Estimation for stochastic volatility model: Quasi-likelihood and asymptotic quasi-likelihood approaches

2.1 The QL Method

Let the observation equation be given by

${\mathit{\zeta }}_t$ is a sequence of martingale difference with respect to ${\mathcal{F}}_t\text{,}{\mathcal{F}}_t$ denotes the $\sigma$ -field generated by ${\mathbf{y}}_t\text{,}{\mathbf{y}}_{t-1}\text{,}\dots \text{,}{\mathbf{y}}_1$ for $t⩾1$ ; that is, $E({\mathit{\zeta }}_t|{\mathcal{F}}_{t-1})$ = $E_{t-1}({\mathit{\zeta }}_t)=$ 0; where ${\mathbf{f}}_t(\mathit{\theta })$ is an ${\mathcal{F}}_{t-1}$ measurable; and $\mathit{\theta }$ is a parameter vector, which belongs to an open subset $\mathrm{\Theta }\in R^d$ . Note that $\mathit{\theta }$ is a parameter of interest. We assume that $E_{t-1}({\mathit{\zeta }}_t{\mathit{\zeta }}_t^{\prime })={\mathbf{\Sigma }}_t$ is known. Now, the liner class ${\mathcal{G}}_T$ of the estimating function (EF) can be defined by $${\mathcal{G}}_T=\left\{{\displaystyle \sum _{t=1}^T}{\mathbf{W}}_t({\mathbf{y}}_t-{\mathbf{f}}_t(\mathit{\theta }))\right\}$$ and the quasi-likelihood estimation function (QLEF) can be defined by

If the sub-estimating function spaces of ${\mathcal{G}}_T$ are considered as follows, $${\mathcal{G}}_t=\{{\mathbf{W}}_t({\mathbf{y}}_t-{\mathbf{f}}_t(\mathit{\theta }))\text{,}t=1\text{,}2\text{,}3\dots \text{,}T\}\text{,}$$ then the QLEF in the space ${\mathcal{G}}_t$ can be defined by

A limitation of the QL method is that the nature of ${\mathbf{\Sigma }}_t$ may not be obtainable. A misidentified ${\mathbf{\Sigma }}_t$ could result in a deceptive inference about parameter $\mathit{\theta }$ . In the next subsection, we introduce the AQL method, which is basically the QL estimation assuming that the covariance matrix ${\mathbf{\Sigma }}_t$ is unknown.

2.2 The AQL method

The QLEF (see (2.1.2) and (2.1.3)) relies on the information of ${\mathrm{\Sigma }}_t$ . Such information is not always accessible. To find the QL when $E_{t-1}({\mathit{\zeta }}_t{\mathit{\zeta }}_t^{\prime })$ is not accessible, Lin (2000) proposed the AQL method.

Suppose, in probability, ${\mathbf{\Sigma }}_{t\text{,}n}$ is converging to $E_{t-1}({\mathit{\zeta }}_t{\mathit{\zeta }}_t^{\prime })$ . Then,

In this paper, the kernel smoothing estimator of ${\mathbf{\Sigma }}_t$ is suggested to find ${\mathbf{\Sigma }}_{t\text{,}n}$ in the AQLEF ((2.2.1)). A wide-ranging appraisal of the Nadaraya–Watson (NW) estimator-type kernel estimator is available in Härdle (1990) and Wand and Jones (1996). By using these kernel estimators, the AQL equation becomes

The estimation of $\mathit{\theta }$ by the AQL method is the solution to (2.2.2). Iterative techniques are suggested to solve the AQL equation (2.2.2). Such techniques start with the ordinary least squares (OLS) estimator ${\widehat{\mathit{\theta }}}^{(0)}$ and use ${\widehat{\mathbf{\Sigma }}}_{t\text{,}n}({\widehat{\mathit{\theta }}}^{(0)})$ in the AQL equation (2.2.2) to obtain the AQL estimator ${\widehat{\mathit{\theta }}}^{(1)}$ . Then, update ${\widehat{\mathit{\theta }}}^{(0)}$ by ${\widehat{\mathit{\theta }}}^{(1)}$ . Repeat this a few times until it converges.

The next section presents the parameter estimation of SVM using the QL and AQL methods.

3.1 Parameter estimation of SVM using the QL method

The stochastic volatility model is given by

The SVM in (3.1.1) can be transformed into a linear model as follows:

Abramovitz and Stegun (1970) showed that if ${\eta }_t\sim N(0\text{,}1)$ , then $E(\ln {\eta }_t^2)=-1.2704$ and $\mathit{Var}(\ln {\eta }_t^2)={\pi }^2/2$ . Now, assume that ${\epsilon }_t=\ln {\eta }_t^2+1.2704$ . Thus, $E({\epsilon }_t)=0$ . However, if ${\eta }_t$ has an unknown distribution, then $E(\ln {\eta }_t^2)=\mu$ and $\mathit{Var}(\ln {\eta }_t^2)={\sigma }_{∊}^2$ . Therefore, let ${∊}_t=\ln {\eta }_t^2-\mu$ . For this scenario, the martingale difference is $$\left(\begin{array}{c}\hfill {∊}_t\hfill \\ \hfill {\delta }_t\hfill \end{array}\right)=\left(\begin{array}{c}\hfill \ln (y_t^2)-h_t-\mu \hfill \\ \hfill h_t-\gamma -\varphi h_{t-1}\hfill \end{array}\right)\text{.}$$

First, to estimate $h_t$ , the QLEF is given by

Given that ${\widehat{h}}_0=0$ , the initial values ${\psi }_0=({\gamma }_0\text{,}{\varphi }_0\text{,}{\sigma }_{{\delta }_0}^2\text{,}{\mu }_0\text{,}{\sigma }_{{∊}_0}^2)$ , and ${\widehat{h}}_{t-1}$ is the QL estimation of $h_{t-1}$ , the QL estimation of $h_t$ is the solation of $G_{(t)}(h_t)=0$ ,

Second, using $\{{\widehat{h}}_t\}$ and $\{y_t\}$ , and considering $\gamma \text{,}\mu$ , and $\varphi$ as unknown parameters, the QLEF can be given by $$G_T(\mu \text{,}\gamma \text{,}\varphi )={\displaystyle \sum _{t=1}^T}\left(\begin{array}{cc}\hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill -h_{t-1}\hfill \end{array}\right){\left(\begin{array}{cc}\hfill {\sigma }_{{∊}_0}^2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\sigma }_{{\delta }_0}^2\hfill \end{array}\right)}^{-1}\left(\begin{array}{c}\hfill \ln (y_t^2)-h_t-\mu \hfill \\ \hfill h_t-\gamma -\varphi h_{t-1}\hfill \end{array}\right)\text{.}$$

The QL estimation of $\mu \text{,}\gamma$ , and $\varphi$ is the solation of $G_T(\mu \text{,}\gamma \text{,}\varphi )=0$ . Therefore,

Further,

$\widehat{\psi }=(\widehat{\mu }\text{,}\widehat{\gamma }\text{,}\widehat{\varphi }\text{,}{\widehat{\sigma }}_{\delta }^2\text{,}{\widehat{\sigma }}_{∊}^2)$ is an updated initial value in the iterative procedure. The initial values $h_0$ and ${\mathit{\psi }}_0$ might be affected by the estimation results of SVM. For an extensive discussion on assigning initial values in the QL estimation procedures, see Alzghool and Lin (2011).

3.2 Parameter estimation of SVM using the AQL method

Consider the SVM given by ((3.1.1)) and ((3.1.2)) and the same argument listed under ((3.1.2)). First, to estimate $h_t$ , the AQLEF sequence is given by $$G_{(t)}(h_t)=(-1\text{,}1){\mathbf{\Sigma }}_{t\text{,}n}^{-1}\left(\begin{array}{c}\hfill \ln (y_t^2)-h_t-\mu \hfill \\ \hfill h_t-\gamma -\varphi h_{t-1}\hfill \end{array}\right)$$

Given ${\widehat{h}}_0=0\text{,}{\mathit{\theta }}^{(0)}=({\gamma }_0\text{,}{\varphi }_0\text{,}{\mu }_0)$ , ${\mathbf{\Sigma }}_{t\text{,}n}^{(0)}={\mathbf{I}}_2$ , and ${\widehat{h}}_{t-1}$ is the AQL estimation of $h_{t-1}$ , the AQL estimation of $h_t$ is the solation of $G_{(t)}(h_t)=0$ ; that is,

Second, using the kernel estimation method, we find $${\widehat{\mathbf{\Sigma }}}_{t\text{,}n}({\mathit{\theta }}^{(0)})=\left(\begin{array}{cc}\hfill {\widehat{\sigma }}_n(y_t)\hfill & \hfill {\widehat{\sigma }}_n(y_t\text{,}{h}_t)\hfill \\ \hfill {\widehat{\sigma }}_n(h_t\text{,}{y}_t)\hfill & \hfill {\widehat{\sigma }}_n(h_t)\hfill \end{array}\right)\text{.}$$

Third, to estimate the parameters $\mathit{\theta }=(\gamma \text{,}\varphi \text{,}\mu )$ , we use $\{{\widehat{h}}_t\}$ and $\{y_t\}$ and the AQLEF sequence $$G_T(\gamma \text{,}\varphi \text{,}\mu )={\displaystyle \sum _{t=1}^T}\left(\begin{array}{cc}\hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill -{\widehat{h}}_{t-1}\hfill \end{array}\right){\widehat{\mathbf{\Sigma }}}_{t\text{,}n}^{-1}\left(\begin{array}{c}\hfill \ln (y_t^2)-{\widehat{h}}_t-\mu \hfill \\ \hfill {\widehat{h}}_t-\gamma -\varphi {\widehat{h}}_{t-1}\hfill \end{array}\right)\text{.}$$

The AQL estimation of $\gamma \text{,}\varphi$ , and $\mu$ is the solation of $G_T(\gamma \text{,}\varphi \text{,}\mu )=0$ . Then ${\mathit{\theta }}^{(0)}=({\gamma }_0\text{,}{\varphi }_0\text{,}{\mu }_0)$ is updated and replaced by the $\widehat{\mathit{\theta }}=(\widehat{\gamma }\text{,}\widehat{\varphi }\text{,}\widehat{\mu })$ , the estimate of $\mathit{\theta }=(\gamma \text{,}\varphi \text{,}\mu )$ . The estimation procedure will be iteratively repeated until it converges.

In the following, the setup for this simulation study is similar to the design used by Rodriguez-Yam (2003). Samples of size T = 500 are taken, and the mean and root mean squared errors (RMSE) for $\widehat{\varphi }\text{,}\widehat{\gamma }$ , $\widehat{\mu }\text{,}{\sigma }_{\delta }$ , and ${\sigma }_{∊}$ are calculated, where N = 1000 independent samples.

In Table 1, QL represents the QL estimate and AQL represents the AQL estimate.

The results in Table 1 confirm that QL and AQL have succeeded in SVM parameter estimation.

The effect of sample size on parameter estimation is considered. Samples of sizes $T=20\text{,}40\text{,}60\text{,}80$ , and 100 were generated.

The results in Table 2 show that the RMSE decreases when the sample size increases.