2. Empirical results and methodology

### 2.1. Data and preliminary findings

The time series data used for modeling volatility in this paper consists of two sets of financial data. The first set includes daily returns of five stock indices: NASDAQ100 (US), Germany (DAX30), Ishares MSCI South Africa index (EZA), Shanghai stock exchange composite index (SSE), and Ishares MSCI Canada index (EWC).2 The second data set includes daily returns of five exchange rates series: British Pound (USD/GBP), Australian Dollar (USD/AUD), Italian Lira (USD/ITL), South Africa Rand (USD/ZAR), and Brazilian Real (USD/BRL).<sup>3</sup> The two data

<sup>1</sup> In general terms, volatility refers to the fluctuations observed in some phenomenon overtime. In terms of modeling and forecasting literature, it means "the conditional variance of the underlying asset return" [17].

<sup>2</sup> Some of the closing price indices were put into US-dollar and some were put into other currencies. For unification of foreign exchange rates, all closing price indices were converted into American US dollar. These closing price indices are obtained from Yahoo Finance (http://finance.yahoo.com).

<sup>3</sup> The exchange rates have been retrieved from the website (http://www.oanda.com).

sets include daily closing prices from August 6, 2001, through December 10, 2013, for all stock indices and from July 1, 2005, to September 17, 2013, for all exchange rate series with a total of 3001 observations for each data set. The estimation process for the two sets of data was run using 2001 observations as in-sample, while the remaining 1000 observations were used for the out-of-sample forecast. Based on the empirical evidence, it is common to assume that the logarithmic return series rt = 100 \* [ln(pt) ln(pt 1)] (where Pt and Pt<sup>1</sup> are the price at the current day and previous day, respectively) is weakly stationary. Table 1 reports the descriptive statistics for all return series. It shows that all data exhibit excess kurtosis (leptokurtosis) and skewness, which represents the nature of departure from normality. The Jarque-Bera (JB) statistics for normality test show that the null hypotheses of normality are strongly rejected for all daily returns of stock and exchange rate series.

### 2.2. Methodology

normal distribution. However, the existence of negative skewness (the third moment of the distribution) has the effect of accentuating the left-hand side of the distribution, which means that a higher probability of decreases given to asset pricing than increases in the market.

The generalized autoregressive conditional heteroscedasticity (GARCH) models, introduced by Engle [5] and Bollerslev [1], allow for time-varying volatility1 but not for time-varying skewness or time-varying kurtosis. Different GARCH models have been developed in the literature to capture dependencies in higher order moments, starting with Hansen [7] who proposed a skew-Student distribution to account for both time-varying excess kurtosis and skewness. A significant evidence of time-varying skewness found [9]. Others [11, 12] found a significant time varying in both skewness and kurtosis, while [3, 15, 16] found little evidence of either. With regard to the frequency of observation, Jondeau and Rockinger [11] found the presence of timevarying skewness and kurtosis in daily but not weekly data, while others including [2, 7, 9] found an evidence of time-varying skewness and kurtosis in weekly and even monthly data. Regarding daily data [4, 12, 18] found an evidence of time-varying skewness and kurtosis in daily data. The chapter employed GARCH(1,1) model as the performance of the model proved compared large number of volatility models; for more details, see Hansen and Lunde [8].

This paper contributes to the literature of volatility modeling in two aspects. First, we jointly estimate time-varying volatility, skewness, and kurtosis assuming Johnson SU distribution for the error term. The method is applied to two different daily returns: stock indices and exchange rates. Second, a new alternative scheme is introduced to generate the sequence of

The rest of the paper is organized as follows. Following this introduction, Section 2 presents the empirical results regarding the estimation of the model. Section 3 compares the models. In Section 4, the new forecasting scheme is presented, while Section 5 gives concluding remarks.

The time series data used for modeling volatility in this paper consists of two sets of financial data. The first set includes daily returns of five stock indices: NASDAQ100 (US), Germany (DAX30), Ishares MSCI South Africa index (EZA), Shanghai stock exchange composite index (SSE), and Ishares MSCI Canada index (EWC).2 The second data set includes daily returns of five exchange rates series: British Pound (USD/GBP), Australian Dollar (USD/AUD), Italian Lira (USD/ITL), South Africa Rand (USD/ZAR), and Brazilian Real (USD/BRL).<sup>3</sup> The two data

In general terms, volatility refers to the fluctuations observed in some phenomenon overtime. In terms of modeling and

Some of the closing price indices were put into US-dollar and some were put into other currencies. For unification of foreign exchange rates, all closing price indices were converted into American US dollar. These closing price indices are

forecasting literature, it means "the conditional variance of the underlying asset return" [17].

The exchange rates have been retrieved from the website (http://www.oanda.com).

the forecasts.

42 Time Series Analysis and Applications

1

2

3

2. Empirical results and methodology

obtained from Yahoo Finance (http://finance.yahoo.com).

2.1. Data and preliminary findings

Preliminary results in the preceding section provided evidence of a significant deviation from normality and obvious leptokurtosis in all daily return series. This suggests specifying GARCH models that capture these characteristics. In presenting these models, there are two distinct equations or specifications, one for the conditional mean and the other for the conditional variance. For the models employed in this paper, the mean equation for all stock return series is the AR(1) model with a constant, and for all exchange rate return series, we used the MA(1) model without a constant. After estimating the mean equation, the next step was to identify whether there is substantial evidence of heteroscedasticity for the daily returns of stock and exchange rate series. Table 2 provides the Ljung-Box statistics of order 20 for ε<sup>2</sup> <sup>t</sup> , ε<sup>3</sup> t and ε<sup>4</sup> <sup>t</sup> , where ε<sup>t</sup> is the error term from the mean equation. The results show that the Ljung-Box


Table 1. Descriptive statistics for daily returns.


Table 2. Ljung-Box statistics with order 20 of ε<sup>2</sup> <sup>t</sup> , ε<sup>3</sup> <sup>t</sup> and ε<sup>4</sup> <sup>t</sup> where ε<sup>t</sup> is the error term for the mean equation for all daily returns of stock and exchange rate series.

statistics on the squared residuals ε<sup>2</sup> <sup>t</sup> , ε<sup>3</sup> <sup>t</sup> , and ε<sup>4</sup> <sup>t</sup> are significant for the presence of time-varying volatility, skewness, and kurtosis for all daily returns of stock and exchange rate series.

### 2.2.1. Distributional assumptions

To complete the basic GARCH specification, an assumption about the conditional distribution of the error term ε<sup>t</sup> is required. The expectation is that the excess kurtosis and skewness displayed by the residuals of conditional heteroscedastic models will be reduced, when a more appropriate distribution is used. The Johnson's SU distribution is resorted to in this study. This distribution has two shape parameters that allow a wide range of skewness and kurtosis levels of the type anticipated, and it is used in financial returns data [4, 18]. The Johnson's SU distribution was derived by Johnson [10] through transformation of a normal variable. Letting z ~ N(0,1) the standard normal distribution, the random variable y defined by the transformation:

$$z = \gamma + \delta \sinh^{-1} \left(\frac{y - \zeta}{\lambda}\right) \tag{1}$$

where sinh�<sup>1</sup> is the inverse hyperbolic sine function defines a Johnson's SU variable. The form of the density of the Johnson's SU distribution, which will be used for the estimation procedure, is that due to Yan [18]:

$$f\_y(y) = \frac{\delta}{\lambda \sqrt{1 + \left(\frac{y-\zeta}{\lambda}\right)^2}} \phi \left[\gamma + \delta \sinh^{-1}\left(\frac{y-\zeta}{\lambda}\right)\right] \tag{2}$$

where y ∈ R, φ is the density function of N(0, 1), ξ and λ > 0 are location and scale parameters, respectively, while γ, δ > 0 can be interpreted as skewness and kurtosis parameters, respectively. The parameters are not the direct moments of the distribution. The first four moments, the mean, variance, third central moment, and fourth central moment, respectively, of the distribution according to Yan [18] are as follows:

$$
\mu = \zeta + \lambda \omega^{1/2} \sinh \Omega \tag{3}
$$

$$
\sigma^2 = \frac{\lambda^2}{2} (\omega - 1)(\omega \cosh 2\Omega + 1) \tag{4}
$$

$$\mu\_3 = -\frac{1}{4}\omega^2(\omega^2 - 1)^2 \left[\omega^2(\omega^2 + 2)\sinh 3\Omega + 3\sinh \Omega\right] \tag{5}$$

$$\mu\_4 = \frac{1}{8} \left(\omega^2 - 1\right)^2 \left[\omega^4 \left(\omega^8 + 2\omega^6 + 3\omega^4 - 3\right) \cosh 4\Omega + 4\omega^4 \left(\omega^2 + 2\right) \cosh 2\Omega + 3\left(2\omega^2 + 1\right)\right] \tag{6}$$

The quantities Ω and ω in the moment formulas are Ω = γ/δ and ω = exp(δ�2). The skewness and kurtosis are jointly determined by the two shape parameters γ and δ. The standardized Johnson's SU innovations exist when ξ = 0 and λ = 1, but the mean and the variance are not 0 and 1, respectively. These can be done by setting the parameters in the following manner:

$$\zeta = -\omega^{1/2} \sinh \Omega \left[ \sqrt{\frac{1}{2} (\omega - 1)(\omega \cosh 2\Omega + 1)} \right]^{-1} \tag{7}$$

$$
\lambda = \left[ \sqrt{\frac{1}{2} (\omega - 1)(\omega \cosh 2\Omega + 1)} \right]^{-1} \tag{8}
$$

#### 2.2.2. Maximum likelihood

statistics on the squared residuals ε<sup>2</sup>

Table 2. Ljung-Box statistics with order 20 of ε<sup>2</sup>

returns of stock and exchange rate series.

2.2.1. Distributional assumptions

Series ε<sup>2</sup>

44 Time Series Analysis and Applications

Stock indices

Exchange rates

dure, is that due to Yan [18]:

f <sup>y</sup>ð Þ¼ y

λ

<sup>t</sup> , ε<sup>3</sup>

<sup>t</sup> , ε<sup>3</sup> <sup>t</sup> and ε<sup>4</sup>

Note. For Ljung-Box statistics, the p-values are reported in parentheses.

<sup>t</sup> , and ε<sup>4</sup>

<sup>t</sup> ε<sup>3</sup>

NASDAQ100 1834.3 (0.000) 305.1 (0.000) 507.1 (0.000) DAX30 2132.9 (0.000) 148.4 (0.000) 676.1 (0.000) SSE 443.2 (0.000) 24.6 (0.216) 52.4 (0.000) EZA 2597.2 (0.000) 305.8 (0.000) 647.8 (0.000) EWC 3614.3 (0.000) 272.1 (0.000) 984.2 (0.000)

USD/GBP 1020.8 (0.000) 98.6 (0.000) 190.6 (0.000) USD/AUD 2525.9 (0.000) 678.2 (0.000) 889.8 (0.000) USD/ZAR 975.5 (0.000) 89.2 (0.000) 39.128 (0.006) USD/ITL 536.2 (0.000) 94.477 (0.000) 77.6 (0.000) USD/BRL 1555.3 (0.000) 406.1 (0.000) 1030.9 (0.000)

standard normal distribution, the random variable y defined by the transformation:

δ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> <sup>y</sup>�<sup>ζ</sup> λ

volatility, skewness, and kurtosis for all daily returns of stock and exchange rate series.

To complete the basic GARCH specification, an assumption about the conditional distribution of the error term ε<sup>t</sup> is required. The expectation is that the excess kurtosis and skewness displayed by the residuals of conditional heteroscedastic models will be reduced, when a more appropriate distribution is used. The Johnson's SU distribution is resorted to in this study. This distribution has two shape parameters that allow a wide range of skewness and kurtosis levels of the type anticipated, and it is used in financial returns data [4, 18]. The Johnson's SU distribution was derived by Johnson [10] through transformation of a normal variable. Letting z ~ N(0,1) the

<sup>z</sup> <sup>¼</sup> <sup>γ</sup> <sup>þ</sup> <sup>δ</sup> sinh�<sup>1</sup> <sup>y</sup> � <sup>ζ</sup>

where sinh�<sup>1</sup> is the inverse hyperbolic sine function defines a Johnson's SU variable. The form of the density of the Johnson's SU distribution, which will be used for the estimation proce-

� �<sup>2</sup> <sup>r</sup> φ γ <sup>þ</sup> <sup>δ</sup> sinh�<sup>1</sup> <sup>y</sup> � <sup>ζ</sup>

λ � �

λ

� � � �

<sup>t</sup> are significant for the presence of time-varying

<sup>t</sup> where ε<sup>t</sup> is the error term for the mean equation for all daily

<sup>t</sup> ε<sup>4</sup>

t

(1)

(2)

Under the presence of heteroscedasticity (autoregressive conditional heteroscedasticity (ARCH) effects) in the residuals of the daily returns of stock and exchange rate series, the ordinary least square estimation (OLS) is not efficient, and the estimate of covariance matrix of the parameters will be biased due to invalid 't' statistics. Therefore, ARCH-type models cannot be estimated by simple techniques such as OLS. The method of maximum likelihood estimation is employed in ARCH models. For the formal exposition of the approach, each realization of the conditional variance ht has the joint likelihood of realization:

$$L = \prod\_{t=1}^{T} \left( \sqrt{\frac{1}{2\pi h\_t}} \right) \exp\left(\frac{-\varepsilon\_t^2}{2h\_t}\right) \tag{9}$$

The log likelihood function is:

$$\log(L) = -\frac{T}{2}Ln(2\pi) - 0.5\sum\_{t=1}^{T} h\_t - 0.5\sum\_{t=1}^{T} \left(\frac{\varepsilon\_t^2}{h\_t}\right) \tag{10}$$

The parameter values are selected so that the log likelihood function is maximized using a search algorithm by computers.

### 2.2.3. Model estimation with time-varying volatility, skewness, and kurtosis

As it was shown in Section 2.2, when the residuals were examined for heteroscedasticity, the Ljung Box test provided strong evidence of ARCH effects in the residuals series, which suggests proceeds with modeling the returns volatility using the GARCH methodology. The model to be estimated in this study is the standard GARCH(1, 1) model with constant shape parameters, and also, we impose dynamics on both shape parameters to obtain autoregressive conditional density (ARCD) models.<sup>4</sup> This allows for time-varying skewness and kurtosis assuming Johnson Su distribution for the error term in the two cases. Before presenting the estimation results obtained with both the stock return series and the exchange rate return series, the four nested models to be estimated are summarized as follows:

For stock return series:

Mean equation

$$
\sigma\_t = \mu + \phi\_1 r\_{t-1} + \varepsilon\_t \tag{11}
$$

$$
\varepsilon\_t = \sqrt{h\_t} z\_{t\nu} z\_t = \sqrt{h\_t} z\_t \sim \text{JSu}\left(\xi\_t, \lambda\_t, \gamma\_t, \delta\_t\right),
$$

Variance equation (GARCH)

$$h\_t = b\_0 + b\_1 \varepsilon\_{t-1}^2 + b\_2 h\_{t-1} \tag{12}$$

Skewness equation

$$
\gamma\_t = c\_0 + c\_1 z\_{t-1} + c\_2 z\_{t-1}^2 + c\_3 \gamma\_{t-1} \tag{13}
$$

Kurtosis equation

$$
\delta\_t = d\_0 + d\_1 z\_{t-1} + d\_2 z\_{t-1}^2 + d\_3 \delta\_{t-1} \tag{14}
$$

For all stock return series, the study is going to use GARCH(1,1) model with a similar specification to that of Hansen [7] for shape parameters (γt, δt) but employs the standardized innovation zt�<sup>1</sup> instead of nonstandardized εt�<sup>1</sup> as in Eqs. (13) and (14).

For exchange rate return series:

Mean equation

$$r\_t = \theta\_1 \varepsilon\_{t-1} + \varepsilon\_t \tag{15}$$

$$
\varepsilon\_t = \sqrt{h\_t} z\_{t\prime} z\_t = \sqrt{h\_t} z\_t \sim \text{JSu}\left(\xi\_t, \lambda\_t, \gamma\_t, \delta\_t\right),
$$

Variance equation (GARCH)

<sup>4</sup> ARCD is the approach, where dynamics imposed on shape parameters and skewness or kurtosis are derived from the time-varying shape parameters.

Volatility Parameters Estimation and Forecasting of GARCH(1,1) Models with Johnson's SU Distributed Errors http://dx.doi.org/10.5772/intechopen.70506 47

$$h\_t = b\_0 + b\_1 \varepsilon\_{t-1}^2 + b\_2 h\_{t-1} \tag{16}$$

Skewness equation

2.2.3. Model estimation with time-varying volatility, skewness, and kurtosis

series, the four nested models to be estimated are summarized as follows:

; δ<sup>t</sup> � �

ht <sup>¼</sup> <sup>b</sup><sup>0</sup> <sup>þ</sup> <sup>b</sup>1ε<sup>2</sup>

<sup>γ</sup><sup>t</sup> <sup>¼</sup> <sup>c</sup><sup>0</sup> <sup>þ</sup> <sup>c</sup>1zt�<sup>1</sup> <sup>þ</sup> <sup>c</sup>2z<sup>2</sup>

<sup>δ</sup><sup>t</sup> <sup>¼</sup> <sup>d</sup><sup>0</sup> <sup>þ</sup> <sup>d</sup>1zt�<sup>1</sup> <sup>þ</sup> <sup>d</sup>2z<sup>2</sup>

innovation zt�<sup>1</sup> instead of nonstandardized εt�<sup>1</sup> as in Eqs. (13) and (14).

; δ<sup>t</sup> � �

For all stock return series, the study is going to use GARCH(1,1) model with a similar specification to that of Hansen [7] for shape parameters (γt, δt) but employs the standardized

ARCD is the approach, where dynamics imposed on shape parameters and skewness or kurtosis are derived from the

For stock return series:

46 Time Series Analysis and Applications

ht

Variance equation (GARCH)

For exchange rate return series:

ht

Variance equation (GARCH)

time-varying shape parameters.

<sup>p</sup> zt � JSu <sup>ξ</sup>t; <sup>λ</sup>t; <sup>γ</sup><sup>t</sup>

Skewness equation

Kurtosis equation

Mean equation

<sup>ε</sup><sup>t</sup> <sup>¼</sup> ffiffiffiffi ht <sup>p</sup> zt, zt <sup>¼</sup> ffiffiffiffi

4

<sup>p</sup> zt � JSu <sup>ξ</sup>t; <sup>λ</sup>t; <sup>γ</sup><sup>t</sup>

Mean equation

<sup>ε</sup><sup>t</sup> <sup>¼</sup> ffiffiffiffi ht <sup>p</sup> zt, zt <sup>¼</sup> ffiffiffiffi

As it was shown in Section 2.2, when the residuals were examined for heteroscedasticity, the Ljung Box test provided strong evidence of ARCH effects in the residuals series, which suggests proceeds with modeling the returns volatility using the GARCH methodology. The model to be estimated in this study is the standard GARCH(1, 1) model with constant shape parameters, and also, we impose dynamics on both shape parameters to obtain autoregressive conditional density (ARCD) models.<sup>4</sup> This allows for time-varying skewness and kurtosis assuming Johnson Su distribution for the error term in the two cases. Before presenting the estimation results obtained with both the stock return series and the exchange rate return

rt ¼ μ þ φ1rt�<sup>1</sup> þ ε<sup>t</sup> (11)

<sup>t</sup>�<sup>1</sup> <sup>þ</sup> <sup>b</sup>2ht�<sup>1</sup> (12)

<sup>t</sup>�<sup>1</sup> <sup>þ</sup> <sup>c</sup>3γ<sup>t</sup>�<sup>1</sup> (13)

<sup>t</sup>�<sup>1</sup> <sup>þ</sup> <sup>d</sup>3δ<sup>t</sup>�<sup>1</sup> (14)

rt ¼ θ1ε<sup>t</sup>�<sup>1</sup> þ ε<sup>t</sup> (15)

$$\gamma\_t = c\_0 + c\_1 z\_{t-1} I\_{z\_{t-1$$

Kurtosis equation

$$
\delta\_t = d\_0 + d\_1 |z\_{t-1}| I\_{z\_{t-1$$

For the exchange rate return series, a specification similar to that of [11] for shape parameters (γt, δt) is used with the exception that it utilizes the standardized innovation zt�<sup>1</sup> instead of nonstandardized εt�<sup>1</sup> as in Eqs. (17) and (18). It also considers the absolute standardized shocks for the shape parameter in Eq. (18), Ghalanos [6]. So, first, we start by estimating the two standard models for the conditional variance: the AR(1)-GARCH(1,1) model (Eqs. (11) and (12)) for the stock return series and MA(1)-GARCH(1,1) model (Eqs. (15) and (16)) for the exchange rate return series. Second, the generalizations of both the standard GARCH and GARCH models with time-varying skewness and kurtosis (GARCHSK) as in Eqs. (11)–(14) for the stock return series and Eqs. (15)–(18) for the exchange rate return series are estimated.

The results for the stock return series are presented in Tables 3 and 4 for both the standard GARCH and GARCHSK models, respectively. As expected, the results indicate high and significant presence of conditional variance, since the coefficient of lagged conditional variance (b2) is high, positive, and significant. Volatility is found to be persistent, since the coefficient of lagged volatility (b1) is positive and significant, indicating that high conditional variance is followed by high conditional variance. The sum of the two estimated coefficients (b<sup>1</sup> + b2) in the estimation process is very close to one, implying that large changes in stock returns tend to be


Table 3. Maximum likelihood estimates of AR(1)-GARCH(1,1) model for stock return series.


Table 4. Maximum likelihood estimates of AR(1)-GARCH(1,1) model with time-varying skewness and kurtosis for stock return series.

followed by large changes, and small changes tend to be followed by small changes. This confirms that volatility clustering is observed in the stock returns series. For the skewness and kurtosis equations, it is found that for all stock return series, days with high conditional skewness and kurtosis are followed by days with high conditional skewness and kurtosis except DAX30 in kurtosis case, since the coefficients for lagged skewness (c3) and for lagged kurtosis (d3) are positive and significant. In summary, there is a significant presence of conditional skewness and kurtosis for all stock return series, since at least one of the coefficients associated with the standardized shocks or squared standardized shocks to (skewness and kurtosis) or to lagged (skewness and kurtosis) is found to be significant.

The results for the five exchange rates are presented in Tables 5 and 6 for GARCH and GARCHSK models, respectively. As expected, the results are the same as in the case of stock return series, i.e., the results also indicate highest significant presence of conditional variance. Volatility is found to be persistent, and volatility clustering is also observed in exchange rate return series. A significant presence of conditional skewness and kurtosis for all exchange rate return series is confirmed, since at least one of the coefficients associated with the standardized

Volatility Parameters Estimation and Forecasting of GARCH(1,1) Models with Johnson's SU Distributed Errors http://dx.doi.org/10.5772/intechopen.70506 49


\*\*Significant at the 1% level.

followed by large changes, and small changes tend to be followed by small changes. This confirms that volatility clustering is observed in the stock returns series. For the skewness and kurtosis equations, it is found that for all stock return series, days with high conditional skewness and kurtosis are followed by days with high conditional skewness and kurtosis except DAX30 in kurtosis case, since the coefficients for lagged skewness (c3) and for lagged kurtosis (d3) are positive and significant. In summary, there is a significant presence of conditional skewness and kurtosis for all stock return series, since at least one of the coefficients associated with the standardized shocks or squared standardized shocks to (skewness and

Parameters NASDAQ100 DAX30 SSE EZA EWC Mean equation μ 0.0155 0.0816\* 0.0555 0.1312\* 0.0851\*

Variance equation b<sup>0</sup> 0.0104\* 0.0167\* 0.0506\* 0.0620\* 0.0250\*

Skewness equation <sup>c</sup><sup>0</sup> 0.0038\* 0.0035\* 0.0015\* 0.0261\* 0.0256\*

Kurtosis equation d<sup>0</sup> 0.0001 0.7193\* 0.9625\* 0.2245\* 0.4362

Log-likelihood 3559.79 3578.15 3620.83 3294.5 3406.96 AIC 3.5728 3.5911 3.6338 4.1344 3.4200

Prob. chi-square (5) 0.2250 0.2518 0.8917 0.9795 0.3698

Table 4. Maximum likelihood estimates of AR(1)-GARCH(1,1) model with time-varying skewness and kurtosis for stock

) 6.942 6.604 1.678 0.7606 5.393

<sup>φ</sup> 0.0567\* 0.0947\* 0.0154 0.0512\* 0.0540\*

b<sup>1</sup> 0.0578\* 0.0717\* 0.1009\* 0.0931\* 0.0762\* b<sup>2</sup> 0.9436\* 0.9239\* 0.8997\* 0.8998\* 0.9183\*

<sup>c</sup><sup>1</sup> 0.00002 0.0083\* 0.0054\* 0.0838\* 0.0163 <sup>c</sup><sup>2</sup> 0.00355\* 0.0037\* 0.0017\* 0.0004 0.0192\* c<sup>3</sup> 0.9939\* 1.0000\* 0.9898\* 0.8661\* 0.9165\*

d<sup>1</sup> 0.9869\* 0.3126\* 0.2684\* 0.4848\* 0.5166\* d<sup>2</sup> 0.0799 0.2929\* 0.0591 0.0000 0.2638\* d<sup>3</sup> 0.8459\* 0.0019 0.5469\* 0.8143\* 0.4358\*

The results for the five exchange rates are presented in Tables 5 and 6 for GARCH and GARCHSK models, respectively. As expected, the results are the same as in the case of stock return series, i.e., the results also indicate highest significant presence of conditional variance. Volatility is found to be persistent, and volatility clustering is also observed in exchange rate return series. A significant presence of conditional skewness and kurtosis for all exchange rate return series is confirmed, since at least one of the coefficients associated with the standardized

kurtosis) or to lagged (skewness and kurtosis) is found to be significant.

ARCH-LM test for heteroscedasticity

48 Time Series Analysis and Applications

Significant at the 5% level.

Statistic (T\*R<sup>2</sup>

return series.

\*

Table 5. Maximum likelihood estimates of MA(1)-GARCH(1,1) model for exchange rate return series.


Table 6. Maximum likelihood estimates of MA(1)-GARCH(1,1) model with time-varying skewness and kurtosis for exchange rate return series.

shocks (either negative or positive) to (skewness & kurtosis) or to lagged (skewness & kurtosis) are found to be significant.

Finally, it is worth noting that from the bottom of Tables 3–6, the value of Akaike information criterion (AIC) decreases monotonically when moving from the simpler model (standard GARCH) to the more complicated ones (GARCHSK) for all return series. Therefore, for all return series analyzed, the GARCHSK model specification seems to be the most appropriate one according to the AIC. Note that the ARCH-LM test statistics for all return series did not exhibit additional ARCH effect. This shows that the variance equations are well specified and adequate.
