MODELING THE EXCHANGE RATE OF THE EURO AGAINST THE DOLLAR USING THE ARCH/GARCH MODELS

Glavni zadatak ARCH i GARCH modela je analiza vremenskih serija u kojima postoji uslovna heteroskedastičnost (vremenski promenljiv varijabilitet, odnosno nestabilnost uslovne varijanse, pojava koja se naziva volatilnost). Cilj ovih modela je da se izračuna neki od pokazatelja volatilnosti, da bi se pomoću tog pokazatelja donosile finansijske odluke. U ovom radu se ispituju performanse uopštenog autoregresionog modela uslovne heteroskedastičnosti (eng. generalized autoregressive conditional heteroscedasticity GARCH) za modeliranje dnevnih promena logaritmovanog deviznog kursa evra prema dolaru. Primenjeno je nekoliko GARCH modela za modeliranje dnevne stope prinosa deviznog kursa evra, sa različitim brojem parametara. Za ocenjene GARCH modele karakteristično je da su dobijeni koeficijenti kvadriranih reziduala na docnji i koeficijenti uz parametar uslovne varijanse u jednačini uslovne varijanse uglavnom naglašeno statistički značajni. Zbir vrednosti ocene ova dva koeficijenta je blizu jedinice, što je tipično za GARCH modele koji se primenjuju na podatke prinosa finansijske aktive. To znači da će šokovi u jednačini uslovne varijanse biti dugotrajni. Velika vrednost zbira ova dva koeficijenta pokazuje da visoka stopa pozitivnog ili negativnog prinosa dovodi do velikih prognoziranih vrednosti varijanse u produženom periodu. Najbolje rezulate u modeliranju stope prinosa evra pokazao je asimetrični EGARCH(1,1) model. Koeficijent asimetrije u jednačini volatilnosti kod ovog modela je negativan, i nije statistički signifikantan. Negativna vrednost ovog koeficijenta sugeriše da pozitivni šokovi manje utiču na uslovnu varijansu u budućem periodu nego negativni šokovi. Asimetrični EGARCH(1,1) model obezbeđuje dokaze o leveridž efektu.


Introduction
The exchange rate fluctuations in the contemporary phase of globalization and liberalization of the world economy have a significant impact on macroeconomic factors such as interest rates, prices, exports and imports, gross domestic product.The more the economies are open, the stronger the impact of the international environment on the exchange rate changes.This, however, does not mean that greater openness implies greater volatility of the exchange rate.Thus, the interest of economic policy for the consideration of the potential volatility of the exchange rate.The participants in trading and investors are motivated for a quantitative assessment of the future exchange rate fluctuations, in order to protect themselves against the exchange rate risk.Financial analysts and econometricians have developed a series of econometric models for the analysis of movements in the exchange rate returns.Among these models, a prominent place belongs to Autoregressive Conditional Heteroskedasticity -ARCH models and Generalized Autoregressive Conditional Heteroskedasticity -GARCH models.ARCH models were developed by Engle (Engle, 1982), and their further generalization as GARCH models was suggested by Bollerslev (1986) and Taylor (1986).To date, a large number of asymmetric GARCH models have been proposed, for example the exponential EGARCH model, formulated by Nelson (Nelson, 1991).GARCH models have been widely employed in financial markets analysis.Some good results in forecasting fluctuations in securities and exchange rates returns have been obtained by means of these models.The aim of this paper is to assess the adequacy of the GARCH models to capture the volatility of the exchange rate of the euro against the dollar.This is significant not only for the direct participants in the foreign exchange trade but also for the balance of payments situation in the Eurozone and the US, but also in other countries.The remainder of this paper is structured as follows.The review of reference literature related to exchange rates modeling is presented in the second part of the paper.The third part is devoted to the methods of research used in this paper.The fourth section presents an analysis of the empirical results, and the conclusion is presented in the fifth part of the paper.

Literature Review
Balaban (2004) compared the characteristics of symmetric and asymmetric GARCH model using a time series of the dollar and the German mark exchange rate returns.The time series were modeled with GARCH (1,1), GJR-GARCH (1,1) and EGARCH (1.1) volatility equations.The author found that the EGARCH model showed better results than GARCH (1,1) model in forecasting the exchange rate movements out of sample.The worst results were produced by the GJR-GARCH model.Pilbeam and Langeland (2015) investigated how suitable one-dimensional GARCH models are for forecasting volatility in exchange rates.The study focused on two periods: the first from 2002 to 2007 and the second from 2008 to 2012.The first period was characterized by lower volatility, the second by higher volatility.The model shows that the future volatility is well estimated by the foreign exchange market.According to the study, GARCH models have shown better results in the periods of low volatility.Alexander and Lazar (2006) examined the possibility of the GARCH (1,1) model to encompass the variations of conditional skewness and kurtosis.The research is an attempt to prove that generic twocomponent GARCH (1,1) models give better results in modeling the exchange rate than the models with three or more components, and that they are better than the symmetric and skewed Student's t-GARCH models.The study evaluated 15 different models which are used for modeling the three major dollar exchange rates.The empirical results of this study do not support the introduction of restrictions on GARCH models.It was also found that the t-GARCH models show good results according to the test specifications of the moment, but they were inferior to the NM-2 models (Normal mixture models that are characterized by normal distribution combined with the GARCH type structure) modeling non-conditional variability.In addition, t-GARCH models have showed the weaker results according to the ACF and VaR criteria.Ghalayini (2014) estimated GARCH (1,1) model in the case of the dollar against the euro exchange rate and came to the conclusion that the sum of the estimated  2010) su analizirali međuzavisnost i prelivanje volatilnosti između tri devizna kursa evra (u odnosu na američki dolar, japanski jen i britansku funtu sterlinga).Primenom metoda realizovane varijanse, ovi efekti su posmatrani u nekoliko vremenskih intervala tokom trgovačkog dana.Dekompozicija varijanse iz ocenjenog VAR modela pokazuje da devizni kurs evra prema dolaru dominira u odnosu na druga dva kursa, kako u pogledu prinosa tako i u prelivanju volatilnosti.Ocenjeno je da šokovi u kretanju deviznog kursa evra u odnosu na sterling i jen marginalno utiču na kurs evra prema dolaru, dok vesti koje se odnose na kurs evra prema dolaru objašnjavaju oko 30% kretanja prinosa i volatilnosti deviznog kursa evra prema sterlingu i jenu.
ARCH modele u osnovi čine dve jednačine: jednačina srednje vrednosti (nivo prinosa posmatrane pojave), kojom se oblikuje bezuslovna varijansa, i jednačina uslovne varijanse (volatilnosti), kojom se opisuje uslovna varijansa prinosa.Za analizu dnevnih podataka korisna je primena GARCH modela, koji uslovnu varijansu prinosa izvode u funkciji kvadrata uslovnog varijabiliteta u izabranom broju prethodnih perioda i kvadrata slučajnih ARCH and GARCH (α + β) coefficient was close to a unit, based on which he concluded that volatility shocks showed persistence, which often occurs in the high frequency financial data.The LM test of autocorrelation rejected the hypothesis of the absence of residual autocorrelation in the model residuals up to the order of two.It was concluded that the model shows the presence of serial correlation, and therefore does not represent a satisfactory framework to capture the correlation in the time series and thus cannot be used for forecasting the exchange rate.Hartwell (2014) used the family of GARCH models to examine the financial volatility as a function of institutional volatility.These models were applied to 20 countries in transition over different time intervals from 1993 to 2012.The results of applied EGARCH and TGARCH models support the view that the more developed and stable institutions prevent the volatility of the financial sector, while institutional volatility directly affects the volatility of the financial sector in transition countries.Hsieh (2012) estimated ARCH and GARCH models for 5 currencies, using the ten-year daily data for various specifications of these models.He also applied a comprehensive set of diagnostic tests.Based on the above tests, the author concluded that ARCH and GARCH models can generally remove the entire heteroskedasticity in price changes in all five currencies.The exponential GARCH model is very well fitted to the Canadian dollar and Swiss franc, and a solid performance was provided for the Germany mark.Juraj Stančík (2006) analyzed the basic factors affecting the volatility of exchange rates of the new EU member states in relation to the euro -the openness of the economy, the "news" factor and the exchange rate regime.Modelling of the exchange rate was performed by TARCH models.The results show that openness has a negative impact on exchange rate volatility.It was also found that the "news" factor significantly affects the volatility of the exchange rate.The effects of these factors vary significantly among the countries.Dumitrescu and Roşca (2015) modeled the volatility of the Romanian, Czech, Hungarian and Polish foreign exchange rates in the period 2005-2014, identifying a robust econometric model.The results confirmed the validity of the GARCH (1,1) model and unconditional volatility.
McMillan and Speight (2010) analyzed the interdependence and volatility spillovers in the three exchange rates of the euro (against the US dollar, Japanese yen and British pound sterling).By applying the variance method, these effects are observed at several time intervals over the trading day.Variance decomposition of the estimated VAR models shows that the exchange rate of the euro against the dollar dominates in relation to the other two rates, both in terms of returns and the spillover of volatility.It is estimated that shocks in the movement of the exchange rate of the euro against the sterling and the yen affect the exchange rate of the euro against the dollar marginally, while the news concerning the euro exchange rate against the dollar account for about 30% of the movement in returns and volatility of the exchange rate of the euro against sterling and the yen.

Research Methodology
The generalized GARCH model proposed by Bollerslev (1986, p. 309) has the following specifications: The model shown in equation ( 1) can be in the most general form marked as GARCH (p, q).In practical applications the model of the first order (p = q = 1) is very popular.The generalized model assumes the following conditions: p≥0, q> 0, α0> 0, αi≥0, wherein i = 1, ..., q, and βi≥0, wherein i = 1, ... , p.The basic weakness of GARCH models is that they assume symmetric effects of positive and negative shocks to volatility.However, the belief that negative shocks in financial time series cause a greater volatility than positive shocks of the same magnitude is widespread in the literature.As regards the series of returns, asymmetric response is attributed to the leverage effect.Therefore, a number of asymmetric GARCH models have been formulated in reference literature.One of them, the exponential GARCH model, will be applied in this paper.
Eksponencijalni GARCH metod je osmislio Nelson (1991).Sledeća specifikacija uslovne varijanse i njeno objašnjenje je prema Eviews 8 Users Guide (2013, str.221, jednačina 25.22): U jednačini (7) w je konstanta (dugoročna srednja vrednost).Parametar α reprezentuje "GARCH" efekat.Parametar β meri perzistenost uslovne volatilnosti bez obzira na to šta se događa na tržištu.Parametar γ meri asimetriju ili efekat leveridža, tako da EGARCH model omogućava testiranje asimetrije.Kad je γ=0, model je simetričan, odnosno pozitivni i negativni šokovi podjednako deluju na volatilnost serije prinosa.U slučaju da je γ<0, pozitivne (dobre) vesti sa tržišta generišu manju volatilnost nego negativni šokovi.Ako of the observed phenomena), which forms the unconditional variance, and conditional variance (volatility) equation, which describes the conditional variance of returns.The GARCH models are useful in the analysis of daily data, and their conditional variance of return shows as a function of conditional square variability in a selected number of previous periods and the square of random shocks in the same number of the previous periods.The most commonly applied is GARCH (1,1) model.In the standard notation GARCH (1,1), the first number in parentheses refers to how many autoregressive lags, or ARCH members of a given model, are included in the equation, while the second number refers to how many moving average lags, which in this model is often referred to as the number of the GARCH terms (Engle, 2001, p. 160).GARCH (1,1) model can be represented by the following two specifications (Equations 2 and 3): where Xt represents the exogenous variables included in the mean equation.ε t is a random error model, which does not have a normal distribution.(Bearing in mind that the authors' calculations in this paper are made by software Eviews, symbols in equations 2 to 8, as well as the describes of GARCH model, are according to the EViews Guide (EViews 8 User's Guide II, 2013, pp.207-208)).The mean equation may contain conditional variance or conditional standard deviation (GARCH-in-Mean models).The conditional variance equation in the specification GARCH (1,1) can be summarized as: where the symbols have the following meanings: is a conditional variance, or forecast variance in the coming period, based on the past information; ω is constant term, is a component of the ARCH model and presents the information about the volatility in the previous period, which was calculated as the lag of squared residuals from the mean equation; is a member of the GARCH model and represents the forecast variance for the last period.The present model is easy to assess and its application has shown good results in forecasting the value of the conditional variance (Engle, 2001, p. 159).
A higher order of the GARCH model, which is referred to as GARCH (p, q) can be assessed if the p or q is greater than one, where p is the order of the moving average ARCH member, and q is the order of the autoregressive GARCH term.GARCH (p, q) variance can be summarized as: When q = 0, GARCH model is reduced to the ARCH model.Within the GARCH (p, q) model, the conditional variance of ε t depends on the squared residuals in the previous p period, and conditional variance in the previous p period.
If conditional variance is included into equation ( 2) (mean equation), GARCH-in-Mean (GARCH-M) model is obtained.The new mean equation is: The parameter next to , designated as λ, measures the risk premium.If the estimated value of the coefficient λ is positive and statistically significant, it means that the higher risk leads to an increase in the level of yields (in our case, this would mean that the euro will appreciate against the dollar).If the estimated conditional standard deviation were included in the mean equation instead of variance, ARCH-M specification could be displayed as equation (6).
In equation (7) w is an intercept term (longterm mean).The parameter α represents the "GARCH" effect.The parameter β measures the persistence in conditional volatility regardless of what is happening in the market.The parameter γ measures the effect of asymmetry or leverage, so the EGARCH model allows the testing of asymmetry.When γ = 0, the model is symmetrical, i.e. positive and negative shocks act on the return volatility equally.In the case of γ <0, the positive (good) news from the market generate less volatility than negative shocks.If γ> 0, positive shocks have a greater impact than negative shocks.The direction of change also affects the yield volatility.The sum of α and γ shows the positive impact of shocks on the return series.This sum will be less than the value of the coefficient α when γ has a negative value, and vice versa.
On the left side of equation ( 7) is the logarithm of the conditional variance.It means that the leverage effect exist in exponential equation, which guarantees the forecast of the conditional variance to be non-negative.The existence of the leverage effect is tested by the hypotheses that γi <0.If γi ≠ 0 the asymmetric effect exist.There are several differences between the specifications of the EGARCH models in Eviews and Nelson's original model.Nelson started from the assumption that e t traces to the generalized error distribution, while Eviews offers a choice between the normal, Student's t-distribution and GED distribution.However, due to the conceptual ease and intuitive interpretation, when using the EGARCH model the conditionally normal error distribution is mainly applied.Second, the Nelson's specification for the log of the conditional variance is a restricted version of the following model: This model is slightly different from the specifications above under (7).Model (8) gives identical estimates as well as the model that uses Eviews except for the intercept term, which depends on the adopted model, and the probability distribution and the order p.For example, as the p = 1 in the model with the normal distribution, the difference would be α 1 (8 Eviews II Users Guide, 2013, p. 221).The Exponential GARCH model has two important advantages over GARCH (1,1) model.First, the EGARCH model show log returns, so that the conditional variance will be positive even if the parameters are negative.Therefore, non-negativity constraint does not need to be introduced on the model parameters.Second, due to the asymmetry, the model includes the so-called leverage effect (this effect is typically interpreted as a negative correlation between negative lag return and volatility).Empirical analysis of the leverage effect has prompted the development of a model with asymmetric volatility, following the positive and negative innovations.The leverage effect was first observed by Black (1976), and it is described the best by news impact curve, introduced by Pagan and Schwert (1990).This curve describes the future volatility as a reaction to good or bad news -in the case of asymmetric GARCH model the curve is asymmetric, so that negative shocks lead to higher volatility than the positive shocks of the same intensity (for more details on the impact of news on volatility see Engle and Ng, 1993).Empirically, it is also found that the error distribution affects the assessment of the asymmetry parameter, but does not affect the volatility of other parameters (Rodríguez and Ruiz, 2012, p. 661).
All models in this paper were estimated using EViews, by the Marquardt algorithm optimization and Bollerslev and Wooldrige (1992) method for standard errors estimates.GARCH model parameters are estimated using the quasi-maximum likelihood -QML (Brooks, 2008, p. 399).The maximum likelihood estimation produces asymptotically more efficient estimates than the estimates which can be obtained by using other methods.

Data and Empirical Results
The data used in this paper are the daily exchange rates of the euro against the dollar in the period from 03.01.2000 to 30.09.2016 (a total of 4291 pieces of data).We used the IMF data on the daily spot exchange rates: https:// www.imf.org/external/np/fin/ert/GUI/Pages/CountryDataBase.aspxThe nominal increase in the exchange rate means appreciation of the U grafikonu 1. se uočava da dnevna stopa prinosa evra prema dolaru oscilira oko nulte vrednosti.Zapaža se prisustvo nekoliko nestandardnih opservacija, a oscilacije su pojačane tokom 2008.godine usled svetske ekonomske krize.

Grafikon 1. Dnevna stopa prinosa evra prema dolaru
Izvor: Obrada autora.euro against the dollar.The baseline data were log transformed.Daily returns of exchange rate are calculated as r t = log(y t ) -log(y t-1 ) x 100, where y t is the level of the prompt exchange rate at the time t, wherein t=1, 2, ..., T. According to this approach, the daily euro appreciation or depreciation against the dollar was obtained as the first difference of the log exchange rate level.The daily euro returns against the dollar are shown in Figure 1.
Figure 1 shows that the daily returns of the euro against the dollar fluctuate around zero.The presence of several non-standard observations can be seen, but fluctuations were intensified in 2008 due to the global economic crisis.
The basic descriptive statistics of the euro returns are set out in Table 1.The mean value of the series does not differ significantly from zero.The coefficient of skewness is -0.039528, which means that the series is skewed to the left.Kurtosis is 5.850363.The value of this ratio in the case of normal distribution is three.So, as the kurtosis is greater than three, the tails of the euro returns are heavier than the normal distribution.The extreme values in the series lead to the formation of heavy tails.Based on the p-value of the Jarque-Bera test statistic, which is zero to six decimal places, the null hypothesis of normality of the euro returns is rejected at the 5% level and the alternative one is accepted according to which the returns are not normally distributed.
According to the both unit root tests, where the only deterministic component is the constant term, it can be concluded that the series of the euro returns against the dollar is stationary.The same conclusion applies if the deterministic components are the constant term and the trend.
The analysis of ordinary and partial correlogram of squared returns indicates that several coefficients are statistically significant (Charts 3 and 4).The first order autocorrelation coefficient value is 0.18, which then oscillates down to the values of 0.04 with 33 lags.Although the value of autocorrelation coefficients is not high, they are statistically significant.The realized p-values for these coefficients are zero to four decimals, based on which the null hypothesis of no ARCH structure is rejected.The partial autocorrelations also highlight that there are more partial autocorrelation coefficients which are statistically significant.According to the estimated coefficient it can  U postupku provere postojanja nestabilne uslovne varijanse prvi korak je da se oceni linearni model, posle čega će se izvršiti testiranje reziduala na postojanje ARCH strukture.Serija prinosa je modelirana u funkciji ARMA (1,1) koja je izabrana arbitrarno u ovoj fazi modeliranja.Postojanje nestabilne varijanse u vremenskoj seriji prinosa evra proverava se pomoću Ljung-Boksove (Ljung-Box) statistike (Q 2 ) i Engleove ARCH statistike.Vrednosti u tabeli 3. dobijene su iz reziduala regresije kojom je serija prinosa evra modelirana kao funkcija konstante i ARMA(1,1).also be seen that the time series of returns has the autoregression structure of variability.Therefore, based on the common and partial correlogram of squared daily returns it can be concluded that this series is autoregressive.
On the basis of high Q 2 values and ARCH statistics we conclude that the series of eurodollar exchange rate returns has an unstable conditional variance.This actually means that in the modeling of the series of euro returns, the family of ARCH models can be applied.4).
The estimated coefficients in the model ARCH ( 5) are statistically significant.Standardized residuals in the estimated model ARCH ( 5) are not autocorrelated.The absence of a correlation can be seen in respect of the squared standardized residuals up to the coefficient with the ordinal number six.The model substantially covers the dynamics of unstable conditional variance.For the ARCH test, the following statement applies: If the value of the test statistic exceeds the critical value of χ2 distribution, the null hypothesis about the absence of autocorrelation is rejected.In the estimated model ARCH (5) JB statistic is less than the original return series.The probability p with Jarque-Bera (JB) statistic is zero to four decimal values, leading to the rejection of the null hypothesis of normal distribution.The deviations from the normal distribution are owing to the increased coefficient of skewness.All obtained values clearly indicate the presence of an unstable variance and thus the ARCH structure.This means that the family of ARCH models can be used for modeling a series of returns.(5) 0,070398 0,018033 3,903897 0,0001 Q(10)=4.74(0,91),Q(20)=12,39(0,90), Q(30)=22,73(0,83), Q 2 (10)=37,99(0,00), Q 2 (20)=138,46(0,00), Q 2 (30)=246,37(0,00), ARCH 10=38,93(0,00), ARCH(20)=133,23(0,00), ARCH(30)=193,41(0,00) Skewness = -0,04 Kurtosis = 3,94 JB=158,33 Note: The numbers in parentheses next to the coefficients in the lower part of the table are p-values.ARCH χ2 test was applied.These notes refer also to Tables 4, 5 and 6.Source: Author by EViews software package.
The main difficulty in the application of the ARCH (q) model is to determine the order of delay q.This problem can be partially solved by using the probability ratio test.In fact, in order to include all dependencies in the equation of conditional variance, the required order of delay of squared errors can be huge.In order to overcome these problems, GARCH models were developed in the reference literature.With a view to assessing the GARCH family models, the maximum likelihood method is used.In this paper we used the Marquardt algorithm numerical optimization, which also represents a modification of the BHHH algorithm (both algorithms are variants of the Gauss-Newton method).The Marquardt algorithm possesses the power of "correction" that quickly pushes up the estimated coefficients to their optimum values (for more details see Press et al., 1992).
The GARCH (1,1) model parameter estimates are presented in table 5.The dummy variables are included in the mean equation for the estimation of the ARCH (5) model.The estimated coefficient of the lagged squared residuals ((ARCH) and the coefficient in the conditional variance equation (GARCH) are highly statistically significant.The sum of these two estimates is close to a unit (which is typical for the estimated GARCH models for financial assets returns).This means that the shocks in the conditional variance equation will exhibit a long memory.The high value of the sum of these two coefficients suggests that the high rates of positive or negative returns lead to a large forecast of the variance value for the extended period.The individual coefficients of the conditional variance are in line with the expectations.The coefficient of constant "C" is very small, the ARCH parameter is about 0.03 (estimated coefficient α 1 ), while the coefficient of lagged conditional variance ("GARCH") is very pronounced (0.97) (estimated coefficient β 1 ).The condition of stability of the GARCH model is α 1 + β 1 <1.Parameter α 1 determines how strongly the changes in returns affect the volatility.The parameter β 1 -variance from the previous period -is the parameter that determines the change in volatility over time.With the restriction β 1 =β 2 =...=β s =0, GARCH(m,s) specification comes down to the autoregressive conditional heteroscedasticity ARCH (m) model.The last column of Table 5 shows the p-value.
Source: Author by EViews software package.

Kovačević R.
Modeliranje deviznog kursa evra prema dolaru pomoću ARCH/GARCH modela Bankarstvo, 2016, vol.45, br. 4 the sum of the estimated coefficients α 1 and β 1 should be less than 1.In Table 5 this condition is met.This table features the values of standard errors, z-statistics (ratio between coefficient and the corresponding standard error), and the p-value.The obtained coefficients are statistically significant, except for the value of the constants in the mean equation.The GARCH process is characterized by the stable mean value and conditional heteroskedasticity, while nonconditional variance is constant.To estimate the equation of GARCH specification, the maximum likelihood method was applied.Standard errors and covariance were calculated by means of the robust Bollerslev-Woodridge method.
The statistical properties of the GARCH (1,1) model are acceptable.Let us recall that the Q statistic refers to the standardized residuals and is used to test the existence of residual serial correlation in the mean equation and check the specifications of this equation.If the mean equation is correctly specified, Q statistics will not be significant.The correlogram of the squared standardized residuals is used to test the remaining ARCH effect in the conditional variance equation and check the specifications of the equation (the standardized residuals are the residuals divided by the relevant assessments of standard deviation ( )). Autocorrelation of squared standardized residuals in the estimated GARCH model (1.1) is significantly lower than the autocorrelation structure of the euro exchange rate returns.The calculated p-values with estimated value of the squares standardized residuals are above 0.05 due to which we can accept the hypothesis that the residuals do not have the ARCH structure.
Source: Author by EViews software package.

Kovačević R.
Modeliranje deviznog kursa evra prema dolaru pomoću ARCH/GARCH modela Bankarstvo, 2016, vol.45, br. 4 The estimated coefficients of EGARCH (1,1) model in the mean equation are statistically significant (except the constant) (Table 6).The leverage effect is a phenomenon when there is a correlation between the past returns and future volatility.The coefficient of the leverage effect will be marked with γ.In the case of γ<0, the positive (good) news from the market generate less volatility in the future than the negative shocks.The coefficient of skewness RESID(-1)/@ SQRT ((GARCH(-1)) in the volatility equation in Table 6 is negative and statistically significant.The reliability of the obtained estimates of the EGARCH(1,1) model will be verified using a series of test specifications, which should confirm how much the estimated model agrees with the time series of euro returns.
According to Table 6, it can be seen that the standardized residuals are not correlated (Ljung-Box test statistics -Q).This conclusion is indicated by the high p-value with the estimated Q statistic (at all lags p> 0.05), which means that one cannot reject the null hypothesis which states: There is no autocorrelation up to the order of k.Since the null hypothesis cannot be rejected, we conclude that there is no residual autocorrelation in the residuals of the estimated model.The correlogram of squared standardized residuals (Q2 statistics) also shows that the squared residuals are not autocorrelated, suggesting that the EGARCH model covers the dynamics of conditional variance in the satisfactory manner.Autocorrelation coefficients (AC) and partial autocorrelation (PAC) in the correlogram of standardized squared residuals are near zero at all lags, hence it can be concluded that there is no conditional autoregression heteroskedasticity in residuals at 0.05 level of significance.
The existence of ARCH effects can be checked by the ARCH statistics (Engle's LM test).This statistic is asymptotically distributed as a chisquare distribution if it cannot reject the null hypothesis that there is no heteroscedasticity in the residuals.The ARCH statistic was calculated by multiplying the coefficient of determination for the model, in which the squared data are assessed in relation to the corresponding number of its previous value, by the volume of the sample.The calculated p-values in Table 6 related to the ARCH statistics are greater than 0.05 at all lags, so that the null hypothesis of the absence of ARCH effects in the residuals cannot be rejected.The sum of the estimated parameters α and β is greater than one, indicating the longlasting impact of shocks to volatility.
Since the calculated JB statistics in Table 6 is greater than the critical value, and thereby the p-value to the fourth decimal is zero, the null hypothesis of normally distributed residuals is rejected at 0.05 level of significance (it should be noted that the quality of the model is not called into question due to deviations of the empirical distribution from the normal distribution, because it is not necessary that the random errors are normal distributed).Moreover, the QQ chart, which features a set of pairs of quantiles of the distribution of the series itself and the quantiles of the normal distribution, shows that there is a disagreement between the empirical and normal distribution on the ends (tails) of the series.As we can see, Figure 5 shows both positive and negative shocks, which cause a deviation of the empirical distribution of residuals from the normal distribution.We noted earlier that the tails of the empirical distribution of financial instruments are often heavier than the tails of the normal distribution (the empirical series of standardized residuals is normally distributed when its third moment is equal to zero, and fourth moment is equal to the value of 3).Although the corresponding quantile couples at the ends (higher values of shocks) do not coincide with the quantiles of the normal distribution, the empirical distribution of residuals in a large part of the series can be approximated by the normal distribution.Given the low value of the coefficient of skewness (-0.05), it can be said that the empirical distribution is symmetrical.However, due to the higher value of the kurtosis, which equals three, there is a deviation from normality.To choose a specification that indicates the best statistical properties, AIC, SC and HQC (GARCH) information criteria were calculated for all models that were estimated in this paper.It is believed that the best specification is owned by the model with the lowest values of information criteria.Based on the calculated values of these criteria, in Table 7 it can be observed that they are the lowest in the EGARCH (1,1) model.It can also be noted that the estimated GARCH specifications are less than the value of the information criteria of the ARCH specification.So, this confirms the expectation that the GARCH models are more economical than the ARCH specifications.In addition to the GARCH (1,1) and EGARCH (1,1) models, the author estimated several others GARCH models.These are: GARCH (1,1)-M, Tarch (1,1) APARCH, Taylor-Schwert PARCH and CGARCH model.However, the estimates of these models are not given in the paper, because they were less successful in modeling the unstable conditional variance.Nevertheless, the values of the information criteria of the estimated models are given in Table 7.
Among the estimated GARCH models, the exponential GARCH model shows the minimum value of AIC and HQC information criteria, while the lowest values of the SC criteria were recorded by the Taylor-Schwert PARCH model.Also, the estimated value of the SC criterion for EGARCH (1,1) model is slightly lower than the estimated value of this criterion for Taylor-Schwert PARCH model.Given that the two criteria give priority to EGARCH (1,1) model, and that in the SC criteria EGARCH (1,1) is slightly inferior to the superior specification of the Taylor-Schwert PARCH model, it was concluded that the EGARCH (1,1) model represents an adequate specification.

Conclusion
The participants in the international financial market and trade-engaged international business are exposed to the foreign exchange risk in the contemporary phase of the world market's globalization and liberalization.The more the economy is open, the stronger the influence of the international environment on the exchange rate changes.Given the large number of factors affecting the exchange rate changes, the forecasts of its potential volatility are important for the macroeconomic projections and the structure of economic policy measures.Investors are particularly exposed to the risk of the exchange rate volatility.Therefore, their interest in the estimates of future exchange rate fluctuations is understandable, since they can help them take the appropriate measures to protect themselves against these risks.To quantitatively model the exchange rate fluctuations, the relevant methods that can respond to this task have been developed in the financial literature.The ARCH and GARCH models play an important role among them.The time series of daily returns of the euro exchange rate against the dollar in this paper is modeled by several ARCH and GARCH models.Several dummy variables were included in the modeling of this series, whereby the extreme fluctuations in the series were removed.Several GARCH models were estimated with a different number of parameters.All estimated models had the satisfactory statistical properties.A common feature of the estimated GARCH models is that the obtained coefficients of the lag squared residuals and coefficients of the conditional variance in the equation of conditional variance are statistically significant.The sum of these two estimates was close to a unit, which is typical for the financial assets returns time series in the estimated GARCH models.It was a signal that the innovations in the conditional variance equation will be long-lasting.Based on the obtained value of the sum of these two coefficients, which were about one, it is estimated that the major changes in the positive or negative returns (shocks) are leading to the higher forecasted value of variance in the extended period.In order to choose the best fitting model, the values of the standard information criterion were estimated.According to these assessments, it was clear that the asymmetrical EGARCH (1,1) model showed the best results in the modeling of the eurodollar exchange rate returns.The coefficient of asymmetry in the conditional variance equation of this model is negative and is not statistically significant.A negative value of this ratio suggests that negative shocks will have a bigger impact on the future conditional variance than the positive innovation.The asymmetric EGARCH (1,1) model has also provided evidence of the leverage effect, showing that the negative information (shocks) generate the greater volatility in the coming period than the positive shocks.The results obtained from these models indicate that the impact of shocks on the volatility of the euro exchange rate against the dollar is very strong and long-lasting.In addition, the volatility of the exchange rate shows the asymmetric effects, whereby the impact of negative shocks is stronger than the impact of positive shocks.The obtained results are very important for investors because they suggest that the degree of cautiousness in dealing with the euro should be increased, particularly in the case of macroeconomic shocks.Further research should examine the asymmetric behavior of other foreign exchange rates in the global economy.

Figure 1 .
Figure 1.The daily euro returns against the dollar

Figure 3 .
Figure 3. Autocorrelation of the euro squared returns against the dollar

Figure 4 .
Figure 4. Partial autocorrelation of squared returns of the euro exchange rate against the dollar

Figure 5 .
Figure 5. QQ diagram of the empirical series against that of the normal distribution model obezbeđuje dokaze o leveridž efektu.

Table 1 .
Descriptive statistics of the daily euro returns Source: Author by EViews software package.

Table 4 .
ARCH(5)with dummy variables in the mean equation

Table 5 .
Parameter estimates of the GARCH (1,1) model with dummy variables in the mean equation

Table 6 .
The parameter estimates of the asymmetric EGARCH (1,1) model with dummy variables in the mean equation

Table 7 .
The model selection based on AIC, SC and HQC information criteria Note: Three information criteria: Akaike, Schwarz and H-QC (Hannan-Quinn) are given in the table.In practice, SC is the most frequently used criterion.Source: Author by EViews software package.