Threshold variable selection of asymmetric stochastic volatility models

Abstract

A threshold stochastic volatility (SV) model is used for capturing time-varying volatilities and nonlinearity. Two adaptive Markov chain Monte Carlo (MCMC) methods of model selection are designed for the selection of threshold variables for this family of SV models. The first method is the direct estimation which approximates the model posterior probabilities of competing models. Using parallel MCMC sampling to estimate these probabilities, the best threshold variable is selected with the highest posterior model probability. The second method is to use the deviance information criterion to compare among these competing models and select the best one. Simulation results lead us to conclude that for large samples the posterior model probability approximation method can give an accurate approximation of the posterior probability in Bayesian model selection. The method delivers a powerful and sharp model selection tool. An empirical study of five Asian stock markets provides strong support for the threshold variable which is formulated as a weighted average of important variables.

Appendix

By Bayesian inference, the posterior conditional distributions of all parameters can be conducted as follows:

• 1. ${\varvec{\phi }} \mid {\varvec{\theta }}_{-{\varvec{\phi }}}, {\varvec{R}}, {\varvec{X}}_1, {\varvec{H}}_n, {\varvec{\lambda }}_n \sim {\varvec{N}} \left( {\varvec{\mu }}_{{\varvec{\phi }}}, {\varvec{\Sigma }}_{{\varvec{\phi }}} \right)$, where ${\varvec{\Sigma }}_{{\varvec{\phi }}} = ({\varvec{Y}}_1^{\prime } {\varvec{Y}}_1 +{\varvec{\Lambda }}_1^{-1})^{-1}$ and ${\varvec{\mu }}_{{\varvec{\phi }}} = {\varvec{\Sigma }}_{{\varvec{\phi }}} ({\varvec{Y}}_1^{\prime } {\varvec{K}}_1 + {\varvec{\Lambda }}_1^{-1} {\varvec{\mu }}_1)$. The hyper-parameters ${\varvec{\mu }}_1$ and ${\varvec{\Lambda }}_1$ are the mean and variance of prior distribution of ${\varvec{\phi }}$. The matrices ${\varvec{K}}_1$ and ${\varvec{Y}}_1$ are

  $$\begin{aligned} {\varvec{K}}_1 \!=\! \left[ \begin{array}{c} c_{d+1}R_{d+1} \\ \vdots \\ c_{t}R_{t} \\ \vdots \\ c_{n}R_n \end{array} \right] \, \text{ and } \, {\varvec{Y}}_1 \!=\! \left[ \begin{array}{ccccccc} c_{d+1} &{} c_{d+1} s_{d+1} &{} c_{d+1} R_d &{} c_{d+1} s_{d+1} R_d &{} c_{d+1} x_d &{} c_{d+1} s_{d+1} x_d \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ c_t &{} c_t s_t &{} c_t R_{t-1} &{} c_t s_t R_{t-1} &{} c_t x_{t-1} &{} c_t s_t x_{t-1} \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ c_n &{} c_n s_n &{} c_n R_{n-1} &{} c_n s_n R_{n-1} &{} c_n x_{n-1} &{} c_n s_n x_{n-1} \\ \end{array} \right] , \end{aligned}$$

  where $c_t=\sqrt{\frac{\nu }{(\nu -2)} \frac{\lambda _{t}}{\sigma ^2_{t}}}$.

• 2. ${\varvec{\alpha }} \mid {\varvec{\theta }}_{-{\varvec{\alpha }}}, {\varvec{R}}, {\varvec{X}}_1, {\varvec{H}}_n, {\varvec{\lambda }}_n \sim {\varvec{N}} \left( {\varvec{\mu }}_{{\varvec{\alpha }}}, {\varvec{\Sigma }}_{{\varvec{\alpha }}} \right) I_{(|\alpha _1|<1)} I_{(|\alpha _1+\beta _1|<1)}$, where ${\varvec{\Sigma }}_{{\varvec{\alpha }}} = ({\varvec{Y}}_2^{\prime } {\varvec{Y}}_2 + {\varvec{\Lambda }}_2^{-1})^{-1}$ and ${\varvec{\mu }}_{{\varvec{\alpha }}} = {\varvec{\Sigma }}_{{\varvec{\alpha }}} ({\varvec{Y}}_2^{\prime } {\varvec{K}}_2 + {\varvec{\Lambda }}_2^{-1} {\varvec{\mu }}_2)$. The hyper-parameters ${\varvec{\mu }}_2$ and ${\varvec{\Lambda }}_2$ are the mean and variance of prior distribution of ${\varvec{\alpha }}$. The matrices ${\varvec{K}}_2$ and ${\varvec{Y}}_2$ are

  $$\begin{aligned} {\varvec{K}}_2 = \left[ \begin{array}{c} h_{d+1} \\ \vdots \\ h_n \end{array} \right] \quad \text{ and } \quad {\varvec{Y}}_2 = \frac{1}{\sqrt{\sigma ^2}} \left[ \begin{array}{cccc} 1 &{} s_{d+1} &{} h_d &{} s_{d+1}h_d \\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ 1 &{} s_n &{} h_{n-1} &{} s_n h_{n-1} \end{array} \right] . \nonumber \end{aligned}$$

• 3. $f\left( r \mid {\varvec{\theta }}_{-r}, {\varvec{R}}, {\varvec{X}}_1, {\varvec{H}}_n, {\varvec{\lambda }}_n \right) \propto L\left( {\varvec{R}} \mid {\varvec{X}}_1, {\varvec{\theta }}, {\varvec{H}}_n, {\varvec{\lambda }}_n \right) I_{(l \le r \le u)}$.

• 4. $Pr\left( d=j| {\varvec{\theta }}_{-d}, {\varvec{R}}, {\varvec{X}}_1, {\varvec{H}}_n, {\varvec{\lambda }}_n \right) = \frac{ L\left( {\varvec{R}}| {\varvec{X}}_1, {\varvec{H}}_n, {\varvec{\lambda }}_n, {\varvec{\theta }}_{-d}, d=j \right) \pi (d=j)}{\sum _{k=1}^{d_0} L\left( {\varvec{R}}| {\varvec{X}}_1, {\varvec{H}}_n, {\varvec{\lambda }}_n, {\varvec{\theta }}_{-d}, d=k \right) \pi (d=k)}$.

• 5. $f\left( \nu \mid {\varvec{\theta }}_{-\nu }, {\varvec{R}}, {\varvec{X}}_1, {\varvec{H}}_n \right) \propto \prod _{t = d_0 + 1}^n \frac{\Gamma (\frac{\nu +1}{2})}{\Gamma (\frac{\nu }{2})\sqrt{(\nu -2)\pi }}\frac{1}{\sqrt{\sigma ^2_t}} \left( 1+\frac{a_{t}^{2}}{(\nu -2) \sigma ^2_t } \right) ^{- \frac{\nu + 1}{2}} \pi (\nu )$.

• 6. $\lambda _t \mid {\varvec{\theta }}, {\varvec{R}}, {\varvec{X}}_1, {\varvec{H}}_n \sim Gamma\left( \frac{\nu +1}{2},\frac{\nu }{2}+\frac{\nu a_t^{2}}{2(\nu -2)\sigma ^2_t} \right)$.

• 7. $f\left( \omega _1 | {\varvec{\theta }}_{-{\varvec{\omega }}}, {\varvec{R}}, {\varvec{X}}_1, {\varvec{H}}_n, {\varvec{\lambda }}_n \right) \propto L({\varvec{R}} | {\varvec{X}}_1, {\varvec{H}}_n, {\varvec{\lambda }}_n, {\varvec{\theta }}) \pi (\omega _1)$.

• 8. $f\left( \sigma ^2 \mid {\varvec{\theta }}_{-\sigma ^2}, {\varvec{R}}, {\varvec{X}}_1 \right) \propto f\left( {\varvec{R}} \mid {\varvec{X}}_1, {\varvec{\theta }} \right) I_{(0 \le \sigma ^2 \le b)}$.

• 9. $h_t \mid h_{t+1}, {\varvec{\theta }}, {\varvec{\rho }}_n, {\varvec{R}}, {\varvec{X}}_1, {\varvec{\lambda }}_n \sim N\left( \mu ^*, \Sigma ^* \right)$, where $\mu ^*$ and $\Sigma ^*$ are computed by Kalman filtering algorithm. It is based on a approximated Gaussian state space model, the approximation of Gaussian distribution is suggested by Kim et al. (1998) with introducing a vector of mixing indicator ${\varvec{\rho }}_n$ for the mixture of normals. For more details, refer to Chen et al. (2008b).