An Additive SARIMA Model for Daily Exchange Rates of the Malaysian Ringgit (MYR) and Nigerian Naira (NGN)
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International Journal of Empirical Finance
Vol. 2, No. 4, 2014, 193-201
An Additive SARIMA Model for Daily Exchange Rates of the
Malaysian Ringgit (MYR) and Nigerian Naira (NGN)
Ette Harrison Etuk1
Abstract
The daily exchange rates of the Nigerian Naira (NGN) and Malaysian Ringgit (MYR) from Wednesday,6 th
November 2013 to Monday, 21st April 2014 are being modeled by Seasonal Autoregressive Integrated
Moving Average (SARIMA) methods. The realization herein referred to as RNER initially had a downward
trend to January 2014 after which the trend became positive. That means that initially the Naira appreciated
before depreciating relatively after January. A 7-day differencing of RNER produced the series called
SDRNER with a fairly horizontal trend. The Augmented Dickey Fuller (ADF) Test declares RNER and
SDRNER as non-stationary and stationary respectively. The correlogram of SDRNER has an autocorrelation
function (ACF) that cuts off at lag 4 and a partial autocorrelation function (PACF) that cuts off at lag 1
suggesting a SARIMA (1, 0, 4)x(0, 1, 0)7 model fit. The residuals of this model, rather than being
uncorrelated, are seasonal with period 7. This seems to invalidate the model. A further but non-seasonal
differencing of SDRNER yields the series called DSDRNER which exhibits a generally horizontal trend. All
along seasonality is not so obvious. However the ACF of DSDRNER shows seasonality of period 7 as well
as the existence of a seasonal moving average component of order one. By Surhatono’s (2011) algorithm a
SARIMA (0, 1, 1)x(0, 1, 1)7 model is fitted. With the non-significance of the last coefficient, an additive
model is suggestive. This additive model is shown to be the best of the three proposed models.
Key words: MYR, NGN, Foreign Exchange Rates, SARIMA models.
1. Introduction
Many economic and financial real life time series are seasonal in nature. That means that they tend
to fluctuate periodically with time. Such series may be modeled by seasonal autoregressive integrated
moving average (SARIMA) models proposed by Box and Jenkins (1976). A few of seasonal time series
which have been modeled by such methods are air travel arrivals ( Chen et al., 2009), crude oil exportation
(Ayinde and Abdulwahab, 2013; Etuk, 2012b), reserve money growth (Suleman and Sarpong, 2012),
malaria mortality rates (Dan et al. 2014), meat exports (Paul et al., 2013), temperature (Khajavi et al, 2012),
relative humidity (Shiri et al., 2011), traffic flow (Williams and Hoel, 2003; Tong and Xue, 2008), tourism
(Chang and Liao, 2010), rainfall (Ibrahim and Dauda, 2012; Ramesh et al., 2013), tuberculosis incidence
(Chowdhury et al. 2013), inflation (Etuk et al, 2012; Otu et al., 2014), and stock price (Lee et al. ,2008; Etuk,
2012a).
The purpose of this write-up is to model the daily exchange rates of the Nigerian Naira (NGN) and the
Malaysian Ringgit (MYR) by SARIMA methods. In section 2 of this write-up, the materials and the
methodology used shall be discussed. Section 3 shall consider the results of the analysis of data and their
discussion. In Section 4 the work shall be concluded.
1
Department of Mathematics/Computer Science Rivers State University of Science and Technology Port Harcourt
Nigeria
© 2014 Research Academy of Social Sciences
http://www.rassweb.com 193E. H. Etuk
2. Materials and Methods
Data
The data for this write-up are 167 daily exchange rates of the Malaysian Ringgit (MYR) and the
Nigerian Naira (NGN) from Wednesday 6th November 2013 to Monday 21st April 2014 obtained from the
website www.exchangerates.org.uk/MYR-NGN-exchange-rate-history.html . It is to be interpreted as the
amount of NGN in one MYR.
Seasonal Arima Modelling
A stationary time series {Xt} is said to follow an autoregressive moving average model of order p and q
designated ARMA(p, q) if it satisfies the following difference equation
Xt - 1Xt-1 - 2Xt-2 - … - pXt-p = t + 1t-1 + 2t-2 + … + qt-q (1)
where the series {t} is a white noise process and the coefficients ’s and ’s are such that the model is
stationary and invertible. The model (1) may be put as
A(L)Xt = B(L)t (2)
where A(L) = 1 - 1L - 2L2 - … - pLp and B(L) = 1 + 1L + 2L2 + … + qLq. Here L is the backward shift
operator defined by LkXt = Xt-k.
For a non-stationary series {Xt}, Box and Jenkins(1976) proposed that differencing of sufficient
order d could make the series stationary. If this dth difference denoted by {dXt} satisfies (1) then {Xt} is
said to follow an autoregressive integrated moving average model of orders p, d and q, denoted by
ARIMA(p, d, q).
Suppose that {Xt} is seasonal of period s, Box and Jenkins(1976) proposed that the series could be
modeled by
A(L)(Ls)dsD Xt = B(L)(Ls)t (3)
Suppose the seasonal autoregressive(AR) and the seasonal moving average(MA) operators (L) and
(L) are polynomials in L of orders P and Q respectively.. Then the series {X t} is said to follow a
multiplicative seasonal autoregressive integrated moving average (SARIMA) model of order (p, d, q)x(P, D,
Q)s. sD is the Dth seasonal difference operator where s is the seasonal difference operator defined by s =
1 - Ls . The coefficients are such that the conditions of stationarity and invertibility are satisfied
Fitting (3) starts with the estimation of the orders p, d, q, P, D, Q and s. The differencing orders d and
D are often chosen such that they sum up to at most 2. Often it is sufficient to put d = D = 1. The seasonality
period s might be suggestive from the time-plot. If not, the lag at which the autocorrelation function (ACF)
of the differenced series shows a significant spike is indicative of it. The non-seasonal AR and MA orders p
and q may be estimated as the cut-off lags of the ACF and the partial autocorrelation function (PACF)
respectively. Similarly the seasonal AR and MA orders P and Q may be estimated as the seasonal cut-off lags
of the ACF and PACF respectively.
Surhatono (2011) has proposed the following SARIMA fitting algorithm:
Fit the SARIMA(0, 1, 1)x(0, 1, 1)s model:
Xt = 1t-1 + st-s + s+1t-s-1
If s+1 is significant the model is multiplicative SARIMA if s+1 = s1 otherwise it is subset.
194International Journal of Empirical Finance
If s+1 is non-significant, the model is an additive SARIMA model.
After order determination the parameters may be estimated by a non-linear optimization technique
like the maximum likelihood or the least squares procedure. A fitted model must be subjected to diagnostic
checking with a view to ascertaining its adequacy. This involves some residual analysis. Uncorrelatedness of
the residuals of a model ts an indication of its goodness-of-fit to the data. In this write-up the software
Eviews is used for all the analytical work.
3. Results and Discussion
The time-plot of RNER in Figure 1 shows an initial decreasing trend up to January 2014. Within this
time period the Naira had a comparative increased value. After this time point, there was a positive trend,
signifying a decline in the value of the Naira. Seasonality is not evident. A plot of the series on weekly basis
shows that weekly maximums tend to appear around Mondays and minimums around Sundays. The
Augmented Dickey Fuller (ADF) Test Statistic value of -1.2 for this series is non-significant given the 10%,
5% and 1% critical values of -2.6, -2.9 and -3.5 respectively. This means that the series RNER is non-
stationary. A 7-day differencing of RNER yields the series SDRNER. Its time-plot of Figure 2 shows a
generally horizontal trend. With an ADF test statistic value of -4.9, it is assumed to be stationary. Its ACF in
Figure 3 cuts off at lag 4 and its PACF in lag 1. That is an indication of a SARIMA(1, 0, 4)x(0, 1, 0) 7 model
summarized in Table 1 as:
SDRNERt - .6188SDRNERt-1 = t + .3155t-1 + .1203t-2 + .4221t-3 + .5002t-4 (3)
The residuals of the model (3) have ACF and PACF in Figure 6 which show that they are seasonal of
order 7. This observation seems to invalidate the model because the residuals should be uncorrelated. A
further non-seasonal differencing of SDRNER yields the series DSDRNER which has a generally
horizontal trend (See Figure 4). Its correlogram in Figure 5 reveals a seasonal tendency of period 7 days and
the involvement of a seasonal moving average component of order one. Using Surhatono’s (2011) algorithm
a SARIMA(0, 1, 1)x(0, 1, 1)7 model is fitted as summarized in Table 2 as:
DSDRNERt = t + .0948t-1 - .8700t-7 - .0197t-8 (4)
Clearly the lag 8 coefficient .0197 is non-significant. That leads to the proposal of an additive SARIMA
model of lags 1 and 7. The summary of the estimation of this model in Table 2 gives the model as:
DSDRNERt = t + .0627t-1 - .8788t-7 (5)
Clearly the additive SARIMA model (5) is the best of the three proposed model in the AIC sense. Its
residuals are uncorrelated (See Figure 7). This indicates its relative adequacy.
4. Conclusion
It may be concluded that daily MYR-NGN exchange rates follow an additive SARIMA model (5). The
model has been shown to be the most adequate of the three proposed models on all counts. This means that
forecasting of the time series might be done on the basis of this model.
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196International Journal of Empirical Finance
197E. H. Etuk
Figure 3: Correlogram Of Sdrner
198International Journal of Empirical Finance
Figure 5: Correlogram Of Dsdrner
Table 1: Estimation Of (1, 0, 4)X(0, 1, 0)7 Sarima Model
199E. H. Etuk
Table 2: Estimation Of The (0, 1, 1)X(0, 1, 1)7 Sarima Model
Table 3: Estimation Of The Additive Sarima Model
200International Journal of Empirical Finance
Figure 6: Correlogram Of The (1, 0, 4)X(0, 1, 0)7 Sarima Residuals
Figure 7: Correlogram Of The Additive Sarima Residuals
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