20120140503019

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

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

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

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

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)

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 *

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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.

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

181
Figure 10: Sample ACF and PACF of the GARCH(1,2) squared residuals
Figure 11: The returns series with two Conditional SD Superimposed

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.

20120140503019

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