A Simulating Model For Daily Uganda Shilling-Nigerian Naira Exchange Rates

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1 Available online at Austrian Journal of Mathematics and Statistics, Vol 1, Issue 1, (2017): ISSN A Simulating Model For Daily Uganda Shilling-Nigerian Naira Exchange Rates Ette Harrison Etuk 1, Yellow Dimkpa Mazi 2, Igboye Simon Aboko 3 1. Department of Mathematics/Computer Science Rivers State University of Science and Technology Port Harcourt, Nigeria 2. Department of Statistics Rivers State Polytechnic Bori, Nigeria 3. Department of Mathematics/Statistics Rivers State College of Arts and Science Port Harcourt, Nigeria Corresponding Author ettetuk@yahoo.com Abstract: The realization of daily Uganda Shilling Nigerian Naira exchange rates from 4 October 2014 to 1 March 2015 shows that the Naira is depreciating relative to the Shilling. Clearly with an upward trend the series is non-stationary. A seven-day differencing yields a series with a flat trend certified stationary by the Augmented Dickey Fuller (ADF)Test. However the correlogram exhibits a sinusoidal pattern of period 7 days which is an indication of seasonality of the same period. A further (but non-seasonal) differencing of the differences yields a series with a flat trend for which the ADF test and correlogram indicate stationarity of period 7 days. With the significance of the partial autocorrelation function value at lag 7 and the comparability of the values at lags 6 and 8, a SARIMA(1,1,0)x(1,1,0) 7 model is suggestive. The estimation of this model shows that the lag1 and the lag 8 coefficients are non-significant. This therefore suggests the SARIMA(0,1,0)x(1,1,0) 7 model which is observed to be the more adequate model. Therefore the time series follows a SARIMA(0,1,0)x(1,1,0) 7 model and may be simulated or forecasted on its basis. Key Words: Ugandan Shilling; Nigerian Naira; Foreign Exchange Rates; SARIMA models Introduction Foreign exchange rates forecasting could be a potent tool for economic and financial management. In this work interest is in the time series modelling of daily exchange rates of Uganda shilling (UGX) and Nigerian Naira (NGN). The approach to be adopted is the seasonal autoregressive integrated moving average (SARIMA) approach. Once a time series is confirmed to show some seasonality of a fixed period the series might be so modelled. The SARIMA technique is still very much in current use after its introduction by Box and Jenkins (1976). A few examples of such usage are given below. Al Tamimi et al. (2008) proposed the use of a SARIMA(1,0,1)x(1,1,1) 12 model to forecast mobile video output. Arumugam and Anithakumari (2013) modelled natural rubber production in India as a SARIMA(2,1,2)x(1,1,1) 12. Dengue virus transmission in Chiang Rai, Thailand has been forecasted using a SARIMA(1,0,0)x(1,0,1) 12 (See Wongkoon et al., 2009). Indian leather export has been predicted on the basis of a SARIMA(0,1,1)x(1,2,1) 12 by Jagathnath et al. (2013). Urrutia et al. (2014) explained the variation in the foreign trade of the Philippines using a SARIMA(5,0,8)x(0,1,1) 12 for the export trade and a SARIMA(7,2,3)x(0,1,1) 12 for the import trade. Abdul-Aziz et al. (2013) modelled and forecasted rainfall pattern in the Ashanti region of Ghana using a SARIMA(0,0,0)x(2,0,1) 12 model. Pelagic fish catch in Malaysia has been modelled by a SARIMA(1,1,0)x(0,0,1) 12 model by Bako et al. (2013). Singh (2013) fitted a SARIMA(0,1,1)x(1,1,1) 12 model to monthly tourist inflow in Bhutan. Etuk et al. (2015) fitted an additive SARIMA with significant lags 1 and 12 to monthly Nigeria Treasury Bill Rates. Zhang et al. (2015) proposed that quarterly Chinese road traffic mortalities could be forecasted on the basis of a SARIMA(0,1,0)x(0,0,1) 4 model. Gikungu et al. (2015) used a SARIMA(0,1,0)x(0,0,1) 4 model to explain the variation in quarterly Kenyan inflation rates. Etuk and Bazinzi (2015) modelled daily exchange rates of UGX and the US dollar as a SARIMA(0,1,1)x(0,,1,1) 7. Materials and methods Data The data used for this study are 149 daily UGX-NGN exchange rates from October 4, 2014 to March 1, 2015 obtained from the website accessed on March 2, The data are to be interpreted as the amount of NGN in one UGX. Box-Jenkins Models: According to Box and Jenkins (1976), a time series {X t} that is stationary 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 X t - 1X t-1-2x t px t-p = t + 1 t t q t-q (1) where { t} is a white noise process and the s and s are constants such that the model is stationary and invertible. Suppose (1) is written as A(L)X t = B(L) t (2)

2 where A(L) is the autoregressive (AR) operator defined by A(L) = 1-1L - 2L pl p and B(L) is the moving average (MA) operator defined by B(L) = 1 + 1L + 2L ql q and L is the backward shift operator defined by L k X t = X t-k. Furthermore p and q are the AR and MA orders respectively. If, however, the time series {X t} is non-stationary Box and Jenkins (1976) proposed that differencing it a sufficient number of times could render it stationary. If differencing the series d times is just enough to render it stationary, then if this d th difference { d X t} of {X t} is ARMA(p, q), then {X t} is said to follow an autoregressive integrated moving average model of order p, d and q designated ARIMA(p, d, q). Here = 1 L. If {X t} is seasonal of period s, Box and Jenkins (1976) further proposed that it could be modelled by A(L) (L s ) d D sx t = B(L) (L s ) t (3) where (L) is the seasonal AR operator and (L) is the seasonal MA operator and s is the seasonal difference operator defined by s = 1 - L s. Suppose the seasonal AR and MA operators are polynomials of degree P and Q respectively. Then model (3) is called a seasonal autoregressive integrated moving average model of order p, d, q, P, D, Q and s denoted by SARIMA(p, d, q)x(p, D, Q) s. Here P and Q are the seasonal AR and MA orders respectively D is the seasonal order of differencing. Box-Jenkins Modelling Box Jenkins (1976) fitting of (3) to data begins with determination of the orders p, d, q, P, D, Q and s. The seasonality period s may be suggestive naturally from knowledge of the seasonal nature of the series. For example, hourly atmospheric temperature is expected to be seasonal of period 24 hours. An examination or the correlogram of the data could also reveal a seasonal nature. The autoregressive orders p and P may be estimated respectively by the non-seasonal and the seasonal cut-off lags of the partial autocorrelation function (PACF). Similarly their respective moving average counterparts, q and Q, may be estimated by the non-seasonal and the seasonal cut-off points of the autocorrelation function (ACF) respectively. It is usually enough to choose the differencing orders such that they add up to at most 2 for stationarity to be attained (Etuk et al., 2015). The estimation of the parameters is usually done by a non-linear optimization approach because of the presence of items of the white noise process in the model. Where more than one model is proposed, model selection criteria including Akaike Information Criterion (AIC) (Akaike, 1974), Schwarz Information Criterion (SIC) (Schwarz, 1978) and Hannan-Quinn Criterion (Hannan and Quinn, 1979) may be used for model discriminant purpose. In the sequel the Eviews 7 software shall be used for the modelling process. The non-linear least error sum of squares criterion is the basis for model identification, estimation and diagnostic checking. 24

3 Figure 3: Correlogram of SDUXNN 25

4 Figure 5: Correlogram of DSDUXNN 26

5 TABLE 1: ESTIMATION OF THE SARIMA(1,1,0)x(1,1,0) 7 MODEL TABLE 2: ESTIMATION OF THE SARIMA(0,1,0)x(1,1,0) 7 MODEL Figure 6: Histogram of the SARIMA(0,1,0)x(1,1,0) 7 residuals 27

6 Results And Discussion The time plot in Figure 1 shows an upward secular trend which depicts that within the time interval under investigation the Naira depreciated relative to the shilling on an overall basis. A 7-daily differencing of the series which we call UXNN yields a series SDUXNN with generally no trend (see Figure 2) and which still shows some seasonality of order 7 days (see Figure 3). A non-seasonal differencing of SDUXNN produces a series DSDUXNN with no trend (see figure 4) and a correlogram which suggests the SARIMA(1,1,0)x(1,1,0) 7 model which on estimation as summarized in Table 1 is given by X t = X t X t X t-8 + t (4) ( ) ( ).0740) ( ) Clearly the first and the last coefficients in model (4) are not statistically significant. This suggests the model estimated as summarized in Table 2 as X t = X t-7 + t (5) ( ) A comparison of models (4) and (5) shows the superiority of the latter model on the grounds of Akaike Information criteria, Schwarz Criterion and Hannan-Quinn Criterion. Conclusion Daily UGX-NGN exchange rates have been observed to follow a SARIMA(0,1,0)x(1,1,0) 7 model. The distribution of its residuals is Gaussian (see Figure 6). This shows that the proposed model is adequate. Simulations or forecasting of the exchange rates may be done on its basis. References Abdul-Aziz, A. R., Anokye, M., Kwame. A., Munyakazi, L. and Nsowah-Nuamah, N. N. N. (2013). Modeling and Forecasting Rainfall Pattern in Ghana as a Seasonal Arima Process: The Case of Ashanti Region. International Journal of Humanities and Social Science, 3(3): Akaike, H. (1974). A new look at the Statistical Model Identification. IEEE Trans. Automatic Control, AC-19, AlTamimi, A., Jain, R. And So-In, C. (2008). SAM: A Simplified Seasonal ARIMA Model for Mobile Video over Wireless Broadband Networks. IEEE International Symposium on Multimedia 2008, Bako, H. Y., Rusiman, M. S., Kane, I. L. and Matias-Peralta, H. M. (2013). Predictive modelling of pelagic fish catch in Malaysia using seasonal ARIMA models. Agriculture, Forestry and Fisheries, 2(3): Box, G. E. P. and Jenkins, G. M. (1976). Time Series Analysis, Forecasting and Control, Holden-Day, San Francisco. Etuk, E. H. and Natamba, B. (2015). Daily Uganda Shilling/United States Dollar Exchhange Rates Modelling by Box-Jenkins Techniques. International Journal of Management, Accounting and Economics, 2(4): Etuk, E. H., Agbam, A. S. and Uchendu, B. A. (2015). A Forecasting Model for Monthly Nigeria Treasury Bill Rates by Box-Jenkins Techniques. International Journal of Life Science and Engineering, 1(1): Hannan, E. J. and Quinn, B. G. (1979). Annals of Statistics, 6, Jagathnath, K. K. M., Nithiyanantha, V. S., Giriyappa, K. and Chandramouli, D. (2013). Identifiication of SARIMA as a Model for forecasting Indian Leather Export. International Journal of Research in Management, 3(6): Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6: Singh, E. H. (2013). Forecasting Tourists Inflow in Bhutan using seasonal ARIMA. International Journal of Science and Research (IJSR), 2(9): Urrutia, J. D., Alano, E. M. D., Aninipot, P. M. R., Gumapac, K. A. and Quinto, J. Q. (2014). Modeling and Forecasting Foreign Trade of the Philippines Using Time Series SARIMA Model. European Academic Research, II(5): Wongkoon, S., Pollar, M., Jaroensutasinee, M. and Taroensutasinee, K. (2009). Prediicting DHF Incidence in Northern Thailand using Time Series Analysis Technique. International Journal of Biological and Medical Sciences, 4(3): Zhang, X., Pang, Y., Cul, M., Stallones, L. and Xiang, H. (2015). Forecasting mortality of road traffic injuries in China using seasonal autoregressive moving average model. Annals of Epidemiology, 25(2):