#### 6.1. Data and Preliminary Results

The empirical analysis focuses on estimating and forecasting the GLMSV model for three sets of exchange rate data, namely YEN/USD, EUR/USD, and GBP/USD. The sample period is from 4 October 2005 to 25 November 2015, giving 2549 observations. We calculated the returns series,

${R}_{t}=\mathrm{log}{P}_{t}-\mathrm{log}{P}_{t-1}$, where

${P}_{t}$ is the closing price on day

t. We use the first

$n=2048$ returns for estimating the GLMSV models, and the remaining 500 series for forecasting. The estimation period includes the global financial crisis.

Table 2 presents descriptive statistics for the whole sample. As our interest is on volatility, we use the mean subtracted returns,

${Y}_{t}={R}_{t}-\overline{R}$.

As a preliminary analysis, we estimated the new generalized fractionally integrated EGARCH (GIEGARCH) model, defined by:

where

$\zeta \left({\xi}_{t}\right)$ is the generalized return, and

$\varphi \left(L\right)$ and

$\theta \left(L\right)$ are defined in

Section 3. Following

Hansen et al. (

2012), we consider the second-order Hermite polynomial for the error term, as:

Assuming that

${\xi}_{t}$ has finite fourth moment, it is straightforward to show

$E\left[\zeta \left({\xi}_{t}\right)\right]=0$ and

$V\left[\zeta \left({\xi}_{t}\right)\right]<\infty $. When

$\eta =1$, the new GIEGARCH(

p,

d,

q;

$\eta $) model reduces to the class of the FIEGARCH(

p,

$2d$,

q) model of

Bollerslev and Mikkelsen (

1996). Following

Bollerslev and Mikkelsen (

1996), we truncate the MA(∞) representation of the GARMA process of log-volatility as:

where

${\tilde{\psi}}_{j}$ is the

jth coefficient of the polynomial

$\tilde{\psi}\left(z\right)={(1-2\eta z+{z}^{2})}^{-d}\varphi {\left(z\right)}^{-1}\theta \left(z\right)$, with

${\tilde{\psi}}_{0}=1$. We calculate the value of

${\tilde{\psi}}_{j}$ by the approximating technique of

McElroy and Holan (

2012) up to

$J=1000$ (see the

Appendix A).

In addition to the FIEGARCH and GIEGARCH models, we estimated the GARCH model with the conditional volatility equation:

as a benchmark.

Table 3 gives the QML estimates of the GARCH model. As a typical result, the estimates of

$\alpha +\beta $ is close to one, indicating a possible long range dependence in volatility.

Table 4 shows the QML estimates of the FIEGARCH(1,

$2d$,0) and GIEGARCH(1,

d,0;

$\eta $) models. For the FIEGARCH model, the estimates of

d indicate that the conditional log-volatility,

$\mathrm{log}{h}_{t}$, has long range dependence. The estimates of

${\gamma}_{1}$ are negative, while those of

${\gamma}_{2}$ are positive. The estimates of

$\varphi $ are located in the interval (−0.25, −0.1). Except for the estimates of

${\gamma}_{1}$ for the EUR/USD return, all parameter estimates are significant at the five percent level. These estimates are similar to the values obtained in the literature.

The estimates of d in the GIEGARCH model are about twice of those for the FIEGARCH model. The estimates of $\eta $ are positive, and the estimates of $\varphi $ are close to one. The estimates of ${\gamma}_{1}$ are negative, while those of ${\gamma}_{2}$ are positive. All parameter estimates are significant at the five percent level. As the estimates of $\eta $ are significantly different from one, the estimates of the Gegenbauer frequency, ${\omega}_{g}=\mathrm{arccos}\left(\eta \right)$, are different from zero.

#### 6.2. Estimates and Forecasts for the GLMSV Model

In the following, we show the empirical results for the GLMSV models as compared with those of the GIEGARCH model.

Table 5 gives the SL estimates of the GLMSV model. The estimates of

d and

$\varphi $ are close to the values of the GIEGARCH model. Compared with the GIEGARCH model, the estimates of

$\eta $ are higher. The estimates of

$\mu $ are different from those of the GIEGARCH model, and the differences may arise from the statistical flexibility of the class of SV models compared with their conditional heteroskedasticity counterparts. All estimates are significant at the five percent level. As the estimates of

$\eta $ are significantly different from one, the estimates of the Gegenbauer frequency

${\omega}_{g}=\mathrm{arccos}\left(\eta \right)$ are different from zero.

As explained previously, we use the last 500 observations for the rolling-window forecasting analysis, based on the approach in the previous section. First, we report the estimates of the Mincer–Zarmowitz (MZ) regression, given by:

where

${\widehat{\sigma}}_{t|t-1}^{2}$ is the one-step ahead forecast of

${\sigma}_{t}^{2}$ on day

t.

Table 6 presents the results of OLS estimation with heteroskedasticity and autocorrelation consistent standard errors. For all three data sets, the GLMSV model has the highest

${R}^{2}$. We should note that the simple GARCH model has the second highest

${R}^{2}$ for two series.

Secondly, we compare the forecasts using the robust and homogeneous loss function suggested by

Patton (

2011), defined by:

where

c is the degree of homogeneity,

h is the proxy of volatility (

$h={Y}^{2}$), and

${\widehat{\sigma}}^{2}$ is the forecast of volatility. The general loss function reduces to the mean squared error (MSE) when

$c=2$, while it is equivalent to the the loss function based on the quasi-log-likelihood (QLIKE) when

$c=0$.

Table 7 shows the average of the loss function for the cases

$c=0,1,2$. For all three series, two of three loss functions chose the GLMSV model, while the remaining one selects the GARCH model.

Thirdly, we calculated the Value-at-Risk (VaR) thresholds, assuming normality of

${\xi}_{t}$. Combined with the one-day-ahead forecasts of log-volatility, we computed the 1 and 5 percent VaR thresholds as

$-2.326{\widehat{\sigma}}_{n+1}^{2}$ and

$-1.645{\widehat{\sigma}}_{n+1}^{2}$, respectively, fixing the sample size as

$n=2048$. In order to assess the estimated VaR thresholds, we use the GMM duration-based tests developed by

Candelon et al. (

2011), which works with the

J-statistic based on the moments defined by the orthonormal polynomials that are associated with the geometric distribution. The conditional coverage test and independence test based on

q orthonormal polynomials have the asymptotic

${\chi}_{q}^{2}$ and

${\chi}_{q-1}^{2}$ distributions under their respective null distributions. The unconditional coverage test is given as a special case of the conditional coverage test, with

$q=1$.

Table 8 shows the percentage of VaR violations and test results for the GARCH, FIEGARCH, GIEGARCH and GLMSV models. For the FIEGARCH model, some of the test statistics are rejected at the five percent significance level. On the other hand, for the GARCH, GIEGARCH, and GLMSV models, the tests do not reject the null hypothesis at the 5% and 1% VaR thresholds, thereby indicating that the estimated VaR thresholds are satisfactory.

By the three kinds of measures of forecasting performance, we found that the GLMSV model always outperforms the FIEGARCH and GIEGARCH models for the period includes the global financial crises. We also found that the simple GARCH model is the second best model.