This art icle was downloaded by: [ Monash Universit y Library] On: 15 Oct ober 2014, At : 21: 36 Publisher: Rout ledge I nform a Lt d Regist ered in England and Wales Regist ered Num ber: 1072954 Regist ered office: Mort im er House, 37- 41 Mort im er St reet , London W1T 3JH, UK Applied Financial Economics Publicat ion det ails, including inst ruct ions f or aut hors and subscript ion inf ormat ion: ht t p: / / www. t andf online. com/ loi/ raf e20 Are OECD stock prices characterized by a random walk? Evidence from sequential trend break and panel data models Paresh Kumar Narayan a a & Russell Smyt h b School of Account ing , Finance and Economics, Grif f it h Universit y b Depart ment of Economics , Monash Universit y , 900 Dandenong Road, Caulf ield East , Vict oria, 3145, Aust ralia c Depart ment of Economics , Monash Universit y , 900 Dandenong Road, Caulf ield East , Vict oria, 3145, Aust ralia E-mail: Published online: 21 Aug 2006. To cite this article: Paresh Kumar Narayan & Russell Smyt h (2005) Are OECD st ock prices charact erized by a random walk? Evidence f rom sequent ial t rend break and panel dat a models, Applied Financial Economics, 15: 8, 547-556, DOI: 10. 1080/ 0960310042000314223 To link to this article: ht t p: / / dx. doi. org/ 10. 1080/ 0960310042000314223 PLEASE SCROLL DOWN FOR ARTI CLE Taylor & Francis m akes every effort t o ensure t he accuracy of all t he inform at ion ( t he “ Cont ent ” ) cont ained in t he publicat ions on our plat form . However, Taylor & Francis, our agent s, and our licensors m ake no represent at ions or warrant ies what soever as t o t he accuracy, com plet eness, or suit abilit y for any purpose of t he Cont ent . Any opinions and views expressed in t his publicat ion are t he opinions and views of t he aut hors, and are not t he views of or endorsed by Taylor & Francis. The accuracy of t he Cont ent should not be relied upon and should be independent ly verified wit h prim ary sources of inform at ion. Taylor and Francis shall not be liable for any losses, act ions, claim s, proceedings, dem ands, cost s, expenses, dam ages, and ot her liabilit ies what soever or howsoever caused arising direct ly or indirect ly in connect ion wit h, in relat ion t o or arising out of t he use of t he Cont ent . This art icle m ay be used for research, t eaching, and privat e st udy purposes. Any subst ant ial or syst em at ic reproduct ion, redist ribut ion, reselling, loan, sub- licensing, syst em at ic supply, or dist ribut ion in any form t o anyone is expressly forbidden. Term s & Condit ions of access and use can be found at ht t p: / / www.t andfonline.com / page/ t erm s- and- condit ions Applied Financial Economics, 2005, 15, 547–556 Are OECD stock prices characterized by a random walk? Evidence from sequential trend break and panel data models Paresh Kumar Narayana and Russell Smythb,* a School of Accounting, Finance and Economics, Griffith University Department of Economics, Monash University, 900 Dandenong Road, Caulfield East, Victoria, 3145, Australia Downloaded by [Monash University Library] at 21:36 15 October 2014 b This paper examines whether stock prices for a sample of 22 OECD countries can be best represented as mean reversion or random walk processes. A sequential trend break test proposed by Zivot and Andrews is implemented, which has the advantage that it can take account of a structural break in the series, as well as panel data unit root tests proposed by Im et al., which exploits the extra power in the panel properties of the data. Results provide strong support for the random walk hypothesis. I. Introduction Much research has focused on the best way to characterize the dynamic properties of economic and financial time series. This is a manifestation of the fact that the random walk hypothesis has farreaching implications for both financial theory and the interpretation of empirical evidence (Fama, 1970; LeRoy, 1982; Malkiel, 2003). If stock prices are mean reversion processes it follows that the price level will return to its trend path over time and that it might be possible to forecast future movements in stock prices based on past behaviour. However, if stock prices follow a random walk process any shock to prices will be permanent. This means that future returns cannot be predicted based on historical movements in stock prices and that volatility in stock markets will increase in the long run without bound (Chaudhuri and Wu, 2003, pp. 575–6). Whether stock prices are best characterized by random walk or mean reverting processes has been the subject of several studies. Fama (1970), DeBondt and Thaler (1985), Fama and French (1988), Lo and MacKinlay (1988), Poterba and Summers (1988), Richards (1997) and Balvers et al. (2000) find evidence of mean reversion in stock prices using different countries and time periods. These findings, however, have been questioned by other authors who argue that studies which have found evidence of mean reversion are not robust or find evidence consistent with stock prices exhibiting a random walk (Richardson and Stock, 1989; Kim et al., 1991; McQueen, 1992; Richardson, 1993). There are few studies that have employed either sequential break or panel data unit root tests to investigate the random walk hypothesis for stock prices. Chaudhuri and Wu (2003) employ the Zivot and Andrews’ (1992) sequential trend break model, which allows for one structural break, to examine the random walk hypothesis for stock prices in 18 emerging markets using monthly data from 1985 to 1997. Chaudhuri and Wu (2003) find that the Zivot *Corresponding author. E-mail: Russell.Smyth@BusEco.monash.edu.au Applied Financial Economics ISSN 0960–3107 print/ISSN 1466–4305 online # 2005 Taylor & Francis Group Ltd http://www.tandf.co.uk/journals DOI: 10.180/0960310042000314223 547 Downloaded by [Monash University Library] at 21:36 15 October 2014 548 and Andrews’ (1992) test rejects the random walk null hypothesis for 10 of the 18 markets which they study. In another article Chaudhuri and Wu (2004) employ a feasible generalized least squares (FGLS) panel data unit root test, which has been proposed by Papell (1997) and O’Connell (1998) among others, to test the random walk hypothesis in 17 emerging markets using monthly data from 1985 to 2002. They again find evidence of mean reversion in stock prices. Balvers et al. (2000) investigated the existence of mean reversion in stock prices for 18 industrialized countries for the period 1969 to 1996 using the FGLS panel unit root approach. Their main finding is that the null hypothesis of a random walk can be rejected. The only other article, of which we are aware, that employs a panel data unit root test is Zhu (1998). Zhu (1998) applies the Levin and Lin (1992) panel data unit root test to investigate the random walk hypothesis for a panel of G7 stock price indices using monthly data from 1958 to 1996. In contrast to Balvers et al. (2000) and Chaudhuri and Wu (2004), Zhu (1998) cannot reject the random walk hypothesis for G7 stock price indices. This study uses daily data for the period 1991 to 2003 to test the random walk hypothesis for the stock price indices of 22 OECD countries. In order to provide a benchmark we begin with the Augmented Dickey–Fuller (ADF) and Phillips–Perron unit root tests. The ADF and Phillips–Perron unit root tests can only reject the unit root null for one country at the 5% level and a second country at the 10% level, providing a strong presumption in favour of the random walk hypothesis. Following this the study proceeds to estimating the Zivot and Andrews (1992) sequential trend break model. Estimating sequential trend break models one can still only reject the random walk hypothesis for one country, although the majority of the break points are statistically significant. It is often argued that unit root tests have low power with short spans of data and therefore failure to reject the unit root null should be treated with caution (Cochrane, 1991; DeJong et al., 1992). Several panel data unit root tests have been developed to exploit the extra power in the panel properties of the data (see Baltagi and Kao, 2000 for a review). These include the Levin and Lin (1992) test, employed by Zhu (1998), FGLS, employed by Chaudhuri and Wu (2004), the t-bar test proposed by Im et al. (2003) and the panel Lagrange multiplier (LM) test with a structural break suggested by Im et al. (2002). This study employs the t-bar test developed by Im et al. (2003) and the LM panel unit root test with one structural break suggested by Im et al. (2002). The t-bar test does not assume P. K. Narayan and R. Smyth that all cross-sectional units converge towards the equilibrium value at the same speed under the alternative and thus is less restrictive than either the Levin and Lin (1992) or FGLS tests. Maddala and Wu (1999) and Karlsson and Lothgren (2000) perform Monte Carlo simulations, which show that in most cases the Im et al. (2003) test outperforms the Levin and Lin (1992) and FGLS tests. The LM panel unit root test has the advantage that it can incorporate a structural break. Using the t-bar test and panel LM test with a structural break one is unable to reject the random walk hypothesis for the 22 countries or for a smaller G7 panel. These results are in contrast to the recent support for mean reversion in the stock indices of emerging markets provided by Chaudhuri and Wu (2003, 2004) using the same methodological approach. II. Univariate Unit Root Tests Without a Structural Break The data used in the paper are the natural logs of daily stock prices (excluding weekends and public holidays) for 22 OECD countries covering the period 1 January 1991 to 4 June 2003 (3242 observations). All stock price indices are expressed in real local currencies, to overcome the possibility that the results reflect fluctuations in exchange rates and/or mean reversion in inflation rates. The data were obtained from Datastream and the start and end dates for the study were dependent on data availability. Table 1 shows each of the countries and the stock price index for each country that was employed in the study. To provide a point of comparison the study started by testing whether stock prices follow a random walk process using the ADF unit root test, based on the auxiliary regressions:
yt ¼  þ yt
1 þ k X j¼1 dj
yt
j þ "t
yt ¼  þ yt
1 þ t þ k X j¼1 dj
yt
j þ "t ð1Þ ð2Þ The ADF auxiliary regression tests for a unit root in yt, where y refers to the stock price series in each country, t ¼ 1, . . . , T is an index of time and
yt
j is the lagged first differences to accommodate serial correlation in the errors, t. Equation 1 tests for the null of a unit root against a mean stationary alternative in yt. Equation 2 tests the null of a unit root against a trend stationary alternative. In Equations 1 and 2 the null and the alternate hypotheses for a unit root in yt are: H0 ¼ 0 and H1 <0. Are OECD stock prices characterized by a random walk? Downloaded by [Monash University Library] at 21:36 15 October 2014 Table 1. Countries and stock price indices employed in the study Country Stock price index used Australia Austria Belgium Canada Finland France Germany Hungary Ireland Italy Japan Korea Mexico Netherlands New Zealand Portugal Spain Sweden ASX ALL ORDINARIES AUSTRIAN TRADED INDEX BEL 20 TSX 200 HEX GENERAL PARIS CAC 40 (16.00 HRS) DAX 30 (16.00 HRS UK) BUDAPEST (BUX) IRELAND SE OVERALL (ISEQ) MILAN COMIT GENERAL NIKKEI 225 STOCK AVERAGE KOREA SE COMPOSITE (KOSPI) MEXICO IPC (BOLSA) AEX INDEX NEW ZEALAND SE ALL PORTUGAL PSI GENERAL IBEX 35 STOCKHOLMSBORSEN ALL SHARE (SAX) SWISS MARKET (16.00 HRS) ISE NATIONAL INDUSTRIALS FTSE 100 (16.00 HRS) Switzerland Turkey United Kingdom United States DOW JONES INDUSTRIALS To select the lag length (k) the ‘general-to-specific’ approach proposed by Hall (1994) is used. This involves starting with a predetermined upper bound kmax. If the last included lag is significant, kmax is chosen. However, if k is insignificant, it is reduced by one lag until the last lag becomes significant. If no lags are significant k is set equal to zero. Ng and Perron (1995) show that the ‘general-to-specific’ approach produces test statistics which have better properties in terms of size and power than when k is selected with information-based criteria such as the Akaike Information Criterion or Schwartz Bayesian Criterion. Following Hayashi (2000, p. 594) kmax ¼ 28 is set according to the formula kmax ¼ int(12(T/ 100)0.25) and the approximate 10% asymptotic critical value of 1.60 is used to determine the significance of the t-statistic on the last lag. The Phillips–Perron test is also based on Equations 1 and 2, but without the lagged differences. While the ADF test corrects for higher order serial correlation by adding lagged difference terms to the right-hand side, the Phillips–Perron test makes a nonparametric correction to account for residual serial correlation. The bandwidths for the Phillips–Perron test were selected with the Newey–West suggestion using Bartlett kernel. The results of both unit root tests are reported in Table 2 with and without a trend. The ADF and Phillips–Perron tests reject the random walk hypothesis for stock prices in Mexico at the 5% level with 549 a trend and the 10% level without a trend. Note that this result differs from Chaudhuri and Wu (2003) who cannot reject the random walk hypothesis for Mexican stock prices applying the ADF and Phillips–Perron unit root tests to monthly data. The Phillips–Perron test also rejects the random walk hypothesis for stock prices in New Zealand at the 10% level without trend. However, for all other countries neither the ADF nor Phillips–Perron unit root tests can reject the random walk null hypothesis. Overall, these findings constitute a benchmark that is consistent with the view that OECD stock prices are characterized by a random walk process. III. Sequential Trend Break Unit Root Test Perron (1989) shows that if there is a structural break, the power to reject a unit root decreases when the stationary alternative is true and the structural break is ignored. There are several events over the period of the study which potentially could have caused a structural break in the stock indices. These include the Asian financial crisis (1997–1998), Russian economic crisis (September 1998), collapse of global technology stocks (April 2000), terrorist attacks on the World Trade Centre, New York and Pentagon, Washington and the ensuing war in Afghanistan (from September 2001) and Enron and WorldCom collapses (February and September, 2002) to name a few. Therefore, failure to find significant evidence of mean reversion in stock prices with the ADF and Phillips–Perron unit root tests could reflect misspecification of the deterministic trend. Perron (1989) proposes a model which imposes the null hypothesis that a given series has a unit root with drift and an exogenous structural break against the alternative of stationarity about a deterministic trend which has an exogenous structural break. However, the problem with imposing an exogenous structural break is that selecting the break point a priori based on an ex post examination or knowledge of the data could lead to an over rejection of the unit root hypothesis (Perron and Vogelsang, 1992). Zivot and Andrews (1992) extend Perron’s (1989) model by endogenizing the break point determination. Zivot and Andrews’ (1992) ‘model A’ and ‘model C’ were applied to investigate whether OECD stock prices are characterized by a random walk. Model A has the following form:
yt ¼  þ yt
1 þ t þ DUt þ k X j¼1 dj
yt
j þ "t ð3Þ P. K. Narayan and R. Smyth 550 Table 2. Augmented Dickey–Fuller (ADF) and Phillips–Perron unit root tests ADF tests Phillips–Perron tests Downloaded by [Monash University Library] at 21:36 15 October 2014 With trend Australia Austria Belgium Canada Finland France Germany Hungary Ireland Italy Japan Korea Mexico Netherlands New Zealand Portugal Spain Sweden Switzerland Turkey United Kingdom United States Without trend (0) (1) (3) (1) (0) (3) (0) (0) (1) (1) (0) (1) (1) (0) (0) (1) (0) (1) (1) (1) (3) (0)
2.580
2.718
0.528
1.633
0.807
0.742
0.238
1.236
0.726
1.270
2.192
2.300
3.646**
0.322
2.535
0.128
0.696
0.352
0.357
1.621
0.686
1.039
2.143
2.255
1.749
1.660
1.051
1.442
1.555
0.790
1.516
1.170
0.833
2.259
2.602***
1.831
2.828***
1.275
1.474
1.629
2.091
1.214
1.967
0.494 With trend (0) (1) (3) (1) (0) (3) (0) (1) (1) (1) (0) (1) (1) (0) (0) (1) (0) (1) (1) (1) (3) (0)
2.544
2.717
0.467
1.762
0.858
0.722
0.184
1.264
0.833
1.236
2.2039
2.243
3.559**
0.418
2.580
0.418
0.706
0.261
0.208
1.827
0.681
0.901 Without trend (11) (8) (13) (12) (7) (22) (8) (10) (14) (11) (5) (4) (9) (12) (9) (21) (4) (3) (12) (16) (14) (16)
2.150
2.194
1.694
1.647
1.055
1.486
1.555
0.797
1.453
1.153
0.683
2.203
2.615***
1.863
2.832***
1.250
1.474
1.564
2.136
1.231
1.939
1.602 (14) (8) (12) (11) (7) (22) (7) (10) (14) (11) (7) (4) (7) (9) (10) (22) (4) (2) (11) (15) (14) (16) Notes: Figures in parentheses are lag lengths for ADF test and bandwiths for Phillips–Perron test. ** Denotes t value is statistically significant at 5%. *** Denotes t value is statistically significant at 1%. Critical values:
3.961 (1%),
3.411 (5%),
3.127 (10%) with trend;
3.432 (1%),
2.862 (5%),
2.567 (10%) without trend. Critical values are from MacKinnon (1991). Model C can be depicted as:
yt ¼  þ yt
1 þ t þ DUt þ DTt þ k X j¼1 dj
yt
j þ "t ð4Þ The null hypothesis in Equations 3 and 4 is that ¼ 0, which implies there is a unit root in yt. The alternative hypothesis is that <0, which implies that yt is breakpoint stationary. Of the other variables, DUt is an indicator dummy variable for a mean shift occurring at time TB, while DT is the corresponding trend shift variable, where: ( 1 if t > TB DUt ¼ 0 otherwise and DTt ¼ ( t
TB if t > TB 0 otherwise To implement the sequential trend break model, some regions must be chosen such that the end points of the sample are not included. The reason is that in the presence of the end points the asymptotic distribution of the statistics diverges to infinity. Zivot and Andrews (1992) suggest the ‘trimming region’ be specified as (0.15T, 0.85T), which is followed. Thus, the break points are selected recursively by choosing the value of TB for which the ADF t-statistic (the absolute value of the t-statistic for ) is maximized. The results for models A and C are reported in Tables 3 and 4. There is no additional evidence from models A and C against the random walk hypothesis. Models A and C suggest that the null hypothesis of a random walk can be rejected for New Zealand, but not for any of the other 21 countries. The result for Mexico, for which the random walk hypothesis is not rejected in the sequential trend break model, is likely to reflect the fact that adding an additional unnecessary structural break lowers the power of the unit root test when the test without a break finds that stock prices in Mexico are mean reverting. In model A the break in the intercept is statistically significant for each country at the 5% level or better. In model C the break in the intercept is statistically insignificant for each country, but the break in the Are OECD stock prices characterized by a random walk? 551 Downloaded by [Monash University Library] at 21:36 15 October 2014 Table 3. Zivot and Andrews (1992) test for unit roots in daily stock prices based on Model A Countries TB USA 18/05/2001 Canada 21/05/2001 France 21/05/2001 Italy 12/12/1996 Japan 04/05/2001 UK 07/06/2001 Germany 29/06/2001 Austria 12/07/1998 Belgium 18/07/2001 Finland 18/12/2000 Hungary 11/01/1996 Ireland 20/07/2001 Netherlands 21/05/2001 Portugal 29/01/2001 Spain 05/09/1996 Sweden 30/01/2001 Switzerland 23/05/2001 Turkey 01/11/2002 Australia 28/06/2001 New Zealand 07/05/1993 Korea 10/05/1996 Mexico 13/07/2000
0.0052 (
3.2652)
0.0042 (
3.1569)
0.0025 (
2.2084)
0.0029 (
2.7627)
0.0074 (
3.6022)
0.0049 (
2.9903)
0.0033 (
2.5719)
0.0051 (
3.2924)
0.0026 (
2.1156)
0.0033 (
3.0279)
0.0048 (
4.0836)
0.0036 (
3.1169)
0.0024 (
1.9229)
0.0020 (
2.6151)
0.0029 (
2.5668)
0.0030 (
2.6360)
0.0032 (
2.5160)
0.0062 (
3.7307)
0.0084 (
3.9926)
0.0110** (
5.1761)
0.0042 (
2.8347)
0.0077 (
4.5440)  k
0.0033*** (
3.4266)
0.0016*** (
2.8969)
0.0033*** (
3.3408) 0.0033*** (3.1171)
0.0038*** (
3.1919)
0.0036*** (
3.2365)
0.0045*** (
3.4022)
0.0020*** (
2.6238)
0.0024*** (
2.5462)
0.0064*** (
4.4408) 0.0073*** (4.3871)
0.0030*** (
3.6269)
0.0039*** (
3.0226)
0.0025*** (
3.3315) 0.0039*** (3.1252)
0.0042*** (
3.5316)
0.0035*** (
3.0051)
0.0085*** (
3.5306)
0.0019*** (
3.1053) 0.0046*** (4.4603)
0.0033** (
2.1682)
0.0029*** (
2.4994) 0 2 6 3 1 7 5 6 4 5 7 7 7 7 7 5 4 7 5 1 4 3 Notes: The 1%, 5% and 10% critical values are
5.34,
4.80 and
4.58 respectively (Zivot and Andrews, 1992). **(***) Denote statistical significance at the 5% and 1% levels respectively. slope is statistically significant for 19 countries at 1% and a further two countries (Hungary and New Zealand) at 10%. The only country in model C for which neither the intercept nor slope is statistically significant is Korea. In model A, for 16 of the 22 countries the structural break is between 2000 and 2002, which was a period of global economic downturn, precipitated by a slowdown in the US economy. This period contains several episodic events including the burst of the internet bubble, a series of revelations about fraudulent accounting practices in the USA, terrorism and wars, which drove stocks lower. In particular, for 13 countries the structural break occurs between May and July 2001 and may be associated with the 11 September 2001 terrorist attacks. In model C the structural break occurs earlier, mainly in the three years 1997–1999, which was a period containing the Asian and Russian P. K. Narayan and R. Smyth 552 Downloaded by [Monash University Library] at 21:36 15 October 2014 Table 4. Zivot and Andrews (1992) test for unit roots in daily stock prices based on Model C Countries TB USA 07/10/1998 Canada 29/11/1999 France 07/10/1998 Italy 27/10/1997 Japan 01/03/1999 UK 07/10/1998 Germany 14/10/1999 Austria 30/14/1993 Belgium 25/12/1997 Finland 15/10/1999 Hungary 01/01/1996 Ireland 27/10/1997 Netherlands 15/10/1999 Portugal 27/10/1997 Spain 11/11/1997 Sweden 15/10/1999 Switzerland 27/10/1997 Turkey 29/10/1999 Australia 25/05/2000 New Zealand 23/06/1993 Korea 06/05/1996 Mexico 14/10/1999 k 
0.0085 (
4.2833)
0.0051 (
3.4472)
0.0045 (
3.3612)
0.0046 (
3.5293)
0.0091 (
4.0959)
0.0096 (
4.6442)
0.0070 (
3.9544)
0.0080 (
4.3917)
0.0073 (
4.0558)
0.0071 (
4.0579)
0.0049 (
4.2090)
0.0050 (
3.4725)
0.0068 (
3.6199)
0.0035 (
3.4737)
0.0056 (
3.6262)
0.0072 (
4.1450)
0.0061 (
3.6031)
0.0079 (
4.1039)
0.0112 (
4.5229)
0.0130** (
5.3364)
0.0046 (
2.9230)
0.0088 (
4.6936) 0.0027 (0.0000) 0.0009 (0.0000) 0.0048 (0.0000) 0.0043 (0.0000) 0.0031 (0.0000) 0.0036 (0.0000) 0.0038 (0.0000) 0.0046 (0.0000) 0.0040 (0.0000) 0.0069 (0.0000) 0.0068 (0.0000) 0.0027 (0.0000) 0.0025 (0.0000) 0.0027 (0.0000) 0.0045 (0.0000) 0.0039 (0.0000) 0.0034 (0.0000) 0.0042 (0.0000) 0.0009 (0.0000) 0.0041 (0.0000)
0.0037 (0.0000) 0.0021 (0.0000) 2.9488*** (
4.3874) 1.5575*** (
3.1044) 3.6697*** (
4.3429) 3.2694*** (
3.4441) 2.5833*** (
3.770) 3.7515*** (
4.9130) 2.7056*** (
4.4248) 3.9350*** (3.3393) 3.4045*** (
4.4609) 2.9779*** (
4.4416) 4.0126* (
1.8222) 2.7257*** (
3.7298) 2.0104*** (
4.1805) 2.7502*** (
4.1989) 3.0901*** (
4.0449) 2.8327*** (
4.5184) 2.8374*** (
3.9925) 1.7136*** (
3.7195) 1.3936*** (
3.6086) 3.7172* (
1.9302)
2.3111 (
0.7814) 1.6119*** (
2.8717) 0 2 6 3 1 7 5 6 4 5 7 7 7 7 7 5 4 7 5 1 4 3 Notes: The 1%, 5% and 10% critical values are
5.57,
5.08 and
4.82 respectively (Zivot and Andrews, 1992). **(***) Denote statistical significance at the 5% and 1% levels respectively. economic crises and the internet bubble of the late 1990s. For six countries the structural break occurs in 1997, for three countries it occurs in 1998 and for eight countries it occurs in 1999. Australia, for which the break occurs in May 2000, is the only country in model C where the break occurs following the collapse of technology stocks. Table 5 reports the growth effects on stock prices of the breaks identified in model C for the 22 OECD countries in the sample. The average growth effects for all the 22 countries are also shown, a group of 15 European countries and the G7 countries on the basis of results obtained from model C. The growth rate in stock prices is calculated for two different periods. The first period corresponds with the period before the structural break (period 1), while the second period covers the period after the structural break (period 2). An interesting feature of the first four columns of Table 5 is that with the exception of Austria, Finland Are OECD stock prices characterized by a random walk? 553 Table 5. Trend breaks and growth rates in stock prices Downloaded by [Monash University Library] at 21:36 15 October 2014 Countries USA Canada France Italy Japan UK Germany Austria Belgium Finland Hungary Ireland Netherlands Portugal Spain Sweden Switzerland Turkey Australia New Zealand Korea Mexico AVERAGE All countries European countries G7 countries Trend breaks TB1 Level changes 07/10/1998 29/11/1999 07/10/1998 27/10/1997 01/03/1999 07/10/1998 14/10/1999 30/04/1993 25/12/1997 15/10/1999 01/01/1996 27/10/1997 15/10/1999 27/10/1997 11/11/1997 15/10/1999 27/10/1997 29/10/1999 25/05/2000 23/06/1993 06/05/1996 14/10/1999 2.9488 1.5575 3.6697 3.2694 2.5833 3.7515 2.7056 3.9350 3.4045 2.9779 4.0126 2.7257 2.0104 2.7502 3.0901 2.8327 2.8374 1.7136 1.3936 3.7172
2.3111 1.6119 and Hungary, the growth rate in stock prices in period 2 has been lower than that in period 1. The average growth rate in stock prices for all countries, the 15 European countries and the G7 countries are all lower in period 2 than period 1. This implies that the structural break in model C, which for most of the countries was in the late 1990s, had a negative impact on world stock prices. Japan exhibits a negative growth rate in stock prices in both periods with the negative growth rate in period 2 higher than that in period 1. In Germany, the UK, Belgium, Netherlands and Sweden, while the growth rate in stock prices in period 1 is positive, the growth rate in stock prices in period 2 becomes negative. Austria is the only country in the sample for which the growth rate in period 1 is negative and it becomes positive in period 2. In the last two columns of Table 5, we report the ratios of the growth rates which provide an indication of the extent of the slowdown in OECD stock markets following the structural break and enable us to gauge the strength of the growth rates in the different periods. On the whole, stock prices for all the 22 countries, 15 European countries and the G7 countries grew much faster in the first period relative Average growth rates Ratios of growth rates Period 1 (P1) Period 2 (P2) P1/P2 P2/P1 0.0567 0.0430 0.0427 0.0417
0.0142 0.0443 0.0650
0.0413 0.0499 0.0667 0.0389 0.0677 0.0773 0.0597 0.0625 0.0731 0.0823 0.2590 0.0378 0.0572 0.0346 0.1057 0.0213 0.0127 0.0154 0.0288
0.0324
0.0014
0.0357 0.0251
0.0035 0.0774 0.1072 0.0213
0.0460 0.0053 0.0191
0.0256 0.0024 0.1487 0.0040 0.0154 0.0091 0.0507 2.6620 3.3858 2.7727 1.4479
0.4383 31.643 1.8207 1.6454 14.257 0.8618 0.3629 3.1784 1.6804 11.264 3.2723 2.8555 34.292 1.7418 9.4500 3.7143 3.8022 2.0848 0.3757 0.2953 0.3607 0.6906
2.2817 0.0316 0.5492 0.6077 0.0701 1.1604 2.7558 0.3146 0.5951 0.0888 0.3056 0.3502 0.0292 0.5741 0.1058 0.2692 0.2630 0.4797 0.0596 0.0660 0.0399 0.0191 0.0185 0.0012 3.1204 3.5676 6.1848 0.3205 0.2803 0.0301 to the second period. The average first period growth rate in stock prices for all countries in the sample was 312% those of the second period rate, for the 15 European countries the first period growth rate in stock prices was 357% those of the second period rate, while for the G7 countries the corresponding difference in the growth rate was 618%. IV. Panel Data Unit Root Test Without a Structural Break A possible reason for the failure of Dickey–Fuller type tests, even those which incorporate trend breaks, to reject the unit root null is the time span of the data. While, using daily data provides a large number of observations the time span of the data is relatively short. This possibility is examined by first employing the t-bar test proposed by Im et al. (2003) in order to exploit the extra power in the panel properties of the data. As discussed in the introduction, it is less restrictive under the alternative hypothesis than either the Levin and Lin (1992) or FGLS tests used in previous studies to test the random walk null P. K. Narayan and R. Smyth 554 Table 6. t-bar panel data model With trend Without trend Critical values All countries G7 countries Critical values Test statistic 1% 5% 10% Test statistic 1% 5% 10% 1.382
0.099
2.54
2.86
2.43
2.66
2.38
2.55
0.179 0.050
1.92
2.27
1.81
2.05
1.75
1.94 Downloaded by [Monash University Library] at 21:36 15 October 2014 Notes: Critical values are from Im et al. (2003, Table 4). hypothesis for stock prices by Zhu (1998) and Chaudhuri and Wu (2004). There are two stages in constructing the t-bar test statistic. The first is to calculate the average of the individual ADF t-statistics for each of the countries in the panel. The second is to calculate the standardized t-bar statistic according to the following formula: pffiffiffiffi pffiffiffiffi t-bar ¼ N ðt
t Þ= t ð5Þ where N is the size of the panel, t is the average of the individual ADF t-statistics for each of the countries and t and t are respectively estimates of the mean and variance of each t i. Im et al. (2003) provide Monte Carlo estimates of t and t and tabulate exact critical values for the t-bar statistic for various combinations of N and T. A potential problem with the t-bar test, involves cross-sectional dependence. When there is crosssectional dependence in the disturbances the t-bar test is no longer applicable. However, Im et al. (2003) suggest that in the presence of cross-sectional dependence, the data can be adjusted by subtracting the cross-sectional means and then applying the t-bar statistic to the transformed data. The standardized de-meaned t-bar statistic converges to a standard normal in the limit.1 Luntiel (2001) and Smyth (2003) show, using data on a panel of real exchange rates and unemployment rates respectively, that the de-meaning procedure does dramatically reduce cross-sectional dependence even in instances where the observed data are highly correlated. Thus, following the suggestion of Im et al. (2003) we de-meaned the data before applying the t-bar test to each of the panels considered. The results for the t-bar test, with and without trend, are reported in Table 6 for a full panel of 22 countries, as well as a G7 panel. The t-bar test is unable to reject the random walk null hypothesis for the full panel or G7 panel. The findings for the panel data model reinforce those with the ADF and Phillips–Perron tests as well as the findings from the sequential trend break model. This result reinforces Zhu’s (1998) earlier findings for a G7 panel using the Levin and Lin (1992) test, but differ from Chaudhuri and Wu’s (2004) finding of mean reversion for 17 emerging markets with the FGLS panel data test. V. Panel Data Unit Root Test with a Structural Break A problem with t-bar test proposed by Im et al. (2003) is that it does not allow for a structural break in the panel data test. To examine the effect of a structural break on the panel the panel LM unit root test suggested by Im et al. (2002) is employed. It begins by explaining the panel LM unit root test used to examine the non-stationarity of the stock price series and then proceed to the results. Consider a model of the form: SPit ¼ 0i Xit þ it , it ¼ i i, t
1 þ "it ð6Þ SP is the stock price series, i represents the crosssection of countries (i ¼ 1, . . . , N ), t represents the time period (t ¼ 1, . . . , T ), it is the error term and Xit is a vector of exogenous variables. The test for the unit root null hypothesis is based on the parameter i, while "it is a zero mean error term that allows for heterogeneous variance structure across cross-section units but assumes no cross-correlations. Parameter i allows for heterogeneous measures of persistence. A structural break in the model is incorporated by specifying Xit as [1, t, Dit, Tit]0 , where Dit is a dummy variable that denotes a mean shift while Tit denotes a trend shift. If a break for country i occurs 1 Im et al. (2003) assume that "it ¼ t þ it where t is a time-specific common effect which indicates the degree of dependence across countries and it are i.i.d. idiosyncratic random effects. While cross-sectional de-meaning will introduce dependence across the de-meaned error terms, the tests will remain asymptotically valid provided that the it are rendered uncorrelated. Are OECD stock prices characterized by a random walk? at TBi, then the dummy variable Dit ¼ 1 if t>TBi, zero otherwise, and Dit ¼ t
TB if t>TBi, zero otherwise. The panel LM test statistic is obtained by averaging the optimal univariate LM unit root t-test statistic estimated for each country. This is denoted as LMi : Downloaded by [Monash University Library] at 21:36 15 October 2014 LMbarNT ¼ N 1X LMi N i¼1 Im et al. (2002) then construct a standardized panel LM unit root test statistic by letting E(LT) and V(LT) denote the expected value and variance of LMi , respectively under the null hypothesis. Im et al. (2002) then compute the following expression: pffiffiffiffi N ½LMbarNT
E ðLT Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi LM ¼ VðLT Þ The numerical values for E(LT) and V(LT) are in Im et al. (2002). The asymptotic distribution is unaffected by the presence of structural breaks and is standard normal. The LM test statistic for the panel of 22 OECD countries turns out to be 1.781, which is greater than the 10% critical value of
1.282, implying that joint null hypothesis of non-stationarity cannot be rejected. The possibility that G7 stock prices, when taken as a panel series, may provide different evidence is also considered. The finding from this exercise revealed a LM test statistic of 1.469, which is also greater than the critical value at the 10% level of significance. Hence, one is unable to reject the joint null hypothesis of nonstationarity for the G7 stock price series. Taken together, these results corroborate the findings from the univariate unit root tests and the Im et al. (2003) panel unit root test without a structural break that the daily stock price series for the OECD countries spanning the period 1991 through to 2003 are non-stationary. VI. Conclusions This study contributes to the literature on the unit root properties of stock prices by applying the Zivot and Andrews (1992) sequential trend break unit root test, Im et al. (2003) t-bar panel unit root test and LM panel unit root test with one structural break to daily data for a sample of OECD countries from January 1991 to June 2003. Overall, the results provide strong support for the random walk hypothesis. With the ADF and Phillips–Perron tests the only countries for which one can reject the random walk null hypothesis are Mexico and New Zealand. With 555 the Zivot and Andrews (1992) sequential trend break model one can only reject the random walk null hypothesis for New Zealand. The location of the structural break in the sequential trend break model is identified and the effects of the breaks in model C on the rate of growth in OECD stock prices examined. It is found that the rate of growth in stock prices before the structural break in model C, which tended to occur in the late 1990s across OECD markets, was generally much higher than the rate of growth in stock prices subsequent to the break. This is consistent with the global economic downturn from mid-2000. Finally, the study was unable to reject the random walk null hypothesis for the full OECD panel or smaller G7 panel with and without a structural break. These results stand in sharp contrast to recent findings by Chaudhuri and Wu (2003, 2004) who apply the sequential trend break model and FGLS panel data test to a sample of emerging markets and find evidence of mean reversion. Future research could examine whether incorporating two structural breaks into the unit root test alters the conclusions. Such studies could employ the sequential trend break unit root test with two breaks suggested by Lumsdaine and Papell (1997), the univariate LM unit root test with two structural breaks proposed by Lee and Strazicich (2003) and the LM panel unit root test with two structural breaks developed by Im et al. (2002). The computational complexities of allowing for two structural breaks mean that this exercise is not practical using daily data with a large number of observations such as in this study. This is because of limitations on computing power, but future studies could explore this possibility using monthly data where the number of observations is more manageable. References Baltagi, B. and Kao, C. (2000) Nonstationary panels, cointegration in panels and dynamic panels: a survey, Advances in Econometrics, 15, 7–51. Balvers, R., Wu, Y. and Gilliland, E. (2000) Mean reversion across national stock markets and parametric contrarian investment strategies, Journal of Finance, 55, 745–72. Chaudhuri, K. and Wu, Y. (2003) Random walk versus breaking trend in stock prices: evidence from emerging markets, Journal of Banking and Finance, 27, 575–92. Chaudhuri, K. and Wu, Y. (2004) Mean reversion in stock prices: evidence from emerging markets, Managerial Finance, 30, 22–31. Cochrane, J. (1991) A critique of the application of unit root tests, Journal of Economic Dynamics and Control, 15, 275–84. DeBondt, W. and Thaler, R. (1985) Does the stock market overreact?, Journal of Finance, 40, 793–805. Downloaded by [Monash University Library] at 21:36 15 October 2014 556 DeJong, D. N, Nankervis, J. C, Savin, N. E and Whiteman, C. H. (1992) The power problems of unit root tests in time series with autoregressive errors, Journal of Econometrics, 53, 323–43. Fama, E. (1970) Efficient capital markets: a review of theory and empirical work, Journal of Finance, 25, 383–417. Fama, E. F. and French, K. R. (1988) Permanent and temporary components of stock prices, Journal of Political Economy, 96, 246–73. Hall, A. D. (1994) Testing for a unit root in time series with pretest data based model selection, Journal of Business and Economic Statistics, 12, 461–70. Hayashi, F. (2000) Econometrics, Princeton University Press, Princeton, NJ. Im, K. S., Lee, J. and Tieslau, M. (2002) Panel LM unit root tests with level shifts, mimeo, Department of Economics, University of Central Florida. Im, K.-S., Pesaran, H. and Shin, Y. (2003) Testing for unit roots in heterogeneous panels, Journal of Econometrics, 115, 53–74. Karlsson, S. and Lothgren, M. (2000) On the power and interpretation of panel unit root tests, Economic Letters, 66, 249–55. Kim, M. J., Nelson, C. R. and Startz, R. (1991) Mean reversion of stock prices: a reappraisal of the empirical evidence, Review of Economic Studies, 58, 515–28. Lee, J., and Strazicich, M. C. (2003) Minimum Lagrange multiplier unit root test with two structural breaks, The Review of Economics and Statistics, 85, 1082–9. Le Roy, S. F. (1982) Expectations models of asset prices: a survey of theory, Journal of Finance, 37, 185–217. Levin, A. and Lin, C.-F. (1992) Unit root tests in panel data: asymptotic and finite sample properties, Department of Economics UCSD Discussion Paper 92-23. Lo, A. W. and MacKinlay, A. C. (1988) Stock market prices do not follow random walks: evidence from a simple specification test, Review of Financial Studies, 1, 41–66. Lumsdaine, R. and Papell, D. (1997) Multiple trend breaks and the unit root hypothesis, Review of Economics and Statistics, 79, 212–18. Luntiel, K. (2001) Heterogeneous panel unit root tests and purchasing power parity, Manchester School Supplement, 42–56. MacKinnon, J. (1991) Critical values for cointegration tests, in Long-run Economic Relationships: Readings in Cointegration (Eds) R. F. Engle and C. W. Granger, Oxford University Press, Oxford, pp. 267–76. P. K. Narayan and R. Smyth Maddala, G. S. and Wu, S. (1999) A comparative study of unit roots with panel data and a new simple test, Oxford Bulletin of Economics and Statistics, 61, 631–51. Malkiel, B. G. (2003) The efficient market hypothesis and its critics, Journal of Economic Perspectives, 17, 59–82. McQueen, G. (1992) Long-horizon mean reverting stock process revisited, Journal of Financial and Quantitative Analysis, 27, 1–18. Ng, S. and Perron, P. (1995) Unit root tests in ARMA models with data dependent methods for the selection of the truncation lag, Journal of the American Statistical Association, 90, 268–81. O’Connell, P. (1998) The overvaluation of purchasing power parity, Journal of International Economics, 44, 1–19. Papell, D. (1997) Searching for stationarity: purchasing power parity under the current float, Journal of International Economics, 43, 313–32. Perron, P. (1989) The great crash, the oil price shock and the unit root hypothesis, Econometrica, 57, 1361–401. Perron, P. and Vogelsang, T. (1992) Non-stationarity and level shifts with an application to purchasing power parity, Journal of Business and Economic Statistics, 10, 301–20. Poterba, J. M. and Summers, L. H. (1988) Mean reversion in stock prices: evidence and implications, Journal of Financial Economics, 22, 27–59. Richards, A. J. (1995) Co-movements in national stock market returns: evidence of predictability but not cointegration, Journal of Monetary Economics, 36, 631–54. Richardson, M. (1993) Temporary components of stock prices: a skeptic’s view, Journal of Business and Economics Statistics, 11, 199–207. Richardson, M. and Stock, J. (1989) Drawing inferences from statistics based on multiyear asset returns, Journal of Financial Economics, 25, 323–48. Smyth, R. (2003) Unemployment hysteresis in Australian states and territories: evidence from panel data unit root tests, Australian Economic Review, 36, 181–92. Zhu, Z. (1998) The random walk of stock prices: evidence from a panel of G7 countries, Applied Economics Letters, 5, 411–13. Zivot, E. and Andrews, D. (1992) Further evidence of the great crash, the oil-price shock and the unit-root hypothesis, Journal of Business and Economic Statistics, 10, 251–70.