þ 1=2
t¼kþ1 ðr2t þ r2t1 . . . þ r2tk Þ
Tk
R2 ¼
1
PT 2
1
3kT
t¼1 r2t
T
ð4:13Þ
The R2 test shares the same sampling distribution as R1. The critical values of
these tests can be obtained by simulating their exact distributions. In a similar
fashion, a signs based variance ratio test is as following:
!
PT
ffi
2
1
2ð2k 1Þðk 1Þ 1=2
t¼kþ1 ðSt þ St1 þ Stk Þ
Tk
1
S1 ¼
ð4:14Þ
PT 2
1
3kT
t¼1 St
T
8
The WRSVR test is used as an alternative test to other variance ratio test like Lo and
MacKinlay (1988). The latter test is found to be biased and right-skewed in finite samples
(Wright 2000; Charles and Darne 2008). For mean-reverting alternatives, the Lo and MacKinlay
test is found to be inconsistent (Deo and Richardson 2003). WRSVR test overcomes these
limitations. In light of non-normal distribution of Indian stock index returns, WRSVR, which is a
nonparametric test, is more appropriate than the conventional one.
4.6 Variance Ratios
75
Under the null hypothesis, yt is a martingale difference sequence (mds) whose
unconditional mean is zero, St is an iid sequence with mean zero constant variance
equal to 1, which takes the value of 1 and -1 with equal probability of . Thus, S1
assumes a zero drift value. The WRSVR test is carried out for the whole sample
and at different k values namely, 2, 5, 10, and 30. The R1 and R2 tests possess
better power properties than the conventional variance ratio tests. Wright (2000)
shows that, if S1 rejects the null, S2 must reject as well. Therefore, only S1 statistics
are reported in the present chapter. The WRSVR is appropriate as we found stock
returns are not normally distributed.
The test statistics (R1, R2, and S1) for the NSE are furnished in Table 4.4. The
results consistently support rejection of the null of iid for CNX Nifty Junior, CNX
Defty, CNX 500, CNX IT. The evidence for CNX Nifty, CNX 100, and CNX
Infrastructure is not consistent. It can be seen from the table that the R1 and R2 test
statistics reject the null of RWH for these index returns at short horizons and as
k values increase, rejection increasingly becomes weaker. It is to be noted that the
mean returns for these indices are higher than the rest. However, the S1 statistics,
which are consistently significant, reject the null of mds for all the selected indices,
indicates potential predictability of stock returns based on the past memory of the
returns. Broadly, it is observed that evidence against RWH for CNX Nifty and
CNX 100 for longer horizons (k = 30) are weaker than for short holding periods
(k = 5, 10). However, in the presence of significant S1 statistics, these indices are
not weak form efficient. Largely, stock returns of the indices traded on NSE exhibit
nonrandom behavior and thus provide space for speculation and resulting excess
Table 4.4 WRSVR test results—NSE: full sample
CNX
CNX
Nifty
CNX
100
Defty
Junior
Nifty
R1
k=
k=
k=
k=
R2
k=
k=
k=
k=
S1
k=
k=
k=
k=
CNX
500
CNX
IT
Bank
Nifty
CNX
Infra
2
5
10
30
3.37*
2.94*
2.11*
0.47
4.48*
4.54*
3.46*
3.75*
5.27*
6.63*
5.63*
5.87*
3.13*
1.42*
0.58
0.20
5.46*
5.43*
5.07*
4.66*
5.27*
6.63*
5.63*
5.87*
2.31*
1.11
0.63
0.39
3.95*
2.47*
1.29
0.79
2
5
10
30
2.43*
1.77*
1.26
0.26
4.21*
3.67*
2.81*
3.65*
5.39*
6.07*
5.03*
5.41*
2.82*
0.95
0.21
-0.13
5.31*
4.57*
4.15*
3.97*
5.39*
6.07*
5.03*
5.41*
2.88*
0.98
0.09
-0.16
3.47*
1.64
0.43
0.23
2
5
10
30
3.18*
3.24*
2.12*
-0.08
4.28*
4.67*
3.67*
3.12*
2.14*
3.85*
3.29*
4.41*
3.76*
3.09*
4.03*
6.22*
5.55*
5.60*
5.35*
6.23*
2.14*
3.85*
3.29*
4.41*
0.58
0.37
0.48
0.80
3.87*
4.44*
4.22*
5.78*
Note Table reports the test statistics for R1, R2, and S1 for holding periods 2, 5, 10 and 30 in panels
one, two, and three, respectively. The R1 and R2 tests null of independent and identical distributions (iid) and S1 tests null of martingale difference sequence (mds). Asterisked values indicate
rejection of the null at 5 % level of significance
76
4 Mean-Reverting Tendency in Stock Returns
Table 4.5 WRSVR test results—BSE: full sample
BSE Sensex
BSE 100
BSE 200
BSE 500
R1
k=
k=
k=
k=
R2
k=
k=
k=
k=
S1
k=
k=
k=
k=
BSE Midcap
BSE Smallcap
2
5
10
30
2.85*
2.40*
1.38
-0.09
3.73*
4.06*
3.31*
2.37*
4.17*
4.31*
3.66*
2.50*
5.56*
5.60*
5.69*
5.12*
7.08*
6.48*
4.91*
3.09*
8.17*
8.78*
7.49*
5.30*
2
5
10
30
2.06*
1.60
0.85
-0.15
2.84*
2.92*
2.35*
2.06*
3.35*
3.16*
2.66*
2.16*
5.03*
4.37*
4.29*
4.30*
6.73*
5.62*
3.72*
2.12*
8.23*
8.19*
6.33*
3.98*
2
5
10
30
2.54*
2.10*
1.21
-0.19
3.35*
3.41*
2.75*
1.79*
3.70*
3.89*
2.88*
1.86*
5.60*
5.35*
4.94*
4.73*
6.88*
7.80*
8.46*
10.29*
7.34*
8.96*
9.57*
12.01*
Note Table reports the test statistics for R1, R2, and S1 for holding periods 2, 5, 10 and 30 in panels
one, two, and three, respectively. The R1 and R2 tests null of independent and identical distributions (iid) and S1 tests null of martingale difference sequence (mds). Asterisked values indicate
rejection of the null at 5 % level of significance
returns. The results for CNX Bank Nifty suggest that the stock returns do follow
random walk at all the holding periods.
The test results for BSE presented in Table 4.5 shows that with the exception of
BSE Sensex, all other indices namely BSE 100, BSE 200, BSE 500, BSE Midcap,
and BSE Smallcap reject iid. The R1 test statistics for BSE Sensex at k = 2 and 5,
and R2 statistics at k = 1, are significant and thus rejects the null of iid. In other
words, rejection of the null is weak as k-value (i.e., holding period) increases. The
iid assumption can be relaxed, as it is difficult to find the iid because of regulatory
and structural changes in the market over a period of years. Therefore, the nonrejection of mds is sufficient to say market is weak form efficient. The results of S1
test, which is robust to heteroscedasticity, are given in the last panel of Table 4.5
reject the null of mds for all the BSE indices. However, similar to R1 and R2
statistics, the S1 statistics for BSE Sensex become weaker as the holding horizon
increases. This suggests that BSE Sensex may be moving toward weak form
efficiency in longer holding periods. It may be because of the existence of
abnormal profits in short horizons, which disappear in longer horizons as the
information begins to reflect in the current returns. Furthermore, it can be inferred
from Table 4.5 that indices having lower market capitalization and liquidity such
as BSE Smallcap and BSE Midcap show stronger rejection of RWH than the
relatively higher market capitalized indices such as BSE 100, BSE 200, and BSE
500.
Structural breaks occurring due to financial and economic events may have
bearing on the variance ratios. To examine such possibilities, the whole sample is
4.6 Variance Ratios
77
Table 4.6 WRSVR test results—NSE: period-I
CNX
CNX
Nifty
CNX
100
Defty
Junior
Nifty
R1
k=
k=
k=
k=
R2
k=
k=
k=
k=
S1
k=
k=
k=
k=
CNX
500
CNX
IT
Bank
Nifty
CNX
Infra
2
5
10
30
2.39*
2.59*
1.73
0.32
1.65
1.96
0.95
0.85
4.46*
4.43*
3.76*
2.26*
1.90*
1.65
1.66
1.99
3.78*
4.21*
3.61*
2.98*
5.15*
6.02*
5.11*
5.76*
3.72*
2.01
1.12
0.53
3.19*
2.22*
1.27
-0.44
2
5
1
30
1.47
1.43
0.92
0.11
0.85
1.00
0.40
0.73
3.36*
3.06*
2.63*
1.73
1.65
1.42
1.40
1.50
4.00*
3.99*
3.17*
3.05*
5.35
5.86
4.87
5.59
4.20*
1.86
0.67
0.15
3.16*
1.71
0.76
-0.75
2
5
10
30
2.80*
2.94*
1.50
-0.45
2.81*
3.42*
2.04
1.49
5.28*
4.99*
3.71*
1.64
3.10*
2.65*
2.83*
4.41*
2.79*
3.33*
3.14*
2.88*
2.02
3.41
2.81
3.75
1.73
1.05
0.85
0.79
2.92*
4.01*
4.19*
5.03*
Note Table reports the test statistics for R1, R2, and S1 for holding periods 2, 5, 10, and 30 in
panels one, two, and three, respectively. The R1 and R2 tests null of identical and independent
distributions (iid) and S1 tests null of martingale difference sequence (mds). Asterisked values
indicate rejection of the null at 5 % level of significance
divided into three sub-period samples. Period-I consists of sample from beginning
to the occurrence of first break. The period between first break and second break is
considered as Period-II and the post-second break period is named as Period-III.9
Then, WRSVR test is carried out on the three different sample periods. The
variance ratio statistics for Period-I are furnished in Tables 4.6 and 4.7 for NSE
and BSE respectively.
The iid and mds for indices namely, CNX Defty, CNX 500, CNX Infrastructure
is rejected at most of the holding periods during period–I and is true even in case
of full sample period. For other indices, namely CNX Nifty, CNX Nifty Junior,
CNX Bank Nifty, null of iid, and mds cannot be rejected, as the statistics are
insignificant. CNX 100 though cannot reject null of iid but still not support weak
form of EMH as S1 test rejects the null of mds. It is to be noted that evidence
against weak form of efficiency for Period-I are either weak or insignificant. In
case of rejection of null of random walk, the statistics are relatively less significant
during Period–I where no structural breaks occurred, than during the full sample
period. The results for BSE Sensex, BSE 100, and BSE 200 are insignificant across
the holding periods (Table 4.7). It is important to note that for the full sample, null
of iid and mds were rejected for these indices. Rest of the indices from BSE during
9
We consider the breaks identified by Lee-Strazicich. The first and second breaks dates found by
Lee-Strazicich are different for the chosen fourteen indices. The WRSVR test carried out each
index separately based on respective break dates. See Table 4.3 for information on break dates.
78
4 Mean-Reverting Tendency in Stock Returns
Table 4.7 WRSVR test results—BSE: period-I
BSE Sensex
BSE 100
BSE 200
R1
k=
k=
k=
k=
R2
k=
k=
k=
k=
S2
k=
k=
k=
k=
BSE 500
BSE Midcap
BSE Smallcap
2
5
10
30
0.84
0.7
0.1
-0.51
1.56
1.8
1.32
0.46
1.98*
1.61
0.83
-0.03
4.37*
4.73*
4.31*
2.88*
5.49*
4.30*
3.07*
0.69
6.12*
5.91*
5.14*
2.60*
2
5
10
30
-0.12
0.1
-0.19
-0.56
0.47
0.88
0.6
0.17
0.95
0.8
0.36
-0.16
3.87*
3.59*
3.09*
2.16*
5.41*
3.60*
2.16*
0.11
6.43*
5.54*
4.29*
1.74
2
5
10
30
0.34
0.11
-0.9
-1.05
1.57
1.86*
1.42
0.9
1.81
1.31
0.14
-0.55
4.73*
4.30*
3.15*
2.37*
5.79*
6.48*
7.53*
9.51*
6.14*
7.12*
7.59*
9.83*
Note: Table reports the test statistics for R1, R2, and S1 for holding periods 2, 5, 10, and 30 in
panels one, two, and three, respectively. The R1 and R2 tests null of independent and identical
distributions (iid) and S1 tests null of martingale difference sequence (mds). Asterisked values
indicate rejection of the null at 5 % level of significance
the Period-I, reject the null of random walk as in case of full sample period.
However, the test statistics are weakly significant for these indices compared to
those reported for the full sample period furnished in Table 4.5. In short, during
the Period-I, where no structural breaks were identified, either stock returns
characterized by random walk or rejection of random walk is significantly weak.
Therefore, market was largely weak form efficient during this period.
Tables 4.8 and 4.9 present test statistics of NSE and BSE, respectively, for the
Period-II and the results are statistically highly significant. The statistical significance of these values are statistically higher than the statistics reported for the
Period-I. It can be inferred that during the Period-II, the stock returns exhibited
nonrandom walk behavior. The values of S1 are less compared to R1 and R2 but
still greater than statistics reported for Period-I. Finally, Tables 4.10 and 4.11
furnish test statistics for Period-III for NSE and BSE respectively. The test statistics show higher significance than those reported for Period-II. Nevertheless,
evidence supports martingale process for BSE Sensex. This indicates that after the
second break, there was a stronger tendency in stock index returns to revert to
trend path. The results for different sub-periods indicate different kinds of behavior
of the stock returns in India. Apparent random walk behavior is observed before
occurrence of structural breaks in the series. Nevertheless, nonrandom walk
behavior is observed in the post-structural breaks periods. It implies that Indian
stock market is not weak form efficient for the whole period and sensitive to the
external shocks.
4.6 Variance Ratios
79
Table 4.8 WRSVR test results—NSE: period-II
CNX
CNX
Nifty
CNX
100
Defty
Junior
Nifty
R1
k=
k=
k=
k=
R2
k=
k=
k=
k=
S1
k=
k=
k=
k=
CNX
500
CNX
IT
Bank
Nifty
CNX
Infra
2
5
10
30
8.64*
11.38*
14.05*
21.44*
19.04*
34.50*
49.80*
84.70*
24.59*
44.67*
64.76*
111.04*
22.04*
39.93*
57.71*
97.71*
14.51*
22.73*
31.03*
51.44*
22.04*
39.93*
57.71*
97.71*
18.61*
33.69*
48.47*
80.65*
17.90*
32.01*
46.03*
76.89*
2
5
10
30
9.79*
13.74*
17.69*
26.86*
16.47*
29.98*
43.28*
73.35*
22.09*
40.16*
58.16*
99.69*
20.33*
36.76*
53.08*
89.93*
15.36*
23.99*
32.56*
52.76*
20.33*
36.76*
53.08*
89.93*
17.34*
31.26*
44.88*
74.77*
16.06*
28.34*
40.63*
66.71*
2
5
10
30
4.99*
4.62*
3.84*
3.03*
2.73*
3.04*
1.49
0.29
2.77*
2.88*
1.42
-0.54
1.63
2.16*
2.51*
3.74*
5.85*
5.66*
5.82*
6.99*
1.63
2.16*
2.51*
3.74*
-0.79
-0.58
-0.50
0.14
1.40
1.95*
1.93
2.22
Note: Table reports the test statistics for R1, R2, and S1 for holding periods 2, 5, 10, and 30 in
panels one, two, and three, respectively. The R1 and R2 tests null of identical and independent
distributions (iid) and S1 tests null of martingale difference sequence (mds). Asterisked values
indicate rejection of the null at 5 % level of significance
Table 4.9 WRSVR test results—BSE: period-II
BSE Sensex
BSE 100
BSE 200
R1
k=
k=
k=
k=
R2
k=
k=
k=
k=
S1
k=
k=
k=
k=
BSE 500
BSE Midcap
BSE Smallcap
2
5
10
30
13.18*
21.45*
29.24*
48.47*
17.09*
29.75*
41.79*
70.16*
11.66*
17.08*
22.09*
34.19*
16.14*
29.09*
41.62*
68.54*
14.44*
25.89*
36.88*
58.48*
11.55*
20.59*
28.98*
43.12*
2
5
10
30
13.07*
22.03*
30.41*
49.89*
15.60*
27.21*
38.28*
63.83*
12.94*
19.54*
25.78*
39.78*
14.87*
26.73*
38.10*
63.08*
13.66*
24.10*
34.26*
53.65*
11.09*
19.55*
27.53*
40.61*
2
5
10
30
3.57*
3.04*
2.24*
1.42
2.76*
3.01*
2.30*
1.70
5.81*
5.79*
5.58*
6.36*
3.74*
4.65*
3.89*
2.51*
3.45*
3.21*
3.08*
3.11*
2.29*
1.86
1.44
0.69
Note Table reports the test statistics for R1, R2, and S1 for holding periods 2, 5, 10, and 30 in
panels one, two, and three, respectively. The R1 and R2 tests null of identical and independent
distributions (iid) and S1 tests null of martingale difference sequence (mds). Asterisked values
indicate rejection of the null at 5 % level of significance
80
4 Mean-Reverting Tendency in Stock Returns
Table 4.10 WRSVR test results—NSE: period-III
CNX
CNX
CNX
CNX
100
Defty
NJ
Nifty
R1
k=
k=
k=
k=
R2
k=
k=
k=
k=
S1
k=
k=
k=
k=
CNX
500
CNX
IT
CNX
Bank
CNX
Infra
2
5
10
30
17.03*
30.77*
44.13*
72.46*
26.69*
46.33*
65.89*
113.70*
25.25*
45.89*
66.57*
114.17*
16.77*
30.28*
43.39*
70.73*
14.50*
22.55*
30.66*
50.66*
25.74*
46.61*
67.48*
115.26*
17.12*
30.88*
44.25*
72.68*
17.37*
31.31*
44.90*
73.46*
2
5
10
30
16.36*
29.51*
42.29*
69.56*
24.46
41.78*
59.21*
101.08*
23.16*
42.12*
61.09*
104.83*
15.75*
28.29*
40.34*
64.85*
15.46*
23.92*
32.51*
52.76*
23.85*
42.60*
61.20*
103.53*
16.26*
29.02*
41.35*
67.77*
16.16*
28.84*
41.27*
66.30*
2
5
10
30
2.09*
2.19*
0.92
-0.34
5.40*
5.24*
4.24*
4.18*
2.77*
2.96*
1.54
-0.54
3.32*
2.85*
3.51*
6.12*
5.85*
5.66*
5.82*
6.99*
2.91*
4.19*
4.02*
5.44*
-0.35
-0.27
-0.12
0.40
1.56
2.06*
2.18*
1.95
Note Table reports the test statistics for R1, R2, and S1 for holding periods 2, 5, 10, and 30 in
panels one, two, and three, respectively. The R1 and R2 tests null of identical and independent
distributions (iid) and S1 tests null of martingale difference sequence (mds). Asterisked values
indicate rejection of the null at 5 % level of significance
Table 4.11 WRSVR test results—BSE: period-III
BSE Sensex
BSE 100
BSE 200
BSE 500
R1
k=
k=
k=
k=
R2
k=
k=
k=
k=
S1
k=
k=
k=
k=
BSE Midcap
BSE Smallcap
2
5
10
30
25.06*
45.47*
65.67*
112.19*
29.27*
52.54*
75.52*
130.14*
11.66*
17.08*
22.09*
34.19*
26.50*
46.99*
67.68*
116.89*
12.62*
22.58*
31.92*
48.80*
15.43*
27.75*
39.64*
63.76*
2
5
10
30
22.79*
41.37*
59.48*
101.54*
25.77*
45.98*
65.77*
112.44*
12.94*
19.54*
25.78*
39.78*
23.74*
41.42*
59.27*
101.40*
12.13*
21.47*
30.31*
45.93*
14.57*
25.90*
36.88*
58.54*
2
5
10
30
0.96
0.73
-0.38
-0.76
4.33*
4.01*
3.20*
2.84*
5.81*
5.79*
5.58*
6.36*
5.54*
5.32*
4.97*
4.52*
2.88*
2.05*
1.65
1.52
2.85*
3.56*
4.00*
5.33*
Note Table reports the test statistics for R1, R2, and S1 for holding periods 2, 5, 10, and 30 in
panels one, two, and three, respectively. The R1 and R2 tests null of identical and independent
distributions (iid) and S1 tests null of martingale difference sequence (mds). Asterisked values
indicate rejection of the null at 5 % level of significance
4.6 Variance Ratios
81
The WRSVR test results clearly rejected null of iid and mds for majority of the
indices for the full sample period. However, evidence against null of random walk
for BSE Sensex and CNX Nifty are weaker in longer holding periods. This can be
attributed to the existence of excess returns in short period and as information
begins to reflect in returns, these profits disappear. Although the results of the
study indicate nonrandom walk behavior of stock returns in India for the whole
period, the sub-sample analysis of stock returns shows that structural events have
bearing on the behavior of stock returns. The Period-I has shown weaker evidence
against weak form efficiency. The results suggest that stock returns exhibited
stronger nonrandom walk behavior during the period characterized by occurrence
of structural breaks majorly due to external events. An increasing mean-reverting
tendency is observed in stock returns after structural breaks. It implies that the
Indian stock market is sensitive to the events especially the events occurred in
external sector.
The mean-reverting tendency in stock returns indicates possibility of predictability of stock returns. It is to be noted that indices such as BSE Sensex, BSE 100,
BSE 200, CNX Nifty, CNX Nifty Junior, which are having higher market capitalization exhibited weak evidence against random walk compared to smaller
indices such as BSE Midcap, BSE Smallcap, CNX Infrastructure, etc. which
provided strong evidence against random walk. However, it is to be noted that the
high cap and liquid index returns do not support the martingale process and thus
are not weak form efficient. The findings of the study suggest an increasing
nonrandom walk tendency in returns in India. Nonrandom walk behavior of stock
returns and vulnerability of stock market to the shock in particular the global
shocks indicate that Indian equity market is still a developing market and it call for
appropriate policy management of external shocks.
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Chapter 5
Long Memory in Stock Returns: Theory
and Evidence
Abstract Long memory is a characteristic of a data generating process, in which
autocorrelation function decays hyperbolically at a slower rate and the underlying
time series realizations display significant temporal dependence at very distant
observations. The issue of long memory though has important theoretical and
practical implications, has not received due importance in India. The present
chapter tests for the presence of long memory in mean of the stock returns by
employing a set of semiparametric tests. A comprehensive data sample from June
1997 to March 2010 is used for the analysis. The findings of the study suggest the
presence of long memory in mean returns. Furthermore, there are no significant
and consistent evidence which could suggest that smaller indices are generally
characterized by the long memory process. It implies a potential prediction of
future returns over a longer period. The use of linear model in the presence of long
memory would result in misleading inferences and this calls for further analysis of
long memory forecasting models.
Keywords ARFIMA
Autocovariance
Covariance stationary
Fractional
integration Hyperbolic decay Long memory Market efficiency Semipatrametric methods
5.1 Introduction
An important aspect of stock market returns that departs from random walk
hypothesis (RWH) is long memory or long-range dependence which gained much
attention over the last one and a half decade. Long memory is a characteristic of a
stationary process in which the underlying time series realizations display significant temporal dependence at very distant observations and autocovariances of
such a process are not absolutely summable. The autocorrelation function of stationary series decays hyperbolically at a slower rate in case of long-range
G. S. Hiremath, Indian Stock Market, SpringerBriefs in Economics,
DOI: 10.1007/978-81-322-1590-5_5, The Author(s) 2014
85
86
5 Long Memory in Stock Returns: Theory and Evidence
dependence. The persistent temporal dependence between distant observations
indicates the possibilities of predictability and hence provides an opportunity for
speculators to forecast future returns based on past information and to make extra
normal returns. Hence, the presence of long memory has an important theoretical
and practical implications. It invalidates the efficient market hypothesis (EMH)
which states that returns are generated by a random walk process so that it is not
possible to predict their future movements based on past information. The assetpricing model would also be invalid in the presence of long memory. Besides,
linear modeling would result in a misleading inference in the presence of long
memory. It is significant to note that perfect arbitrage is not possible when returns
exhibit a long-range dependence (Mandelbrot 1971). Furthermore, the derivative
pricing models which are based on Brownian motion and martingale process
become inappropriate in the presence of long memory in stock returns.1
The issue of long memory though has important implications for theory of
finance and practical applications, has not received much attention in India. In
view of the significance of long memory in financial time series, it is felt more
appropriate to examine the issue in the Indian context. Accordingly, the objective
of the present chapter is to examine the issue of long memory in Indian stock index
returns. The study uses data of 14 index series from June 1997 to March 2010
traded on the National Stock Exchange (NSE) and Bombay stock exchange (BSE),
the major exchanges in India.2 For empirical testing, the study carries out a set of
sophisticated time series tests, such as Geweke and Porter-Hudak (GPH) semiparametric, Robison’s Gaussian semiparametric and bias-reduced technique of
Andrews and Guggenberger. The rest of the chapter is organized as follows.
Section 5.2 gives a brief introduction of theory of long memory. Review of previous empirical work is presented in Sect. 5.3. In Sect. 5.4, testing methods
employed are explained. The empirical results are presented in Sect. 5.5 and the
last section provides the concluding remarks.
5.2 Theory of Long Memory
5.2.1 Meaning and Definitions
There are various definitions of long memory. According to McLeod and Hipel
(1978), a covariance stationary time series, Rt is said to exhibit long memory if
1
X
k¼1
1
jwðkÞj ¼ 1
ð5:1Þ
For an excellent and comprehensive discussion on theory of long memory see Beran et al.
(2013) and for long memory issues in Economics, see Teyssiere and Kirman (2007).
2
For details on selected sample, please see Chap. 1, Table 1.2.
5.2 Theory of Long Memory
87
where wðkÞ is the autocorrelation at lag k. This infinite sum condition suggests that
correlation at a very distant lags cannot be ignored. Long memory is usually
defined in terms of time domain and frequency domain. In time domain, a stationary discrete series Rt said to exhibit long memory if its autocovariances decay
hyperbolically. In symbols
wðkÞ k2d1 f1 ðkÞ;
k!1
ð5:2Þ
where d is the long memory parameter and f1 ð:Þ is a slowly varying function.
In frequency domain, a stationary stochastic discrete time series Rt is defined by
its spectral density function. This is represented as in the following Eq. (5.3)
f ðxÞ x2d f2 ð1=jxjÞ;
x!1
ð5:3Þ
for x in a neighborhood of zero and f1 ð:Þ is a slowly varying function. Following
Palma (2007), an alternative definition of long memory based on Wold decomposition can be given as
uj jd1 f3 ð jÞ;
j[0
ð5:4Þ
where f3 is a slowly varying function. Palma (2007) noted that further conditions
are required to be imposed to make these definitions necessarily equivalent.3
The long memory models have been in existence in Physical Sciences such as,
Geophysics. Hurst (1951) developed a rescaled range statistic (R/S) to study
long-range dependence in river flows. Mandelbrot (1972) applied the R/S test,
which compares the range of partial sums of deviation from the sample mean,
rescaled by the sample standard deviation, to stock returns. Later, Mandelbrot
and Van Ness (1968), Granger and Joyeux (1980), Hosking (1981) developed the
stochastic models which explain dependence over a long period. Granger and
Joyeux (1980) and Hosking (1981) introduced fractional differencing in autoregressive integrated moving average (ARIMA) framework. In other words, they
developed a fractional differencing model which allows a fractional value in
integration order of the ARIMA model. ‘‘The fractionally differenced process can
be regarded as a halfway house between the I(0) and I(1) paradigms’’ (Baillie
1996). The model is known as autoregressive fractionally integrated moving
average (ARFIMA) model. The fractional parameter can be estimated from the
data. This is one of the important models which is employed to examine long
memory properties of the times series realizations. The ARFIMA model has
special long memory properties which give extra potential in long run forecasting
(Granger and Joyeux 1980).
3
Further discussion on these conditions can be found in Palma (2007).
88
5 Long Memory in Stock Returns: Theory and Evidence
5.2.2 ARFIMA Model
Granger and Joyeux (1980), and Hosking (1981) propose the autoregressive
fractionally integrated moving average (ARFIMA) model. Following Palma
(2007), a time series fyt g follows ARFIMA (p, d, q) process if
/p ðBÞyt ¼ hq ðBÞð1 BÞd et
ð5:5Þ
where /p ðBÞ ¼ 1 þ /1 B þ þ /p Bp and hq ðBÞ ¼ 1 þ h1 B þ þ hq Bq are,
respectively, autoregressive and moving average polynomials of orders p and q,
and B is back shift operator. It is assumed that the /(B) and h(B) have no common
roots (1 - B)–d, is fractionally differencing operator defined by binomial
expansion.
ð1 BÞd ¼
1
X
j¼0
nj B j ¼ nðBÞ
ð5:6Þ
where
nj ¼
Cðj þ dÞ
Cðj þ 1ÞCðdÞ
ð5:7Þ
where C denotes the gamma function. For d \ d = 0, -1, -2… and {et} is a
white noise sequence with finite variance.
The parameter d determines the memory process. If d [ 0, the process exhibits
long memory. If d = 0, the process has short memory and when d \ 0, the process
is called anti-persistent and displays negative memory. If d [ –0.5, the ARFIMA
process is invertible and has linear Wold representation and if d \ 0.5, it is
covariance stationary. Therefore, if 0 \ d \ 0.5, the process is stationary and
exhibit long memory.4 Various methods are used in empirical work to estimate the
Hurst exponent and fractional parameter. The Mandelbrot’s rescaled range (R/S)
statistic, modified R/S test proposed by Lo (1991), parametric and semiparametric
testing methods5 are used to explore long memory process in returns.
5.3 Review of Previous Work
The first systematic empirical study of long memory was conducted by Greene and
Fielitz (1977). They employed Hurst’s (1951) rescaled (R/S) statistic on 200 individual stocks on the New York stock exchange (NYSE) and found that the US stock
returns contain long memory. Later, Aydogan and Booth (1988) who find no
4
If the estimated d value is greater than 0.5 but less than 1, it still indicates mean reversion. It is
also known as nonstationary long memory.
5
For a review on long memory econometric methods, see Baillie (1996).
5.3 Review of Previous Work
89
evidence of long memory concludes that the results obtained from R/S statistic are
subject to the underlying restrictive assumptions of the R/S test. Peters (1989)
assesses randomness of S & P stock prices. Using the percentage of stock price, he
reported biased random walk (long-term dependence). He attributes the observed
persistence to market sentiments prevailed in the past. Later, pointing out the inappropriateness of the use of the percentage of the price in his previous paper, Peters
(1992) uses logarithmic returns. Nevertheless, the findings of the study are in perfect
agreement with the previous study. Findings of Mills (1993), however, do not support
the presence of long memory in the UK stock returns; whereas Goetzmann (1993)
shows that the NYSE and the LSE stock returns are characterized by long memory.
Lo (1991) challenges the findings of Greene and Fielitz (1977) and questions
the R/S method of Mandelbrot (1972). He demonstrates that in the presence of
short run dependence in the form of heteroscedasticity, R/S test significantly
becomes a biased estimator. A modified R/S test which is robust to non-normality
and heteroscedasticity was proposed by Lo (1991). He provides contrary evidence
of nonexistence of long memory in the US stock returns. Lo’s (1991) modified R/S
test subsequently became one of the popular tests employed in empirical research
to detect long-range dependence. Using both classical and modified R/S tests,
Ambrose et al. (1993) reject the presence of long-range dependence. Chow et al.
(1995) conducting similar tests reject long-range dependence in series and conclude that the random walk model validly describes stock market returns behavior.
Similarly, Barkoulas and Baum (1996) conduct analysis of seven sectors and 30
companies included in the Dow Jones industrial index (DJIA). They find consistent
evidence in support of long-range dependence in the US stock indices. However,
the study reports fractional dynamics in the individual returns series. In his
investigation of long memory in five markets of Europe, the US, and Japan;
Jacobsen (1996) identifies nonexistence of long-term dependence with the
exception of Germany and Italy. Hiemstra and Jones (1997) find presence of long
memory process confined to only a tiny segment of stocks. A study by Blasco and
Santamaria (1996) which covers stock returns of IGBM index of Spain, and
sectoral indices for the period 1990-1993 observes long memory in time series
and but during extremely long periods, find weak evidence of long memory.
The modified R/S statistics proposed by Lo (1991) has a complicated asymptotic distribution when the null is true (Lobato and Savin 1998b). Furthermore, it is
difficult to distinguish between short and long memory in Lo’s framework. Baillie
(1996) provides simulation evidence which is not favorable to Lo’s (1991)
approach. Researchers hence attempted to estimate fractional integration through
parametric and semiparametric approaches. Cheung and Lai (1995) in addition to
modified R/S analysis, employ GPH test to explore whether the findings of Lo
(1991) are unique to the US or stock returns of other countries would also exhibit
such dependence. The study made use of Morgan Stanley International Capital
indexes of 17 countries including the US. The R/S test results resoundingly rejects
long-term dependence in stock returns of all the 17 markets considered for the
study, while the GPH test provided evidence of long memory only for five
countries. The findings of the study are consistent with the findings of Lo (1991).
90
5 Long Memory in Stock Returns: Theory and Evidence
Nagasayu (2003) documents long memory in Japanese stock market and concluded that financial reforms could not improve efficiency as long-range dependence was detected in the post reform period also. Using Hurst (1951) exponent,
modified R/S statistic, GPH and Robinson’s frequency domain tests; Sadique and
Sivapulle (2001) identify long memory in stock returns of Korea, Malaysia, Singapore, and New Zealand. The parametric test of Lee and Schmidt (1996), and
semiparametric methods carried out by Henry (2002) indicates the presence of
long memory in stock returns of Germany, Japan, South Korea, and Taiwan.
Another test of long memory proposed by Lobato and Robinson (1998a) was
applied on daily data on individual stocks in the DJIA by Lobato and Savin
(1998b). The results suggest no significant long memory process in the returns
series. The work by Caporale and Gil-Alana (2004) show no evidence against the
null of absence of long memory for S & P index returns. Hence, they recommend a
standard model of the first difference as an appropriate model for stock returns than
the fractionally integrated model. Gil-Alana (2006) while refuting the persistence
in stock returns of Amsterdam, Frankfurt, Hong Kong, London, Paris, Singapore,
and Japan, concludes that returns followed a unit root process. However, Tolvi
(2003) reported that three of six indices of OECD countries exhibit long-range
dependence. He suggested that the outliers should be taken into account as
otherwise potential outlier biases results. According to Tolvi (2003) October/
November of 1987 (Great Market Crash) was found to be an important outlier to
bias the results of the study. Furthermore, Grau-Carles (2005) probe the issue of
long memory in the US market by using returns series of S & P 500 and DJIA and
found no significant long-range dependence.
It may be noted from the foregoing review that most of the studies, unsurprisingly, have focused on well-developed markets. However, it is interesting to
see whether stock returns of emerging equity markets, which are supposed to
possess frictions exhibit long memory properties. Since they are relatively less
developed than their developed counterparts, it is believed that stock returns of the
emerging markets may be characterized by the long memory process. Using
parametric and semiparametric estimation procedures, Limam (2003) investigates
long memory properties in 14 markets ranging from developed markets, like the
USA, the UK, and Japan, to emerging ones including the Arab markets. The study
suggests that long memory is more persistent in thin markets rather than welldeveloped markets. Further, the study attributed the long memory process
observed in Arab countries to the peculiar characteristics and environment of these
economies such as weakness of regulatory framework, lack of transparency,
openness to foreign investors etc.
The proposition of the presence of long memory in emerging markets, however,
has not remained unchallenged. Brazil did not exhibit long memory patterns,
irrespective of post-Real Plan6 (Resende and Teixera 2002). The evidence for the
6
Brazil introduced structural reforms, known as Real stabilization plan in 1994 keeping
objective to stabilize macroeconomic uncertainties.
5.3 Review of Previous Work
91
Brazilian stock market drew further support from Cavalcate and Assaf (2005).
They establish that process of equity prices cannot be explained by differences in
institutions and information flows. This view was confounded by evidence from
China, the most important emerging market. Cajueiro and Tabak (2006), after
examining Chinese data, documents strong evidence of long-range dependence in
Share B and weak evidence in Share A series. They attributed information
asymmetry and liquidity as the factors responsible for observed discrepancies.
The review of previous work shows that the issue of long memory remains
unresolved. Earlier studies probing long memory in a return series largely
employed R/S test and thereafter Lo’s (1991) test became a popular test of long
memory. Later studies exploiting ARFIMA model estimated fractional integration
through various parametric and semiparametric methods. Previous work largely
focused on well developed markets. Many studies pointed out that the Great crash
of 1987 has altered the time series properties of returns in developed markets. The
thinness of the market is cited as an important factor inducing long memory in
emerging markets. Some of the studies held that informational flow/asymmetry
explains long memory in developing markets. Although evidence from emerging
markets are mixed but relatively these markets, as reported in empirical research,
indicate long memory process in the mean returns. This view provides the necessary background and motivation for the present study to detect long memory in
one of the fastest emerging markets like India. With the exception of the study by
Nath (2001), there is no empirical work in India on long memory in stock returns.
Nath’s (2001) study was based on a conventional test which has restrictive
assumptions and data was confined to NSE Nifty returns. The present study has
made many improvements to analyze the issue. The study uses semiparametric
methods which are known for better power properties and consistent estimation. It
is perhaps the first study to use the Andrews and Guggenberger (2003) test of long
memory. As mentioned, the study uses updated and disaggregated data both from
NSE and BSE. The sample characteristics make the results of the present study
robust and reduce the risk of overemphasizing the generality of the findings.
5.4 Testing Methods
The test of long memory in the present study is carried by utilizing the ARFIMA
model. To estimate fractional integration, Geweke and Porter-Hudak (1983)
semiparametric test, Robinson’s (1995) Gaussian semiparametric test and,
Andrews and Guggenberger (2003) test are employed. Parametric tests require
correct specification of p and q for consistent estimation of fractional integration.
Therefore, Robinson (2003) suggests semiparametric tests. A brief description of
the tests carried out in this study is given here.
92
5 Long Memory in Stock Returns: Theory and Evidence
5.4.1 Geweke and Porter-Hudak Semiparametric Test
Geweke and Porter-Hudak (1983) (GPH) proposes a semiparametric approach to
estimate the d. GPH test is simple in application and robust to non-normality.
Under the assumption that the spectral density of stationary process may be
written as
ffi2d
k
f ðkÞ ¼ f0 ðkÞ 2 sin
ð5:8Þ
2
The following regression method is considered for parameter estimation.7
Taking logarithms on both sides of (5.8) and evaluating the spectral density at the
Fourier frequencies kj = 2pj/n, we have
ffi
ffi
f 0 kj
kj
:
ð5:9Þ
log f ðkj Þ log f0 ð0Þ d log 2 sin
þ log
f0 ð0Þ
2
On the other hand, the logarithm of the periodogram I (kj) may be written as
ffi
I ð kJ Þ
log I kj ¼ log
ð5:10Þ
þ log f ðkj Þ
f ðkj Þ
Now, combining (5.9) and (5.9), it can be written as
ffi
I kj 2 sin k2
kj
log I kj ¼ log f0 ð0Þ d log 2 sin
þ log
f0 ð0Þ
2
ð5:11Þ
By defining yj = log I (kj), a = log f0 (0), b = -d, xj = log [2 sin(kj/2)]2, and
(
2d )
I kj 2 sin k2
ǫj ¼ log
;
ð5:12Þ
f 0 ð 0Þ
The following regression equation can be obtained
yj ¼ / þbxj þ ǫj:
ð5:13Þ
In theory, one could expect that for frequencies near zero (that is, for j = 1…
m with m n
f kj f0 ð0Þ½2 sinðkj =2Þ2
ð5:14Þ
So that
7
Discussion of the test is based on Palma (2007).
5.4 Testing Methods
93
" #
I kj
ǫj log :
f kj
The least squares estimate of the long memory parameter d is given by
Pm
j¼1 xj x ðyj yÞ
b
ð5:15Þ
dm ¼
Pm
2
j¼1 ðxj xÞ
Pm
y
P
j¼1 j
where x ¼ m
:
j¼1 xj =m and y
m
The bandwidth m must be chosen such that for T ? ?, m ? ?, m/T ? 0. The
estimates are sensitive to the number of special ordinates from periodogram of
returns (m). The GPH in the present study is performed choosing values m = T0.50,
T0.55, and T0.60.
5.4.2 Robinson’s Gaussian Semiparametric Test
Robinson (1995) suggests a Gaussian semiparametric estimate of the self-similarity parameter H. It is assumed that the spectral density of the time series,
denoted by f(.), behaves as
f ðkÞ Gk12H
as
k ! 0þ
ð5:16Þ
for G [ (0,?) and H [ (0,1). The self-similarity parameter H relate to the long
b , is obtained
memory parameter d by H = d + 1/2. The estimate for H, denoted by H
through minimization of the function
b ðH Þ ¼
RðH Þ ¼ log G
v
1X
log kd
v d¼1
ð5:17Þ
P
b ðH Þ ¼ 1 v k2H1 I ðkd Þ. The discrete averaging is
with respect to H, where G
d¼1 d
v
carried out over the neighborhood of zero frequency and, m is assumed to be
tending to infinity much more slowly than does T under asymptotic theory. The
Gaussian semiparametric proposed by Robinson (1995) is consistent under mild
conditions and is asymptotically normal.
5.4.3 Andrews and Guggenberger Bias-Reduced Test
Andrews and Guggenberger (2003) develop a bias-reduced log periodogram
method to estimate long memory parameter. The method is the same as that of the
GPH estimator except that it includes frequencies to the power 2k for k = 1, 2…, r,
for some positive integer r, as additional regressors in the pseudo regression model
94
5 Long Memory in Stock Returns: Theory and Evidence
that yields GPH estimators. This estimation method eliminates the first- and higher
order biases of GPH estimator.
The fundamental frequencies for a sample size n denoted as
hni
2pj
for j ¼ 1; . . .;
kj ¼
ð5:18Þ
2
n
The estimator dbr (of the long memory) is defined to be the least squares estimator of the coefficient on -2 log ki in a regression of log of the periodogram.
The AGBR adds regressors k2j k3j . . .; k2r
j to the regression model. When r = 0, dr
is asymptotically equivalent to the standard GPH. Andrews and Guggenberger
(2003) suggest that bias-reduced log-periodogram estimator performs well for the
small values of r such as r = 1 and r = 2. The simulation results of Nielsen and
Frederiksen (2005) demonstrates that the test not only outperforms semiparametric
tests but also the correctly specify time domain parametric methods.
5.5 Empirical Evidence
The GPH test of long memory assumes relevance as it is robust to non-normality
and returns in the present study found non-normal distribution (see Chap. 2,
Table 2.1). The GPH test is performed on the daily stock returns of 14 indices and
the results are reported in Table 5.1. The number of special ordinates from periodogram of returns (m) to include in the estimation of d must be chosen judiciously as otherwise they produce an inaccurate estimation of d. The value of d is
estimated choosing m = T0.50, T0.55 and T0.60. It is clearly evident from the table
that long-range dependence structure exists in most of the stock indices. The
values for all stock indices are positive and range between lowest 0.021 for CNX
Bank Nifty, to highest 0.228 for CNX Infrastructure. However, the value of d for
CNX Nifty is negative at m = 0.5 and 0.55 and for CNX Defty it is negative at m =
0.5. This may be due to the sensitivity of the test to the chosen ordinates. Broadly,
the results indicate long memory in stock returns.
The test statistics of Robinson’s Gaussian semiparametric estimates of d are
provided in Table 5.2. The value of d is estimated using T0.50, T0.55, and T0.60. The
results obtained from RGSE are quite different from GPH test results. The value of
the fractional differencing parameter d is within the theoretical value. However,
the estimated d reported for BSE Sensex, BSE 200, CNX 500, and CNX 100 at
T0.75, and for BSE 100, Bank Nifty and CNX Infrastructure at T0.75 and T0.80 is
negative in this case. These negative values of d suggest anti-persistence. The
remaining stock return series are characterized by the long memory process.
Furthermore, the AGBR test is employed on stock returns of all 14 indices from
the NSE and the BSE. The AGBR test substantially mitigates the finite sample
bias. In other words, it eliminates the first and higher order biases of GPH. The
value of d is estimated with r = 1 and r = 2 and the results are furnished in
5.5 Empirical Evidence
95
Table 5.1 GPH estimates of ‘d’
Index returns
GPH estimator
m = 0.50
m = 0.55
m = 0.60
S & P CNX Nifty
CNX Nifty Junior
S & P CNX Defty
CNX 100
CNX 500
BSE Sensex
BSE 100
BSE 200
BSE 500
BSE Midcap
BSE Smallcap
CNX IT
CNX Bank Nifty
CNX Infrastructure
-0.119
0.039
-0.022
0.195
0.006
0.068
0.045
0.038
0.021
0.164
0.093
0.051
0.014
0.228
-0.050
0.075
0.034
0.171
0.023
0.076
0.075
0.061
0.130
0.114
0.054
0.052
0.086
0.207
0.034
0.161
0.089
0.143
0.077
0.102
0.114
0.101
0.165
0.108
0.111
0.066
0.075
0.103
Note Value in each cell of the table represents fractional integration, d, estimated by GPH
semiparametric method. The values of ‘d’ obtained by choosing m = T0.50 , T0.55 and T0.60 ,
T. m is special ordinates from periodogram of returns. The positive values in the table are
significant at 5 % level
Table 5.2 RGSE estimates of ‘d’
Index returns
S & P CNX Nifty
CNX Nifty Junior
S & P CNX Defty
CNX 100
CNX 500
BSE Sensex
BSE 100
BSE 200
BSE 500
BSE Midcap
BSE Smallcap
CNX IT
CNX Bank Nifty
CNX Infrastructure
RGSE
0.50
0.55
0.60
-0.007
0.023
0.010
-0.006
-0.001
-0.018
-0.011
-0.173
0.033
0.069
0.103
0.021
-0.053
-0.002
0.020
0.057
0.036
0.011
0.009
0.015
-0.007
-0.233
0.055
0.065
0.115
0.014
-0.004
-0.007
0.020
0.077
0.036
0.044
0.018
0.025
-0.08
-0.35
0.075
0.130
0.196
0.018
0.050
0.039
Note The values given in the table are the estimates of ‘d’ computed following RGSE method.
These values are obtained by conducting tests with power, 0.50, 0.55 and 0.60. The positive
values in the table are significant at 5 % level
96
5 Long Memory in Stock Returns: Theory and Evidence
Table 5.3 AGBR estimates of ‘d’
Index returns
r=1
r=2
S & P CNX Nifty
CNX Nifty Junior
S & P CNX Defty
CNX 100
CNX 500
BSE Sensex
BSE 100
BSE 200
BSE 500
BSE Midcap
BSE Smallcap
CNX IT
CNX Bank Nifty
CNX Infrastructure
-0.074
-0.032
0.0379
0.031
0.132
0.141
0.097
0.116
0.048
0.080
0.096
0.036
0.095
0.145
0.040
0.008
0.123
0.264
0.389
0.180
0.112
0.143
-0.008
0.205
0.028
0.074
0.371
-0.020
Note The biased reduction estimation is performed with bandwidth m equal to square root of the
number of observations. Andrews and Guggenberger (2003) suggest small values of r for better
performance of the estimation. Accordingly test is performed with r = 1, and 2, r being the nonnegative integer. The positive values in the table are significant at 5 % level
Table 5.3. It can be observed from the table that the value of d is less than 0.5 and
thus indicate long memory process in the mean. The value of d is ranging between
0.04 for CNX Nifty to 0.38 for CNX 500. Nevertheless, the negative value of the
fractional parameter is evident from Table 5.3 for CNX Nifty and CNX Nifty
Junior at r = 1, and BSE 500 and CNX Infrastructure at r = 1 and r = 2. The results
from AGBR thus are in consonance with the results obtained from GPH and RGSE
which also indicated long memory.
The empirical results provide mixed evidence of long memory in the mean of
the stock returns of different indices traded on the NSE and the BSE. The antipersistence evidence is not consistent for tests conducted. However, results
broadly indicate stationarity process.
5.6 Concluding Remarks
The present chapter attempted to examine the issue of long memory in the Indian
stock market. The study employed three tests namely, Geweke and Porter-Hudak
(1983), Gaussian semiparametric test of Robinson (1995), and Andrews and
Guggenberger (2003) for eight NSE indices and six BSE indices. Mixed evidence
for indices is provided by the GPH and RGSE tests. The test results of AGBR are
quite definite and suggest long-range dependence. The findings of the study largely
suggest the existence of long memory in mean returns of the most of the indices.
Furthermore, there are no significant and consistent evidence which could suggest
5.6 Concluding Remarks
97
that smaller indices are generally characterized by the long memory process. It
may be inferred from the findings that stock returns in India are not characterized
by the random walk process. It implies rejection of weak form efficient in case of
Indian stock market. The tendency of mean-reversions indicates the possibility of
prediction and speculative abnormal profits in these two premier exchanges. This
has a practical implication for market participants. It implies a potential prediction
of future returns over a longer period. The use of linear model in the presence of
long memory would result in misleading inferences. This calls for examination of
long memory forecasting models which can generate speculative profits.
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Chapter 6
Long Memory in Stock Market Volatility
Abstract Long memory in variance or volatility refers to a slow hyperbolic decay
in autocorrelation functions of the squared or log-squared returns. The conventional volatility models extensively used in empirical analysis do not account for
long memory in volatility. This chapter revisits the Indian stock market by using
the fractionally integrated generalized autoregressive conditional heteroscedasticity (FIGARCH) model. For empirical modeling, daily values of 14 indices from
the National Stock Exchange (NSE) and Bombay Stock Exchange (BSE) from
June 1997 to March 2010 are used. The results of the study confirm the presence of
long memory in volatility of index returns. This shows that FIGARCH model
better describes the persistence in volatility than the conventional GARCH models. Against the evidence of fractional behavior of volatility in Indian stock market,
it is essential to factor the long memory in derivative pricing and value at risk
models.
Keywords Long memory Fractional integration Volatility FIGARCH
Emerging markets Adaptive market hypothesis Heteroscedasticity Leptokurtic
Quasi maximum likelihood
6.1 Introduction
The rapid growth of Indian economy in the recent past and the increasing importance of the Indian equity market in the global finance have attracted the attention of
investors across the globe. At the same time, the stability of the stock market, which
is exposed to global environment, is crucial from the point of view of investors, and
it is an indispensable part of public policy. Thin-trading, high volatility and various
frictions generally characterize the stock markets of emerging economies. The
volatility is a measure of risk exposure and hence volatility forecasting is significant
for the economic agents. In other words, volatility is an indicator of vulnerability of
financial markets and the economy. The volatility forecasting has also been
G. S. Hiremath, Indian Stock Market, SpringerBriefs in Economics,
DOI: 10.1007/978-81-322-1590-5_6, The Author(s) 2014
99
100
6 Long Memory in Stock Market Volatility
essential for option pricing, value at risk modeling, and portfolio management and
investment strategies. Hence, there has been increasing interest among researchers,
investors, and practitioners to understand the behavior of the Indian stock volatility.
Modeling long memory in volatility has gained much importance in recent
years due to its practical implications. In empirical studies, absolute returns,
squared and log-squared returns were used as proxies of returns volatility. A large
volume of literature focuses on modeling volatility. The unconditional volatility
models which assume that volatility would be constant are the oldest ones found in
the literature. Later, scholars have recognized the fact that volatility cannot be
constant as it evolves over time and shocks persist for a longer time. Hence,
several conditional volatility models have been proposed to capture the volatility
persistence properties in conditional variance. Autoregressive conditional heteroscedasticity (ARCH) and generalized ARCH (or GARCH) proposed by Engle
(1982) and Bollerslev (1986) respectively, are the most popular among them.
However, these models do not capture long memory in volatility. Sometimes,
autocorrelation of the returns decays at a slower rate. The slow mean-reverting
hyperbolic rate decay in the autocorrelation functions of squared, log-squared
returns is defined as long memory in variance or volatility process.
Granger and Joyeux (1980), Hosking (1981) have introduced a model of fractional difference in the mean process which is known as autoregressive fractionally
integrated moving average (ARFIMA). On similar lines, Baillie et al. (1996)
propose a fractionally integrated GARCH (or FIGARCH) model which introduces
a fractional difference operator in the conditional variance function. The presence
of long memory in the conditional variance masks the true dependence structure.
Further, perfect arbitrage is not possible when returns display a long-range
dependence (Mandelbrot 1971). The derivative pricing models, which are based on
Brownian motion and martingale process, also become inappropriate in the presence of long-range dependence. The value at risk models which use short memory
as input possibly leads to incorrect inferences.
The issue of long memory has important implications for the theory of finance
and also significant for practical applications. Nevertheless, there is lack of work
on this issue in case of India. In the light of this backdrop, the present chapter
examines the presence of long memory in volatility in the Indian stock returns by
using FIGARCH model. This is perhaps the first study, which examines the issue
of long memory in volatility in the Indian context and thus extends the literature on
Indian stock market volatility. The Indian economy has registered a tremendous
growth in the recent past and the financial sector reforms coupled with market
microstructure changes have given much impetus for the growth of the stock
market. The economy in the past decade has not only witnessed rapid growth, but
also faced financial crisis at different points of time leading to erratic fluctuations
in stock prices. In this context, this study which uses updated and disaggregate data
set covering the period of such structural changes assumes relevance. The multiple
choice of the indices from the National Stock Exchange (NSE) and Bombay Stock
Exchange (BSE) helps to assess the sensitivity of empirical results with respect to
their different composition.
6.1 Introduction
101
The rest of the chapter is organized as follows: Sect. 6.2 presents a brief review
of the previous work on long memory in volatility primarily from the emerging
markets. The methodology followed in the study is described in Sects. 6.3 and 6.4
discusses the empirical results. The last section presents the important observations
of the chapter.
6.2 Review of Previous Work
There are several studies which have focused on long memory in volatility in the
developed markets, particularly the USA (see, Ding et al. (1993); Crato and Lima
(1994); Ding and Granger (1996); Andersen and Bollerslev (1997); Granger et al.
(1997); Comte and Renault (1998); Lobato and Savin (1998); Andersen et al.
(2003); Gurgul and Wojtowicz (2006)). However, there has been little focus on the
issue of long memory in the context of emerging markets barring a few studies in
the recent past, which have provided some evidence of long memory in volatility.
This section provides a brief review of previous work, particularly recent studies
on emerging markets. Cavalcante and Assaf (2005) report strong dependence in
the absolute and squared returns series of Brazilian market during the period
1997–2002. The MENA markets, namely, Egypt, Jordon, Morocco and Turkey,
exhibit significant long memory in volatility, but long memory was not because of
sudden shifts in variance (Assaf 2007). This view draws support from Kang and
Yoon (2008) who argue that the long memory in volatility is inherent in the data
generating process and not because of any shocks. In contrast, Korkmaz et al.
(2009) prove that unfiltered index returns in Turkey display strong evidence of
long memory but after treating structural breaks properly, the results show weak
evidence. The study thus puts that long memory in volatility is the result of the
occurrence of structural breaks.
The studies from Turkey provide evidence of long memory in returns volatility.1 Killic (2004) set out to examine the issue of long memory volatility process in
Turkey. Using data of the period 1988–2003, the study indicates presence of long
memory volatility process. Kasman and Torun (2007) extend the data period up to
2007, and report long memory both in mean and variance. These findings of long
memory characterization of volatility in Turkey draw further support from DiSario
et al. (2008). The study of Floros et al. (2007) based on data covering the period
from 1993 to 2006 of Bolsa de Valores de Lisboa Porto (BVLP) stock exchange
conclude that long memory characterizes the Portuguese stock market volatility.
However, data for sub-period, from 2002 to 2006, show weaker evidence of long
memory in volatility. Floros et al. (2007) attributes such evidence to merger of
BVLP with Euronext 2002.
1
Studies relating to Turkey used data on Istanbul Stock Exchange.
102
6 Long Memory in Stock Market Volatility
The empirical findings of long memory in volatility in African markets are
mixed. Jefferis and Thupayagale (2008) offer evidence of long memory in volatility in South Africa and Zimbabwe, whereas they find no such evidence in
Botswana. Illiquidity and trading conditions in these markets are cited as factors
responsible for the evidence of long memory. Against the backdrop of economic
reforms in South Africa, McMillan and Thupayagale (2008) investigate the issue.
For the purpose, the study divides the data into the pre and post-reform period. The
results suggest long memory in volatility for both pre and postreform period. They
conclude that the behavior of stock returns in South Africa continued to be driven
by risk.
The evidence of long memory in volatility process from emerging markets is
mixed. However, there is no comprehensive study of long memory in volatility in
India, which is one of the fastest growing emerging markets. In this backdrop, this
chapter is devoted to examining long memory in the Indian stock market volatility.
6.3 Data and Methodology
The chapter, like previous chapters, uses the daily values of eight indices traded on
NSE and six on BSE from June 1997 to March 2010 (see Chap 1, Table 1.2).
Squared returns or absolute returns, which are used as a measure of volatility,
sometimes have autocorrelations that decay at a slow hyperbolic rate. The conventional ARCH models are incapable of capturing the slow decay of autocorrelation
function in the conditional variance because shocks in the GARCH process decays
quickly at an exponential rate. Granger and Joyeux (1980), and Hosking (1981)
propose the autoregressive fractionally integrated moving average (ARFIMA) for
the mean process.2 Robinson (1991) extends ARFIMA process for the variance to
model the volatility. Based on the framework of ARFIMA, a fractionally integrated
GARCH or FIGARCH model is proposed by Baillie et al. (1996). Hence, this study
uses fractional integrated GARCH or FIGARCH model, which captures a slow
hyperbolic rate of decay for the lagged squared innovation in the conditional variance
function. A brief description of the model is given here. The standard GARCH (p, q)
model in ARMA for squared errors can be written as
½1 aðBÞ bðBÞe2t ¼ x þ ½1 bðBÞmt
ð6:1Þ
where B is the back shift operator, aðBÞ; bðBÞ are polynomials in Band mt
e2t r2t is mean zero serially uncorrelated error, e2t is the squared error of the
GARCH process and r2t is its conditional variance. Thus, the fmt g process is
integrated as the ‘‘innovations’’ for the conditional variance. All the roots of the
polynomials ½1 aðBÞ bðBÞ and ½1 bðBÞ are constrained to lie outside the
2
For details see Chap 2, Sect 5.2.2.
6.3 Data and Methodology
103
unit circle in order to ensure stability and covariance stationary of the fet g process.
When autoregressive lag polynomial, 1 aðBÞ bðBÞ contains a unit root, the
model becomes integrated GARCH or IGARCH model of Engle and Bollerslev
(1986). This is given by
UðBÞð1 BÞe2t ¼ x½1 þ bðBÞmt
ð6:2Þ
where UðBÞ ¼ 1 d ðBÞ PðBÞ. Similar to ARFIMA process for the mean, by
introducing a difference operator ð1 BÞd in Eq. (6.2), fractionally integrated
GARCH or FIGARCH (p q d) model can be specified as
UðBÞð1 BÞd e2t ¼ x þ ½1 bðBÞmt
ð6:3Þ
where UðBÞ and bðBÞ are polynomial in B of orders p and q respectively, and b’s,
x and d are parameters to be estimated. In Eq. (6.3), mt is a mean-zero, serially
uncorrelated process, and 0 \ d \ 1. The FIGARCH captures a slow hyperbolic
rate of decay for the autocorrelations of et . The FIGARCH model reduces to
GARCH when d ¼ 0 and to the IGARCH when d ¼ 1.
6.4 Empirical Results
This section discusses the empirical results. The daily closing values of indices
both from the NSE and the BSE are presented in Fig. 6.1. It is evident from the
figure that most of the indices followed the same pattern. A slowly increasing
uptrend growth can be observed in index values which reached the highest peak in
mid-2007. This was a period when BSE Sensex and CNX Nifty touched highest
benchmark. In post mid-2007, there has been a downward slope in daily values and
the slope is significantly steep. The downward slope is steeper for BSE 200 and
CNX 500. The daily closing values of CNX IT registered sudden uptrend during
1999–2000 and thereafter stock prices for CNX IT are fluctuating, and almost a
straight line can be seen since 2004. The graphical representation of daily stock
returns of indices is presented in Fig. 6.2 for further understanding volatility
persistence. The occurrence of tranquil and volatile periods is clearly evident from
Fig. 6.2. This indicates volatility clustering in Indian stock market which is a
stylized fact of stock returns.
The descriptive statistics for the 14 index returns are given in Table 2.1 of
Chap 2. The table shows that BSE 200 has the highest standard deviation,
followed by CNX IT indicating high volatility, and lowest is of CNX Nifty and
BSE Sensex. The volatility has especially increased during 2007 and 2008 for all
stock index returns indices (see Fig. 6.2). The stock returns of all 14 indices are
negatively skewed implying the returns are flatter to the left compared to a normal
distribution (see Table 2.1).
The objective of this chapter is to examine whether Indian stock market
volatility exhibit long memory. To achieve this purpose, the presence of long
104
6 Long Memory in Stock Market Volatility
memory in variance is tested by estimating FIGARCH model of Baillie et al.
(1996) by using quasi-maximum likelihood estimate (QMLE), which is a consistent method.3 For a comparison purpose, GARCH (1, 1) model is estimated and
the results of GARCH (1, 1) estimation for the NSE and the BSE are presented in
Table 6.1. It is evident from the table that the ARCH (lagged squared residuals, a)
and GARCH (lagged conditional variance, b) coefficients are statistically significant for all the index returns traded on the NSE and the BSE. The significant
S & P CNX DEFTY
6000
6000
5000
Value of Index
Value of Index
S & P CNX NIFTY
7000
5000
4000
3000
2000
1000
4000
3000
2000
1000
0
0
250 500 750 1000 1250 1500 1750 2000 2250 2500 2750
250 500 750 1000 1250 1500 1750 2000 2250 2500 2750
Days (Observations)
Days (Observations)
CNX 100
7000
12000
6000
Value of Index
Value of Index
CNX NIFTY JUNIOR
14000
10000
8000
6000
4000
2000
5000
4000
3000
2000
1000
0
0
250 500 750 1000 1250 1500 1750 2000 2250 2500 2750
200
400
12500
5000
10000
4000
3000
2000
1000
1000
1200
1400
7500
5000
2500
0
0
250
500
750
1000 1250 1500 1750 2000 2250
250
500
Days (Observations)
22500
20000
17500
15000
12500
10000
7500
5000
2500
750 1000 1250 1500 1750 2000 2250 2500 2750
Days (Observations)
BSE SENSEX
BSE200
3000
Value of Index
Value of Index
800
BSE 100
6000
Value of Index
Value of Index
CNX 500
2500
2000
1500
1000
500
0
250
500
750 1000 1250 1500 1750 2000 2250 2500 2750
250
Days (Observations)
BSE 500
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
500
750 1000 1250 1500 1750 2000 2250 2500 2750
Days (Observations)
14000
Value of Index
Value of Index
600
Days (Observations)
Days (Observations)
BSE SMALLCAP
12000
10000
8000
6000
4000
2000
0
250
500
750
1000
1250
1500
1750
Days (Observations)
2000
2250
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300
Days (Observations)
Fig. 6.1 Daily closing index values
3
Baillie et al. (1996) also have shown that QMLE method performs better than other methods to
estimate the model.
6.4 Empirical Results
BSE MIDCAP
12000
100000
10000
Value of Index
Value of Index
105
8000
6000
4000
CNX IT
75000
50000
25000
2000
0
0
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300
250 500 750 1000 1250 1500 1750 2000 2250 2500 2750
Days (Observations)
Days (Observations)
CNX BANK NIFTY
12000
7000
6000
Value of Index
10000
Value of Index
CNX INFRASTRUCTURE
8000
6000
4000
2000
5000
4000
3000
2000
1000
0
0
250
500
750
1000
1250
1500
Days (Observations)
1750
2000
2250
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300
Days (Observations)
Fig. 6.1 (continued)
coefficients demonstrate volatility clustering effect and consequently imply that the
conditional variance might change over time. The significant GARCH coefficient
indicates that the conditional variance depends on its own lagged values.
^ is close to unity for the stock index returns of both
The persistent estimate ^
aþb
the NSE and the BSE, indicating a highly persistent tendency for the volatility
response to shocks (see Table 6.1). The results confirm to the tendency that large
(small) returns, positive or negative, would lead to large (small) change. The Ljung
and Box (1978) Q statistics in Table 6.1 give the impression that the model adequately describes the volatility persistence. Furthermore, since the sum of the
coefficients is very close to unity, one can infer that IGARCH model better
describes the volatility persistence. Baillie et al. (1996) cautioned that such results
may lead one to infer that IGARCH model provide a satisfactory description of the
volatility process. However, it may not be the case if the shocks decay hyperbolically at a slower rate. To examine this possibility, the study estimates FIGARCH
model and the results are reported in Table 6.2. It can be seen from the table that the
fractional difference parameter, d is significantly within the theoretical value and
thus indicates the long memory characterization of Indian stock market volatility.
The FIGARCH model becomes a covariance stationary GARCH model when
d = 0 and the model becomes non-stationary GARCH when d = 1. Thus the
major merit of FIGARH (0 \ d \ 1) model is that it sufficiently allows the
intermediate range of persistence. More importantly, a one sided t test for d = 1.0
against 1.0 in FIGARH model clearly rejects the IGARCH null hypothesis against
FIGARCH model estimated here. Thus, the FIGARCH model adequately
describes the persistence of shocks in variance.
The results thus clearly suggest that most of the stock index returns display long
memory volatility. It implies that the shocks to conditional variance decays at a
slower rate hyperbolically. Furthermore, the significant results of long memory in
volatility found in returns show that conventional models such as GARCH and
6 Long Memory in Stock Market Volatility
0.10
0.10
0.05
0.05
Index Returns
Index Returns
106
0.00
-0.05
-0.10
-0.15
0.00
-0.05
-0.10
-0.15
Days (Observations)
0.10
0.10
0.05
0.05
Index Returns
Index Returns
Days (Observations)
0.00
-0.05
-0.10
-0.15
0.00
-0.05
-0.10
-0.15
22
Days (Observations)
Days (Observations)
0.10
0.100
0.075
0.05
0.050
0.025
0.00
-0.000
-0.025
-0.05
-0.050
-0.075
-0.10
-0.100
-0.15
-0.125
Days (Observations)
0.100
0.10
0.075
0.05
0.050
0.025
0.00
-0.000
-0.025
-0.05
-0.050
-0.075
-0.10
-0.100
-0.125
-0.15
BSE 500
BSE SMALLCAP
0.100
0.075
0.075
0.050
0.025
Index Returns
Index Returns
0.050
0.025
-0.000
-0.025
-0.050
-0.075
-0.000
-0.025
-0.050
-0.075
-0.100
-0.100
-0.125
-0.125
250
500
750
1000
1250
1500
1750
2000
2250
100
200
300
400
500
600
700
800
Days (Observations)
Days (Observations)
BSE MIDCAP
CNX IT
0.100
900
1000
1100
1200
1300
0.10
0.075
0.05
Index Returns
Index Returns
0.050
0.025
-0.000
-0.025
-0.050
-0.075
0.00
-0.05
-0.10
-0.100
-0.125
-0.15
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
250
500
750
Days (Observations)
CNX BANK NIFTY
1250
1500
1750
2000
2250
CNX INFRASTRUCTURE
0.15
0.15
0.10
0.10
0.05
0.05
Index Returns
Index Returns
1000
Days (Observations)
-0.00
-0.05
-0.10
-0.00
-0.05
-0.10
-0.15
-0.15
-0.20
-0.20
200
400
600
800
1000
1200
1400
1600
Days (Observations)
Fig. 6.2 Daily log index returns
1800
2000
2200
100
200
300
400
500
600
700
800
Days (Observations)
900
1000
1100
1200
1300
6.4 Empirical Results
107
Table 6.1 Estimates of GARCH model for NSE and BSE index returns
Index
Mean
C
a
b
Q (20)
S & P CNX Nifty
CNX Nifty Junior
S & P CNX Defty
CNX 100
CNX 500
BSE Sensex
BSE 100
BSE 200
BSE 500
BSE Midcap
BSE Smallcap
CNX IT
CNX Bank Nifty
CNX Infrastructure
0.001
(0.00)
0.001
(0.00)
0.001
(0.00)
0.001
(0.00)
0.001
(0.00)
0.001
(0.00)
0.001
(0.00)
0.001
(0.00)
0.001
(0.00)
0.002
(0.00)
0.002
(0.00)
0.001
(0.00)
0.001
(0.00)
0.001
(0.00)
0.000
(0.00)
0.000
(0.00)
0.00
(0.00)
0.000
(0.01)
0.000
(0.00)
0.000
(0.00)
0.000
(0.00)
0.000
(0.00)
0.000
(0.00)
0.000
(0.02)
0.000
(0.03)
0.000
(0.01)
0.000
(0.01)
0.000
(0.05)
0.149
(0.00)
0.165
(0.00)
0.150
(0.00)
0.149
(0.00)
0.163
(0.00)
0.138
(0.00)
0.157
(0.00)
0.161
(0.00)
0.170
(0.00)
0.193
(0.00)
0.208
(0.00)
0.129
(0.00)
0.103
(0.00)
0.162
(0.00)
0.829
(0.00)
0.821
(0.00)
0.828
(0.00)
0.839
(0.00)
0.824
(0.00)
0.843
(0.00)
0.824
(0.00)
0.821
(0.00)
0.815
(0.00)
0.804
(0.00)
0.769
(0.00)
0.863
(0.00)
0.880
(0.00)
0.837
(0.00)
66.40
(0.00)
129.80
(0.00)
64.88
(0.00)
46.11
(0.00)
97.26
(0.00)
72.87
(0.00)
94.65
(0.00)
99.1
(0.00)
97.98
(0.00)
116.3
(0.00)
166.30
(0.00)
41.13
(0.00)
72.49
(0.00)
60.11
(0.00)
Q2 (20)
10.92
(0.94)
20.96
(0.39)
10.03
(0.96)
16.40
(0.69)
13.09
(0.87)
17.06
(0.64)
17.28
(0.63)
18.43
(0.55)
17.33
(0.63)
21.87
(0.34)
22.81
(0.29)
14.80
(0.19)
24.81
(0.20)
14.13
(0.82)
Note The table reports GARCH (1, 1) estimates for indices from the NSE and the BSE. C denotes
intercept in the variance equation, a is estimated lagged squared residual (ARCH coefficient), and
b, the lagged variance (GARCH coefficient). The Q (20) and Q2 (20) refer to the Ljung-Box
portmanteau tests for serial correlation in the standardized and squared standardized residuals up
to 20 lags. The values in the parentheses represent corresponding significance level. The ARCH
and GARCH coefficients of all index returns are significant at 1 % level
IGARCH models are not capable of capturing such slow rate of decay in autocorrelation. The relative size hypothesis which states that small indices substantially exhibit long memory has not found support from the empirical evidence of
the present study, as long memory properties are found in most of the series.
The stock returns on both NSE and BSE display long memory in volatility. The
evidence of long memory in volatility indicates the persistence of shocks for a
longer period. Poon and Granger (2003) point out that long memory in volatility
implies that the shock to volatility process would have a long-lasting impact. This
highlights the importance of treating long memory in volatility in monetary policy
108
6 Long Memory in Stock Market Volatility
Table 6.2 FIGARCH estimates for NSE and BSE index returns
Index
Mean
C
b
d
S & P CNX Nifty
CNX Nifty Junior
S & P CNX Defty
CNX 100
CNX 500
BSE Sensex
BSE 100
BSE 200
BSE 500
BSE Midcap
BSE Smallcap
CNX IT
CNX Bank Nifty
CNX Infrastructure
0.001
(0.00)
0.001
(0.00)
0.001
(0.00)
0.001
(0.00)
0.001
(0.00)
0.001
(0.00)
0.001
(0.00)
0.001
(0.00)
0.001
(0.00)
0.002
(0.00)
0.002
(0.00)
0.001
(0.00)
0.001
(0.00)
0.001
(0.00)
0.000
(0.03)
0.000
(0.05)
0.000
(0.08)
0.00
(0.53)
0.00
(0.36)
0.000
(0.07)
0.000
(0.13)
0.000
(0.06)
0.000
(0.10)
0.000
(0.12)
0.000
(0.06)
0.000
(0.05)
0.00
(0.42)
0.000
(0.00)
0.326
(0.00)
0.214
(0.00)
0.261
(0.00)
0.708
(0.00)
0.362
(0.00)
0.356
(0.00)
0.219
(0.00)
0.311
(0.00)
0.299
(0.00)
0.227
(0.00)
0.263
(0.00)
0.274
(0.00)
0.249
(0.00)
0.521
(0.00)
0.471
(0.00)
0.476
(0.00)
0.447
(0.00)
0.839
(0.00)
0.530
(0.00)
0.474
(0.00)
0.478
(0.00)
0.441
(0.00)
0.480
(0.00)
0.488
(0.00)
0.514
(0.00)
0.451
(0.00)
0.390
(0.00)
0.675
(0.00)
Q (20)
Q2 (20)
69.50
(0.00)
131.66
(0.00)
66.11
(0.00)
48.22
(0.00)
98.76
(0.00)
78.03
(0.00)
97.39
(0.00)
102.01
(0.00)
99.39
(0.00)
114.80
(0.00)
158.7
(0.00)
143.46
(0.00)
74.76
(0.00)
64.12
(0.00)
11.43
(0.93)
21.44
(0.37)
11.03
(0.94)
16.81
(0.66)
13.26
(0.86)
16.66
(0.67)
17.22
(0.63)
18.72
(0.53)
17.67
(0.60)
20.48
(0.42)
20.36
(0.43)
23.24
(0.39)
17.97
(0.58)
14.93
(0.77)
Note The table reports FIGARCH (1, d, 0) estimates for indices from the NSE and the BSE.
C denotes intercept in the variance equation, the d represent fractional difference in the variance
equation. The Q (20) and Q2 (20) refer to the Ljung-Box portmanteau tests for serial correlation
in the standardized and squared standardized residuals up to 20 lags. The values in the parentheses represent corresponding significance level. The d values are significant at 1 % level
measures. The evidence of study indicate that long memory models such as
FIGARCH is preferable to conventional models for modeling volatility.
6.5 Concluding Remarks
The purpose of this chapter was to investigate empirically the presence of long
memory in the volatility of the Indian stock market, in the light of several macroeconomic and market microstructure changes. The study has empirically found
substantial evidence of fractional integration which shows the existence of long
6.5 Concluding Remarks
109
memory in Indian stock market volatility. In other words, there exists a tendency
for the volatility response to shocks to display a long memory as shocks hyperbolically decay at a slow rate. Further, the evidence of long memory in volatility
across the indices suggests that FIGARCH model adequately describes the persistence than the conventional ARCH class models. Therefore, in the light of
present evidence, long memory models such as FIGARCH is appropriatefor volatility forecasting. Future research could focus on factoring the long memory
volatility in derivatives pricing and value at risk modeling, and carry out a comparative analysis.
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Summary and Conclusion
Summary and Major Findings
The past two decades have witnessed important policy reforms aimed at
liberalisation and globalisation of the Indian economy. To achieve an efficient,
transparent and vibrant financial sector in general and stock market in particular,
several financial sector reforms, changes in market microstructure and trading
practices were introduced. The Capital Issues (Control) Act 1947 was repealed and
pricing of financial assets was liberalized. As a part of market reforms, new stock
exchange was established, and the existing stock exchanges were demutualized
and exchanges adopted screen-based automated trading. The National Stock
Exchange (NSE) and Bombay Stock Exchange (BSE) launched several new
financial products and SEBI was set up as the regulator of capital market. As
results of these reforms, Indian stock market has registered a notable growth in
terms of listed companies, trading volume and emerged as one of the favourite
destination of investment. Against the back backdrop of these reforms and
changes, a study of behaviour of stock returns, particularly, analysis of efficiency
of stock market in a liberalized environment assumes significance.
Various schools of thought have theorized the behaviour of stock returns. The
Neo-classical School of Finance proposes a theory of efficient market or efficient
market hypothesis (EMH) based on rational expectation and no-trade argument.
Eugene Fama, one of the main architects and advocates of the theory, provided
strong theoretical foundations and a framework to test the EMH empirically. In an
informationally efficient market, prices quickly absorb new information and reflect
all the available information instantly in such a way that such price processing
mechanism does not provide extra normal returns. In other words, there is no
possibility of predictability of returns by using the history of returns and a simple
buy and hold strategy would do well in such an informationally efficient market.
The vital functions of stock market such as optimal allocation of capital and
facilitation of climate conducive to investment would have adverse effects if
market were inefficient. Therefore, the study of efficiency assumes importance.
The large body of research conducted in the last three decades itself reflects the
importance of the informational efficiency of stock market. Various methods are
G. S. Hiremath, Indian Stock Market, SpringerBriefs in Economics,
DOI: 10.1007/978-81-322-1590-5, The Author(s) 2014
111
112
Summary and Conclusion
employed in empirical studies to test different forms of market efficiency. Random
walk hypothesis is considered one of the effective and convenient ways to test
weak form of efficiency. In an efficient market, returns are expected to respond
randomly to new information and therefore it is not possible to predict future
returns based on past memory of prices. The early studies of 1960s and 1970s
supported the view that stock returns follow a random walk. There was a paradigm
shift in post 1987 studies, which reported nonlinear dynamics in stock returns. The
conventional tests of market efficiency found to be incapable of capturing such
dynamics. Concomitant to this, long memory properties of stock returns have
gained particular attention over the last decade in finance.
In the light of the above factors, the main purpose of the present volume was to
examine the returns behaviour in Indian equity market in the changed market
environment. The book primarily focused on weak form of efficiency. In this work,
the random walk hypothesis was empirically tested and the volume addressed issues
such as nonlinear serial dependence mean reversion, and long memory in stock
returns. The data used in the study consists of daily stock index returns of NSE and
BSE, the major stock exchanges in India. Eight indices including three sectoral
indices from the NSE and six indices from the BSE were chosen. The study has made
improvements from previous studies in terms of the application of sophisticated
tests, updated, comprehensive and disaggregated dataset, addressing issues which
have not received due importance in previous research on Indian stock market.
This study empirically tested whether stock returns in India follow a random
walk. Towards this end, data on major indices during the period June 1997–March
2010 are analysed using both parametric and non-parametric tests, some of which
are not employed in previous studies in India. The results from parametric tests
offered mixed evidence. The parametric test results suggest significant rejection of
random walk hypothesis in case of smaller stock indices with lower market
capitalization and liquidity. The evidence of rejection of random walk behaviour in
stock returns of large cap and high liquid indices are weaker as the investment
horizon increases. Non-parametric tests, which are considered appropriate when
returns are non-normal, have shown rejection of hypothesis that increments are
independent and identically distributed for the selected index returns and these
results are not sensitive to the composition of index. The rejection of random walk
at longer horizon implies that the information in short-horizon is not instantly
reflected in returns and thus provides opportunity for excess returns to those who
have access to information. Later, as time horizon increases, trading strategies of
those who had access to such information began to reflect in prices leading the
market towards efficiency.
Nonlinear dependence in returns directly contrasts the EMH since such
dependence structure provides potential opportunities for prediction. In view that
there has not been much empirical work in the case of India, the present study has
applied a set of nonlinearity tests which have different power against different
classes of nonlinear process, to uncover nonlinear dependence in stock returns of
selected indices. The tests results provide strong evidence of nonlinear serial
dependence in stock returns for full sample period. However, the windowed test
Summary and Conclusion
113
procedure applied in the study shows a nonlinear structure that is not consistent
throughout the full sample period but confined to a few sub-periods thus
suggesting episodic nonlinear dependence surrounded by long periods of pure
noise. Furthermore, it is found that both negative and positive events were
associated with these nonlinear dependence periods, but negative events had a
significant effect. The episodic presence of nonlinear dependence implies that
certain events induce such nonlinear dependence. The major events identified were
uncertainties in international oil prices, volatile exchange rates, turbulent world
markets, sub-prime crisis, global economic meltdown and political uncertainties,
especially border tensions. Though the nonlinear dependence found in stock
returns indicates predictability of stock returns, investors find it difficult to exploit
such dependence to forecast, because it is not present throughout the sample period
but just confined to a few periods. The episodic dependence in returns indicates
that investors take time to learn about shock and adjust their trading strategies.
The mean-reversion hypothesis is tested as an alternative explanation for the
behaviour of stock returns to random walk behaviour. The conventional unit root
tests results may mislead in the presence of structural breaks. Therefore, multiple
structural breaks tests are carried out and two significant structural breaks in each of
the index series are found. The test results have shown rejection of null of unit root,
thus clearly indicating trend-stationary process. The study identified the events
associated with significant structural break dates. The dot.com bubble burst and
consequent recession in the USA, bad monsoons, international oil shocks, volatile
exchange rates, sub-prime crisis and global economic meltdown, fluctuations in
foreign institutional investment, political uncertainties including border tensions are
the major events identified around significant trend breaks. The study found that
smaller cap indices were more vulnerable to external shocks than large cap indices.
The long memory in stock returns is important because it explains the returns
behaviour. To detect long memory in mean returns, the study has carried out multiple
semi-parametric tests. The study has largely found the presence of long memory in
mean returns. The anti-persistence evidence observed in index returns is not
consistent. The findings of the study did not support the relative size proposition. In
the same fashion, this study endeavoured to detect long memory in volatility. The
model estimates indicate strong evidence of long memory in volatility. In other
words, this study has found that the FIGARCH model better describes the persistence
of volatility than the conventional models of volatility. The evidence of long memory
in both mean and volatility suggests that using linear modelling would result in
misleading inferences. The evidence of long memory suggests proper factoring of
long memory volatility in derivative pricing and risk management models.
Implications of the Study
To conclude, study largely suggests rejection of random walk hypothesis in Indian
stock market. This implies that Indian equity market is not weak form efficient.
114
Summary and Conclusion
The results indicate no significant difference in the behaviour of index returns
between the NSE and BSE. Nevertheless, the stock indices having higher liquidity
and market capitalization prove to be less inefficient than small indices on both the
exchanges. Furthermore, the small indices with less liquidity appear to be more
vulnerable to external shocks. Sectorwise, there has not been much difference. In
the light of the present evidence, it is clear that policy reforms aimed at improving
the efficiency have not brought the desired results.
In view of the above discussion, some policy implications are proposed here.
The evidence of existence of potential excess returns in short horizons of
investment calls for policy measures aiming at proper dissemination of
information to the participants. This has further support from the episodic
nonlinear dependence, which suggests that investor takes time to respond to the
events. Further, to improve the performance of small indices having lesser
liquidity, it is important to improve the liquidity of smaller stocks. Encouraging
retail trading and promoting the mutual funds in the Indian market may achieve
this. RBI’s initiative for financial literacy and NSE’s certificate courses for
financial education are welcoming steps in this direction. External events have
always created panic in the Indian stock market. Whenever there were some
shocks , there was net outflow of FIIs. In the light of this, policy measures aiming
at an appropriate management of external sector and global events need to be
initiated in order to improve the immunity of stock market towards ill effects of
global shocks. There is also need of optimal regulation of FIIs and pressing for
further disclosures from the FIIs. In light of the current empirical evidences, before
hastening for the third generation reforms, a pause for a holistic review of financial
sector reforms is important at this crucial juncture.
The limitations of the study highlight scope for further research. The study
indicated stronger rejection of market efficiency and vulnerability of Indian stock
market to external shocks. The interaction between market microstructure
variables and market efficiency indicators may throw further light. A further
investigation of sources of long memory and a causal analysis of inefficiencies
would provide useful information for policy measures.
The empirical evidence presented in this book resoundingly rejects the EMH.
From the theoretical perspective, there are no convincing explanations for nonrandom walk behaviour in stock returns. According to Andrews Lo, perfect
efficient market is difficult to find in real world and he advocates the engineering
notion of ‘‘relative efficiency’’ of market as a useful concept. Using the evolution
of human behavioural principles, Lo proposes adaptive market hypothesis,
according to which market efficiency evolves over a period. In this framework,
rational EMH co-exists with behavioural models in an intellectually consistent
manner. Future research on Indian stock market could focus on these aspects.
About the Author
Gourishankar S Hiremath is Assistant Professor of Economics & Finance at Indian
Institute of Technology Kharagpur (India). He previously worked at Indian
Institute of Technology Jodhpur, Gokhale Institute of Politics and Economics,
Pune, and ICFAI Business School-Hyderabad. He holds a PhD in Financial
Economics from University of Hyderabad, India and his areas of specialisation
include Indian Capital Market, International Finance, Financial and Commodity
Derivatives, and Applied Time Series Econometrics. He has presented his papers
both in National and International conferences and published research papers in
some leading journals. He has done research for National Bank for Agriculture and
Rural Development (NABARD) and Climate Works Foundation, New Delhi. He is
member of Panel of Experts, Young Entrepreneurs Incentive Scheme of Rajasthan
Financial Corporation sponsored by the Council of State Industrial Development
and Investment Corporations of India.
G. S. Hiremath, Indian Stock Market, SpringerBriefs in Economics,
DOI: 10.1007/978-81-322-1590-5, The Author(s) 2014
115
Index Description
BSE Sensex
BSE Sensex represents large and financially sound 30 companies across key
sectors. It accounts for about 45 % of total market capitalization on BSE.
BSE 1001
BSE 100 index is made up of 100 companies listed on five important stock
exchanges in India. The scripts included are of those companies that have been
traded more than 95 % trading days and figured in final 200 ranking.2 BSE 100
stocks represent about 73 % of market capitalization.
BSE 200
Equity shares of 200 selected companies from the specified and non-specified lists
of BSE constitute BSE 200 index. It represents 82.70 % of market capitalization
on BSE.
BSE 500
BSE 500 constitutes about 94 % of market capitalization on BSE. It covers major
20 industries of the company. The stocks which are included in BSE 500 are those
which have traded 75 % days and figured in top 750 companies in final ranking.
1
BSE 100 was formerly known as BSE National Index.
BSE arrives at this ranking base on 3 months full market capitalization of stock and liquidity
which are given 75 and 25 % of weight respectively.
2
G. S. Hiremath, Indian Stock Market, SpringerBriefs in Economics,
DOI: 10.1007/978-81-322-1590-5, The Author(s) 2014
117
118
Index Description
BSE Midcap
This index constitutes medium-sized stocks and represents about 16 % of total
market capitalization on BSE.
BSE Smallcap
It accounts for about 6 % of market capitalization and made up of small-sized
stocks.
S & P CNX Nifty
It represents most liquid and well-diversified 50 stocks traded at NSE representing
22 sectors of the economy. Its percentage to total market capitalization is about
65 % on NSE.
S & P CNX Defty
CNX Defty is nothing but CNX Nifty, measured in dollars. This index is to
facilitate FIIs and off-shore fund enterprises.
CNX Nifty Junior
CNX Nifty Junior consists of next 50 liquid stocks excluded from CNX Nifty and
represents about 10 % of total market capitalization on NSE.
CNX 100
Diversified 100 stocks representing 35 sectors of the economy constitute CNX 100
index. It represents 75 % of total market capitalization on NSE
Index Description
119
CNX 500
CNX 500 equity index is broad-based index and accounts 95 % of total market
capitalization. The companies included are disaggregated into 72 industry indices.
CNX IT
Companies that have more than 50 % of their turnover from IT-related activities
are compressed in CNX IT. The CNX IT Index stocks represent about 80.33 % of
the total market capitalization of the IT sector as on March 31, 2010. Companies
included in CNX IT have at least 90 % trading days and ranked less than 500
based on market capitalization. This index accounts 14 % of total market
capitalization on NSE.
CNX Bank Nifty
The most liquid and large market capitalised 12 Indian Banking stocks traded on
NSE comprises CNX Bank Nifty. The CNX Bank Index stocks represent about
87.24 % of the total market capitalization of the banking sector and about 8 % of
the total market capitalization on NSE.
CNX Infrastructure
CNX Infrastructure index includes 25 stocks of companies belonging to Telecom,
Power, Port, Air, Roads, Railways, Shipping and other Utility Services providers.
CNX Infrastructure Index constituents represent about 21.43 % of the total market
capitalization on NSE.
Index
A
Abnormal returns, 1, 2, 4, 19
Adaptive market hypothesis, 114
ADF unit root, 22, 90
AGBR test, 94
Allocation of resource, 2, 111
Anomalies, 20
Anti-persistence, 94, 96
Arbitrage, 5, 6, 86, 100
ARCH model, 102
ARFIMA, 87, 88, 91, 100, 102, 103
Asian financial crisis, 56, 100
Asymmetric, 0
Asymmetry, 91
Asymptotic, 89, 93
Autocorrelation, 13, 41, 43, 85, 87, 100, 102,
103, 107
Autocovariance, 13, 85, 87
Autoregressive, 87, 88, 100, 102, 103
Autoregressive integrated moving average
(ARIMA), 87
B
Behavior, 89
Behavioral School, 6
Behaviour of stock returns, 37, 111, 113
Bicorrelations, 44, 46
Binomial expansion, 88
Bispectrum, 43, 45
Bombay Stock Exchange (BSE), 9, 12, 14, 15,
41, 43, 47, 51, 52, 54, 55, 62, 67, 70–73,
76–78, 81, 86, 91, 94, 96, 101–105
Broack, Dechert, Sheinkman, LeBaron (BDS),
42, 44, 46, 47
Brownian motion, 7, 86, 100
C
Capital market, 8–10
Chow and Denning Test, 89
Competitive market, 2, 3, 5
Conditional heteroscedasticity, 42, 51, 100
Conditional variance, 100, 102, 105
Correlation, 44, 65, 87, 102
Covariance stationary, 86, 88, 103, 105
Credit, 8, 52, 55
Crisis, 8, 12, 52, 55, 72, 100
D
Daily prices, 103
Daily returns, 43, 103
Daily values, 15, 102, 103
Dependence, 41, 42, 44, 89, 91, 100, 101
Derivative pricing, 13, 86, 100
Deterministic, 65
Dickey-Fuller, 61
Difference stationary, 67, 73
Distribution, 6, 7, 103
Dot com bubble, 11
E
Economic reforms, 8, 62, 102
Efficiency, 9, 13–15, 47, 73, 76, 81, 90
Efficient equity market, 2
Efficient market hypothesis, 1, 3, 5, 6, 41, 59,
60, 73, 86
Efficient Market Theory, 1, 2, 2, 3, 5, 6
Emerging market, 10, 11, 42, 53, 61, 62, 90,
91, 101, 102
Endogenous, 59, 61, 66
Episodic, 1, 42, 44, 47, 56
G. S. Hiremath, Indian Stock Market, SpringerBriefs in Economics,
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122
Equilibrium return, 3
Equity market, 2, 9–11, 13, 14, 53, 55, 73, 81,
90, 99
Events, 2, 41–43, 47, 51–56, 60, 66,
73, 76, 81
Excess returns, 3, 5, 76, 81
Exchange rate, 42, 52, 59, 62
Exogenous, 60, 64
Expected returns, 3, 7
External events, 59, 73, 81
External shocks, 73, 78, 81
F
Financial system, 1, 8
Forecast, 3, 13, 42, 56, 86, 99
Foreign Direct Investment (FDI), 53
Foreign Institutional Investors (FIIs), 9, 10, 52,
72, 73
Foreign investors, 90
Fractional integration, 89, 91, 95, 109
Fractionally differenced, 100
Fractionally integrated generalized autoregressive conditionalheteroskedasticity
(FIGARCH), 100, 102–105, 108
Frequency domain, 43, 87, 90
Frictions, 90, 99
Fundamental School, 6
G
Gaussian, 91, 93, 94
Generalized autoregressive conditional
heteroskedasticity (GARCH), 100,
102–105, 107
Geweke Porter-Hudak semiparametric test
(GPH), 86, 89, 90, 92–94
Global economic meltdown, 41, 55,
56, 67, 72
Global meltdown, 59, 72
Global recession, 55, 59
Globalization, 7
Great Depression, 60
H
Heteroscedastic, 25, 26, 32–34
Heteroscedasticity, 76, 89, 100
Hinich bicorrelation test, 47
History of stock returns, 3
Homoscedastic, 25, 26, 32–34
Hurst, 87, 88, 90
Hyperbolic, 13, 100, 102, 103, 107
Index
I
IID, 75–77, 81
Independence, 7, 41, 42, 44
Independent and identically distribution, 2, 7,
45, 112
India, 8–14, 42, 43, 52, 53, 55, 61, 62, 71–73,
78, 81, 86, 91, 100, 102
Information, 1–6, 46, 53, 59, 60, 63, 76, 81,
86, 91
Information asymmetry, 91
Informational efficiency, The, 12, 13
Informationally efficient, 2
Informationally efficient market, 2, 5
Insider trading, 21
Integrated generalized autoregressive conditional heteroskedasticity (IGARCH), 103,
105, 107
Integration, 2, 89, 91
International oil prices, 41, 52, 53, 55, 56, 59,
67, 72
Investors, 2, 3, 5, 6, 9, 10, 14, 42, 90, 99, 100
Irrational, 6
K
Kurtosis, 29, 30
L
Lagrange Multiplier, 61, 62
Large cap, 72
Lee-Strazicich test, 62, 67, 71, 72
Leptokurtic, 0
Liberalization, 10
Linear dependence, 13, 41, 44
Liquidity, 1, 8, 11, 14, 53, 55, 59, 72,
73, 76, 91
Ljung and Box, 105
Lo and MacKinlay Test, 74
Long horizons, 37, 75, 76
Long memory, 1, 13, 16, 91, 93, 94, 96,
100–102, 104, 105, 107, 108
Long memory in volatility, 13, 14, 16,
100–102, 107
Long-range dependence, 13, 14, 85–87, 89, 90,
94, 100
M
Macroeconomic, 4, 6
Market capitalization, 2, 8, 10, 11, 14, 22, 37,
55, 59, 67, 72, 76, 81
Market crash, 42, 90
Index
Market microstructure, 1, 7, 9, 13, 14, 43,
51, 62, 73, 100
Martingale hypothesis, 42
Martingale process, 78, 81, 86, 100
McLeod and Li test, 43–47
Mean reversion, 13, 15, 20, 59–62, 65, 66, 73,
100
Memory, 13, 75, 85–91, 100–102, 105, 107,
108
Microstructure, 1, 7, 9, 12, 13, 43, 51, 62, 73,
100, 108, 111, 114
Mid cap, 67
Moving average, 6, 87, 88, 100, 102
Multiple variance ratios, 26
N
National Stock Exchange (NSE), 9, 19, 20, 35,
37, 41, 43, 46, 51, 52, 55, 62, 72, 73, 75,
77, 78, 86, 91, 94, 96, 100
Noise, 20, 44, 46, 47, 51, 56, 88
Nonlinear, 13
Nonlinear dependence, 1, 13, 15, 28, 41–45,
47, 51, 55, 56
Nonlinear dynamics, 112
Non-normality, 37, 41, 89, 92, 94
Non parametric tests, 37
Non-stationary, 59, 105
Normal distribution, 25, 27, 28, 30, 94, 103
Normality, 25, 27, 28, 30, 74, 86, 93, 103, 111
Null hypothesis, 25, 30, 35, 44, 46, 65,
74, 75, 105
O
Oil prices, 54, 72
Oil shock, 60
P
Paradigm shift, 42, 112
Parametric, 15, 19, 28, 34, 35, 37, 65,
88–91, 94
Periodogram, 92–94
Persistence, 7, 13, 43, 89, 90, 100, 103,
105, 107
Portfolio management, 100
Portmanteau, 24, 27, 43, 44
PP test, 65
Predictable/Predictability, 2, 3, 20, 21, 86
Price, 2, 3, 5–8, 19–21, 24, 37, 54, 60,
61, 72, 89, 91, 100, 103
Private incentive, 5
123
Q
Q statistic, 105
Q test, 0
Quasi-maximum likelihood estimate, 104
R
Random walk, 7, 13, 19–26, 30, 32, 35, 37, 42,
43, 59, 60, 62, 73, 76–78, 81, 89
Random walk hypothesis, 7, 13, 15, 19,
46, 60, 85
Random walk model, 6, 20, 21, 23, 89
Random walk process, 2, 21, 24, 86
Rational, 5, 6, 21
Rational expectation, 3, 20
Reforms, 1, 8–10, 12, 13, 23, 52, 56, 62, 73,
90, 100
Regulatory, 9, 19, 37, 42, 59, 61, 76, 90
Relative efficiency, 14
Relative size hypothesis, 107
Relative size proposition, 113
Rescaled range statistic, 87
Reserve Bank of India (RBI), 52–55, 66
Retail trading, 73
Returns, 1–7, 13–15, 19–25, 27, 28, 30, 32, 34,
35, 37, 41–47, 51, 53–56, 59, 60, 62, 65,
66, 72, 73, 75, 78, 81, 87–91, 93, 94, 96,
100–103, 105, 107
Risk management, 9
Robinson’s Gaussian semiparametric estimation (RGSE), 94, 96
Runs test, 23, 27, 34, 35
Runs tests, 41
S
Scam, 41, 52, 55, 56
Sector, 1, 7–9, 12, 14, 15, 22, 28, 37, 53, 72,
73, 81, 89, 100
Sectoral, 9, 22, 30, 32, 35, 89
Securities and Exchange Board of India
(SEBI), 9, 53, 54, 66, 72
Semi-parametric, 113
Semi-strong form efficiency, 4
Serial dependence, 21, 22, 32, 42, 43
Serially correlated, 20
Short horizons, 37, 75, 76
Size distortions, 22, 26, 61
Skewness, 44, 45
Small cap, 55, 67
South East Asia, 52
Spectral, 21, 23, 87, 92, 93
Stationarity, 61, 66, 67, 72, 73, 96
124
Stationary, 43, 44, 59, 60, 65, 72, 85, 87, 92,
103
Stochastic process, 24
Stock market, 6, 11–15, 19, 22, 23, 37, 42, 43,
53, 54, 59, 61, 78, 81, 85, 89–91, 99–105
Strong Form Efficiency, 4
Structural breaks, 1, 13, 16, 59, 62, 66, 67,
71–73, 76–78, 81, 101
Structural breaks, 60
Stylized fact, 30, 42, 60, 103
Sub-prime crisis, 10, 41, 55, 56, 59, 67, 72, 73
T
Technical analysis, 6
Technical school, 6
Thin trading, 21–23
Time domain, 87, 94
Time series, 13, 15, 20, 23, 41, 44, 59, 61,
85–87, 89, 91, 93
Time varying, 21
Transaction costs, 5
Trend, 55, 59, 61–65, 67, 72, 73, 78, 103
Trend-stationary, 63
Tsay test, 44–46
Turnover ratio, 10, 11
Index
U
Unconditional heteroscedasticity, 7
Unit root, 60–65, 67, 72, 73, 90, 103
V
Value at risk, 13, 100
Variance, 7, 13, 22, 24, 25, 42, 64, 74, 75, 88,
100–102
Variance ratio, 21, 22, 24–26, 32, 35, 41, 60,
73–76
Volatility, 10, 13, 30, 32, 52, 53, 99–103, 105,
107, 108
Volatility forecast, 99
W
Weak form of efficiency, 4, 77
Windowed, 42, 43, 51, 52, 55
Wright test, 74, 75
Z
Zivot-Andrew test, 66, 67, 71