20120140503019

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20120140503019

  1. 1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 3, March (2014), pp. 173-182, © IAEME 173 A TIME SERIES MODEL FOR THE EXCHANGE RATE BETWEEN THE EURO (EUR) AND THE EGYPTIAN POUND (EGP) Taha Abdelshafy Abdelhakim Khalaf Department of Electrical Engineering, Assiut University, Assiut, Egypt, 71516 ABSTRACT In this paper, we introduce a time series model that is capable of characterizing the exchange rate of the Euro to the Egyptian Pound (EUR/EGP). Since the exchange rate is considered as a financial time series, the traditional autoregressive integrated moving average (ARIMA) model would not be sufficient to model the data series. Financial time series often exhibit volatility clustering or persistence. Therefore, a model which captures the changes in the variance is required. In this paper, we adopt the general autoregressive conditional heteroskedastic (GARCH) model to fit the data. The analysis show that GARCH(1,2) captures the heteroskedasticity of the data. I. INTRODUCTION The analysis of experimental data that have been observed at different points in time leads to new and unique problems in statistical modeling and inference. The obvious correlation introduced by the sampling of adjacent points in time should not be neglected in order to have a good model that represents the data [1]. One approach, advocated in the landmark work of Box and Jenkins, develops a systematic class of models called autoregressive integrated moving average (ARIMA) models to handle time-correlated modeling and forecasting. However ARIMA models work very well with most of time series it does not correctly fit the financial time series. That is because the set of ARIMA model tries to fit the conditional means of a stationary time series (i.e., changes in variance has to be alleviated) however the financial time series often exhibit volatility clustering or persistence. Therefore a model which is able to capture the heteroskedasticity of the data and fit the conditional variances is required. Time series also often exhibit volatility clustering or persistence. In volatility clustering, large changes tend to follow large changes, and small changes tend to follow small changes. The changes from one period to the next are typically of unpredictable sign. Large disturbances, positive or negative, become part of the information set used to construct the variance forecast of the next period’s disturbance. In this way, large shocks of either sign can persist and influence volatility forecasts for INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING AND TECHNOLOGY (IJARET) ISSN 0976 - 6480 (Print) ISSN 0976 - 6499 (Online) Volume 5, Issue 3, March (2014), pp. 173-182 © IAEME: www.iaeme.com/ijaret.asp Journal Impact Factor (2014): 7.8273 (Calculated by GISI) www.jifactor.com IJARET © I A E M E
  2. 2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 3, March (2014), pp. 173-182, © IAEME 174 several periods. Volatility clustering suggests a time series model in which successive disturbances are uncorrelated but serially dependent. Recent problems in finance have motivated the study of the volatility, or variability, of a time series. Although ARMA models assume a constant variance, models such as the autoregressive conditionally heteroskedastic or ARCH model, first introduced by Engle (1982), were developed to model changes in volatility. These models were later extended to generalized ARCH, or GARCH models by Bollerslev (1986). In this project, we would like to find a good model that fits the time of exchange rate EUR/EGP. The daily exchange rate for the year 2008 is considered in this work. The source of the data is [4]. The rest of the report is organized as follows. Section 2 introduces the pre-estimation analysis in order to find the good model that fits the data. Some models are suggested based on the pre-estimation analysis and theye are presented in Section 3. In Section 4, I compare between the suggested models and select the best model that fits the data. Finally, conclusions are drawn in Section 5. II. PRE-ESTIMATION ANALYSIS When estimating the parameters of a composite conditional mean/variance model, we may occasionally encounter some problems problems such as: 1- Estimation may appear to stall, showing little or no progress; 2- Estimation may terminate before convergence.3- Estimation may converge to an unexpected, suboptimal solution. In order to avoid many of these difficulties it’s better to select the simplest model that adequately describes the data, and then performing a pre-fit analysis. This pre-estimation analysis includes 1- Plot the return series and examine the ACF and PACF; 2- Perform preliminary tests, including McLeod-Li test and the Ljung-Box test. Figure 1: Original Series Plot: Exchange rate EUR/EGP Figure 2 shows the time series plot of the exchange rate EUR/EGP. Since GARCH models assume a return series, the original exchange rate series has to be converted first to the returns. If x is the original exchange rate series then the returns series r is given by one of the following equations. Day DailyExchRate 0 100 200 300 7.07.58.08.5
  3. 3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 3, March (2014), pp. 173-182, © IAEME 175 1 1 = − −− t tt t x xx r (1) ( ) 100log= ×tt xr (2) Figure 2: Return series of the exchange rate Figure 3: Sample ACF and sample PACF of the returns series
  4. 4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 3, March (2014), pp. 173-182, © IAEME 176 In this work, (2) is used to calculate the return series. Figure 2 shows the plot of the return series. From the original series and the return series plots, we notice that the series is heteroskedastic, meaning that its variance varies with time. We also notice that the return series shows volatility clustering. In order to test the data for the conditional means model, we plot the sample autocorrelation function (ACF) and the sample partial autocorrelation function (PACF). Figure 2 shows the sample ACF and sample PACF plots of the returns series. From the shown figure, it is clear that the return series does not exhibit any correlation between data points and there is no real indication that we need to use any correlation structure in the conditional mean. To ensure that there is no conditional mean model required, the “tsdiag” function is used to test the (0,0,0)ARIMA model. Figure 2 shows the standardized residuals (returns in this case), correlation of the residuals, and results for the Ljung-Box test. These results confirms that the return series matches the characteristics of the white noise. Now, we check the returns series for the conditional variance model. Figure 2 shows the results of the McLeod-Li test when applied to the return series. We notice that all p-values are less than the 5% threshold. Figure 2 shows the sample ACF and sample PACF of the squared series. Although the returns themselves are largely uncorrelated, the variance process exhibits some correlation. From Figure 2 and Figure 2, we conclude that a conditional variance model is required to fit the exchange rate series. Figure 4: Diagnostics of the (0,0,0)ARIMA model of the returns series
  5. 5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 3, March (2014), pp. 173-182, © IAEME 177 Figure 5: McLeod-Li test of the the return series. Figure 6: Sample ACF and sample PACF of the squared returns series. III. Estimating Model Parameters The presence of heteroskedasticity, shown in the previous analysis, indicates that GARCH modeling is appropriate. The ),(GARCH qp model is defined by the following two equations tttt wr 1|= −σ (3)
  6. 6. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 3, March (2014), pp. 173-182, © IAEME 178 2 1= 2 1| 1= 0 2 1| = jtj q j ititi p i tt r−−−−− ∑∑ ++ ασβασ (4) where tr is the returns series, 1| −ttσ is the variance of the returns at time t , and ( )0,1Nwt : . In this section, we will try different values of p and q and estimate the coefficients { }q ii 0= α and { }p jj 0= β . Small lags for p and q are common in empirical applications. Typically, (1,1)GARCH , (2,1)GARCH , or (1,2)GARCH models are adequate for modeling volatilities even over long sample periods (see Bollerslev, Chou, and Kroner [2]). Since small lags are preferable, we will start with the simple ARCH(1) model first. The ARCH(1) model only consider 0α and 1α . The estimated coefficients and their standard errors are stated in Table 1. The estimated coefficients of (1,1)GARCH , (1,2)GARCH , and (2,1)GARCH models are listed in Tables 2, 3, and 4 respectively. We notice that, the mean value µ is not statistically significant in the all proposed models. Table 1: ARCH(1) Model estimated coefficients Coefficient Estimate Std. Error t value Pr( |>|t ) Significance µ 0.001551− 0.009362 -0.166 0.868 0α 0.146527 0.006397 22.905 162< −e *** 1α 0.370867 0.043667 8.493 162< −e *** Table 2: GARCH(1,1) Model estimated coefficients. Coefficient Estimate Std. Error t value Pr( |>|t ) Significance µ 0.006190− 0.008462 0.732− 0.464447 0α 0.010761 0.002838 3.793 0.000149 *** 1α 0.153134 0.026422 5.796 096.8 −e *** 1β 0.805974 0.033381 24.144 162< −e *** Table 3: GARCH(1,2) Model estimated coefficients Coefficient Estimate Std Error t value Pr( |>|t ) Significance µ 0.005041− 0.008511 0.592− 0.553610 0α 0.011252 0.002971 3.788 0.000152 *** 1α 0.168217 0.027507 6.115 109.64 −e *** 1β 0.489888 0.130730 3.747 0.000179 *** 2β 0.297427 0.125888 2.363 0.018145 *
  7. 7. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 3, March (2014), pp. 173-182, © IAEME 179 Table 4: GARCH(2,1) Model estimated coefficients Coefficient Estimate Std. Error t value Pr( |>| t ) Significance µ 036.252 −− e 038.488 −e 0.737− 0.461 0α 021.079 −e 037.253 −e 1.487 0.137 1α 011.531 −e 022.643 −e 5.792 096.97 −e *** 2α 081.000 −e 027.882 −e 071.27 −e 1.000 1β 018.059 −e 011.045 −e 7.708 141.27 −e *** IV. MODEL SELECTION AND POST-ESTIMATION ANALYSIS Figure 7: McLeod-Li test of the ARCH(1) residuals In this section, we select one of the models proposed in the previous section. First, we check the ARCH(1) model by applying the McLeod-Li test to its residuals. Figure 4 shows the results of the McLeod-Li test of the ARCH(1) residuals. It is clear that the ARCH(1) didn’t capture the heteroskedasticity of the data very well and therefore this model is not accepted. The residuals of the other three models passed the McLeod-Li (we will only show the results for the selected model). In order to select one of the other three models, the AIC values are listed in Table 5. Based on the AIC values and the significance of the coefficients, the GARCH(1,2) model is adopted. The results of the Ljung–Box test, the McLeod-Li test, and the sample autocorrelations of the squared residuals are shown in Figures 6, 7, and 8 respectively. From the shown figures, we see that the GARCH(1,2) model is a good fit for the returns series of the exchange rate. Figure 9 shows the returns series plot with two Conditional SD Superimposed. It clear that the selected models captures the heteroskedasticity very well. To benefit from the fitted model, we use it to predict 20 days ahead. Figure 10 shows the last 120 points of the return series together with the 20 predicted points.
  8. 8. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 3, March (2014), pp. 173-182, © IAEME 180 Table 5: AIC values for the proposed GARCH models GARCH(1,1) GARCH(1,2) GARCH(2,1) AIC 1.125236 1.123964 1.126617 Figure 8: Diagnostics of the GARCH(1,2) Figure 9: McLeod-Li test of the GARCH(1,2) residuals
  9. 9. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 3, March (2014), pp. 173-182, © IAEME 181 Figure 10: Sample ACF and PACF of the GARCH(1,2) squared residuals Figure 11: The returns series with two Conditional SD Superimposed
  10. 10. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 3, March (2014), pp. 173-182, © IAEME 182 Figure 12: Prediction with confidence intervals V. CONCLUSIONS In this project, we studied he characteristics of the time series of the exchange rate Euro to Egyptian pound. We found that the return series has the same correlation properties as that of the white noise and therefore, we chose not to fit a conditional mean model. The squared return series has dependence between its points and the GARCH(1,2) model is proved to be a good fit for the series. It was also shown that the selected model captures the heteroskedasticity very well. The fitted model is used to predict 20 days ahead. REFERENCES [1] Robert Shumway and David Stoffer, Introduction to Time Series and Its Applications with R examples, Second Edition, Springer. [2] T. Bollerslev, R. Y. Chou, and K. F. Kroner “ARCH Modeling in Finance: A Review of the Theory and Empirical Evidence.” Journal of Econometrics. vol. 52, 1992, pp. 5–59. [3] Matlab Econometric and Financial Toolboxs [4] http://www.oanda.com/currency/historical-rates [5] H. J. Surendra and Paresh Chandra Deka“Effects of statistical properties of dataset in predicting performance of various artificial intelligence techniques for urban water consumption time series,” IAEME International Journal of Civil Engineering and Technology (IJCIET), vol. 3, no. 2, pp.426-436.

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