DATA TRANSFORMATION AND ARIMA MODELS: A STUDY OF
EXCHANGE RATE OF NIGERIA NAIRA TO US DOLLAR
Supported by
A. A. Adetunji1*, A.O. Adejumo2 and A.J. Omowaye3
1
Department of Statistics, Federal Polytechnic, IIe-Oluji, Ondo State, Nigeria
2
Department of Statistics, University of Ilorin, Kwara State, Nigeria
3
Department of Mathematical Science, Federal University of Technology, Akure, Ondo State, Nigeria
*Corresponding author: adecap4u@yahoo.com;
Abstract:
Exchange rate of any country’s currency goes a long way in affecting various economic activities and it
ensures effective and efficient planning. In order to assist different policy makers in Nigeria in purposeful
prediction by identifying and validating the usage of essential model, the yearly average exchange rate of
Nigeria naira to US dollar from 1960 to 2015 is examined. ARIMA (0,0,0 to 2,2,2) were sequentially
examined using Square Root Transformation (SRT), Natural Log Transformation (NLT) and original series
without transformation (WT). NBIC, RMSE, MAE, and Ljung-Box Q are used as selection criteria among all
the competing models within and among different transformations. ARIMA(1,0,0) when SRT is utilized is
found to provide optimal output with stationary-R2 of 0.976; coefficient of determination (R2) of 97.3%;
NBIC of 4.888 and Ljung-Box Q P-value of 0.981. Hence, the recommended model for forecasting of
average yearly exchange rate of Nigeria naira to US dollar.
Keywords: Time series, ARIMA model, natural log transformation, square root transformation.
Introduction
The rate of exchange of a country’s currency is the relative
price which measures the value of a domestic currency in
terms
of
another
currency,
(https://en.wikipedia.org/wiki/Exchange_rate). Because of
inherent structural transformations required, exchange rate
policies in developing and under-developed countries are
usually sensitive and controversial. When there is a very
high disparity in the balance of trade of any country, the
exchange rate is usually affected. The effect becomes so
obvious and negative when such country is a consuming
nation rather than a producing one. Guitan (1976) reported
that the “for any currency depreciation towards promoting
balance of trade to succeed, it must depend on switching
demand in proposer direction and the economy must have
capacity to meet additional demand by ensuring supply of
more goods”. Effectiveness and efficiency of an economy
are usually determined by fluctuations of exchange rate.
Hence, attaching importance to planning economic
policies based on the predictions of exchange rate is
necessary.
Exchange rate policy in Nigeria has gone through
numerous transformations since her independence when
there was a fixed parity with the British Auctions System
(BAS) as against the former auctions done once two weeks
which assured a relative steady supply of foreign
exchange. The Central Bank of Nigeria (CBN) introduced
the Wholesale Dutch Auction System (WDAS) in 2006
with intention to liberalize the money market, reduce the
arbitrage premium between the Bureaus de Change (BdC)
operators and the interbank officials. The purpose of the
introduction is to consolidate gains recorded when CBN
was using the Retail Dutch Auction System (RDAS) and
also to deepen the foreign exchange market in order to
reveal a realistic exchange rate of the naira. This process
gave room for dealers that are authorized to deal in foreign
exchange using their respective accounts before selling to
their customers.
One of the leading demands of modern time series analysis
and forecasting is exchange rate prediction. The exchange
rates are naturally non-stationary, deterministically
chaotic, and noisy, Box and Jenkins (1994). It fluctuates
299
and requires adequate statistical technique that can
consider adequately represent the variability. In order to
have a better understanding of the underlying process, the
fluctuations are usually examined with a class of structural
time series models with intention of obtaining estimates
that are more efficient. The importance of optimal model
of any economic variable cannot be over emphasized. Both
developed and developing countries need these models for
effective management of often limited resources and
effective planning. Usually, there are always sets of
competing models that may be seen to be equally effective
and efficient when being applied to a particular data set.
The interest then is to find out which particular one among
these models will give be the best and most efficient
(optimal) taking into consideration all the essential factors.
Different approaches have been developed for forecasting
time series data and there are competitions among these
methods on efficiency and minimal error while
forecasting. Among widely used techniques is the
Autoregressive Integrated Moving Average (ARIMA)
where a time series is expressed in terms of its past values
and lagged values of error term. There are variations of
ARIMA models that can be employed depending on the
nature of the data to be analyzed. If there are multiple time
series data, then the Xt can be assumed to be vectors and a
(Vector ARIMA) VARIMA model may be appropriate.
When a seasonal effect is suspected in the model, a
Seasonal ARIMA (SARIMA) model can be used. If the
there is a suspicion that the series exhibits a long-range
dependence, then the Autoregressive Fractionally
Integrated Moving Average model (ARFIMA) which is
also called a Fractional ARIMA (FARIMA) model may be
used.
This paper varies parameters of ARIMA model under
different transformations with the purpose of observing
their efficiency in purposeful forecasting. In economic
time series, transformation is often considered to stabilize
the variances of the series, hence, this research compares
various results forecasting based on the original series
(0,0,0 to 2,2,2) with both its square root transformations
(SRT) and natural log transformations (NLT). For NLT,
let Xt = log Yt be the natural logarithm of the time series
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April, 2016 Vol. 1 No. 1 – e-ISSN: 24085162; p-ISSN: 20485170 pp 299-306
Data Transformation and Arima Models: A Study of Exchange Rate of Nigeria Naira to US Dollar
data, Xt is then used to generate an ARIMA model while
=
for the SRT.
Little or no attention has been given to effect of data
transformation when researching into exchange rate of
Naira to Dollar, Onasanya & Adeniji (2013) and Nwakwo
(2014). Granger and Newbold (1976) opined that optimal
forecast may not be obtained when an instantaneous nonlinear transformation is applied to a variable while
Lϋtkepohl and Xu (2009) stated that substantial reduction
in error may be committed in forecast Mean Squared Error
(MSE) if the log transformation can lead to a more stable
variance of a series of interest but warned that forecasting
prediction may be damaged when the log transformation is
applied and it does not make the variance more
homogeneous.
Application of ARIMA models in diverse studies of
interest is inexhaustible. Various researches had been
carried out for different scenario using the Box-Jenkins
approach. While forecasting the exports of Pakistan’s
South Asian Association of Regional Cooperation
(SAARC), Shafaqat (2012) applied the Box-Jenkins
methodology of univariate ARIMA model and found
ARIMA (1,1,4) as most appropriate model among other
tested ARIMA models. The study revealed that exports
from Pakistan to SAARC will be on the increase in a few
years and hence the need for Pakistani government to
invest into those sectors in which the country has export
potential to the SAARC countries.
While predicting next day process of electricity for Spain
and California (Contreras et al., 2003) used ARIMA
model and it was observed that the Spanish model requires
5 h to forecast for future prices which opposes 2 h needed
with the Californian model. Tsitsika et al. (2007) adopted
ARIMA model in forecasting pelagic fish production. The
ARIMA (1,0,1) and ARIMA (0,1,1) were adjudged to be
optimal while Datta (2011) used ARIMA to forecast
inflation rate in Bangladesh. The result of his analysis
showed that ARIMA (1,0,1) is the best model that fits the
inflation rate of Bangladesh.
ARIMA had also been applied in healthcare studies.
Sarpong (2013) studied Maternal Mortality Ratios (MMR)
in a Kumasi Teaching Hospital for 11 years. The result
showed that the hospitals MMR was relatively stable with
a very alarming average quarterly MMR of 9.677 per
thousand live births which is almost twice the ratio
obtained in the whole of Ghana (4.51 per thousand). AIC
value of 581.41 made the researcher to conclude that
ARIMA (1,0,2) is the most adequate model for forecasting
quarterly MMR at the hospital (Liv et al., 2011) as well
utilized ARIMA model to forecast hemorrhagic fever
incidence with renal syndrome in China, ARIMA (0,3,1)
model was found to be the best for predictive purpose.
Albayrak (2010) applied same model to forecast the
production and consumption of primary energy in Turkey.
With intention of obtaining forecast values for the average
daily price of share of Square Pharmaceuticals Limited
(SPL), Jiban et al. (2013) examined ARIMA model by
observing the conditions for the stationarity of the data
series using ACF and PACF plots, and later used DickeyFuller test statistic and Ljung-Box Q-statistic. The result
showed that the time series data is not stationary even after
log-transformation but the series became stationary after
taking the first difference of the log-transformation.
RMSE, AIC and MAPE are used to select the most fitted
ARIMA model, they concluded that the best model that
nest describes the series is ARIMA (2,1,2). While using
some measures such as: MAPE, RMSE and MAE for
300
selection of appropriate model (Emang et al., 2010) as
well made use of ARIMA model in forecasting chipboard
and moulding export demand in Malaysia. Rahman (2010)
constructed an ARIMA model to forecast the production
of rice in Bangladesh using MAPE, MSE, MAE, RMSE
and R2 as selection criteria.
In Nigeria, researchers had utilized ARIMA models for
various purposes. Badmus and Ariyo (2011) used this
model to forecast the production and area of maize from
Nigerian. Adams, Akano, and Asemota (2011) also used
this model to forecast generation of electricity power from
Nigeria. They concluded ARIMA (3,2,1) is the best model.
While applying ARIMA Model on rate of exchanging
Naira to Dollar for a period of thirty years (1982-2011),
Nwakwo (2014) concluded that AR(1) was the preferred
model for purposeful prediction.
From various works of researchers, little effort had been
given to effect of data transformation on forecasting and
overall usefulness of ARIMA model. Hence, this paper is
aimed at observing efficiency of ARIMA (p,q,d) under
different transformations and using various measures like
MAE, RMSE, and R2 as selection criteria. This is expected
to improve quality of decision by those involved in
monetary policies formulation as it affects exchange rate.
Materials and Methods
Autoregressive integrated moving average (ARIMA)
model is a general form of an autoregressive moving
average (ARMA) model. The model is fitted to time series
data with primary aim of having a better understanding of
the series and to predict its future values, especially when
the series shows signs of non-stationarity. The nonstationarity is often reduced by applying an initial
differencing step (integrated). ARIMA models that are
non-seasonal are usually denoted with ARIMA (p,d,q)
where p implies the order of the AR model, d is the
differencing degree and q represents the moving average
order, Box and Jenkins (1994).
ARIMA model has a major advantage over majority of
time series modelling since it utilizes data on the time
series of interest only. This usually serves most when
dealing with multivariate models where different factors
might have affected the quality of the input variables.
Although arguments in using ARIMA models among
researchers persists, ARIMA models has been proved to be
relatively robust most especially when dealing with shortterm forecast. Glassman and Stockton (1987) verified the
robustness of ARIMA models for short-term forecasting.
Autoregressive (AR) process
An AR process requires each value of a series to be a
linear function of value preceding them. Hence, in an AR
of order 1, only the first preceding value is utilized as a
function of the current value. AR(1) denotes the first order
AR scheme while AR(2) denotes its second order.
Suppose that the variable Zt is a linear function of any
preceding variable Zt-1, the model for an AR(1) can be
written as:
= +
+ … … … (1)
where ~
(0, )
For an AR(2), the model becomes
= +
+
+⋯+
+ … … (2)
(0, ) and
is the coefficient of first
where ~
order AR while is the coefficient for pth order AR.
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April, 2016 Vol. 1 No. 1 – e-ISSN: 24085162; p-ISSN: 20485170 pp 299-306
Data Transformation and Arima Models: A Study of Exchange Rate of Nigeria Naira to US Dollar
Differencing
Procedure for differencing involves calculating series of
sequential changes in the values of the time series data. It
is usually used when there is a systematic change in the
mean of the observation as the time changes. Differencing
often ensures that a series that is not stationary becomes
stationary with homogeneous variance. Differencing a
series once requires calculating the periodic change once
and to do it twice needs the calculation to be done twice.
Moving average (MA)
This is also known as the rolling average. It is usually
applied in analysing financial data and can as well be used
like a generic smoothing operation. MA series can be
obtained for any time series data and are usually used to
smoothen short-term fluctuations and therefore highlights
a longer-term cycles.
Let the model Zt be defined as:
Z =θ+ε + β ε
… … … (3)
where θ is constant and ε ~NIID(0, σε ).
Zt is a constant added to a MA of the current and error
terms in the past. Hence, Zt is said to follow MA (1)
process i.e. a moving average of order 1. But if Zt is
denoted with:
Z =θ+ε + β ε + β ε
… … … . (4)
thenZt is said to follow a MA(2) process.
Generally, Zt follows a MA(q) process if
= + + &
+ &
+ ⋯ + &' ' … … (5)
ARIMA Model Selection, Checking and Validation
Model Selection
When an attempt is being made to use ARIMA model for
predictive purpose, the first step is to identify the model
that best explains the model. Such model should have
smallest values of parameters and should be good enough
to adequately explain the model. In ARIMA (p,d,q), p and
q must not be more than 2, Al–Wadia (2011). Therefore,
checking the NBIC (Normalized Bayesian Information
Criterion) of the model is only limited to p, q and d values
2 or less. According to Al–Wadia (2011), the model that
has the least NBIC value should be given preference.
Another criterion often used to measure goodness of fit of
a model is the Akaike Information Criteria (AIC). When
two or more models are competing, the one that has the
least AIC is generally considered to be closer with real
data, Yang (2005). However, Anderson (2008) opined that
AIC does not usually penalize complexity of a model
heavily as the BIC (Bayesian Information Criterion) does.
Checking the model
Appropriate lag (the value of p) is usually identified using
the autocorrelation function (ACF) and partial
autocorrelation function (PACF). The PACF provides
more information on the behaviour of the time series while
the ACF provides information on the correlation between
observations in a time series at different time apart. Both
ACF and PACF suggest the model to be built. Generally,
the ACF and the PACF has spikes at lag k and cuts off
after lag k at the non-seasonal level. The order of the
model can be identified by the number of spikes that are
significant. It must be noted however that both ACF and
PACF only suggest on where to build the model, it is very
essential to obtain different models around the suggested
order and criteria like Akaike Information Criterion (AIC),
Akaike (1974) or Bayesian Information Criterion (BIC),
301
Schwarz (1978) can then be used to select the best among
the competing models.
The AIC and BIC are obtained using:
/00
AIC = 2k – 2 log(L) = 2) + n log . 2 … … . . . (6)
1
[https://en.wikipedia.org/wiki/Akaike_information_criterio
n]
BIC = -2 Log(L) + k Log(n) = 4 log(
5
) + ) log(4) … . . . (7)
[https://en.wikipedia.org/wiki/Bayesian_information_criter
ion]
k is the number of parameters in the model; L is the value
maximized for the likelihood function for the estimated
model; n is the number of observation i.e. the sample size;
RSS is the residual sum of squares of the estimated model
and is the error variance.
Model validity
In order to select the best among competing models, it is
essential to compute some statistics that would ensure that
the final model to be selected has the least variance. These
criteria are compared for three periods viz, estimation
period, validation period and total period. With respect to
this research, the following selection criteria are used: (a)
Mean Absolute Error (MAE), and (b) Root Mean Square
Error (RMSE)
Mean absolute error (MAE)
This is the mean of the absolute deviation of predicted and
observed values and it is obtained using;
789 = :
CD
;
<=>
−
B
@5A ;
… … … (8)
Root mean square error (RMSE)
It is the square root of the sum of square of the differences
between the predicted values and the observed values
dividing by their number of observation (t). It is given by:
1
F7G9 = H :I
B
CD
<=>
−
@5A J
… … … (9)
When comparing models, the best one is the one with the
least error whether MAE or RMSE.
Properties of a good ARIMA model
The following characteristics are considered in this
research before the best among all competing models is
selected.
(i) Stationary- It must have a relatively high stationary-R2
value, usually in excess of 0.95
(ii) Invertible- Its MA coefficient must not be
unreasonable high
(iii) Parsimonious- It must utilize small number of
coefficient as possibly needed to explain the time series
data
(iv) Its residuals must be statistically independent
(v) It must fit the time series data sufficiently well at the
stage of estimation
(vi) Its MAE and RMSE must not be unnecessarily high
(vii) Sufficiently small forecast errors
Diagnostic checking
This is essential after the selection of a particular ARIMA
model having estimated its parameters. The model’s
adequacy is verified by analyzing the residuals. The model
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April, 2016 Vol. 1 No. 1 – e-ISSN: 24085162; p-ISSN: 20485170 pp 299-306
Data Transformation and Arima Models: A Study of Exchange Rate of Nigeria Naira to US Dollar
is accepted if the residuals are found to be white noise,
hence, the model selection procedure is restarted
The conformity of white noise residual of the model fit
will be judge by plotting the ACF and the PACF of the
residual to see whether it does not have any pattern or we
perform Ljung Box Test on the residuals. The null
hypothesis is:
H0: There is no serial correlation
H1: There is serial correlation
The test statistics of the Ljung box is;
1
NO
LM = 4(4 + 2 :
… PQ … … 10
4?)
CD
Here, n is the sample size, m is the lag length and ρ is the
sample autocorrelation coefficient.
The decision: if the LB is less than the critical value of X2,
then the null hypothesis is not rejected. This implies a
small value of Ljung Box statistics will be in support of no
serial correlation or i.e. the error term is normally
distributed. This is concerned about the model accuracy.
Results and Discussions
The time plot shows that the exchange rate of Naira to
dollar was relatively stable from 1960 to 1985 after when
there was an obvious increase trend in the rate. A
significant increment in the exchange rate was observed in
the year 1998 to 1999 which kept on increasing till 2004
when a brief downward trend was observed till 2009.
However, the rate jumped up significantly from 2009 to
2010 and the increment is sustained till 2015.
Chart 1: Time plot of yearly exchange rate of the naira to US dollar from 1960 - 2015
Autocorrelation function
Since the autocorrelation coefficient (Table 1) starts at a
very high value at lag 1 (0.942) and declines rapidly as the
lag lengthens, this indicates that exchange rate (Naira to
Dollar) is a non-stationary series. This is supported by the
auto-correlogram (Chart 2) that follows with most of the
point falling outside the control limit and the point falling
above the positive side of the chart (no randomness),
hence the series is not stationary. This table shows various
values obtained for autocorrelations of exchange rate of
Nigeria naira to US dollar at the first 16 lags. The value of
autocorrelation function for lag i, i = 1 to 16 is obtained
using:
NRS ,RSTU
302
∑1D W ? WX
∑1D
W ? WX
W
C
? W̅
∑1D W ? W̅
… … … 11
Table 1: ACF of exchange rate of naira to dollar
Lag
Autocorrelation
Std. Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
.942
.887
.827
.764
.701
.638
.573
.524
.468
.412
.341
.267
.193
.122
.055
-.006
.130
.129
.128
.127
.125
.124
.123
.122
.120
.119
.118
.116
.115
.114
.112
.111
LB Statistic
Value
Sig.
52.413
.000
99.694
.000
141.635 .000
178.111 .000
209.390 .000
235.836 .000
257.580 .000
276.125 .000
291.293 .000
303.270 .000
311.658 .000
316.932 .000
319.751 .000
320.899 .000
321.140 .000
321.144 .000
FUW Trends in Science & Technology Journal ftstjournal@gmail.com
April, 2016 Vol. 1 No. 1 – e-ISSN: 24085162; p-ISSN: 20485170 pp 299-306
Data Transformation and Arima Models: A Study of Exchange Rate of Nigeria Naira to US Dollar
Chart 3: Auto-correlogram of the first differenced
exchange rate (naira to dollar) for 16 lags
Chart 2: Auto-correlogram of the original exchange rate
(naira to dollar) for 16 lags
Table 3 and Chart 3 show that the series is stationary after
the first difference since most of the points hover around
zero and show randomness. This suggests that is will be
essential to difference the original series at least once for
predictive purpose. It can also be observed from the chat
that almost all the points fall within the control limit.
When no transformation was made (Table 4a) there are
disparities on the efficiency of various competing models
ARIMA (2,0,2) has the best stationary-R2; ARIMA (1,1,1)
and ARIMA (2,1,2) have the best R2; ARIMA (1,1,1) has
the least RMSE; ARIMA (2,2,2) has the least MAE; while
ARIMA (1,0,0) has the most desirable Ljung-Box Q
statistics. However, when various competing models are
considered across the board by rating their efficiency,
ARIMA (1,0,0) and ARIMA (2,0,2) best explained the
series with higher preference for the former since it has
lower Normalized Bayesian Information Criteria (NBIC)
of 5.044 and Ljung-Box Q Significant value of 0.956.
Table 2: ACF (First Differenced)
Lag
Autocorrelation
Std. Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
.066
.074
.042
.000
-.031
.032
.002
-.026
-.081
.337
-.020
.017
-.001
-.055
-.047
.078
.131
.130
.129
.128
.126
.125
.124
.122
.121
.120
.118
.117
.116
.114
.113
.112
LB Statistic
Value
Sig.
.252
.615
.573
.751
.678
.878
.678
.954
.736
.981
.801
.992
.801
.997
.847
.999
1.295
.998
9.196
.514
9.223
.601
9.243
.682
9.243
.754
9.476
.799
9.647
.841
10.132 .860
Table 3: Summary table of competing models
Transformation
Model Type
R2
0.968
0.971
Statistics
RMSE
MAE
11.179
6.020
10.948
4.816
LB Q(18)
Statistics
Sig.
8.444
0.956
10.659
0.713
None
ARIMA (1,0,0)
ARIMA (2,0,2)
Stationary R2
0.968
0.971
Square Root
ARIMA (1,0,0)*
ARIMA (1,0,1)
ARIMA (2,0,0)
ARIMA (2,0,1)
0.976
0.976
0.976
0.980
0.973
0.973
0.973
0.973
10.341
10.436
10.436
10.494
4.079
3.959
3.960
3.852
4.888
4.978
4.978
5.061
7.180
7.333
7.352
9.343
0.981
0.966
0.966
0.859
Natural Log
ARIMA (2,0,1)
ARIMA (2,0,2)
0.988
0.985
0.970
0.968
11.145
11.511
4.493
4.613
5.181
5.318
11.482
7.449
0.718
0.916
303
Normalized BIC
5.044
5.218
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April, 2016 Vol. 1 No. 1 – e-ISSN: 24085162; p-ISSN: 20485170 pp 299-306
Data Transformation and Arima Models: A Study of Exchange Rate of Nigeria Naira to US Dollar
Table 4a: ARIMA models with various statistics (No Transformation)
Model Type
ARIMA (0,0,0)
ARIMA (0,0,1)
ARIMA (0,1,0)
ARIMA (0,1,1)
ARIMA (1,0,0)*
ARIMA (1,0,1)
ARIMA (1,1,0)
ARIMA (1,1,1)
ARIMA (1,1,2)
ARIMA (1,2,0)
ARIMA (1,2,1)
ARIMA (1,2,2)
ARIMA (2,0,0)
ARIMA (2,0,1)
ARIMA (2,0,2)*
ARIMA (2,1,0)
ARIMA (2,1,1)
ARIMA (2,1,2)
ARIMA (2,2,0)
ARIMA (2,2,1)
ARIMA (2,2,2)
Stationary R2
0.731
0.889
0.061
0.061
0.968
0.968
0.061
0.114
0.074
0.253
0.471
0.470
0.968
0.966
0.971
0.062
0.074
0.109
0.325
0.471
0.482
R2
0.731
0.889
0.972
0.972
0.968
0.968
0.972
0.974
0.973
0.959
0.971
0.971
0.968
0.966
0.971
0.972
0.973
0.974
0.963
0.971
0.971
Statistics
RMSE
MAE
32.285
28.563
20.927
17.489
10.388
4.357
10.487
4.335
11.179
6.020
11.274
5.875
10.487
4.334
10.291
4.856
10.621
4.147
12.873
4.666
10.937
3.961
11.061
3.993
11.313
5.830
11.795
4.826
10.948
4.816
10.588
4.306
10.622
4.215
10.524
4.085
12.357
4.638
11.053
3.913
11.041
3.855
Normalized BIC
7.093
6.298
4.827
4.919
5.044
5.133
4.919
4.954
5.090
5.332
5.080
5.176
5.140
5.295
5.218
5.011
5.090
5.144
5.324
5.175
5.246
LB Q(18)
Statistics
Sig.
258.235
0.000
163.327
0.000
12.007
0.847
12.059
0.797
8.444
0.956
8.868
0.919
12.061
0.796
11.509
0.777
11.989
0.680
22.185
0.178
12.195
0.730
12.274
0.658
8.799
0.921
7.939
0.926
10.659
0.713
12.050
0.741
11.770
0.696
10.033
0.760
17.385
0.361
12.092
0.672
12.393
0.575
Table 4b: ARIMA models with various statistics (Square Root Transformation)
Model Type
ARIMA (0,0,0)
ARIMA (0,0,1)
ARIMA (0,1,0)
ARIMA (0,1,1)
ARIMA (1,0,0)*
ARIMA (1,0,1)*
ARIMA (1,1,0)
ARIMA (1,1,1)
ARIMA (1,1,2)
ARIMA (1,2,0)
ARIMA (1,2,1)
ARIMA (1,2,2)
ARIMA (2,0,0)*
ARIMA (2,0,1)*
ARIMA (2,0,2)
ARIMA (2,1,0)
ARIMA (2,1,1)
ARIMA (2,1,2)
ARIMA (2,2,0)
ARIMA (2,2,1)
ARIMA (2,2,2)
Stationary R2
0.814
0.922
0.041
0.041
0.976
0.976
0.041
0.041
0.082
0.241
0.477
0.483
0.976
0.980
0.976
0.041
0.083
0.088
0.318
0.477
0.479
R2
0.856
0.933
0.970
0.970
0.973
0.973
0.970
0.970
0.972
0.943
0.968
0.969
0.973
0.973
0.972
0.970
0.972
0.972
0.953
0.968
0.969
Statistics
RMSE
MAE
23.636
19.418
16.307
10.809
10.885
4.454
10.980
4.409
10.341
4.079
10.436
3.959
10.979
4.409
11.088
4.418
10.826
4.349
15.177
5.994
11.414
4.699
11.390
4.567
10.436
3.960
10.494
3.852
10.734
4.223
11.088
4.418
10.803
4.388
10.809
4.168
13.902
5.804
11.522
4.705
11.590
4.693
Normalized BIC
6.469
5.799
4.920
5.011
4.888
4.978
5.011
5.103
5.128
5.661
5.165
5.235
4.978
5.061
5.178
5.103
5.124
5.198
5.560
5.258
5.344
LB Q(18)
Statistics
Sig.
262.513
0.000
156.960
0.000
9.200
0.955
9.311
0.930
7.180
0.981
7.333
0.966
9.311
0.930
9.309
0.900
9.472
0.852
19.974
0.276
9.349
0.898
9.110
0.872
7.352
0.966
9.343
0.859
7.651
0.907
9.310
0.900
9.335
0.859
9.486
0.799
13.623
0.627
9.440
0.853
9.508
0.797
Table 4c: ARIMA models with various statistics (Natural Log Transformation)
Model Type
ARIMA (0,0,0)
ARIMA (0,0,1)
ARIMA (0,1,0)
ARIMA (0,1,1)
ARIMA (1,0,0)
ARIMA (1,0,1)
ARIMA (1,1,0)
ARIMA (1,1,1)
ARIMA (1,1,2)
ARIMA (1,2,0)
ARIMA (1,2,1)
ARIMA (1,2,2)
ARIMA (2,0,0)
ARIMA (2,0,1)*
ARIMA (2,0,2)*
ARIMA (2,1,0)
ARIMA (2,1,1)
ARIMA (2,1,2)
ARIMA (2,2,0)
ARIMA (2,2,1)
ARIMA (2,2,2)
304
Stationary R2
0.873
0.950
0.018
0.055
0.988
0.988
0.059
0.067
0.069
0.189
0.421
0.422
0.988
0.988
0.985
0.063
0.073
0.070
0.275
0.406
0.407
R2
0.470
0.840
0.936
0.937
0.962
0.960
0.938
0.944
0.942
0.849
0.949
0.949
0.960
0.970
0.968
0.940
0.945
0.942
0.889
0.940
0.941
Statistics
RMSE
MAE
45.279
21.098
25.089
11.737
15.865
8.093
15.780
7.920
12.225
5.332
12.661
5.538
15.647
7.868
15.094
7.711
15.468
7.816
24.632
8.929
14.386
6.994
14.880
7.028
12.705
5.322
11.145
4.493
11.511
4.613
15.547
7.839
15.107
7.197
15.593
7.771
20.347
8.254
15.801
7.969
15.877
7.865
Normalized BIC
7.769
6.660
5.674
5.736
5.223
5.365
5.719
5.720
5.842
6.630
5.628
5.730
5.371
5.181
5.318
5.779
5.795
5.931
6.321
5.890
5.973
LB Q(18)
Statistics
Sig.
294.520
0.000
186.648
0.000
17.779
0.470
16.089
0.518
17.576
0.416
14.485
0.563
16.404
0.495
16.076
0.448
15.735
0.400
30.726
0.022
16.026
0.451
15.114
0.443
14.343
0.573
11.482
0.718
7.449
0.916
16.677
0.407
16.552
0.346
16.660
0.275
22.225
0.136
16.433
0.354
15.784
0.327
FUW Trends in Science & Technology Journal ftstjournal@gmail.com
April, 2016 Vol. 1 No. 1 – e-ISSN: 24085162; p-ISSN: 20485170 pp 299-306
Data Transformation and Arima Models: A Study of Exchange Rate of Nigeria Naira to US Dollar
From various results obtained when square root
transformation (Table 4b) was used ARIMA (1,0,0);
ARIMA(1,0,1); ARIMA (2,0,0) and ARIMA (2,0,1) are
competing models with all having a relatively higher
stationary-R2 and R2 when compared with those obtained
when no transformation was made on the original series.
The models are also having lower NBIC and Ljung-Box Q
statistics in comparison with those obtained when original
series was used. However, ARIMA (1,0,0) and ARIMA
(2,0,1) give best explanation to the series among the
competing models and they both performed better than
those obtained when no transformation was made to the
series.
Under natural log transformation, ARIMA (2,0,1) and
ARIMA (2,0,2) outperformed all other models. Both have
best stationary-R2 in comparison with square root
transformation and original series. Between the two
however, ARIMA (2,0,1) relatively perform better than
ARIMA (2,0,2).
Conclusions
For prediction purpose, ARIMA model offers a good
technique because its strength is in the fact that it is a
suitable method for any time series with any pattern of
change and it does not require the researcher to choose the
value of any parameter a priori. However, its requirement
of a long time series is a limitation. Like so many other
methods, it does not assure a perfect forecast but it
performs relatively better compared to competing models
when dealing with time series data.
With the result from several tentative ARIMA models
entertained and under different transformations, it is
obvious that there is no any parameter combination (under
respective transformations) that generally stands out
among the rest. With all aforementioned expected
characteristics of a very good ARIMA model, which
include among other; stationarity, parsimoniousness,
“acceptable” RMSE, MAE, relatively small forecast error,
least NBIC (Normalized Bayesian Information Criterion),
the most suitable model that relatively perform very well
in comparison with all other models is ARIMA(1,0,0)
when square root transformation is utilized.
The model has stationary R2 of 0.976; coefficient of
determination (R2) of 97.3%; NBIC of 4.888 and LjungBox Q P-value of 0.981. Hence, the recommended model
for forecasting of average yearly exchange rate of Nigeria
naira to US dollar. This research has provided empirical
forecasts of the exchange rate in Nigeria. The exchange
rate of Nigeria naira to US dollar is on the increasing side
on the long run. ARIMA model has been shown to be
more effective and efficient when data transformation is
employed. A continuous depreciation of exchange rate of
any country will make import more expensive. This in
turns will negatively affect the entire economy across all
the value chains. Therefore, countries must strive to reduce
import and policies towards improving volume of export
must be encouraged in order to have a favourable balance
of trade occurs and hence a positive balance of payment.
The policy implication of this research is for those
involved in formulating foreign exchange policies to
always compare various transformations of competing
models before deciding on the final choice of the model to
be adopted for prediction purpose.
305
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April, 2016 Vol. 1 No. 1 – e-ISSN: 24085162; p-ISSN: 20485170 pp 299-306