modeling the volatility of exchange rate currency …

ECONOMIA INTERNAZIONALE / INTERNATIONAL ECONOMICS 2019 Volume 72, Issue 2 – May, 209-230

Author::::

CHAIDO DRITSAKI

Department of Accounting and Finance, Western Macedonia University of Applied Sciences, Kila, Kozani, Greece

MODELING THE VOLATILITY OF EXCHANGE RATE

CURRENCY USING GARCH MODEL

ABSTRACT In this paper, we study GARCH models with their modifications in order to study the volatility of

Euro/US dollar exchange rate. Given that there are ARCH effects on exchange rate returns

Euro/US dollar, we estimated ARCH(p), GARCH(p,q) and EGARCH(p,q) including these effects

on mean equation. These models were estimated with maximum likelihood method using the

following distributions: normal, t-student and generalized error distribution. The log likelihood

function was maximized using Marquardt’s algorithm (1963) in order to search for optimal

parameter of all models. The results showed that ARIMA(0,0,1)-EGARCH(1,1) model with

generalized error distribution is the best in order to describe exchange rate returns and also

captures the leverage effect. Finally, for the forecasting of ARIMA(0,0,1)-EGARCH(1,1) model

both the dynamic and static procedure is used. The static procedure provides better results on

the forecasting rather than the dynamic.

KeywordsKeywordsKeywordsKeywords: Exchange Rate, Volatility, ARIMA-GARCH Models, Forecasting

JEL ClassificationJEL ClassificationJEL ClassificationJEL Classification: C22, C32, C53

RIASSUNTO

Modelli di volatilità del tasso di cambio con l’utilizzo di un modello GARCH

In questo articolo viene esaminata la volatilità del tasso di cambio Euro/dollaro USA tramite un

modello GARCH. Accertato che ci sono degli effetti ARCH sul tasso di cambio Euro/dollaro USA,

sono state effettuate delle stime ARCH (p), GARCH (p,q) e EGARCH(p,q) includendo questi

effetti su un’equazione minima. Questi modelli sono stati stimati col metodo della massima

somiglianza utilizzando queste distribuzioni: normale, t-student e distribuzione generalizzata di

errore. La funzione logaritmica di verosomiglianza è stata massimizzata usando l’algoritmo di

Marquardt (1963), al fine di cercare dei parametri ottimali per tutti i modelli. I risultati hanno

evidenziato che il modello ARIMA (0,0,1)-EGARCH(1,1) con distribuzione generalizzata di

210 C. Dritsaki

www.iei1946.it © 2019. Camera di Commercio di Genova

errore è la migliore per descrivere i rendimenti del tasso di cambio e cogliere gli effetti leva.

Infine, nell’utilizzo del modello di previsione ARIMA(0,0,1)-EGARCH(1,1) sono state usate sia la

procedura dinamica che quella statica. Quest’ultima ha fornito risultati migliori sulla previsione

rispetto a quella dinamica.

1. INTRODUCTION Issues concerning exchange rate are of great interest to researchers in modern economic theory.

Foreign exchange rate is regarded as the value of a currency in relation to another currency and

is fluctuated in relation to the demand and supply of currencies. Exchange rate is of vital

importance in the volume of trade and investment and its volatility decreases the volume of

international trade as well as foreign investment. Also, it is used as a fundamental economic

variable taken into consideration by investors of foreign currency, exporters and governments

for policy making. It is well known that exchange rate is under the authority of central banks in a

large extent, as well as under other financial institutions.

Modeling and forecasting of exchange rate has been widely discussed after the collapse of the

agreement for fixed currencies between industrial countries in the mid of 20th century – Bretton

Woods agreement. Thus, learning about currency volatility contributes in designing new

strategies about investment formulation.

The volatility on exchange rates influences the competitiveness of exports and imports, debt

payments of countries, as well as international investment portfolios. Moreover, the volatility of

exchange rates has impact on business economic cycles and capital flows, determining trade

terms with other countries thus directing the economic life of each country. The volatility of

exchange rate can also have consequences on international trade and influence balance of

payments of a country as well as government policy making. Consequently, forecasting the

volatility of exchange rate is of great importance for various groups, including the market and

also the decision makers.

During the last decades there has been an extended discussion as far as the exchange rate

volatility is concerned. As a consequence, several models have been developed in order to

examine this volatility. The models that are often applied in measuring the instability of

exchange rates are Autoregressive Conditional Heteroskedasticity-ARCH developed by Engle

Modeling the volatility of exchange rate currency using GARCH model 211


(1982) and Generalised ARCH-GARCH models developed by Bollerslev (1986) and Taylor (1986)

respectively.

This paper tries to develop and examine the characteristics of exchange rate volatility on

Euro/US dollar using monthly data from August 1953 until January 2017. The remainder of the

paper is organized as follows: Section 2 provides a brief literature review. Section 3 presents the

analysis of methodology. Section 4 summarizes the data and the descriptive statistics. The

empirical results are provided in Section 5 and Section 6 proposes the forecasting results.

Finally, the last section offers the concluding remarks.

2. LITERATURE REVIEW After the collapse of the agreement of fixed currencies, exchange rate is a significant issue to

analyze, as its impact to countries’ trade balance, price levels and output is quite important.

Given its role to economy, many researchers focus on the forecasting of exchange rates both on

developed and developing countries using various methodologies, not only in a fundamental but

also in a technical level.

The ARCH model for the exchange rate was first applied by Hsieh (1988) in order to examine

daily data for five exchange rates. The results of his research support the view that if there is not

a linear correlation on data but a non-linear, then the model’s form is multiplicative and not

additive. So, he concludes that the generalized ARCH model (GARCH) can explain one part of

the non-linearity of exchange rates.

Most scientists paid more attention on bilateral exchange rates such as the paper of Mundaca

(1991), Johnston and Scott (2000), Yoon and Lee (2008), Abdalla (2012) and others.

During last years, many researchers have dealt with the forecasting of exchange rate volatility

both on developed and emerging markets such as Sandoval (2006), Vee et al. (2011), Antonakakis

and Darby (2012), Miletic (2015), Epaphra (2017) and others. Specifically, Sandoval (2006)

examined the exchange rates of seven countries in Asia and Latin America regarding the US

dollar. Using GARCH, GJR-GARCH and EGARCH he found that four out of seven exchange rates

that follow asymmetric models are included in developed countries. Furthermore, forecasting

on symmetric models showed better results than that of asymmetric models.

212 C. Dritsaki


Vee et al. (2011) studied the forecast of exchange rate volatility of US Dollar/Mauritian Rupee.

For the estimation they used daily data from the period 30 June 2003 until 31 March 2008 and

the symmetric GARCH(1,1) model with Generalized Error Distribution (GED) and the

Student’s-t distribution. The results of their paper showed that Generalized Error Distribution

(GED) gives better results for exchange rate out-of-sample forecasts.

Antonakakis and Darby (2012) examine daily data from November 8, 1993 until December 29,

2000 for four exchange rates against the US dollar, such as the Botswana pula (BWP), Chilean

peso (CLP), Cyprus pound (CYP) and Mauritius rupee (MUR). Applying ARCH, GARCH,

EGARCH, IGARCH, FIGARCH and the HYGARCH models, they conclude that the IGARCH

model gives better results for out-of-sample forecast for all the examined exchange rates.

Miletić (2015) in his paper examines the hypothesis which refers that the exchange rate in

emerging markets is more sensitive in negative crises than positive ones. In order to study the

involved risk, he used daily data of exchange rate for the currencies of Hungary, Romania, Serbia,

Great Britain, Japan and European Union against US dollar, as well as the symmetric and non-

symmetric GARCH models. The results of forecast showed that the symmetric models, both on

developed and emerging markets (except the rate of Romania), have better return on the

exchange rates of all examined currencies.

Finally, Epaphra (2017) in his paper uses non-linear series for the forecasting of daily data on the

exchange rate of Tanzania (TZS/USD) from 4 January 2009 until 27 July 2015. Using the

symmetric GARCH model and the asymmetric EGARCH model, he concludes that the

symmetric GARCH model gives better results on forecasting but the asymmetric EGARCH

presents leverage effect implying a higher next period conditional variance than negative shocks

of the same sign.

3. THEORETICAL BACKGROUND Taking into account the papers of Mandelbrot (1963), Fama (1965) and Black (1976), many

researchers found that the characteristics of exchange rate returns follow a non-linear time

dependence, while according to Friedman and Vandersteel (1982), the percentage of exchange

rate follow a leptokurtic distribution. Moreover, in their papers argue that small and big



variances on exchange rates returns are clustered during time and are distributed symmetrically

with fat tails. The characteristics of these data follow the normal distribution and are included in

an ARCH model, introduced by Engle (1982) or in a GARCH model developed by Bollerslev

(1986).

While GARCH models can isolate the excessive kurtosis on returns, they cannot deal with the

asymmetry of distribution. To cope with this problem, researchers made modifications on

GARCH model, taking into account distributions’ asymmetry. Exponential GARCH (EGARCH)

is considered a non-linear model that deals with asymmetry and is developed by Nelson (1991).

3.1 ARIMA Model ARIMA is one of the type of models in the Box-Jenkins methodology. Box-Jenkins (1976)

approach on time-series analysis is a methodology where we try to find an ARIMA(p,d,q) model

which best describes the stochastic process where the sample is derived. The methodology

consists of four stages, the model identification, estimation, diagnostic testing and finally

forecasting. In the first stage of identification, data are transformed in order to have a stationary

series. The process of stationarity is the foundation in formulating an ARIMA(p,d,q) model. The

ARIMA (p,d,q) can be expressed as:

( )( ) ( ) ( ) tqtd

p eLyLL ϑµϕ =−−1 (1)

where

( ) ∑=

−=p

i

iip LL

1

1 ϕϕ and ( ) ∑=

−=q

j

jjq LL

1

1 ϑϑ are polynomials in terms of L of degree p and q.

ty is the time series, and te is the random error at time period t, with µ is the mean of the

model.

d is the order of the difference operator.

pϕϕϕ ,...,, 21 and qϑϑϑ ,...,, 21 are the parameters of autoregressive and moving average terms

with order p and q respectively.

L is the difference operator defined as ( ) tttt yLyyy −=−=∆ − 11 .

214 C. Dritsaki


3.2 ARCH(p) Model Engle (1982) developed the Autoregressive Conditional Heteroscedasticity (ARCH) model for

testing the volatility of financial series. The basic ARCH model consists of two equations, a

conditional mean equation and a conditional variance equation. Both equations should be

estimated simultaneously given that variance is a mean equation. The mean equation estimate

the conditional mean of the examined variable. The variance equation estimates this process as a

typical autoregressive process. Both equations form a system that is estimated together with

maximum likelihood method. So, ARCH model is an autoregressive process (AR) and can be

written as follows:

ttt z σε = where tz is white noise

∑=

−+=p

iitit

1

22 εαωσ (2)

where 0>ω , 0≥iα and 0>i .

3.3 GARCH(p,q) Model Bollerslev (1986) extended the ARCH model in a new one that allows the errors of variance to

depend on its own lags, as well as lags of the squared errors. In other words, it allows the

extension of conditional variance to follow an Auto Regressive Moving Average (ARMA) process.

The GARCH model can be expressed as:


∑∑=

−=

− ++=q

jjtj

p

iitit

1

2

1

22 σβεαωσ (3)

We assume that for every 0≥p and 0>q , the parameters are unknown and since the variance

is positive, then the following relations must be positive too 0≥ω , and 0≥iα for every i=1,…,q

and 0≥jβ for j=1,…,p. If the parameters are constrained such that 111

<+ ∑∑==

q

jj

p

ii βα , they imply a

weak stationarity. If 0q = , then GARCH model is becoming an ARCH model.



3.4 EGARCH(p,q) Model The ARCH/GARCH models analyzed previously, focus mainly on the size of conditional variance

on the returns of exchange rate and ignore information about the direction, if it is positive or

negative. So, these models do not explain the leverage effect presented in time series. For testing

this asymmetric volatility, various models have been developed. The EGARCH model is regarded

one of these models that captures asymmetric responses of varying variance to shocks at the

same time keeps the variance positive. The EGARCH model developed by Nelson (1991) can be

expressed as:


kt

ktq

j

r

kkjtj

p

i it

itit

−

−

= =−

= −

− ∑ ∑∑ +++=σεγσβ

σεαωσ

1 1

2

1

2 loglog (4)

where 2tσ is the conditional variance, ω , iα , jβ , and kγ are parameters to be estimated. On

parameters ω , iα and kγ there are no restrictions. However, the parameter jβ should be

positive and less than 1 in order to have stationarity. Moreover, kγ parameter is an indicator of

leverage effect meaning asymmetry and must be negative and statistically significant.

3.5 Estimation of the GARCH Model For the estimation of GARCH models, the maximum likelihood method is used. This method

enables the rates of return and variance to be jointly estimated. Parameters’ estimation on

logarithmic function of maximum likelihood are obtained through nonlinear least squares using

Marquardt’s algorithm (1963). The logarithmic function of maximum likelihood is computed

from the conditional densities of the prediction errors and is provided in the following form:

[ ] [ ]∑=

−=T

tttt zDyL

1

2 )(ln21

)]),((ln[),(ln θσυθθ (5)

where θ is the vector of the parameters that have to be estimated for the conditional mean,

conditional variance and density function, tz denoting their density function, )),(( υθtzD , is

the log-likelihood function of )]([ θty , for a sample of T observation. The maximum likelihood

estimator θ for the true parameter vector is found by maximizing (5) (see Dritsaki, 2017).

216 C. Dritsaki


3.5.1 Conditional Distributions Logarithmic function of maximum likelihood used for parameters’ estimation on volatility

models for all theoretical distributions are the following (see Dritsaki, 2017 :

• Normal distribution

[ ]

++−= ∑ ∑= =

T

t

T

tttt zTyL

1 1

22 )ln()2ln(2

1),(ln σπθ (6)

where θ is the vector of the parameters that have to be estimated for the conditional mean,

conditional variance and density function, T is observations.

• Student-t distribution

[ ] ∑=

−+++−

−−

Γ−

+Γ=T

1t

2t2

tt2

z1ln)1()ln(

2

1)]2(ln[

2

1

2ln

2

1lnT),y(Lln

υυσυπυυθ (7)

where

∫∞ −−=Γ0

1)( dxxe x υυ is the gamma function and υ is the degree of freedom.

• Generalized error distribution

[ ] ( ) ( )∑=

−

−

Γ−+−−

=T

tt

tt

zyL

1

21 ln211

ln)2ln(121

ln),(ln συ

υλλ

υθυ

(8)

where

2/1

/2

3

1

2

Γ

Γ= −

υ

υλ υ

3.6 Diagnostic Checking of the ARIMA-GARCH Model The diagnostic tests of ARIMA-GARCH models are based on residuals. Residuals’ normality test

is employed with Jarque-Bera (1980) test. Ljung and Box (1978) (Q-statistics) statistic for all

time lags of autocorrelation is used for the serial correlation test. Also, for the conditional

heteroscedasticity test we use the squared residuals of autocorrelation function.



3.7 Forecast Evaluation On ARIMA-GARCH models, we use both the static and dynamic forecast. The dynamic forecast

is known as n-step ahead forecast and uses the actual lagged value of Y variable in order to

calculate the first forecasted value. The one-step ahead forecast of 1+tY based on an ARIMA-

GARCH model is defined as:

( ) ∑ ∑= =

−+−+−+ ++=Ε=p

i

q

jjtjititttt YYYYY

1 111011 ,...,)1(ˆ εθφφ (9)

where the sε follow the stated GARCH model.

For the evaluation of effectiveness on forecasting, we use two statistical measures namely Mean

squared error (MSE) and Mean absolute error (MAE).

Mean squared error (MSE) computes the squared difference between every forecasted value and

every realized value of the quantity being estimated, and finds the mean of them afterwards.

Mean squared error (MSE) has the following formula:

( )∑=

−=n

iii YY

nMSE

1

2ˆ1 (10)

where

iY is the vector of observed values of the variable being predicted.

iY is the vector of n predictions.

Mean absolute error (MAE) computes the mean of all the absolute, instead of squared, forecast

errors. The formula is the following:

∑=

−=n

iii YY

nMAE

1

ˆ1 (11)

4. DATA The data used on this model of exchange rate return are monthly and refer to the Euro/US dollar

exchange rate. The data span is from August 1953 until January 2017, a total of 763 observations.

All data come from http://fxtop.com/en/historical-exchange-rates.php. Monthly percentage

return of exchange rate is the first difference from natural logarithm of exchange rate and is

given from the following equation:

218 C. Dritsaki


( ) ( )[ ]11

lnln*100ln*100 −−

−=

= tt

t

tt XX

X

XR (12)

where

tR is the monthly percentage return to the exchange rate;

tX is the exchange rate at time t.

The average monthly values of exchange rates and their returns are presented in diagrams 1 and

2 respectively.

DIAGRAM 1 - Average Monthly Values of Exchange Rates of the Euro/US dollar

From diagram 1, we can see that average monthly values of the exchange rate of Euro/US dollar

present a random walk.

DIAGRAM 2 - Average Monthly Values of Return of The Exchange Rate of the Euro/US dollar



From diagram 2 we can see that average monthly values of the Euro/US dollar exchange rate are

stationary. Also, the variance seems to be unstable, thus we conclude that the exchange rate

returns show volatility.

For autocorrelation test on average monthly returns of Euro/US dollar, we use autocorrelation

coefficients, as well as partial autocorrelation coefficients. Autocorrelation function defines the

q rank of Moving Average model – MA(q) while partial autocorrelation function defines p rank

of autocorrelation model – AR(p). The graph of autocorrelation coefficients is the correlogram.

Also, for autocorrelation testing on average monthly returns of Euro/US dollar, we use the

Ljung-Box test (1978), which is presented on the following relationship:

∑=

−+=

m

k

kLB kn

nnQ1

2ˆ)2(

ρ∼ χ2

m (13)

Following Tables 1 and 2 present the correlograms and we test if there is autocorrelation on

average monthly returns of Euro/US dollar, as well as the ARCH effect, on the correlogram of

average square monthly returns.

TABLE 1 - Correlogram of Average Monthly Returns of the Exchange Rate Euro/US dollar

220 C. Dritsaki


From the results of Table 1 we point out that the Ljung-Box statistic (Q-statistics) for all time

lags of autocorrelation function are statistically significant meaning that there is serial

correlation on the average monthly returns of the exchange rate Euro/US dollar. Furthermore,

partial autocorrelation coefficients show a long run dependence among data (time lags of high

order). This result, according to Bollerslev (1986), features GARCH models as the most suitable

for the data of the Euro/US dollar exchange rate.

TABLE 2 - Correlogram of Average Square Monthly Return of Euro/US dollar

The results of Table 2 show that Ljung-Box (1978) statistic (Q-statistics) for all time lags of

square residuals of autocorrelation function are statistically significant, thus there is an ARCH

effect on the return of Euro/US dollar exchange rate.

Continuing, the descriptive statistics of the return on Euro/US dollar exchange rate are

presented.



TABLE 3 - Descriptive Statistics of Average Monthly Return on the Euro/US dollar Exchange Rate

EuroEuroEuroEuro////USUSUSUS dollar dollar dollar dollar

Mean 0.003742

Median 0.001572

Maximum 6.812899

Minimum -8.143697

Std. Dev. 2.116776

Skewness -0.017459

Kurtosis 4.385232

Jarque-Bera 60.96277

Probability 0.0000

Observations 762

The results of Table 3 show that average monthly returns of the Euro/US dollar exchange rate do

not follow the normal distribution. Also, asymmetry’s coefficient shows that the distribution of

exchange rate returns is left asymmetric (-0.017), is leptokurtic (k=4.385), and has heavy tails

(see Diagram 3).

DIAGRAM 3 - Normal Density Graphs of the Average Monthly Return of the Euro/ US dollar Exchange Rate

Continued, we test the stationarity of the average monthly returns of the Euro/ US dollar

exchange rate using Dickey-Fuller (1979, 1981) and Phillips-Perron (1998) tests.

222 C. Dritsaki


TABLE 4 - Stationarity Test of Average Monthly Returns of the Euro/ US dollar Exchange Rate

VariableVariableVariableVariable ADFADFADFADF PPPP----PPPP

C C,T C C,T

REURUS -19.946(0)* -19.933(0)* -20.016[1]* -20.004[1]*

Notes: 1. *, ** and *** show significant at 1%, 5% and 10% levels respectively. 2. The numbers within parentheses followed by ADF statistics represent the lag length of the dependent variable used to obtain white noise residuals. 3. The lag lengths for ADF equation were selected using Schwarz Information Criterion (SIC). 4. Mackinnon (1996) critical value for rejection of hypothesis of unit root applied. 5. The numbers within brackets followed by PP statistics represent the bandwidth selected based on Newey West (1994) method using Bartlett Kernel. 6. C=Constant, T=Trend.

The results in Table 4 show that the average monthly returns of the Euro/US dollar are

stationary in their levels on both tests.

After detecting stationarity, we determine the form of the ARMA(p,q) from the correlogram of

Table 1. Parameters p and q can be defined from partial autocorrelation coefficients and

autocorrelation coefficients, respectively, compared to the critical value

072.0762

22 ±=±=±n

. Therefore, we can see that p value is between 10 ≤≤ p and q value

is between 10 ≤≤ q .

So, Table 5 provides with the following values:

TABLE 5 -Model Comparison between AIC, SIC and HQ Tests

ARIMA model AIC SC HQ

REURUS

((((0000,0,,0,,0,,0,1111)))) 4.4.4.4.232232232232 4.4.4.4.244244244244 4.4.4.4.236236236236

(1,0,0) 4.239 4.251 4.244

(1,0,1) 4.234 4.252 4.241

The results from the above Table indicate that according to Akaike (AIC), Schwartz (SIC) and

Hannan-Quinn (HQ) criteria, ARIMA (0,0,1) model is the most suitable, as far as the mean

monthly returns for the Euro/US dollar is concerned.



TABLE 6 ---- Estimation of ARIMA(0,0,1) Model

After the estimation of the above model, we test for the existence of conditional

heteroscedasticity (ARCH(p) test) from the squared residuals of the above model. Table 7 gives

these results.

TABLE 7 - ARCH(p) Test

224 C. Dritsaki


From the results of the above Table, we conclude that autocorrelation and partial

autocorrelation coefficients are statistically significant. Consequently, the null hypothesis for

the absence of ARCH or GARCH procedure is rejected.

5. EMPIRICAL RESULTS Since there are ARCH effects on the returns of the Euro/US dollar exchange rate, we can proceed

with the estimation of ARCH(p), GARCH(p,q) and EGARCH(p,q) models. The estimation of the

parameters is accomplised using Marquardt’s algorithm (1963). The parameters (coefficients) of

estimated models and the test of normality, autocorrelation and conditional heteroskedasticity

of the residuals are provided in Table 8. A higher log-likelihood value (LL) yields a better fit.

Table 8 provides the estimations of all models and the standard errors of the parameters

(coefficients) together with the value of log-likelihood function, as well as the normality test,

autocorrelation test and conditional heteroskedasticity test. From the Table below we can see

that the ARIMA(0,0,1)-EGARCH(1,1) model with the generalized error distribution (GED) is the

most suitable because all the coefficients are statistically significant in 1% level of significance,

the log-likelihood value is the highest and there is no problem in autocorrelation and the

conditional heteroscedasticity. In addition, β1 coefficient is positive and less than one showing

the stationarity of the model. Also, γ1 coefficient is negative and statistically significant

indicating leverage result (asymmetry). Thus, this model can be used for forecasting.



TABLE 8 - Estimated ARIMA(0,0,1)-ARCH(1), ARIMA(0,0,1)-GARCH(1,1),

ARIMA(0,0,1)-EGARCH(1,1)

ARIMA(0,0,1)ARIMA(0,0,1)ARIMA(0,0,1)ARIMA(0,0,1)----ARCH(1)ARCH(1)ARCH(1)ARCH(1)

Parameter Normal t-Student GED ω 3.327(0.000) 1.059(0.999) 0.165(0.000) α1 0.198(0.000) 0.492(0.999) 1.436(0.000)

D.O.F=2.001(0.000) PAR=0.403(0.000) LL -1604.01 -1352.00 ----1313131346464646....88887777 Jarque-Bera 136.12(0.000) 1368.25(0.000) 3198(0.000) ARCH(1) 1.450(0.228) 0.163(0.685) 0.623(0.429) Q2( 10) 10.364(0.322) 2.655(0.976) 5.910(0.749)

ARIMA(0,0,1)ARIMA(0,0,1)ARIMA(0,0,1)ARIMA(0,0,1)----GARCH(1,1)GARCH(1,1)GARCH(1,1)GARCH(1,1) Parameter Normal t-Student GED

ω 0.012(0.000) 0.001(0.161) 0.003(0.000) α1 0.081(0.000) 0.434(0.107) 0.748(0.000)

β1 0.923(0.000) 0.560(0.000) 0.664(0.000)

D.O.F=2.246(0.000) PAR=0.656(0.000) LL -1385.388 -1136.38 ----1181181181184444....66666666 Jarque-Bera 31749.92(0.000) 6812.99(0.000) 6097.4(0.000) ARCH(1) 0.479(0.488) 0.025(0.873) 0.003(0.951) Q2( 10) 0.482(0.487) 0.082(0.344) 0.082(0.997)

ARIMA(0,0,1)ARIMA(0,0,1)ARIMA(0,0,1)ARIMA(0,0,1)----EGARCH(1,1)EGARCH(1,1)EGARCH(1,1)EGARCH(1,1) Parameter Normal t-Student GED

ω 0.010(0.000) 0.001(0.166) -0.003(0.000) α1 0.099(0.000) 0.159(0.120) 0.178(0.000)

β1 0.923(0.000) 0.564(0.000) 0.666(0.000)

γ1 -0.059(0.235) -0.403(0.543) -0.082(0.000)

D.O.F=2.243(0.000) PAR=0.655(0.000) LL -1380.42 -1136.18 ----1111111174747474....77777777 Jarque-Bera 6269.45 (0.000) 5940.12(0.000) 1597(0.000) ARCH(1) 0.012(0.910) 0.003(0.981) 0.011(0.916) Q2( 10) 9.164(0.422) 1.379(0.992) 1.568(0.997) Notes: 1.Values in parentheses denote the p-values. 2.LL is the value of the log-likelihood.

6. FORECASTING For the forecasting of ARIMA(0,0,1)-EGARCH(1,1) model on the returns of Euro/US dollar

exchange rate, we use both the dynamic (n-step ahead forecasts) and static (one step-ahead

forecast) procedure. The dynamic procedure computes forecasting for periods after the first

226 C. Dritsaki


sample period, using the former fitted values from the lags of dependent variable and ARMA

terms. The static procedure uses actual values of the dependent variable. In the following

diagram, we present the criteria for the evaluation of forecasting the returns of Euro/US dollar

exchange rate, using the dynamic and static forecast respectively.

DIAGRAM 4 - Dynamic and Static Forecast of Euro/US dollar

The above diagram indicates that the static procedure gives better results rather than the

dynamic (Mean Squared Error and Mean Absolute Error are lower in the static rather than the

dynamic process).

7. DISCUSSION AND CONCLUSION This paper focuses on the formation of a model for the Euro/US dollar exchange rate. Due to the

fact that exchange rate is regarded as a financial time series that may present volatility, it is more

suitable to use models from GARCH family. More specific, the non-linear ARIMA(0,0,1)-

ARCH(1), ARIMA(0,0,1)-GARCH(1,1) and ARIMA(0,0,1)-EGARCH(1,1) models were used in

order to register the volatility of exchange rate. Furthermore, the leverage effect is captured

from the estimation of ARIMA(0,0,1)-EGARCH(1,1) model, showing that positive shocks cause

lower volatility in relation to negative ones. Finally, the forecast of ARIMA(0,0,1)-EGARCH(1,1)

model is evaluated, using both the dynamic and static procedure.



So, economic policy makers should forecast the future values of exchange rates using the

equivalent models. The instability of exchange rate is an uncertainty measure in an economic

environment for each country that should be forecasted. The main policy implication from the

results of this paper is that since exchange rate instability may increase transaction costs and

reduce the gains to international trade, the insight of exchange rate volatility estimation and

forecasting is important for asset pricing and risk management.

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