ECONOMETRIC MODELLING OF EURO, BRITISH POUND AND JAPANESE YEN EXCHANGE RATES AGAINST DOLLAR - MULTIVARIATE GARCH APPROACH

U ovom radu se istražuje uporedna volatilnost vodećih svetskih valuta (evro, britanska funta i japanski jen) prema američkom dolaru. U istraživanju je primenjen dijagonalni multivarijantni GARCH BEKK model. Pomoću ovog modela ocenjena je uslovna korelacija, kao i uslovna varijansa i kovarijansa u sistemu od ove tri valute. Rezultati empirijske analize pokazuju značajno slaganje u kretanju deviznih kurseva, posebno između evra i britanske funte. To zapravo znači da postoji i međusobno prenošenje volatilnosti deviznih kurseva između valuta sa većim stepenom korelacije. Veća stabilnost ili nestabilnost jedne valute prenosi se i na volatilnost druge valute. Ovi nalazi utiču na valutnu diversifikaciju portfolija i upravljanje rizikom.


Introduction
The euro as the second global currency, after the US dollar, came under the pressure of the public debt crisis of the Eurozone.According to the data from the BIS Triennial Central Bank Survey of Foreign Exchange on Trade, in April 2016 the average daily turnover of the euro accounted for 31.3% of all foreign exchange transactions, whereas the dollar accounted for 87.6% (any foreign currency transaction consisting of the two currencies, so the total sum, according to the BIS report, is 200%).The share of the British pound and the Japanese yen is considerably lower.Although after the introduction of the euro in 1999 it was expected to become a serious rival to the dollar in the role of the leading world's reserve currency, this has not yet become true.With the outbreak of the global economic and financial crisis in 2008, the euro has lost a part of its former share in the international foreign exchange trade.The European sovereign debt crisis has shaken the position of the euro in global foreign exchange trading and amplified the business uncertainty in the global market.In this environment, forecasting exchange rates can be more difficult, which discourages investors.Given the significance of the euro in the international monetary system, it is important to examine its volatility and compliance with the volatility of other currencies.After all, the various models of risk management show that the exchange rate volatility is a leading indicator of the exchange rate risk.It facilitates decision-making with regard to portfolio diversification, determining asset prices and options, and serves the purposes of hedging implementation.The currency crises in the international monetary system show that the contagion of one currency can be quickly transferred to the global financial market.This means that the movement in one currency can significantly affect the value of other currencies, despite the impact of fundamental factors.This paper contributes to the literature that studies the exchange rate volatility comovements in several ways.First, by examining the impact of the 2008 financial crisis on the joint movement of the exchange rates of the key currencies in the world economy: the euro, British pound and Japanese yen against the US dollar.Secondly, the paper applies the multivariate GARCH model, which is more flexible compared to the conventional models.The reference literature recognizes several different multivariate GARCH models.These models include: VECH, diagonal VECH, DCC and BEKK model (for a more detailed discussion, see Kroner and Ng (1998)).In this paper, the diagonal BEKK model (Baba-Engle-Kraft-Kroner representation) (Engle and Kroner, 1995) is applied for a time series of the exchange rate of the euro, the Japanese yen and the British pound against the US dollar, in order to model the conditional variance (conditional heteroskedasticity) of yields.Changes in the exchange rates against the dollar are seen as returns of that currency.Since the variance in the time series of exchange rates basically expresses the level of currency risk, the goal of the model is to determine the rate of volatility and thereby indirectly the level of risk incurred by these currencies.Conditional variance will be modelled using the diagonal BEKK model.The third contribution of this paper relates to the estimation of the model based on the data on exchange rates until the end of 2016, which allows for a more rigorous evaluation and a more complete analysis of the dynamics of the exchange rate volatility among the observed currencies.The remainder of the paper is organized as follows: the second part provides an overview of the reference literature.The third section describes the methodology and data, while the fourth part presents the results of the empirical research.The fifth section presents the final estimates.

Literature Review
Pérez-Rodríguez (2006) applied the Dynamic Conditional Correlation (DCC) -GARCH model to examine the interdependence of daily conditional volatility of the euro, the pound and the yen against the dollar in the period 1999-2004.The paper presents evidence of significant volatility spillovers between these currencies, with the existence of a significant correlation between the euro and the pound.A similar analysis was conducted by Kitamura (2010)  volatility spillover of the euro to the pound and the franc.It was also concluded that the pound and the franc are highly integrated in the euro market.Antonakakis (2012) examined the co-movements and volatility spillovers of the euro, British pound, Swiss franc and Japanese yen against the dollar in the period before and after the introduction of the euro.Based on the results of the dynamic correlation, variance decomposition, generalized VAR and the newly introduced index of spillovers, Antonakakis found that there are significant comovements and volatility spillovers between the yields observed from the four currencies.This author used a new version of the generalized spillover index developed by Diebold and Yilmaz (2012).Kühl (2009) examined the agreement in the movement of foreign exchange rates in the euro/ dollar and the British pound/dollar and came to the conclusion about their greater agreement than suggested by the fundamental factors.From this he concluded that non-fundamental factors play an important role in modeling the exchange rate.Boero, Silvapulle and Tursunalieva (2011) revealed the existence of asymmetry in the dynamics of joint movements in the exchange rates of the euro, pound and yen during the appreciation and depreciation of the dollar in the period 1994-2007.Asymmetric volatility of the exchange rates of the Australian dollar, British pound and Japanese yen against the US dollar was also identified in Wang and Yang (2009).Using the exchange rates of the US dollar, Japanese yen and British pound against the euro in 2002-2006, McMillan and Speight (2010) found that the exchange rate dollar/ euro dominates over the other two exchange rates, both in terms of yields and volatility spillovers.Enlarging the Eurozone increases the role of the euro in the foreign exchange market.Morales and O'Donnell (2007) investigated the spillover volatility of stocks and exchange rates in Italy, Spain and Portugal in the period 1996-1998 (the period before the introduction of the euro) and in the period 1999-2001 and 2002-2006 (the period after the introduction of the euro).They found that in all countries there was no significant spillover volatility of shares on the exchange rates before the introduction of the euro.However, in the period after the introduction of the euro they found a significant spillover volatility of shares on the exchange rates in all countries, for all currencies, except for Portugal in 2002-2006.Tamakoshi and Shigeyuki (2014) applied the multivariate asymmetrical DCC GARCH model to study the interdependence of the exchange rates of the euro, British pound and Swiss franc against the US dollar.The aim of this research was to evaluate the effects of financial disturbances on the dynamic interrelationships between these currencies.The findings indicate the increased interdependence in the case of common movements in the exchange rates observed in the phase of currency appreciation compared with the phase of depreciation.The results also confirm that the crisis has led to the redirection of funds into the Swiss franc, which proved to be a safe currency in the times of crisis.
Caporale, Ali and Spagnolo (2015) examined the influence of the exchange rates uncertainty on the different components of portfolio flows, primarily bond and equity markets, as well as the dynamic link between the exchange rate volatility and the dynamics of these two types of capital movements.They estimated the Bivariate GARCH-BEKK-in-mean model using the data on the bilateral relations between the US on one side, and Australia, Great Britain, Japan, Canada, Sweden and the euro zone, on the other side.The observation period is 1988: 01-2011: 12.The results show that the effect of exchange rate uncertainty on equity flows is negative in the euro area, the UK and Sweden, and positive in Australia, while in the case of bonds it is negative in all countries, except Canada.
BEKK model was first suggested by Baba, Engle, Kraft and Kroner (1989), and the idea was again thoroughly explained by Engle and Kroner (1995)
U razvijenoj formi, model sa dve serije podataka, se može prikazati kao kod (Rosi, 2010): pieces of data for each time series).The data was downloaded from the internet address of the IMF: https://www.imf.org/external/np/fin/ert/GUI/Pages/CountryDataBase.aspx.In the analysis we used three time series of data on the daily spot exchange rate.All three time series of the prompt exchange rate show what the value of one euro, British pound and Japanese yen is expressed in dollars.Thus, the nominal increase in the exchange rate implies the appreciation of the respective currencies against the dollar.The baseline data were logarithmically transformed.The daily return of foreign exchange rate is calculated as rt = log (yt) -log (yt-1) x 100 where yt is the level of a prompt exchange rate at the time t, where t = 1, 2, ..., T. Thus, the daily appreciation or depreciation of the exchange rate of the euro, yen and British pound against the dollar was obtained as the first difference log exchange rate level.The model is estimated by the software package Eviews, using Marquart algorithm optimization and Bollerslev-Wooldrige correction of estimate standard errors.These time series are used in similar empirical studies.For example, Pérez-Rodríguez (2006) used a DCC model for investigating the interdependence of the daily conditional volatility of the euro, pounds and yen against the dollar in the period 1999-2004.while Antonakakis (2012) analyzed the joint movement and mutual spillover of the yield volatility of the euro, the British pound, the Swiss franc and the Japanese yen against the dollar in the period before and after the introduction of the euro.Our research covers the time period in which there has been a significant strengthening of the dollar against other currencies since the second half of 2014, and the sample is broader and includes more recent data.Therefore, the results of our research are partially comparable to previous findings.
To examine the characteristics of the comovements of foreign exchange rates, we employed the diagonal BEKK model.We will briefly describe the BEKK model in the format of the two assets, and then we apply it to the data for three foreign exchange rates.The BEKK model (Engle and Kroner, 1995) solves the VECH model problems and ensures that the matrix H is always a positive definite.Unlike the VECH model, whose deficiency is that it is not certain whether the semi-definite covariance matrix will be positive, the BEKK model guarantees the positive definite structure.
In a developed form, a model with two data series, can be expressed as (Rosi, 2010): where H t is the matrix of conditional variance of dependent variables, A, B, and C are NxN parameter matrices.In this model, the constant matrix is divided into two matrices C and C', where C is the upper triangular matrix.Decomposition of the constant in two triangular matrices is made to ensure positive semi-definiteness of H t .A and B matrices are diagonal.The matrix A includes ARCH effects.Parameters a ij of the matrix A measures the degree of innovation in markets i and j.Matrix B is focused on GARCH effects and parameters b ij represents the conditional volatility between the markets, or persistence of the news.So, the coefficients that quantify the impact of news from the foreign exchange market of one currency on its conditional variance are estimated by the BEEK model.The diagonal matrices A and B were obtained by imposing the restrictions, so that the number of parameters in the model is reduced.When the diagonal BEEK model decomposes into its one-dimensional equations, they can be represented as:  where Θ is a vector which indicates all the unknown parameters should be assessed, N is the number of series in the system (preferably 2 in the above example), and T is the number of observations (Brooks, 2008, pp. 435).H matrix is always a positive definite.We estimated the MGARCH BEKK model of Engle and Kroner (1995).The model parameters were estimated through the maximum likelihood.

Empirical Results and Discussion
The daily rate of euro/dollar exchange rate returns In the original series of euro yields, there are several non-standard observations hence we assessed the time series of reur (daily yield rate of the euro against the dollar), depending on the constants and somewhat artificial variables, which are defined as to take the only non-zero values of the actual rates of return for the following dates: 22/09/2000, /2016.In the new series, which we labeled as resid01 (Fig. 1), it can be seen that non-standard observations are eliminated, so that a revised daily rate of return will be used in further analysis.We did the same with the other two time series.For adjusted series we will conduct the unit root test and present the empirical statistics.The daily rates of return of the exchange rate euro/dollar in the reporting period oscillate around the zero value.Oscillations are not uniform.The highest degree of variability was observed in the second half of 2008, which is due to the increased uncertainty in the financial markets due to the outbreak of the global financial crisis.Increased fluctuations were registered in 2015 and 2016.
In order to check whether the corrected daily return series are stationary, we apply the ADF and KPSS unit root test (Table 1 and 2).If we take into account only the constant, it can be concluded that the series are stationary.The same conclusion is reached if the stationarity test includes the trend and the constant.

Dnevna stopa prinosa britanska funta/dolar
U originalnoj seriji prinosa britanske funte takođe postoji nekoliko nestandardnih opservacija, zbog čega smo ocenili vremensku seriju rgbp (dnevna stopa prinosa britanske funte u odnosu na dolar) u zavisnosti od konstante i nekoliko veštačkih promenljivih, koje smo definisali tako da uzimaju jedine nenulte vrednosti stvarnih stopa prinosa za sledeće datume: The existence of unstable conditional variance is checked by Q 2 and ARCH statistics.High levels of Q 2 statistics suggest the existence of autoregression heteroscedasticity.The p-values are equal to zero, so the hypothesis about the lack of ARCH structure is rejected.Thus, the series RESID01 possesses the unstable conditional variance.The skewness α3 = -0.067,that is less than zero, which means that the distribution is skewed to the left.Kurtosis α4 = 4.110, therefore greater than 3.This means that the tails of the distribution are heavier than the tails of the normal distribution (series are normally distributed if α 3 =0 and α 4 =3).Based on the obtained values of skewness and kurtosis, it can be concluded that the empirical distribution of daily returns RESID01 deviates from the normal distribution.Based on the JB test statistics and Q-Q diagrams, it can also be concluded that the empirical distribution of the rate of return of the euro/dollar exchange rate deviates from the normal distribution (Fig. 2).ARCH test also confirms the existence of an unstable variance, or autoregression heteroscedasticity up to 20 lags.The null hypothesis about the lack of ARCH effect is rejected at 5% level of significance.The obtained values of the test statistics indicate the presence of an unstable variance.
The empirical yield distribution can be approximated by a normal distribution if the corresponding quantile couples are on a straight line.This is not the case here.And according to the JB statistics (212.5),we can conclude that the calculated value is greater than the critical value for the Chi-square, so the null hypothesis about the residuals being normally distributed is rejected.Based on these findings, the dynamics of this series can be estimated by the GARCH model.

The daily rate of the British pound to US dollar exchange rate returns
In the original yield series of the British pound there are also several non-standard observations, which is why we estimated the rgbp time series (daily return rate of the British pound against the dollar), depending on the constants and somewhat artificial variables, which are defined so as to take the only nonzero values of the actual rate of return for the following dates:   We have checked the stationarity of the series by applying the ADF and KPSS unit root test and found that it was stationary (Table 1 and Table 2).Based on the JB test statistics and Q-Q diagrams, we have concluded that the empirical distribution of the exchange rate British pound/dollar returns deviates from the normal distribution (Table 4 and Fig. 4).
The ARCH test also confirms the existence of an unstable variance, or autoregression heteroscedasticity.The dynamics of this series can also be assessed using the GARCH model.

The daily rate of Japanese yen to US dollar exchange rate returns
In the original return series of the Japanese yen there are also several non-standard observations, which is why we estimated the rjpy time series (daily return rate of the Japanese yen against the dollar), depending on the constants and somewhat artificial variables, which are defined so as to take the only nonzero values of the actual rate of return for the following dates:  Radi provere da li korigovana serija dnevnih prinosa poseduje stacionarnu reprezentaciju, primenili smo ADF i KPSS test jediničnog korena (Tabela 1 i Tabela 2).Ako se uzme u obzir samo konstanta, zaključuje se da je serija stacionarna.Do istog zaključka se dolazi ako se u testiranje stacionarnosti uključe trend i konstanta.Na osnovu Žark-Bera (JB) test statistike i Q-Q dijagrama (gde je zapaženo odstupanje parova kvantila od prave dijagonalne linije) zaključili smo da empirijska raspodela stope prinosa deviznog kursa japanski jen/dolar odstupa od normalne raspodele.Primenom ARCH testa utvrđeno je postojanje autoregresione heteroskedastičnosti. Dinamika ove serije takođe se može oceniti primenom GARCH modela.

Rezultati i diskusija
Rezultati primenjenog dijagonalnog BEKK modela daju se u tabeli 6.In order to check whether the corrected daily return series are stationary, we applied the ADF and KPSS unit root test (Table 1 and 2).If we take into account only the constant, it can be concluded that the series is stationary.The same conclusion is reached if the stationarity test includes the trend and the constant.Based on the JB test statistics and Q-Q diagram (where deviation of quantile pairs from the straight diagonal line were observed), we have concluded that the empirical distribution of Japanese yen/dollar exchange rate returns deviates from the normal distribution.Using the ARCH test, we have found the presence of autoregression heteroscedasticity.The dynamics of this series can also be assessed using the GARCH model.
Bearing in mind the previous considerations, we have concluded that all three series are characterized by variability in their returns.Due to the instability of conditional variance, the series can be modelled using GARCH models.Previously, it was pointed out that several variants of multivariate GARCH model are used in the reference literature.We have opted for the diagonal BEEK model respecting its advantages over the VECH model.
In the variance equation there are coefficients for the constant matrix C, ARCH and GARCH.The estimated parameters of the volatility equation are derived using the ML-ARCH (Marquardt) method with normal and t distribution.The convergence was achieved after 11 iterations (normal distribution), or 9 iterations (Student's t distribution).The total number of balanced observations was 13056.Bollerslev-Wooldridge robust standard errors and covariance for the normal distribution have been applied.The coefficients of the matrix A and B indicate innovations in each market and the persistence or gradual decline (weakening) of the impact of news, respectively.A necessary condition for the stability of the variance is that the sum of the coefficients A and B is less than 1.If the sum is close to 1, the process slightly oscillates around the mean value, and shows the effects of long memory in the series.
The estimated coefficients in the above model are statistically significant.The significance of these parameters indicates a strong GARCH (1,1) process, which runs the conditional variance of returns of the three observed currencies.In other words, the conditional variance is under the dominant influence of its own past shocks and volatility (one lag), and partly under the influence of the news that came from the markets of other currencies.
The coefficients a 11 , a 22 , a 33 quantify the impact of news in the euro-dollar, pounddollar and yen-dollar market, respectively, and all coefficients are positive and statistically significant.This means that the news or shocks in one market are important for the future volatility in this market.The a 33 is the largest estimated coefficient, followed by the a 22 coefficient in the second place, and coefficient a 11 in the third place.This means that dealers in Japan must pay greater attention to the news coming from that market compared to the British traders paying attention to the news arriving from the pound market.
The coefficients of B matrix indicate the persistence of news.High values of the coefficient b 11 , b 22 and b 33 indicate that the incoming news in every market last at least one day.Since this is a diagonal matrix, it is not possible to assess the spillover effects of shocks and volatility between the two markets, and their impact on the conditional variance.Coefficient b 11 has the highest value.This means that the shocks on the euro market last longer than in the pound and yen market in terms of future market volatility.The minimum persistence is on the yen market b 33 .This means that the yen has become a very stable currency after the introduction of the euro.The British pound has also become more stable.This can be used as an argument explaining why the United Kingdom has not adopted the euro.
Most empirical studies based on the daily data reject the assumption of the normal distribution of shocks, so it is useful to estimate the coefficients by means of the multivariate Student's t distribution.This adds another parameter to the assessment model (i.e.degrees of freedom parameter).When this parameter strives to infinity, Student's distribution strives towards the normal distribution.The estimated degrees of freedom were at 8.812 with a standard error of 0.60.
The significant highly estimated coefficients b nn indicate a high persistence of news on all three foreign exchange markets.Coefficient b 11 has the highest value.This means that the shocks from the euro market are more persistent in terms of future volatility than in the other two markets.Therefore, the traders in the Eurozone must pay more attention to the news coming from that market in relation to the traders in other markets.It was found that the lowest persistency is in the yen market, which indicates the stability of that currency.
The conditional correlation of all three time series in this paper is estimated using the diagonal BEKK model (Fig. 7).
The estimated conditional correlation between the euro and the British pound (Fig. 7) shows an unstable structure, although it appears the oscillations centralized around the stabilized level after 2008.The correlation ranges from 0.2 to 0.8.The correlation between the euro and the Japanese yen after 2008 oscillates around a slightly growing trend, and it ranges from -0.3 to 0.6.Looking at the correlation between the British pound and the Japanese yen, we can see that the oscillations stabilized around the growing trend after 2008.The Residual Correlation Matrix is shown in Table 7.
Table 7 shows that the market expectations in terms of the exchange rates return movements have the same direction and that they are positively correlated among the three observed currencies.The value of the correlation coefficients varies between 0.12 for the relationship between the British pound and the Japanese yen to 0.61 for the correlation between the euro and the British pound.The high correlation between the European currencies is associated with the faster transmission of currency contagion within the regions than between the currency regions (for a more detailed analysis see Glick and Rose, 1998).The evidence of this phenomenon is the experience of the Asia's 1997 currency crisis.The correlation matrix of residuals confirms the existence of cointegration in the movement of foreign exchange rates of the euro, British pound and Japanese yen against the dollar.The applied multivariate BEKK model gives a possibility of analyzing the effects of conditional covariance between the observed time series, i.e. the ability to perceive the volatility dynamics of the analyzed foreign exchange rates.
The coefficients of the System Residual Portmanteau Autocorrelation Tests (multivariate normal distribution) are statistically significant.Since all p values of Q statistic equal 0, the Null Hypotheses is dismissed at the level of significance of 5%.It is not realistic to expect the model on the whole to eliminate the serial correlation of standardized residuals because the high daily returns are leptokurtic.
The estimated coefficients of asymmetry (skewness-α 3 ) individually for the euro and the British pound are less than zero, which means that the distribution is skewed to the left.The estimates of kurtosis α 4 for all three currencies individually are higher than 3.This means that the tails of the distribution are heavier than the tails of the normal distribution.Thus, the empirical distribution deviates from the normal distribution.Based on the statistics of JB, the system residual normality tests (normal distribution) rejects the null hypothesis.The absence of normal distribution is still a difficulty, although we used the Bollerslev-Wooldridge robust standard errors and covariance method because the estimator is consistent but not asymptotically efficient.The same conclusions apply to t-distribution.

Conclusion
In this paper we examined the exchange rate volatility comovements of the euro, the pound and the yen against the dollar in the period 03.01.2000 to 30.12.2016 (a total of 4353 pieces of data for each time series).This paper contributes to the literature that studies the volatility comovements of exchange rates in several ways.First, we examined the impact of the financial crisis of 2008 on the joint movement of the exchange rates of key currencies in the world economy: the euro, British pound and Japanese yen versus the US dollar.Secondly, the methodological framework we used included the multivariate diagonal BEKK GARCH model as a more flexible variant of GARCH models.Using this model, the conditional variance of the exchange rate returns of the euro, the Japanese yen and the British pound versus the US dollar is modelled in this paper.The changes in exchange rates versus the dollar in the time series are seen as returns of that currency.The paper estimated the dynamics of volatility in exchange rates, and we found that the conditional volatility in all three series was reinforced during the global financial crisis of 2008.According to the estimated conditional correlations, the highest degree of correlation between the euro and the British pound was observed in the paper, which is in line with the fact that the currency contagion is more of an intra-regional than an inter-regional phenomenon.The experience of the 1997 Asian currency crisis can be seen as the evidence of this phenomenon.The correlation matrix of residuals confirms the existence of cointegration in the movement of foreign exchange rates of the euro, British pound and Japanese yen versus the dollar.The conclusion can be drawn that the applied multivariate BEKK model opens up the possibility of analyzing the effects of conditional covariance between the observed time series, i.e. the ability to perceive the volatility dynamics of the analyzed foreign exchange rates.

Figure 3 .
Figure 3.The daily rate of the British pound/dollar exchange rate returns (non-standard observations were removed)

Figure 5 .
Figure 5.The daily rate of the Japanese yen/dollar exchange rate returns (non-standard observations were removed)

Podaci i metodologija istraživanja
which used the data for the period from 2 April to 31 August 2006 and established the existence of significant Bankarstvo, 2017, Vol.46, Issue 4

Table 6 .
Parameterization of the diagonal BEKK (1,1) model Note: Prob.-p-values.Estimation Method: ARCH Maximum Likelihood (Marquardt).The diagonal elements (a i ) include the ARCH effects, while the diagonal elements (b i ) show GARCH effects.()denote p-values.Q() is the Ljung-Box portmanteau test statistics for serial correlation in the univariate residuals.C is an indefinite matrix; A1 is a diagonal matrix; B1 is a diagonal matrix.The Q statistics up to 30 lags is a test for the null hypothesis that there is no autocorrelation up to 30 lags.Source: Author's calculation by Eviews