A GARCH MODEL FOR EFFECTIVE EXCHANGE RATES

Name: Serena Darino Enrollment number: 064597

Contents

Contents .......................................................................................................................................... 2 Introduction .................................................................................................................................... 3 1.

Graphs and summary statistics ................................................................................................. 4

2.

Data analysis ............................................................................................................................ 8 2.1.

Correlogram...................................................................................................................... 8

2.2.

Testing for unit roots ......................................................................................................... 9

2.3.

Cointegration .................................................................................................................. 13

2.4.

Testing for non-linearity.................................................................................................. 14

2.5.

Model selection............................................................................................................... 15

3.

ARCH-GARCH model .......................................................................................................... 17

4.

Diagnostic checking ............................................................................................................... 21

5.

4.1.

Representation ................................................................................................................ 21

4.2.

Residual graphs ............................................................................................................... 22

4.3.

Correlograms .................................................................................................................. 23

4.4.

Remaining ARCH effects................................................................................................ 24

4.5.

Testing for normality ...................................................................................................... 25

Forecasting ............................................................................................................................ 27

Conclusions .................................................................................................................................. 33

Introduction Volatility in exchange rates is, during the last years, at the centre of the theoretical and empirical economic and financial analysis. The first reason is represented by the phenomenon’s growing importance, thanks to the increased international financial markets’ integration. The second reason is that volatility in exchange rates has a relevant influence on the assets’ riskiness. Models based upon the past observed variance are unable to capture the accentuated variability that characterizes these kind of data. Considering also that, in the agents’ decisional processes, it is not the unconditional variance that enters, but the conditional variance, which is a measure of the time uncertainty of the evaluated variable. Furthermore, volatility is not constant but used to change over time1. Empirical analysis show that exchange rates series are characterized by heteroskedasticity, nonlinearity

and

non-normality.

The

GARCH

(generalized

autoregressive

conditionally

heteroskedastic) family models let us to deal with these data features since it does not assume that the variance is constant over time and which describes how the variance of the errors evolves. In this way it is possible to study volatility in exchange rates. Data used in this report come from the statistical data warehouse of the European Central Bank. They are nominal effective exchange rates (that is a summery measure of the external relative strength of a currency against the Euro), monthly observations from January 1999 to November 2015 (total amount of 203 observations for each series), weighting aggregation method, not seasonally adjusted. The three strongest and more traded currencies are examined here: the US Dollar, the UK Pound Sterling and the Japanese Yen. The following analysis is developed complying with the Chris Brooks’ Introductory Econometric for Finance and using the EViews software to compute estimations. You will find, through the test, in italic in brackets, the names of the corresponding outputs in the associated EViews workfile. I first make a graphical and statistical analysis of the data, testing for the main basic assumption; then I proceed to the model estimation. After this, I run some diagnostic tests to check out the model’s accuracy and then I perform a brief forecasting exercise.

1

E. Rossi, 1995.

1. Graphs and summary statistics2 The starting point is represented by the effective exchange rates series of the three strongest currencies against Euro: Japanese Yen, US Dollar and UK Pound Sterling3. Observing the line graph and the summary statistics of each series we can see that the relative strength of UK Pound Sterling (uk_pound_ster_linegraph and uk_pound_ster_stat) rises starting from 2007. It reaches the maximum values during 2009 (the absolute maximum value of 0.91966 on March 2009) and then it decreases slightly till today, when it approximately settles around the mean value. The standard deviation, in the amount of 0.093908, is low enough, so the observations are not characterized by great jumps in average. UK pound sterling .95 .90 .85 .80 .75 .70 .65 .60 .55 2000

2002

2004

2006

2008

2010

2012

2014

20

Series: UK_POUND_STERLING Sample 1999M01 2015M11 Observations 203

16

12

8

Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis

0.738084 0.700450 0.919660 0.589330 0.093908 0.297874 1.694123

Jarque-Bera Probability

17.42611 0.000164

4

0 0.60

2 3

0.65

0.70

0.75

Page â€œdata_analysisâ€? in the EViews workfile. From forexitalia24.com

0.80

0.85

0.90

Regarding US Dollar (us_dollar_linegraph and us_dollar_stat), from the absolute minimum of 0.8532 on June 2001, it grows up to 1.577 on July 2008. After this, the series shows several drifts to end with a considerable decrease during 2014 and the last months of 2015. The standard deviation is not so high, but a little bigger than the UK Pound Sterlingâ€™s one. US dollar 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 2000

2002

2004

2006

2008

2010

2012

2014

20

Series: US_DOLLAR Sample 1999M01 2015M11 Observations 203

16

12

8

4

Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis

1.218957 1.267300 1.577000 0.853200 0.179045 -0.448346 2.341505

Jarque-Bera Probability

10.46864 0.005330

0 0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

Finally, about Japanese Yen (jap_yen_linegraph and jap_yen_stat) we can see that the distribution is characterized by a significant higher standard deviation (in the amount of 17.795); the maximum relative strength of the currency is reached on July 2008 with a value of 168.45 while the minimum is touched on October 2000 with a value of 92.74, but we must consider that a lot of local minimum and maximum points are present throughout all the distribution.

Japanese yen 170 160 150 140 130 120 110 100 90 2000

2002

2004

2006

2008

2010

2012

2014

24

Series: JAPANESE_YEN Sample 1999M01 2015M11 Observations 203

20

16

12

8

4

Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis

127.8773 130.3400 168.4500 92.74000 17.79511 0.176312 2.460292

Jarque-Bera Probability

3.515527 0.172430

0 100

110

120

130

140

150

160

170

Examining the skewness and kurtosis values we can see that UK Pound Sterling and Japanese Yenâ€™s distributions have a positive skewness, so they have the tail on the right side longer than the left one, even if they are not so greater than zero so they are not so different from the normality; US Dollar instead has a negative skewness value, hence its left side tail is longer than the right side. All the kurtosis values are less than 3 (the normal distributionâ€™s kurtosis): the distributions are platykurtic, with a lower, wider peak around the mean and thinner tails. However, the Japanese Yen has the smallest differences from the normal distribution. In fact, the associated probability of the Jarque-Bera test for Japanese Yen leads not to reject the null hypothesis of a normal distribution; on the other hand, for both US Dollar and UK Pound Sterling we reject the null even at 1% level.

Now, we can observe the line graph with the normalized values of the effective exchange rates of the three series (yen_dollar_ster_normgr) to check out any similar path. 2

1

0

-1

-2 2000

2002

2004

2006

2008

2010

2012

2014

Japanese yen US dollar UK pound sterling

It shows that, for almost the first part of the sample, US Dollar and Japanese Yen follow a similar trend. During 2008 and 2009 they start to differ from one another and in the second part of the sample we observe that US Dollar and UK Pound Sterlingâ€™s drifts are more similar, while Japanese Yen gradually detaches. All the three series are characterized by evident volatility, more highlighted during 2007-2008 (the years of the financial crisis) and during 2012-2013 for Japanese Yen only.

2. Data analysis4 I can now proceed to analyze some specific characteristics of the series in order to come to the most appropriated model to describe the data. 2.1.

Correlogram

The observation of the seriesâ€™ correlogram let us to consider the autocorrelation function (ACF) and the partial autocorrelation function (PACF). I choose to plot the correlogram of the raw series (level) and I specify the highest order of lag in the amount of 12 (since I have monthly observations). All the three correlograms (jap_yen_correlogram, uk_pound_ster_correlogram and us_dollar_correlogram) show a geometrically decaying acf and a pacf that cuts off close to zero at lag 3. In fact, in all the three cases, only the first two pacf coefficients are significant. The QStatistics and their p-values lead to the rejection of the null hypothesis that there is no autocorrelation up to order 12.

4

Page â€œdata_analysisâ€? in the EViews workfile.

This should suggests to estimate an AR(2) seeing as we have a decaying acf and two non zero pacf coefficients. 2.2.

Testing for unit roots

But the autocorrelation functions suggest also the presence of non-stationary and this is confirmed by the evidence that all the acf coefficients are positive and decrease slightly while in the partial autocorrelation functions the first coefficients have an high value but the other ones are not so different from zero. To verify non-stationary I can test for unit roots. There are a large number of options in EViews concerning the test and we can also choose among various methods and criteria

for determining the optimum lag length in the test. I opt for conducting (on the raw data series, with a constant but no trend in the test equation): the Augmented Dickey-Fuller (ADF) test, with up to 12 lags of the dependent variable, leaving the default Schwarz Info Criterion to select the optimum length; the Phillips-Perron (PP) test and the Kwaitkowski-Phillips-Schmidt-Shin (KPSS) test, with the default spectral estimation method of Bartlett kernel and the Newey-West bandwidth. The results are as follows:

In the ADF test we can see that the SIC selects 1 lag as optimum length for all the three series; the tStatistic values let us not to reject the null hypothesis of a unit root in all the cases since they are all not more negative than the test critical values at 1%, 5% and 10% level. In the PP test we have that the adopted bandwidth is different for each series: 2 for UK Pound Sterling, 5 for US Dollar and 3 for Japanese Yen. However, in every case, the adjusted t-Statistic values are not more negative than all the test critical values, so we do not reject the null hypothesis of a unit root again. Finally, the KPSS test (with a bandwidth of 11 for all the series) leads to reject the null of stationary for UK Pound Sterling and US Dollar only. The Japanese Yen LM-Statistic value is less than the asymptotic critical values at 1%, 5% and 10% level, so in this case we cannot reject the null hypothesis of stationary. Considering the tests for unit rootâ€™s results, it could be necessary to differentiate the three series in order to obtain the stationary. I repeat the ADF test, the PP test and the KPSS test with the same characteristics described above, but this time I conduct them on the first differences of the data.

In this case we reject the null hypothesis of a unit root for all the three series in both the ADF test and the PP test. The KPSS test leads to the non-rejection of the null hypothesis of stationary regarding UK Pound Sterling and US Dollar, but it leads to the rejection of the null in the Japanese Yen case. So, according to the ADF and PP tests, the series are all non-stationary. The KPSS test agrees with them except concerning the Japanese Yen. Non-stationary of the data means that the persistence of shocks will always be infinite, standard regression techniques cannot be applied (the result could be a â€œspurious modelâ€?), the standard assumptions for asymptotic analysis are not valid and it is not possible to validly undertake hypothesis tests about the regression parameters. 2.3.

Cointegration

The data, therefore, contain a unit root and are thus I(1) (that is the series must be differenced once before they become stationary, then they are said to be integrated of order 1). Now, if a linear combination of these variables is stationary, the set of them is defined cointegrated. In this case I should test for cointegration because pure first difference models have no long-run solution. Made sure that the variables are all non-stationary in their levels form, I can proceed in the cointegration test within the Johansen VAR framework. I choose to examine the sensitivity of the result to the type of the specification used among the possible options by selecting to summarize all the five sets of allowed assumptions. The upper part of the results (johansen_cointegration) is the following:

Date: 01/03/16 Time: 16:19 Sample: 1999M01 2015M11 Included observations: 190 Series: JAPANESE_YEN UK_POUND_STERLING US_DOLLAR Lags interval: 1 to 12 Selected (0.05 level*) Number of Cointegrating Relations by Model Data Trend: Test Type Trace Max-Eig

None None Linear Linear Quadratic No Intercept Intercept Intercept Intercept Intercept No Trend No TrendNo Trend Trend Trend 0 0 0 0 0 0 0 0 0 0

*Critical values based on MacKinnon-Haug-Michelis (1999)

The results across the five types of model and the types of test (Trace and Max-Eig tests) agree upon the non-cointegration between the vectors. 2.4.

Testing for non-linearity

Financial theory suggests that there could be a non-linear relationship between the variables concerning exchange rates series. In particular, exchange rates are characterized by volatility clustering: the tendency for volatility to appear in bunches, that is large changes are expected to follow large changes and small changes to follow small changes. There can be also the probability of leverage effects: the tendency for volatility to rise more following a large fall then following a rise of the same strength. Linear structural models are unable to explain all these features. First of all I should testing for non-linearity on statistical grounds. I can start running a Brock-DechertScheinkman (BDS) independence test on the raw data to see if there is anything there. I opt to specify the distance as a fraction of pairs (the default option since this method is the most invariant to different distributions of the underlying series); the value used in calculating the distance in the amount of 0.7, that is a good starting point for the default method; the maximum correlation dimension is set equal to 12 and I choose not to use bootstrapping. The test has as its null hypothesis that the data are pure noise. The p-values associated to the BDS-Statistics for all the three series lead to the rejection of the null hypothesis.

The rejection of the null hypothesis implies that there is remaining structure in the series which could include hidden non-linearity, hidden non-stationary or other types of structure missed. To examine in depth the presence of non-linearity in the data I can test for ARCH effects. The test is calculated by regressing the squared residuals on a constant and 12 lags (the number of lags to include is chosen arbitrarily). I first estimate a linear model: I opt for an AR(2) considering acf and pacf structure and series identification (jap_yen_ar2, uk_pound_ster_ar2, us_dollar_ar2). Then I run a Heteroskedasticity test on the residuals, selecting the ARCH test type. Regarding the outputs, both the F-version and the LM-Statistic are very significant for UK Pound Sterling and US Dollar. This suggests the presence of ARCH effects in Pound and Dollar data. Observing the test results for Japanese Yen, instead, we cannot reject the null hypothesis of homoskedasticity.

2.5.

Model selection

Until here we observe, first of all, that the series are not normally distributed and they are characterized by a robust volatility in their trend. From the autocorrelation and partial autocorrelation functions we can exclude ARMA and ARIMA models because the three exchange

rates series have not a geometrically decaying pacf. At this point the AR(2) could seem the most appropriate model to fit the data (since there is a geometrically decaying acf and a number of non zero points of pacf equal to 2, that determine the AR order). Financial theory should not suggest that there could be a two-way relationship between more exchange rates series, so we do not consider simultaneous equations systems like 2SLS and IV methods. Going on with the analysis, we observe that the series are not-stationary but not cointegrated so it does not make sense using a VAR or a VECM model. Furthermore, the presence of non-linearity lead us to test the presence of ARCH effects with the rejection of the null hypothesis of homoskedasticity. All these results make us to estimate a non-linear model of the GARCH family.

3. ARCH-GARCH model5 The non-linear ARCH (autoregressive conditionally heteroskedastic) model let us to deal with heteroskedasticity. It is unlikely that in financial time series the variance of the error will be constant over time, and hence it makes sense to consider a model that does not assume that the variance is constant and which describes how the variance of the errors evolves. Furthermore, it can capture volatility clustering for which the current level of volatility tends to be correlated with its level during the immediately previous period. Obviously, this model is used only for UK Pound Sterling and US Dollar data, since it is not appropriate in order to describe Japanese Yen series. A more widely employed model is the GARCH (generalized autoregressive conditionally heteroskedastic) model, because it is more parsimonious and avoids overfitting, considering also that a GARCH(1,1) model can be written as a restricted infinite order ARCH model. To estimate the model it is necessary to specify both the mean and the variance equations. I enter the dependent variable and the constant term C for the mean equation and I leave the default option for the variance equation. I need also to specify the number of ARCH and GARCH terms. For UK Pound Sterling both are set equal to one, that is one lag for the squared errors and one lag for the conditional variance (a GARCH(1,1) model). For US Dollar the ARCH term is set equal to one while the GARCH term equal to zero (a GARCH(0,1) model). These choices are adopted looking at the minimization of the Information Criteriaâ€™s values and the maximization of the log-likelihood. Among the available options in EViews, I check the Heteroskedasticity Consistent Covariance only for UK Pound Sterling, since I suspect that the residuals could not be conditionally normally distributed. This option compute the quasi-maximum likelihood, covariances and standard errors using the method described by Bollerslev and Wooldridge and only the estimated covariance matrix will be altered (the parameters estimates will be unchanged). For both, the selective iterative method to achieve the convergence is the Marquardt algorithm. These are the results (uk_pound_sterl_garch11 and us_dollar_garch01):

Dependent Variable: UK_POUND_STERLING Method: ML - ARCH (Marquardt) - Normal distribution Date: 01/04/16 Time: 14:27 Sample: 1999M01 2015M11 Included observations: 203 Convergence achieved after 48 iterations Bollerslev-Wooldridge robust standard errors & covariance Presample variance: backcast (parameter = 0.7) GARCH = C(2) + C(3)*RESID(-1)^2 + C(4)*GARCH(-1)

5

Page â€œgarch_modelâ€? in the EViews workfile.

Variable

Coefficient

Std. Error

z-Statistic

Prob.

C

0.680545

0.000665

1024.066

0.0000

3.095987 11.21413 -1.682333

0.0020 0.0000 0.0925

Variance Equation C RESID(-1)^2 GARCH(-1) R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

5.45E-05 1.153345 -0.127046 -0.377277 -0.377277 0.110208 2.453440 314.5970 0.014868

1.76E-05 0.102847 0.075518

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter.

0.738084 0.093908 -3.060069 -2.994784 -3.033657

Dependent Variable: US_DOLLAR Method: ML - ARCH (Marquardt) - Normal distribution Date: 01/04/16 Time: 15:06 Sample: 1999M01 2015M11 Included observations: 203 Convergence achieved after 53 iterations Presample variance: backcast (parameter = 0.7) GARCH = C(2) + C(3)*RESID(-1)^2 Variable

Coefficient

Std. Error

z-Statistic

Prob.

C

1.308746

0.004720

277.3046

0.0000

4.055111 3.064007

0.0001 0.0022

Variance Equation C RESID(-1)^2 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

0.000670 0.972242 -0.252734 -0.252734 0.200397 8.112141 171.3859 0.022530

0.000165 0.317311

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter.

1.218957 0.179045 -1.658974 -1.610011 -1.639166

The outputs are divided into two sections: the upper part provides the standard output for the mean equation, the lower part contains the coefficients, standard errors, z-Statistics and p-values for the coefficients of the variance equation. Starting from the UK Pound Sterling, in the conditional variance equation, the coefficient on the lag squared residual is highly significant but the one on the lagged conditional variance is not. As is typical of GARCH model estimates, the sum of the coefficients on the lagged squared error and lagged conditional variance is very close to unity (approximately 1.02), indicating that volatility shocks are quite persistent. The variance intercept term is very small, the ARCH parameter is grater then 1 and the GARCH parameter has a negative sign. The bottom panel of the output presents the standard set of regression statistics using the

residuals from the mean equation. In this case measures as RÂ˛ could be meaningless because there are no regressors in the mean equation (here RÂ˛ is negative). Finally, in this case the log-likelihood could be not appropriate because we have used the heteroskedasticity consistent covariance option in the GARCH estimation, which calculate the quasi-maximum likelihood (QML). What is about the GARCH(0,1) model on US Dollar series, we have a statistically significant ARCH term (which is close to unity even if its standard error is not too low) and a small variance intercept. Here, the sign of the parameters are as expected. In this case the convergence is achieved after more iterations than the previous GARCH model (53 iterations against 48 for UK Pound Sterling model). Now, it could be useful to analyze the GJR model for possible asymmetries, the EGARCH model to avoid non-negativity constraints of the conditional variance and the GARCH-M model to take account of riskâ€™s influence and then to compare all these variations with the models estimated above. To estimate the GJR model for US dollar we need to change the threshold order number from 0 to 1 in the main menu screen for GARCH estimation. To estimate an EGARCH model, indeed, we need to select the EGARCH model estimation method. For US Dollar (us_dollar_gjr and us_dollar_egarch), the asymmetry terms in both models (the GJR(0,1) and the EGARCH(0,1)) are not statistically significant and furthermore the information criteria are bigger than the previous GARCH model ones. The GARCH-M(0,1) model (us_dollar_garchm), with the conditional logarithm of the variance term in the mean, shows that the estimated parameter on the mean equation has a negative sign and it is highly statistically significant. Information criteria too are smaller than above (more negative) and the log-likelihood is bigger. Thus we can conclude that, for this effective exchange rate series, there is feedback from the logarithm of the conditional variance to the conditional mean. The ARCH term in the variance equation has a positive sign, it is close to unity and it is still highly significant.

Talking about the UK Pound Sterling series, I remake all the passages described above, estimating a GJR(1,1) model, an EGARCH(1,1) model and a GARCH-M(1,1) model with the conditional logarithm of the variance term in the mean (uk_pound_sterl_gjr, uk_pound_sterl_egarch and uk_pound_sterl_garchm). We can observe almost the same tendency seen for US Dollarâ€™s models: in the GJR and EGARCH models, the asymmetry terms are not significant and both the models are less appropriate according to information criteria. On the other hand we have a log(GARCH) term in the mean equation that is very significant even if it is quite small. Here it appears with a positive sign as well as both the ARCH and GARCH terms in the variance equation. The last two coefficients are significant again. This model appears to capture dataâ€™s features better than the first one.

We can conclude that there is some feedback from the conditional variance to the conditional mean, specially for US Dollar model, where increased risk (given by an increase in the conditional variance) leads to a decrease in the mean, that is a reduction of the relative strength of the currency against Euro. In the UK Pound Sterling model, the conditional logarithm of the variance has a too small value to consider a relevant influence of the conditional variance on the conditional mean.

4. Diagnostic checking6 Once our models have been estimated, I proceed to perform some diagnostic tests in order to verify the plausibility of the hypothesis that make up the models’ basis. 4.1.

Representation

The followings are the representations for UK Pound Sterling’s GARCH-M(1,1) and US Dollar’s GARCH-M(0,1) respectively, available in the views: Estimation Command: ========================= ARCH(ARCHM=LOG,H,DERIV=AA) UK_POUND_STERLING C Estimation Equation: ========================= UK_POUND_STERLING = C(1)*LOG(GARCH) + C(2) GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*GARCH(-1) Substituted Coefficients: ========================= UK_POUND_STERLING = 0.00992398696085*LOG(GARCH) + 0.768281354478 GARCH = 3.4215421254e-05 + 1.01066713411*RESID(-1)^2 + 0.144122942586*GARCH(-1)

Estimation Command: ========================= ARCH(1,0,ARCHM=LOG,DERIV=AA) US_DOLLAR C Estimation Equation: ========================= US_DOLLAR = C(1)*LOG(GARCH) + C(2) GARCH = C(3) + C(4)*RESID(-1)^2 Substituted Coefficients: ========================= US_DOLLAR = -0.0143569236623*LOG(GARCH) + 1.21697942983 GARCH = 0.000694250034735 + 0.964028322033*RESID(-1)^2

The two representations display the estimation command as well as the estimation equation and the substituted coefficients equation for the mean and variance specification. Summarizing, for UK Pound Sterling series we have a GARCH-M(1,1) model in which the logarithm of the conditional variance enters in the mean equation multiplied for a low coefficient (added to the constant). In the variance equation we have a small constant, the ARCH term, approximately equal to one, that multiplies the square of the residuals of the previous period, and the GARCH term, still positive, for

6

Page “garch_model” in the EViews workfile.

the one lag of the conditional variance. The ARCH term measures the volatility observed in the previous period whereas the GARCH term measures the forecasted variance from the last period. For US Dollar series we have a GARCH-M(0,1) model with the constant and the logarithm of the conditional variance in the mean equation too. In the variance equation, here, we have the constant and the ARCH term only. So this model depends entirely on the influence of the volatility of the previous period that has a high, positive coefficient. 4.2.

Residual graphs

Here the graphs of residual, actual and fitted errors of both models (uk_poundsterl_residgraph and us_dollar_residgraph): 1 UK Pound Sterling 1.0 0.9 0.8 .2

0.7 0.6

.1

0.5

.0

-.1

-.2 2000

2002

2004

2006

Residual

2008 Actual

2010

2012

2014

Fitted

2 US Dollar 1.6

1.4 .4 1.2 .2 1.0 .0 0.8 -.2

-.4 2000

2002

2004

2006

Residual

2008 Actual

2010 Fitted

2012

2014

As we can observe, the errors fitted by the models are much less volatile than the actual errors. The difference between the actual and fitted errors (the blue line) stands around zero throughout the period 2003-2007 for the UK Pound Sterling and in some points only for the US Dollar. So it shows that they are far away from the real trends. 4.3.

Correlograms

Among the residual diagnostic tests we have the Correlogram-Q-Statistic that shows the correlogram (autocorrelations and partial autocorrelations) of the standardized residuals. It is used to test remaining serial correlation in the mean equation and to check the specification of the mean equation (uk_poundsterl_correlogram and us_dollar_correlogram). If the mean equation is well specified, all Q-Statistics should not be significant.

Unfortunately, in both case all the coefficients are high significant till the 12th lag. This means that something is missed in the specification of the mean equation. Through the correlogram of the squared standardized residuals I can test for remaining ARCH effects in the variance equation and to check its specification. As above, if the variance equation is correctly specified, all the Q-Statistics should not be significant (uk_poundsterl_resid2corr and us_dollar_resid2corr).

For the UK Pound Sterling model all the coefficient are not significant so we do not reject the null hypothesis of no autocorrelation. This let us to affirm that, in this case, the variance equation is well specified. About the US Dollar model, only the coefficient on the third lag is not significant. All the others p-values are less than 0.05 but not less than 0.01 so they are significant but not high statistically significant. Looking back to the estimation outputs in fact, we observe that the DurbinWatson values are very close to zero (approximately 0.02 and 0.03 for UK Pound Sterling and US Dollar respectively), that is there is positive autocorrelation in the residuals and so something is missed in the specification of the variance equation. 4.4.

Remaining ARCH effects

However, according to the ARCH LM residual diagnostic test (run with 12 lags), in both models we do not reject the null hypothesis of homoskedasticity in the variance equations, as we can see below (uk_poundsterl_archlm and us_dollar_archlm):

The p-values associated to both the F-version and the Ď‡Â˛-version are greater than 0.05 and so the statistic values are not significant. After this we can conclude that there are no ARCH effects left in the standardized residuals and the variance equation could be correctly specified. 4.5.

Testing for normality

Moving to the histogram of the normality test over the standardized residuals, we see a nonnormality behavior in the data (uk_poundsterl_stat and us_dollar_stat): the kurtosis value is less than 3 and the skewness is positive for US Dollar and negative for UK Pound Sterling. The JarqueBera test, effectively, leads to the rejection of the null hypothesis of normality. 3 UK Pound Sterling 36

Series: Standardized Residuals Sample 1999M01 2015M11 Observations 203

32 28

Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis

24 20 16 12 8

Jarque-Bera Probability

4

0.350594 0.796137 1.873206 -1.283871 0.856994 -0.534607 1.850069 20.85454 0.000030

0 -1.0

-0.5

0.0

0.5

1.0

1.5

4 US Dollar 50

Series: Standardized Residuals Sample 1999M01 2015M11 Observations 203

40

Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis

30

20

-0.280671 -0.757638 2.585972 -1.889315 0.962926 0.652646 2.136828

10

Jarque-Bera Probability

20.71320 0.000032

0 -2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

We must recall that the conditional normality assumption for the standardized residuals is essential for the maximization of the likelihood function. The upshot is that these GARCH-M models are

able to capture some although not all the features in the unconditional distributions of the effective exchange rates. However, even if the conditional normality assumption does not hold, the parameter estimates are still consistent if the equations for mean and variance are correctly specified, but usual standard error estimates are inappropriate.

5. Forecasting7 Last but not least, I proceed to estimate forecasting. Forecasting is an attempt to determine the values that a series is likely to take. Determining the forecasting accuracy of a model is an important test of its adequacy. Consider that, sometimes, even if a model violates the basic assumptions or contains insignificant parameters, it is affirmed to be irrelevant if the model produces accurate forecast. The dynamic forecast plot (to calculate multi-step forecasts starting from the first period in the forecast sample), produced for US Dollar over the whole sample (us_dollar_df), is given below: 1.8

Forecast: US_DOLLAR_DFORECAST Actual: US_DOLLAR Forecast sample: 1999M01 2015M11 Included observations: 203 Root Mean Squared Error 0.185616 Mean Absolute Error 0.142740 Mean Abs. Percent Error 13.19223 Theil Inequality Coefficient 0.074106 Bias Proportion 0.084012 Variance Proportion 0.911304 Covariance Proportion 0.004684

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The dynamic forecast shows a completely flat forecast structure for the mean because of the formulation of the conditional mean equation. The forecast of variance provides the basis of the standard error bands that are given by the dotted red lines around the conditional mean forecast. Since the conditional variance forecasts decrease rapidly as the forecast horizon increases, the standard error bands shrink slightly. In the box on the right we have the forecast evaluation statistics for the conditional mean. Here we find some useful information like the decomposition of the

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Page “forecasting” in the EViews workfile.

forecast error: the mean squared forecast error can be decomposed into a bias proportion, a variance proportion and a covariance proportion. The bias proportion measures the extent to which the mean of the forecasts is different from the mean of the actual data (whether the forecasting is biased). The variance component measures the difference between the variation of the forecasts and the variation of the actual data. The covariance component captures any remaining unsystematic part of the forecast errors. Accurate forecasts should be unbiased and also have a small variance proportion, so that most of the forecasting error should be attributable to the covariance. In this case, unfortunately, most of the forecast error is caused by the difference between the variation of the forecasts and the variation of the actual data (the variance proportion is close to unity). In any case, the root mean squared error and the mean absolute error are low enough. The static forecast plot (to calculate a sequence of one-step-ahead forecasts, rolling the sample forwards one observation after each forecast to use actual rather than forecasted values for lagged dependent variables) for US Dollar over the whole sample is the following (us_dollar_sf): 2.4

Forecast: US_DOLLAR_SFORECAST Actual: US_DOLLAR Forecast sample: 1999M01 2015M11 Included observations: 203 Root Mean Squared Error 0.178259 Mean Absolute Error 0.136437 Mean Abs. Percent Error 12.59687 Theil Inequality Coefficient 0.070871 Bias Proportion 0.129306 Variance Proportion 0.748661 Covariance Proportion 0.122033

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The static forecast shows a fall of the variance forecast even if these graphs are characterized by much more volatility. This also results in more variability in the standard error bars around the conditional mean forecast. Nevertheless, the forecast of the mean lies always inside the confidence interval. The root mean squared error, the mean absolute error and the mean absolute percent error

too are all less than the ones in the dynamic forecast. But in this case the forecast is a little more biased than above, even if the variance proportion is lightly decreased and a smaller portion of the forecast error is attributable to the covariance portion. Here the results for UK Pound Sterling model (dynamic and static forecast over the whole sample): 8,000,000

Forecast: UK_POUNDSTERL_DFORECAST Actual: UK_POUND_STERLING Forecast sample: 1999M01 2015M11 Included observations: 203 Root Mean Squared Error 0.126522 Mean Absolute Error 0.112833 Mean Abs. Percent Error 15.92663 Theil Inequality Coefficient 0.079218 Bias Proportion 0.768723 Variance Proportion 0.006122 Covariance Proportion 0.225156

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The dynamic forecast (uk_pounsterl_df) shows a completely flat forecast structure for the mean, as well as the US Dollarâ€™s one. The forecast of variance is flat too and rises only at the end of the sample. This leads to straight error bars that lie on the conditional mean to widen, then, at the end of the sample. In the right box we can see how the most of the forecast error is caused by the bias proportion while the variance proportion is almost zero. This means that we have a large difference of the mean of the forecasts from the mean of the actual data, even if the mean absolute error and the root mean squared error are small.

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Forecast: UK_POUNDSTERL_SFORECAST Actual: UK_POUND_STERLING Forecast sample: 1999M01 2015M11 Included observations: 203 Root Mean Squared Error 0.090316 Mean Absolute Error 0.073253 Mean Abs. Percent Error 9.659205 Theil Inequality Coefficient 0.062102 Bias Proportion 0.096759 Variance Proportion 0.636225 Covariance Proportion 0.267016

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The static forecast (uk_poudsterl_sf) of the variance shows a relevant volatility and a flattening at the middle of the sample. This also results in more variability in the standard error bars around the conditional mean that get close to the mean in the middle part of the sample. In this case the root mean squared error, the mean absolute error and the mean absolute percent error too are greater than the ones in the dynamic forecast. But here the error forecast is mostly due to the variance proportion while the covariance proportion is almost equal. The latter forecast is less biased than the former.

We can estimate forecasting for all the sub-samples we desire. To make an example, these are the dynamic and static forecasts for both series over the last two years (us_dollar_df2, us_dollar_sf2, uk_poundsterl_df2 and uk_poundsterl_sf2):

Finally, even if we kept out the other variants of the GARCH model before, we could estimate forecasting for those models too and then to compare them with the selected model. This because some models can produce an accurate forecasting even if, in their estimation, they cannot capture all the data features. To make just an example, the GARCH(1,1) model for UK Pound Sterling (uk_forecast3) produces a more unbiased dynamic forecast (but we donâ€™t have the variance and the covariance proportion) with smaller mean errors:

Conclusions Summing up, I chose to analyze how the past volatility of a currency influenced its current value. To do this, I imported the monthly effective exchange rates of the currency as a measure of its relative strength against the Euro. The Japanese Yen, the US Dollar and the UK Pound Sterling showed themselves to be the three most traded currencies on the forex market, hence I opted to conduct the survey with these three series. All of them were characterized by an evident volatility, more highlighted during 2007-2008 (the years of the financial crisis) and during 2012-2013 for Japanese Yen only. For almost the first part of the sample, US Dollar and Japanese Yen followed a similar path. During 2008 and 2009 they started to differ from one another and in the second part of the sample we observed that US Dollar and UK Pound Sterlingâ€™s trend were more similar, while Japanese Yen gradually detached. Furthermore all the three series were far away from the normal distribution: they were skewed and platykurtic, even if for the Yen we could not reject the null hypothesis of normality in the JarqueBera test. The three currencies had a geometrically decaying ACF and a number of non-zero PACF coefficients equal to 2. The results of the ADF test, the PP test and the KPSS test led to the rejection of the stationary null hypothesis for the Dollar and the Pound but not for the Yen. Since pure first differences models have no long-run solution, I tested for cointegration (within the Johansen VAR framework), non-linearity (with the BDS independence test) and for ARCH effects. The series were not cointegrated but they had not a linear structure and there were the presence of ARCH effects in the US Dollar and UK Pound Sterling data. For this reason any linear model could not be estimated and hence the GARCH model appeared the most appropriate. Following the maximization of the log-likelihood function and the minimization of the information criteria as measures to compare the variations of the GARCH family models, I estimated a GARCH-M(0,1) for the Dollar and a GARCH-M(1,1) for the Pound, since there were visible feedback from the logarithm of the conditional variance to the conditional mean. Even if I succeeded in removing ARCH effects, after the estimation of the models there were still some problems. The fitted residuals showed lesser volatility but they were autocorrelated and they did not follow a normal distribution. This could be attributed to a non-correctly specified mean equation for both the US Dollar and the UK Pound Sterling. Forecasting accuracy was medial: both models produced forecastings with low mean errors but a bit biased or with a large variance proportion which determined the forecast error.

The conclusion is that these GARCH-M models are able to capture some although not all of the features in the unconditional distributions of the series. These models could be improved, maybe, with a better specified mean equation, considering the real effective exchange rates instead of the nominal ones, providing with seasonal adjustment or a better structured sample, added to solid theoretical grounds. Hence, we can consider this report as a starting point for a more valid and accurate analysis.

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