A Model for Daily Exchange Rates of the Naira and the XOF by Seasonal ARIMA Methods

Transcription

1 Euro-Asian Journal of Economics and Finance ISSN: (print) ISSN: (online) Volume: 2, Issue: 3 (July 2014), Pages: Academy of Business & Scientific Research A Model for Daily Exchange Rates of the Naira and the XOF by Seasonal ARIMA Methods Ette Harrison Etuk 1 *, Pius Sibeate 2, and Love Cherukei Nnoka 3 1. Department of Mathematics/Computer Science Rivers State University of Science and Technology NIGERIA 2. Department of Planning, Research and Statistics Rivers State Ministry of Education Port Harcourt NIGERIA 3. Department of Mathematics/Statistics Rivers State School of Arts and Science Port Harcourt NIGERIA The daily exchange rates of the Nigerian Naira (NGN) and West African CFA franc (XOF) from Thursday, 14 th March, 2013 to Saturday, 23 rd November 2013 are being modeled by Seasonal Autoregressive Integrated Moving Average (SARIMA) methods. This realization of the time series, referred to as NXOF, is a generally decreasing one, reflecting the relative depreciation of the Naira within the period of interest. As expected the Augmented Dickey Fuller (ADF) Tests adjudge it to be non-stationary. There is indication that NXOF is seasonal of period 7 days, there being a tendency for weekly maximums around Mondays and minimums around Sundays. A seasonal (i.e. 7-day) differencing produces a series SDNXOF with an overall horizontal trend. A non-seasonal differencing of SDNXOF yields a series DSDNXOF with an overall horizontal trend. Both SDNXOF and DSDNXOF are adjudged stationary by the ADF test. By the autocorrelation functions of SDNXOF and DSDNXOF two SARIMA models are suggestive: the (1, 0, 5)x(0, 1, 0)7 and the (0, 1, 1)x(0, 1, 1)7. Residual analysis of the models reveals that the latter is the more adequate model. Keywords: XOF, Naira, Foreign Exchange Rates, SARIMA models. INTRODUCTION Time series modeling of foreign exchange rates between two currencies has engaged the interest of many researchers in recent times a few of whom are Olowe (2009), Appiah and Adetunde (2011), Etuk and Igbudu(2013), Etuk and Nkombou (2014) and Etuk (2014). The idea is to provide basis for forecasting such rates as the need arises. Such economic and financial time series are observed to often show some seasonal tendencies. Often the seasons could be identified. For instance, monthly rainfall data invariably fluctuates periodically according to the climatic yearly seasons. Hourly temperature is likely to fluctuate periodically according to a daily pattern; peaks appearing in the daytime and troughs at night. Etuk (2012) observed that daily Nigerian naira US dollar exchange rates had a tendency of being maximum on Fridays and minimum on Mondays. Martinez et al. (2011) observed that the monthly numbers of dengue tended to maximize in the rainy seasons and minimize in the dry seasons. *Corresponding author: Ette Harrison Etuk Department of Mathematics/Computer Science Rivers State University of Science and Technology NIGERIA. ettetuk@yahoo.com, ettehetuk@gmail.com 203

2 A model for daily exchange rates of the Naira and the XOF by seasonal ARIMA methods Etuk et al. Such seasonal time series may be modeled by seasonal autoregressive integrated moving average (SARIMA) methods. Instances abound where authors modeled such series by SARIMA methods of recent. A few of such works are Kaushik and Singh (2008), Longanathan and Ibrahim (2010), Nirmala and Sundaram(2010), Validat et al. (2011), Ahmad (2012), Mahsin et al. (2012), Lee et al. (2012), Bako et al. (2013), Osarumwense (2013) and Singh (2013). The relative advantage of using SARIMA models in modeling such seasonal series over other approaches has been highlighted in the literature ( Etuk, 2014). In section 2 of this write-up, the materials and the methodology employed shall be highlighted. Section 3 shall consider the results of the data analysis and their discussion. Section 4 is devoted to the concluding remarks. MATERIALS AND METHODS Data The data for this write-up are 258 daily exchange rates of the Naira and the XOF from Thursday 14 th March 2013 to 23 rd November 2013 obtained from the website XOF-exchange-rate-history.html. It is to be interpreted as the amount of XOF in one NGN. Seasonal Arima Modelling A stationary time series {Xt} is said to follow an autoregressive moving average model of order p and q denoted by ARMA(p, q) if it satisfies the difference equation X t - 1X t-1-2x t px t-p = t + 1 t t q t-q (1) In this equation the series { t} is a white noise process and the coefficients s and s are such that the model is stationary as well as invertible. Suppose we put (1) in the form A(L)X t = B(L) t (2) where A(L) = 1-1L - 2L pl p and B(L) = 1 + 1L + 2L ql q. Here L is the backshift operator such that L k X t = X t-k. If however {X t} is not stationary, Box and Jenkins(1976) proposed that differencing of sufficient order d of the series could make it stationary. Suppose that this d th difference denoted by { d X t} satisfies (1) then {X t} is said to follow an autoregressive integrated moving average model of orders p, d and q, designated ARIMA(p, d, q). Suppose that {X t} in addition shows some seasonality of period s, Box and Jenkins(1976) proposed that the series could be modeled by A(L) (L s ) d s D X t = B(L) (L s ) t (3) Here (L) and (L) are respectively the seasonal autoregressive and the seasonal moving average operators. They are polynomials in L. Suppose they are of order P and Q respectively. s D is the D th seasonal difference operator where s = 1 - L s. The coefficients are such that the conditions of stationarity and invertibility are satisfied. 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. Fitting (3) begins with the determination of the orders p, d, q, P, D, Q and s. The differencing orders d and D are often chosen such that d + D < 3. Often it is enough to put d = D = 1. The period of seasonality s if not naturally suggestive from the time-plot might be so from the correlogram as the lag at which the autocorrelation function (ACF) of the differenced series shows a significant spike. The non-seasonal autoregressive (AR) and moving average (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. After order determination the parameters may be estimated by a non-linear optimization technique like the maximum likelihood or the least squares procedure. In this write-up the software Eviews is used for all the analytical work. RESULTS AND DISCUSSION The time-plot of NXOF in Figure 1 shows a generally decreasing time series. This shows that 204

3 Euro-Asian j. econ. financ. ISSN: (print); (online) Volume: 2, Issue: 3, Pages: the relative value of the Naira was decreasing constantly within the time period. An inspection of the time series on weekly basis reveals that there is a tendency for the weekly maximums to appear on Mondays and minimums on Sundays. This supports a 7-day seasonality hypothesis. A 7- day differencing yields the generally horizontal series SDNXOF. With a test statistic of -1.7 and for NXOF and SDNXOF respectively, and the 1%, 5% and 10% critical values of -3.5, -2.9 and -2.6 respectively, NXOF is adjudged non-stationary and SDNXOF as stationary by the ADF Test. However the ACF of SDNXOF of Figure 4 does not seem to support the stationarity hypothesis. Otherwise a SARIMA(1, 0, 5)x(0, 1, 0) 7 is suggestive. A further non-seasonal differencing of SDNXOF yields the series DSDNXOF which as well has a horizontal trend. With a test statistic value of -6.2, it is adjudged stationary by the ADF Test. Moreover its ACF in Figure 5 corroborates the stationarity hypothesis. Furthermore from the correlogram, seasonality of 7-day periodicity and the involvement of a seasonal moving average component of order one are evident. The correlogram also reveals a SARIMA(0, 1, 1)x(0, 1, 1) 7 autocorrelation structure since in the ACF the spikes at lags 6 and 8 are comparable. Two models are therefore proposed and fitted 1) A SARIMA(1, 0, 5)x(0, 1, 0) 7 given by SDNXOF t SDNXOF t-1 = t t t t t-5 + t 2) A SARIMA(0, 1, 1)x(0, 1, 1) 7 (4) DSDNXOF = t t t-8 + t (5) FIGURES 1-7 HERE TABLES 1 & 2 HERE Estimation of the model (4) is summarized in Table 1 whereas that of the latter model (5) is summarized in Table 2. The correlogram of the residuals of the former in Figure 6 shows that the residuals rather than being uncorrelated are seasonal of period 7 days. This invalidates the model (4). On the other hand the residuals of the second model (5) are uncorrelated (See Figure 7). Hence the SARIMA(0, 1, 1)X(0, 1, 1) 7 model is considered to be adequate. CONCLUSION It may be concluded that daily Naira-NXOF exchange rates follow a SARIMA(0, 1, 1)x(0, 1, 1) 7 model. The model has been shown to be adequate. This means that forecasting of the time series might be done on the basis of this model. REFERENCES Ahmad MI (2012). Modelling and forecasting Oman crude oil prices using Box-Jenkins techniques. International Journal of Trade and Global Markets, 5(1): Appiah ST, Adetunde IA (2011). Forecasting Exchange Rate Between the Ghana Cedi and the US Dollar Using Time Series Analysis. African Journal of Basic & Applied Sciences, 3(6): Bako HY, Rusiman MS, Kane IL, Matias-Peralta HM (2013). Predictive modeling of pelagic fish catch in Malaysia using seasonal ARIMA models. Agriculture, Forestry and Fisheries, 2(3): Box GEP, Jenkins GM (1976). Time Series Analysis, Forecasting and Control, Holden-Day: San Francisco. Etuk, EH (2012). A seasonal ARIMA Model for Daily naira-us dollar Exchange Rates. Asian Journal of Empirical Research, 2(6): Etuk EH (2014). Modelling of Daily Nigerian naira-british Pound Exchange Rates using SARIMA Methods. British Journal of Applied Science and Technology, 4(1): Etuk EH, Igbudu R (2013). A Sarima Fit to Monthly Nigerian Naira-British Pound Exchange Rates. Journal of Computations & Modelling, 3(1): Etuk, EH, Nkombou BW (2014) Monthly XAF- USD Exchange Rates Modeling by SARIMA Techniques. Euro-Asian Journal of Economics and Finance, 2(1):

4 A model for daily exchange rates of the Naira and the XOF by seasonal ARIMA methods Etuk et al. Kaushik I, Singh SM (2008). Seasonal ARIMA Model for forecasting of Monthly Rainfall and Temperature. Journal of Environmental Research and Development, 3(2): Lee MH, Rahman NHA, Surhatono, Latif MT, Nor ME, Kamisan NAB (2012). Seasonal ARIMA for Forecasting Air Pollution Index: A Case Study. American Journal of Applied Sciences, 9(4): Longanathan N, Ibrahim Y (2010). Forecasting International Tourism Demand in Malaysia Using Box Jenkins Sarima Application. South Asia Journal of Tourism and Heritage, 3(2); Mahsin M, Akhter Y, Begum M (2012). Modelling Rainfall in Dhaka Division of Bangladesh Using Time Series Analysis. Journal of Mathematical Modelling and Application, 1(5): Martinez EZ, Soares da Silva EA, Fabbro ALD (2011). A Sarima forecasting model to predict the number of cases of dengue in Campinas, State of Sao Paulo, Brazil. Revista da Sociedade Brasileira de Medicina Tropical, 44(4): Nirmala M, Sundaram SM (2010). A Seasonal Arima Model for forecasting monthly rainfall in Tamilnadu. National Journal on Advances in Building Sciences and Mechanics, 1(2): Olowe RA (2009). Modelling Naira/Dollar Exchange rate Volatility: Application of Garch and Assymetric Models. International Review of Business Research Papers, 3(3): Osarumwese OI (2013). Applicability of Boxjenkins SARIMA Model in Rainfall Forecasting: A Case Study of Port Harcourt South South Nigeria. Canadian journal on Computing in Mathematics, Natural Sciences, Engineering and Medicine, 4(1): 1-4. Singh EH (2013). Forecasting Tourist Inflow in Bhutan using Seasonal ARIMA. International Journal of Science and Research, 2(9): Vahdat SF, Sarraf A, Shamsnia A, Shahidi N (2011). Prediction of monthly mean inflow to DezDam reservoir using time series models (Box-Jenkins) International Conference on Environment and Industrial Innovation, IPCBEE, 12(2001) IACSIT Press, Singapore,

5 Euro-Asian j. econ. financ. ISSN: (print); (online) Volume: 2, Issue: 3, Pages: APPENDIX TABLE 1: ESTIMATION OF THE SARIMA (1, 1, 5)X(0, 1, 0) MODEL 207

6 A model for daily exchange rates of the Naira and the XOF by seasonal ARIMA methods Etuk et al. TABLE 2: ESTIMATION OF THE SARIMA(0, 1, 1)X(0, 1, 1) 7 MODEL 208

7 Euro-Asian j. econ. financ. ISSN: (print); (online) Volume: 2, Issue: 3, Pages:

8 A model for daily exchange rates of the Naira and the XOF by seasonal ARIMA methods Etuk et al. 210

9 Euro-Asian j. econ. financ. ISSN: (print); (online) Volume: 2, Issue: 3, Pages:

10 A model for daily exchange rates of the Naira and the XOF by seasonal ARIMA methods Etuk et al. FIGURE 4: CORRELOGRAM OF SDNXOF 212

11 Euro-Asian j. econ. financ. ISSN: (print); (online) Volume: 2, Issue: 3, Pages: FIGURE 5: CORRELOGRAM OF DSDNXOF 213

12 A model for daily exchange rates of the Naira and the XOF by seasonal ARIMA methods Etuk et al. FIGURE 6: CORRELOGRAM OF THE (1, 0, 5)X(0, 1, 0) 7 SARIMA RESIDUALS 214

13 Euro-Asian j. econ. financ. ISSN: (print); (online) Volume: 2, Issue: 3, Pages: FIGURE 7: CORRELOGRAM OF THE (0, 1, 1)X(0, 1, 1) 7 SARIMA RESIDUALS 215