Etuk and Amadi /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 6, 2014, pp.89-97 A Model for the Forecasting of Daily South African Rand and Nigerian Naira Exchange Rates 2 Ette Harrison Etuk1 Eberechi Humphrey Amadi Department of Mathematics / Department of Mathematics / Computer Science, Rivers State Computer Science, Rivers State University of Science and University of Science and Technology, P. Harcourt, Nigeria Technology, P. Harcourt, Nigeria [email protected] [email protected] Abstract Daily South African Rand / Nigerian Naira exchange rates are analyzed by seasonal autoregressive integrated moving average (SARIMA) methods. The actual realization considered herein and called ZNER spans from 20th March 2014 to 15th September, 2014, a 180-day interval. Its time plot shows a generally negative secular trend which depicts the relative depreciation of the rand within the time interval. A seven-point differencing of the series yields a series SDZNER with a very slightly negative trend and an autocorrelation structure of a seasonal series of period 7 days. A non-seasonal differencing of the differences yields the series DSDZNER with a horizontal trend and a correlogram showing a seasonal nature of period 7days and the involvement of a seasonal moving average component of order one. There is also an indication of a seasonal autoregressive component of order two. It is noteworthy that Augmented Dickey Fuller (ADF) tests consider ZNER as non- stationary (p < .01). SDZNER is also considered non-stationary (p < .05). Only DSDZNER is considered stationary. A close look reveals the involvement of a SARIMA(0, 1, 1)X(0, 1, 1)7 component. On the overall, the models that are suggestive are (1) SARIMA(0, 1, 1)X(2, 1, 1)7 and (2) SARIMA(1, 1, 1)X(1, 1, 1)7. The later model is found to be the more adequate one on all counts. Keywords: South African Rand, Nigerian Naira, Sarima Modelling, Time Series Analysis, Foreign Exchange Rates. 1. INTRODUCTION: Foreign exchange rates have to do with the price of a country’s currency in terms of another country’s currency. International economic relations are hinged on the relative strengths of the partner nations’ currencies. This write-up is concerned with the modeling of the daily exchange rates of the South African Rand (ZAR) and the Nigerian Naira (NGN). This is with a view to providing a model which may be used to forecast the exchange rates. The modeling approach adopted is the seasonal autoregressive moving average (SARIMA) technique. The SARIMA approach to time series modeling was introduced specifically for the modeling of series that are seasonal in nature. Economic and financial time series are mostly observed to be seasonal. They have mostly been modeled using SARIMA techniques. Researchers that have adopted this approach to model some of such series are Brida and Risso (2011), Fannoh et al. (2012), Abdelghani et al. (2013), Arumugam and Anithakumari (2013), Lira(2013), AquilBurney et al. (2006), Li et al. (2013), Gharbi et al. (2011), Hassan et al. (2013), Khajavi et al. (2012), Oduro- Gyimah et al. (2012), Mombeni et al. (2013), to mention only a few. Where seasonal time series are involved SARIMA models have been observed to outdo other types of models in forecasting performance. See for instance Dagowa and Alade (2013), Sabri (2013), Qiao et al. (2013), Etuk (2014), and so on. © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “IJMSET promotes research nature, Research nature enriches the world’s future” 89 Etuk and Amadi /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 6, 2014, pp.89-97 Many such exchange rates have been observed in the past to exhibit seasonal tendencies to necessitate and justify the application of such methods on them. A few of such cases are Etuk and Igbudu (2013), Etuk and Nkombou (2014), Appiah and Adetunde(2011). 2. MATERIALS AND METHODS: The data used for this work are 180 ZAR-NGN daily exchange rates from Thursday, 20th March, 2014 to Monday, 15th September, 2014 retrievable from the website www.exchangerates.org.uk/ZAR-NGN-exchange-rate-history.html . It is to be interpreted as the amount of NGN in one ZAR. 2.1. SARIMA MODELS: Suppose {Xt} is a stationary time series. It is said to follow an autoregressive moving average model of order p and q, and denoted by ARMA(p, q), if Xt - a1Xt-1 - a2Xt-2 - … - apXt-p = et + b1et-1 + b2et-2 + … + bqet-q (1) where {et} is a white noise process and the a’s and b’s are constants such that model (1) is both stationary and invertible. Model (1) may be written as A(L)Xt = B(L)et (2) 2 p 2 q k where A(L) = 1 - a1L - a2L - … - apL and B(L) = 1 + b1L + b2L + … + bqL and L Xt = Xt-k. For stationarity and invertibility of the model A(L) and B(L) must have zeroes that lie outside the unit circle respectively. Suppose the time series {Xt} is not stationary Box and Jenkins (1976) proposed that differencing of the series to an appropriate order may make the series stationary. Suppose that the th minimum order of differencing of the series for stationarity is d. The d differences of {Xt} shall be d d denoted by {Ñ Xt} where Ñ = 1 – L. If {Ñ Xt} satisfies equation (1) or (2) {Xt} is said to follow an autoregressive integrated moving average model of order p, d and q denoted by ARIMA(p, d, q). If {Xt} is seasonal of period s, Box and Jenkins (1976) further proposed that the series might be modeled by s d D s A(L)F(L )Ñ Ñ s Xt = B(L)Q(L )et (3) s where F(L) and Q(L) are polynomials in L. Ñs = 1 – L and D is the order of seasonal differencing. Then {Xt} is said to follow a multiplicative seasonal autoregressive integrated moving average model of order p, d and q denoted by SARIMA(p, d, q)X(P, D, Q)s model. 2.3. SARIMA MODEL FITTING: The fitting of the SARIMA model (3) starts with the determination of the orders p, d, q, P, D, Q and s. The seasonal period s may be suggestive from knowledge of the nature of the series as with monthly rainfall or hourly atmospheric temperature. The time plot and the correlogram could indicate a seasonal nature and therefore the value of s. The non-seasonal and the seasonal cut-off lags of the partial autocorrelation function (PACF) estimate the non-seasonal and the seasonal autoregressive (AR) orders p and P respectively. Similarly the non-seasonal and the seasonal cut-off lags of the autocorrelation function (ACF) estimate the non-seasonal and the seasonal moving average (MA) orders q and Q respectively. The non-seasonal and the seasonal differencing orders d and D are © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “IJMSET promotes research nature, Research nature enriches the world’s future” 90 Etuk and Amadi /International Journal of Modern Sciences and Engineering Technology (IJMSET) ISSN 2349-3755; Available at https://www.ijmset.com Volume 1, Issue 6, 2014, pp.89-97 often chosen such that the model is not too complex. In fact, often d + D < 3. Before and after differencing the series is tested for stationarity. This is done by the Augmented Dickey Fuller (ADF) test. Parameter estimation is invariably by the application of a nonlinear optimization technique like the least squares technique or the maximum likelihood technique. This is because of the presence in the model of values of the white noise process. After model estimation the fitted model is subjected to diagnostic checking to ascertain its goodness-of-fit to the data. This is done by residual analysis. Uncorrelatedness and/or, better still, the normality of the residuals indicates model adequacy. 3. RESULTS AND DISCUSSION: The time plot of the 180-point series ZNER in Figure 1 shows a negative secular trend. This indicates the relative decline in value of the ZAR. A 7-day differencing of ZNER yields the series SDZNER which has a slightly negative trend (See Figure 2) and a correlogram in Figure 3 which shows it is non-stationary. A non-seasonal differencing of SDZNER yields the series DSDZNER which has a horizontal trend (See Figure 4) and a correlogram in Figure 5 which suggests its stationarity. The ADF tests have the following data: ZNER test statistic is -2.80, SDZNER test statistic is -3.18 and that of DSDZNER is -9.30; the 1%, 5% and 10% critical values are -3.47, -2.88 and -2.56 respectively. That means that at p < 0.01 both ZNER and SDZNER are non-stationary and DSDZNER stationary. Moreover the correlogram of DSDZNER suggests the involvement of a seasonal MA component of order one and the involvement of a seasonal AR component of order two. It is also evident from the correlogram that the following models are possible: 1) SARIMA(0, 1, 1)X(2, 1, 1)7 and 2) SARIMA(1, 1, 1)X(1, 1, 1)7. The estimation of the SARIMA(0,1,1)X(2, 1,1)7 model of Table 1 yields: Xt + .9670Xt-7 + .4252Xt-14 = et - .0513et-1 + .4468et-7 - .1875et-8 (4) (±.1244) (±.0771) (±.0808) (±.1368) (±.0817) That of the SARIMA(1,1,1)X(1,1,1)7 model of Table 2 yields: Xt - .8922Xt-1 + .0234Xt-7 + .0147Xt-14 = et - .9596et-1 - .9482et-7 + .9132et-8 (5) (±.0409) (±.0826) (±.0824) (±.0175) (±.0182) (±.0221) where X represents DSDZNER in (4) and (5).