Multivariate Wishart stochastic volatility and changes in regime

Abstract

This paper generalizes the basic Wishart multivariate stochastic volatility model of Philipov and Glickman (J Bus Econ Stat 24:313–328, 2006) and Asai and McAleer (J Econom 150:182–192, 2009) to encompass regime-switching behavior. The latent state variable is driven by a first-order Markov process. The model allows for state-dependent (co)variance and correlation levels and state-dependent volatility spillover effects. Parameter estimates are obtained using Bayesian Markov Chain Monte Carlo procedures and filtered estimates of the latent variances and covariances are generated by particle filter techniques. The model is applied to five European stock index return series. The results show that the proposed regime-switching specification substantially improves the fit to persistent covariance dynamics relative to the basic model.

Appendix

1.1 Full conditional distributions: basic WMSV model

The basic WMSV model is outlined in Eqs. (1), (2) and (5). The joint prior distribution is assumed to factor into the product of marginal prior distributions given by

• 1. a Wishart prior $\pi _{A^{-1}}(Q_0,\gamma _0)$ for $A^{-1}$ with scale matrix $Q_0$ and d.o.f. parameter $\gamma _0$;

• 2. a uniform prior $\pi _{d}(0,1)$ on [0, 1] for d;

• 3. a gamma prior $\pi _{\nu }(\alpha _0,\beta _0)$ for $\nu -k$ with shape parameter $\alpha _0$ and scale parameter $\beta _0$.

Denote the augmented parameter vector by $\theta ^{\text {aug}}=(\theta ',\text {vech}(\Sigma _1)',\ldots ,\text {vech}(\Sigma _T)')$ and the vector of remaining model parameters for each parameter block $\theta _k$ by $\theta ^{\text {aug}}_{-}=(\theta _1,\ldots ,\theta _{k-1},\theta _{k+1},\ldots ,\theta _K)'$. The full conditional distributions are obtained as follows:

(more details on the derivation of the full conditional densities are given in an additional web-appendix available at http://www.wisostat.uni-koeln.de/de/institut/mitarbeiter/gribisch/)

Full conditional distribution of $\Sigma _t^{-1}$:

For notational convenience suppressing dependence on model parameters, the kernel of the full conditional distribution of $\Sigma _t^{-1}$ is obtained as

$$\begin{aligned} p(\Sigma _t^{-1}|\theta ^{\text {aug}}_{-})\propto & {} {\mathcal W}_k^{\kappa }(\Sigma _t^{-1}|\tilde{\nu },\tilde{S}_{t}) \times f(\Sigma _t^{-1}), \end{aligned}$$

(34)

where ${\mathcal W}_k^{\kappa }(\Sigma _t^{-1}|\cdot )$ denotes a Wishart kernel in $\Sigma _t^{-1}$ and $\tilde{\nu }= \nu (1-d)+1, \tilde{S}_{t} = (S_{t}^{-1}+\xi _t \xi _t')^{-1}, f(\Sigma _t^{-1}) = \exp \{ -0.5\ \text {tr}[S_{t+1}^{-1}\Sigma _{t+1}^{-1}] \}, S_t = \Sigma _{t}^{-d/2}A\Sigma _{t}^{-d/2}$. The full conditional distribution of $\Sigma _t^{-1}$ is known up to an integrating constant. Hence, the Metropolis–Hastings (MH) algorithm¹⁶ is applied to obtain samples from $p(\Sigma _t^{-1}|\theta ^{\text {aug}}_{-})$. The proposal density is ${\mathcal W}_k(\nu ,\tilde{S}_{t})$.

Full conditional distribution of $A^{-1}$:

The full conditional distribution of $A^{-1}$ is Wishart since

$$\begin{aligned} p(A^{-1}|\theta ^{\text {aug}}_{-})\propto & {} \pi _{A^{-1}}(Q_0,\gamma _0) \times {\mathcal W}_k^{\kappa }(A^{-1}|\gamma ,U), \end{aligned}$$

(35)

where $U^{-1} = \nu \sum _{t=1}^{T} \Sigma _{t-1}^{d/2}\Sigma _{t}^{-1} \Sigma _{t-1}^{d/2}$ and $\gamma = T\nu +k+1$. Hence

$$\begin{aligned} p(A^{-1}|\theta ^{\text {aug}}_{-})\propto & {} |A^{-1}|^{(\gamma _0+\gamma -2k-2)/2}\ \exp \{ -0.5\ \text {tr}[(Q_0^{-1}+U^{-1})A^{-1}] \}. \end{aligned}$$

(36)

Therefore, $A^{-1}|\theta ^{\text {aug}}_{-} \sim {\mathcal W}_k(\tilde{\gamma }, \tilde{U})$, where $\tilde{U}^{-1} = Q_0^{-1}+U^{-1}$ and $\tilde{\gamma } = \gamma _0+\gamma -k-1$.

Full conditional distribution of $\nu$ and d:

The full conditional distributions of the parameters $\nu$ and d are not obtained in closed form and simulation is carried out by applying the Metropolis–Hastings algorithm. Since $\nu > k$ and $d \in (0,1)$, truncated normal proposal densities are used where mean and variance are given by the optimum and the corresponding Hessian obtained after numerically optimizing the posterior distribution’s density kernel.

The respective kernels are obtained as

$$\begin{aligned} p(d|\theta ^{\text {aug}}_{-})\propto & {} \exp \left\{ \mathrm{d}\psi -0.5\ \text {tr}\left[ Q(d)A^{-1}\right] \right\} \end{aligned}$$

(37)

$$\begin{aligned} p(\nu |\theta ^{\text {aug}}_{-})\propto & {} \exp \{ (\alpha -1)\ln (\nu -k)-\beta (\nu -k) \}\nonumber \\&\times \left( \frac{ |\nu A^{-1}|^{\nu /2} }{ 2^{\nu k/2}\prod _{j=1}^k \Gamma \left( (\nu -j+1)/2\right) } \right) ^T \nonumber \\&\times \prod _{t=1}^T |Q_t^{-1}|^{\nu /2} \exp \{ -0.5\ \text {tr}\left[ Q^{-1}A^{-1}\right] \}, \end{aligned}$$

(38)

where $\psi = -\frac{\nu }{2} \sum _{t=1}^{T}\ln (|\Sigma _{t-1}^{-1}|), Q(d) = \sum _{t=1}^{T} \nu \Sigma _{t-1}^{d/2}\Sigma _{t}^{-1}\Sigma _{t-1}^{d/2}, Q_t^{-1}=\Sigma _{t-1}^{d/2} \Sigma _t^{-1}\Sigma _{t-1}^{d/2}$ and $Q^{-1} = \nu \sum _{t=1}^{T} \Sigma _{t-1}^{d/2} \Sigma _t^{-1} \Sigma _{t-1}^{d/2}$.

1.2 Full conditional distributions: Markov switching MWSV model

The MS WMSV model is outlined in Eqs. (10), (11) and (14). The joint prior distribution is assumed to factor into the product of marginal prior distributions. Given the state sequence $s=(s_1,s_2,\ldots ,s_T)'$, the derivation of the full conditional distributions for $\Sigma _t^{-1}, A_1, A_2, \nu$ and d is analogous to the illustrations of the previous section, except that we have to condition on $A_{s_t}\ \forall t\in \{1,\ldots ,T\}$ instead of A.

Full conditional distribution of $s=(s_1,s_2,\ldots ,s_T)'$:

Denoting $\underline{\Sigma }_t^{-1}=\{\Sigma _1^{-1},\ldots ,\Sigma _t^{-1}\}$ and exploiting the Markov property of $s_t$, the full conditional density of the state vector s can be factorized as

$$\begin{aligned} p(s|\theta ^{\text {aug}}_{-})= & {} P(s|\underline{\Sigma }_T^{-1},\theta ) \nonumber \\= & {} P(s_T|\underline{\Sigma }_T^{-1},\theta )\times P(s_{T-1}|s_T,\underline{\Sigma }_T^{-1},\theta )\times \cdots \times P(s_1|s_2,\underline{\Sigma }_T^{-1},\theta )\nonumber \\= & {} P(s_T|\underline{\Sigma }_T^{-1},\theta )\times P(s_{T-1}|s_T,\underline{\Sigma }_{T-1}^{-1},\theta )\times \cdots \times P(s_1|s_2,\underline{\Sigma }_1^{-1},\theta ).\quad \quad \quad \end{aligned}$$

(39)

The conditional probabilities

$$\begin{aligned} P(s_t|s_{t+1},\underline{\Sigma }_t^{-1},\theta )=\frac{P(s_{t+1}|s_t) \times P(s_t|\underline{\Sigma }_t^{-1},\theta )}{P(s_{t+1}|\underline{\Sigma }_t^{-1},\theta )} \end{aligned}$$

(40)

are obtained by the “Hamilton filter” which—given a starting value for $P(s_{0}|\underline{\Sigma }_{0}^{-1},\theta )$ (e.g. stationary probabilities, see Hamilton 1994, p. 683)—proceeds recursively in five steps $\forall t \in \{1,\ldots ,T\}$:

$$\begin{aligned}&I \qquad P(s_t,s_{t-1}|\underline{\Sigma }_{t-1}^{-1},\theta )=P(s_t|s_{t-1}) \times P(s_{t-1}|\underline{\Sigma }_{t-1}^{-1},\theta ) \end{aligned}$$

(41)

$$\begin{aligned}&{ II} \qquad P(s_t|\underline{\Sigma }_{t-1}^{-1},\theta ) =\sum _{s_{t-1}} P(s_t,s_{t-1}|\underline{\Sigma }_{t-1}^{-1},\theta ) \end{aligned}$$

(42)

$$\begin{aligned}&{ III} \qquad f(\Sigma _t^{-1},s_t|\underline{\Sigma }_{t-1}^{-1},\theta )=f(\Sigma _t^{-1}|s_t,\Sigma _{t-1}^{-1},\theta ) \times P(s_t|\underline{\Sigma }_{t-1}^{-1},\theta ) \end{aligned}$$

(43)

$$\begin{aligned}&{ IV} \qquad f(\Sigma _t^{-1}|\underline{\Sigma }_{t-1}^{-1},\theta )=\sum _{s_t} f(\Sigma _t^{-1},s_t|\underline{\Sigma }_{t-1}^{-1},\theta )\end{aligned}$$

(44)

$$\begin{aligned}&V \qquad P(s_t|\underline{\Sigma }_t^{-1},\theta )= \frac{f(\Sigma _t^{-1},s_t|\underline{\Sigma }_{t-1}^{-1},\theta )}{f(\Sigma _t^{-1}|\underline{\Sigma }_{t-1}^{-1},\theta )}. \end{aligned}$$

(45)

The whole state sequence $s=(s_1,s_2,\ldots ,s_T)'$ can then be sampled backward recursively based on Eq. (39).

Full conditional distributions of $e_1$ and $e_2$:

Using beta prior distributions $\pi _{e_i}(\alpha _{i,0},\beta _{i,0}), i\in \{1,2\}$, the kernel of the full conditional distribution of $e_i$ is obtained as

$$\begin{aligned} p(e_i|\theta ^{\text {aug}}_{-})\propto & {} e_i^{\alpha _{i,0}-1}(1-e_i)^{\beta _{i,0}-1} \times e_i^{g_i}(1-e_i)^{h_i}, \end{aligned}$$

(46)

where $g_i$ denotes the number of switches from state i to state $i-$ (not state i) and $h_i$ denotes the number of periods where the state does not change. The full conditional distribution of $e_i$ is therefore beta with parameters $\alpha _{i}=\alpha _{i,0}+g_i$ and $\beta _{i}=\beta _{i,0}+h_i, i\in \{1,2\}$.