Skew exponential power stochastic volatility model for analysis of skewness, non-normal tails, quantiles and expectiles

Abstract

This paper proposes a unified framework to analyse the skewness, tail heaviness, quantiles and expectiles of the return distribution based on a stochastic volatility model using a new parametrisation of the skew exponential power (SEP) distribution. The SEP distribution can express a wide range of distribution shapes through two shape parameters and one skewness parameter. Since the asymmetric Laplace and skew normal distributions are included as special cases, the proposed model is related to quantile regression and expectile regression. The efficient and simple Markov chain Monte Carlo estimation methods are also described. The proposed model is demonstrated using the simulated data and real data on daily return of foreign exchange rate.

Appendices

Appendix 1: SEP distribution and double two-piece family

Rubio and Steel (2013) proposed the double two-piece (DTP) family. Let $f(y|\delta ,\sigma ,\theta )=\frac{1}{\sigma }f(\frac{y-\delta }{\sigma }|\theta )$ be the probability density function for the family of of continuous, unimodal, symmetric distribution where $\delta \in \mathbb {R}$ is the mode and the location parameter, $\sigma >0$ is the scale parameter, and $\theta$ is the shape parameter. Then the probability density function of the DTP family is given by

$$\begin{aligned} f_{DTP}(y|\delta ,\sigma _1,\sigma _2,\theta _1,\theta _2)= & {} \frac{2\epsilon }{\sigma _1}f\left( \frac{y-\delta }{\sigma _1}\Big |\theta _1\right) I(y<\delta )\nonumber \\&+\frac{2(1-\epsilon )}{\sigma _2}f\left( \frac{y-\delta }{\sigma _2}\Big |\theta _2\right) I(y\ge \delta ), \end{aligned}$$

(9)

where $I(\cdot )$ is the indicator function and

$$\begin{aligned} \epsilon =\frac{\sigma _1f(0|\theta _2)}{\sigma _1f(0|\theta _2)+\sigma _2f(0|\theta _1)}. \end{aligned}$$

The density (9) is continuous, unimodal with mode at $\delta$, and the amount of mass to the left of its mode is given by $\epsilon$.

By letting $\sigma _1=\frac{\sigma }{(1-p)^{1/\theta _1}}$ and $\sigma _2=\frac{\sigma }{p^{1/\theta _2}}$, the probability density function of the SEP distribution (1) becomes the parameterisation of Naranjo et al. (2012):

$$\begin{aligned} f_{SEP}(y|\delta , \sigma _1,\sigma _2, \theta _1,\theta _2)= & {} c^{-1} \left\{ \begin{array}{ll} \exp \left\{ -\left| \frac{y-\delta }{\sigma _1}\right| ^{\theta _1} \right\} , \ &{}\quad \text {if} \quad y<\delta , \\ \exp \left\{ -\left| \frac{y-\delta }{\sigma _2}\right| ^{\theta _2} \right\} , \ &{}\quad \text {otherwise}, \\ \end{array} \right. \nonumber \\ c= & {} \sigma _1\Gamma (1+1/\theta _1) + \sigma _2\Gamma (1+1/\theta _2). \end{aligned}$$

(10)

It can be immediately seen that (10) belongs to the DTP family based on the exponential power distribution given by (6).

Appendix 2: MCMC methods for SEPSV

1.1 Single-move sampler with MH updates for $\psi _0$ and $\psi _1$

Using the mixture representation for the SEP distribution (7), the return equation can be rewritten as

$$\begin{aligned} y_t|u_{1t}&\sim U\left( \psi _0+\psi _1y_{t-1}-\frac{e^{h_t/2}}{s_{\varvec{\theta }}}\left( \frac{u_{1t}}{1-p}\right) ^{1/\theta _1}, \psi _0+\psi _1y_{t-1}\right) \ \quad \text {with probability } \pi _1,\nonumber \\ y_t|u_{2t}&\sim U\left( \psi _0+\psi _1y_{t-1}, \psi _0+\psi _1y_{t-1}+\frac{e^{h_t/2}}{s_{\varvec{\theta }}}\left( \frac{u_{2t}}{p}\right) ^{1/\theta _2}\right) \ \quad \text {with probability } \pi _2, \end{aligned}$$

(11)

where $u_1\sim Ga(1+1/\theta _1, 1)$, $u_2\sim Ga(1+1/\theta _2, 1)$, and $\pi _1$ and $\pi _2$ are defined in (2).

The sampling scheme described here samples the values of $\psi _0$, $\psi _1$, $\theta _1$, $\theta _2$, p, $\mu$, $\phi$, $\tau ^2$, $\{u_{1t}\}_{t=1}^T$, $\{u_{2t}\}_{t=1}^T$, and $\{h_t\}_{t=1}^T$ alternately from their full conditional distributions (FCDs). The single-move sampler samples each $h_t$ at a time from its FCD.

Sampling of $\psi _0$, $\psi _1$, $\theta _1$, $\theta _2$, and p We integrate out the latent variables $u_{1t}$ and $u_{2t}$ (see, e.g., Park and van Dyk 2008) and implement the Metropolis–Hastings (MH) algorithm which samples the following three blocks: $\psi _0$, $\psi _1$, and ${\varvec{\theta }}=(\theta _1,\theta _2,p)$. The acceptance probability of the MH algorithm for $\psi _0$ is given by

$$\begin{aligned} \min \left\{ 1,\frac{\prod _{t=1}^Tf_{{\textit{SEP}}}(y_t|\psi _0^*+\psi _1y_{t-1},s_{{\varvec{\theta }}}^{-1} e^{h_t/2}, \theta _1,\theta _2,p)\pi (\psi _0^*)q(\psi _0|\psi ^*)}{\prod _{t=1}^Tf_{{\textit{SEP}}}(y_t|\psi _0+\psi _1y_{t-1},s_{{\varvec{\theta }}}^{-1} e^{h_t/2}, \theta _1,\theta _2,p)\pi (\psi _0)q(\psi _0^*|\psi _0)}\right\} , \end{aligned}$$

where $\psi _0^*$ is the proposed value, $q(\cdot |\cdot )$ is the proposal density, and $\pi (\cdot )$ is the prior density. Similarly, the acceptance probabilities are given by

$$\begin{aligned} \min \left\{ 1,\frac{\prod _{t=1}^Tf_{{\textit{SEP}}}(y_t|\psi _0+\psi _1^*y_{t-1},s_{{\varvec{\theta }}}^{-1} e^{h_t/2}, \theta _1,\theta _2,p)\pi (\psi _1^*)q(\psi _1|\psi _1^*)}{\prod _{t=1}^Tf_{{\textit{SEP}}}(y_t|\psi _0+\psi _1y_{t-1},s_{{\varvec{\theta }}}^{-1} e^{h_t/2}, \theta _1,\theta _2,p)\pi (\psi _1)q(\psi _1^*|\psi _1)}\right\} , \end{aligned}$$

for $\psi _1$, and

$$\begin{aligned} \min \left\{ 1,\frac{\prod _{t=1}^Tf_{{\textit{SEP}}}(y_t|\psi _0+\psi _1y_{t-1},s_{{\varvec{\theta }}^*}^{-1} e^{h_t/2}, \theta _1^*,\theta _2^*,p^*)\pi ({\varvec{\theta }}^*)q({\varvec{\theta }}|{\varvec{\theta }}^*)}{\prod _{t=1}^Tf_{{\textit{SEP}}}(y_t|\psi _0+\psi _1y_{t-1},s_{{\varvec{\theta }}}^{-1} e^{h_t/2}, \theta _1,\theta _2,p)\pi ({\varvec{\theta }})q({\varvec{\theta }}^*|{\varvec{\theta }})}\right\} , \end{aligned}$$

for ${\varvec{\theta }}$, where $\psi _1^*$ and ${\varvec{\theta }}^*$ are the proposed values. We use the normal distribution centred at the current state as the proposal distribution. The variance of the proposal distribution is chosen such that the acceptance rate is between 0.2 and 0.4.

Sampling of $u_{1t}$ and $u_{2t}$ The FCD of $u_{1t}$ is the exponential distribution with mean 1 truncated to the region in which the value of $u_{1t}$ satisfies the mixture representation as

$$\begin{aligned} u_{1t}|-\sim Exp(1) I\left[ (1-p)\left( \frac{s_{\varvec{\theta }}(\psi _0+\psi _1y_{t-1}-y_t)}{e^{h_t/2}}\right) ^{\theta _1}<u_{1t}\right] ,\quad \text {if}\quad y_t<\psi _0+\psi _1y_{t-1}, \end{aligned}$$

(12)

for $t=1,\dots ,T$ where $u_{1t}|-$ denotes $u_{1t}$ conditional on all other variables. The value of $u_{2t}$ is sampled in a similar manner from

$$\begin{aligned} u_{2t}|-\sim Exp(1) I\left[ p\left( \frac{s_{\varvec{\theta }}(y_t-\psi _0-\psi _1y_{t-1})}{e^{h_t/2}}\right) ^{\theta _2}<u_{2t}\right] ,\quad \text {if}\quad y_t\ge \psi _0+\psi _1y_{t-1}. \end{aligned}$$

(13)

See also Naranjo et al. (2012, 2015).

Sampling of $h_t$ Utilising the mixture representation for the SEP distribution, the FCD for $h_t$ is the truncated normal distribution. Therefore, no MH type update is required.

The FCD of $h_t$ is given by

$$\begin{aligned} h_t|-\sim TN_{(\underline{h_t},\infty )}(m_t,s_t), \quad t=1,\dots ,T, \end{aligned}$$

where $TN_{(a,b)}(\mu ,\sigma )$ denotes the normal distribution with mean $\mu$ and variance $\sigma$ truncated on the interval (a, b),

$$\begin{aligned} m_t= & {} \left\{ \begin{array}{ll} \mu +\phi (h_2-\mu )-\frac{\tau ^2}{2}, \ &{}t=1, \\ \mu +\frac{\phi ((h_{t-1}-\mu ) + ( h_{t+1} -\mu )) -\tau ^2/2}{1+\phi ^2}, \ &{}\quad t = 2,\dots ,T-1, \\ \mu + \phi (h_{T-1}-\mu )-\frac{\tau ^2}{2}, \ &{}\quad t = T \end{array} \right. \\ s_t= & {} \left\{ \begin{array}{ll} \tau ^2, \ &{}t=1, T \\ \frac{\tau ^2}{1+\phi ^2}, \ &{}\quad t = 2,\dots ,T-1, \\ \end{array} \right. \\ \underline{h_t}= & {} \left\{ \begin{array}{ll} 2\left[ \log (s_{\varvec{\theta }})+\log (\psi _0+\psi _1y_{t-1}-y_t) +\frac{1}{\theta _1}\left( \log (1-p) - \log u_{1t} \right) \right] , &{}\quad \text {if} \ y_t<\psi _0+\psi _1y_{t-1},\\ 2\left[ \log (s_{\varvec{\theta }})+\log ( y_t-\psi _0-\psi _1y_{t-1}) + \frac{1}{\theta _2}\left( \log p-\log u_{2t}\right) \right] , &{}\quad \text {otherwise.} \\ \end{array} \right. \end{aligned}$$

A similar method can be found, e.g., in Choy et al. (2008), where the SV model with Student’s t distribution was estimated using the scale mixture of uniforms representation.

Sampling of $\mu$ The FCD of $\mu$ is given by

$$\begin{aligned} \mu |-\sim N(m_1, M_1), \end{aligned}$$

where

$$\begin{aligned} m_1= & {} M_1\left[ \frac{1-\phi ^2}{\tau ^2}h_1 + \frac{1-\phi }{\tau ^2}\sum _{t=2}^T(h_t-\phi h_{t-1})+ \frac{m_0}{M_0}\right] ,\\ \quad \text {and}\quad M_1= & {} \left[ \frac{(T-1)(1-\phi )^2}{\tau ^2} + \frac{(1-\phi ^2)}{\tau ^2} + \frac{1}{M_0}\right] ^{-1}. \end{aligned}$$

Sampling of $\tau ^2$ The FCD of $\tau ^2$ is given by

$$\begin{aligned} \tau ^2|-\sim IG\left( \frac{n_1}{2}, \frac{S_1}{2}\right) , \end{aligned}$$

where

$$\begin{aligned} n_1\!=\!n_0 + T, \quad \text {and}\quad S_1\!=\!S_0 + (h_1-\mu )^2(1-\phi ^2) \!+\! \sum _{t=2}^T(h_t-\mu -\phi (h_{t-1}-\mu ))^2. \end{aligned}$$

Sampling of $\phi$ The FCD of $\phi$ is given by

$$\begin{aligned} \pi (\phi ^*|-)\propto \pi (\phi )\times \sqrt{1-\phi ^2} \times N( k_1, t_1), \end{aligned}$$

where

$$\begin{aligned} k_1= \frac{t_1}{\tau ^2} \sum _{t=2}^T ( h_t-\mu )( h_{t-1}-\mu ), \quad \text {and} \quad t_1 = \frac{\tau ^2}{\sum _{t=2}^{T-1}( h_t-\mu )^2}. \end{aligned}$$

We can sample from this FCD using the independence MH algorithm with the $N(k_1,t_1)$ proposal distribution and the acceptance probability is given by

$$\begin{aligned} \min \left\{ 1,\frac{\pi (\phi ^*)\sqrt{1-\phi ^{*2}}}{\pi (\phi )\sqrt{1-\phi ^2}}\right\} . \end{aligned}$$

1.2 Single-move sampler with Gibbs updates for $\psi _0$ and $\psi _1$

An alternative sampling method draws the values of $\psi _0$ and $\psi _1$ from the FCDs derived directly from the mixture representation (11) as Naranjo et al. (2012, 2015). The rest of the steps for $\theta _1$, $\theta _2$, p, $\mu$, $\phi$, $\tau ^2$, $\{u_{1t}\}_{t=1}^T$, $\{u_{2t}\}_{t=1}^T$, and $\{h_t\}_{t=1}^T$ is identical to the sampler described previously.

Sampling of $\psi _0$ The FCD of $\psi _0$ is the truncated normal distribution given by

$$\begin{aligned} \psi _0|-\sim TN_{(\underline{\psi _0}, \overline{\psi _0})}(v_0, V_0), \end{aligned}$$

where

$$\begin{aligned} \underline{\psi _0}= & {} \max \left\{ \max _{t:u_{2t}>0}\left\{ y_t-\psi _1y_{t-1}-\frac{e^{h_t/2}}{s_{\varvec{\theta }}}\left( \frac{u_{2t}}{p}\right) ^{1/\theta _2}\right\} ,\quad \max _{t:u_{1t}>0,\#\left\{ u_{2t}>0\right\} =0}\left\{ y_t-\psi _1y_{t-1}\right\} \right\} , \\ \overline{\psi _0}= & {} \min \left\{ \min _{t:u_{1t}>0}\left\{ y_t-\psi _1y_{t-1}+\frac{e^{h_t/2}}{s_{\varvec{\theta }}}\left( \frac{u_{1t}}{1-p}\right) ^{1/\theta _1}\right\} ,\quad \min _{t:u_{2t}>0,\#\left\{ u_{1t}>0\right\} =0}\left\{ y_t-\psi _1y_{t-1}\right\} \right\} . \end{aligned}$$

Sampling of $\psi _1$ The FCD of $\psi _1$ is given by

$$\begin{aligned} \psi _1|-\sim U(\underline{\psi _1}, \overline{\psi _1}) \end{aligned}$$

where

$$\begin{aligned} \underline{\psi _1}= & {} \max \left\{ \max _{t:u_{1t}>0,y_{t-1}<0 }\left\{ \frac{y_t-\psi _0+\frac{e^{h_t/2}}{s_{\varvec{\theta }}} \left( \frac{u_{1t}}{1-p}\right) ^{1/\theta _1}}{y_{t-1}}\right\} ,\right. \\&\left. \max _{t:u_{2t}>0,y_{t-1}>0}\left\{ \frac{y_t-\psi _0-\frac{e^{h_t/2}}{s_{\varvec{\theta }}}\left( \frac{u_{2t}}{p}\right) ^{1/\theta _2}}{y_{t-1}}\right\} , -1\right\} , \\ \overline{\psi _1}= & {} \min \left\{ \min _{t:u_{1t}>0, y_{t-1}>0}\left\{ \frac{y_t-\psi _0+\frac{e^{h_t/2}}{s_{\varvec{\theta }}}\left( \frac{u_{1t}}{1-p}\right) ^{1/\theta _1}}{y_{t-1}}\right\} , \right. \\&\left. \min _{t:u_{2t}>0,y_{t-1}<0}\left\{ \frac{y_t-\psi _0-\frac{e^{h_t/2}}{s_{\varvec{\theta }}} \left( \frac{u_{2t}}{p}\right) ^{1/\theta _2}}{y_{t-1}}\right\} , 1\right\} . \end{aligned}$$

1.3 Block sampler

A block sampler (Shephard and Pitt 1997; Watanabe and Omori 2004; Omori and Watanabe 2008) samples $(h_s,\dots ,s_{s+m-1})$ in blocks in stead of sampling each $h_t$ at a time. Let $\alpha _t=h_t-\mu$ and $\psi _t=\psi _0+\psi _1y_{t-1}$. We divide $(\alpha _1,\dots ,\alpha _T)$ into $K+1$ blocks, $(\alpha _{k_{i-1}+1},\dots ,\alpha _{k_i})$ for $i=1,\dots ,K+1$, with $k_0=0$, $k_{K+1}=T$, and $k_i-k_{i-1}\ge 2$. How to determine K and $k_i$ are described below.

The posterior distribution of ${\varvec{\eta }}=( \eta _s,\dots ,\eta _{s+m-1})$ conditional on $\Xi =(\alpha _s, \alpha _{s+m}, \psi _0,\psi _1,\theta _1,\theta _2,p,\mu ,\tau ^2,\phi , \mathbf{y})$ is given by

$$\begin{aligned} \pi ({\varvec{\eta }}|\Xi ) \propto \prod _{t=s}^{s+m-1}f_{{\textit{SEP}}}(y_t|\psi _t, e^{h_t/2}s^{-1}_{\varvec{\theta }},\theta _1,\theta _2,p) \prod _{t=s}^{s+m-1}f(\eta _t) \times f(\alpha _{s+m}), \end{aligned}$$

(14)

where

$$\begin{aligned} f(\eta _t)=\left\{ \begin{array}{ll} \exp \left\{ -\frac{-(1-\phi ^2)\eta _1^2}{2\tau ^2}\right\} , &{}\quad \text {if}\quad t=1, \\ \exp \left\{ -\frac{\eta _t^2}{2\tau ^2}\right\} , &{} \quad \text {otherwise},\\ \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} f(\alpha _{s+m+1})=\left\{ \begin{array}{c@{\quad }l} \exp \left\{ -\frac{(\alpha _{s+m+1}-\phi \alpha _{s+m})^2}{2\tau ^2}\right\} , \quad &{}\text {if} \quad s+m<T,\\ 1\quad &{}\text {if} \quad s+m=T. \end{array} \right. \end{aligned}$$

We sample ${\varvec{\eta }}$ from (14) using the acceptance–rejection (AR) MH algorithm. To construct a proposal distribution, consider the second-order Taylor expansion of

$$\begin{aligned} \ell _t= & {} \log f_{{\textit{SEP}}}(y_t|\psi _t,e^{h_t/2}s^{-1}_{\varvec{\theta }},\theta _1,\theta _2,p) \\= & {} \text {const} - \frac{\alpha _t}{2}-e^{-\frac{\theta _1\alpha _t}{2}}g_1(y_t)I(y_t<\psi _t) -e^{-\theta _2\alpha _t/2}g_2(y_t)I(y_t\ge \psi _t), \end{aligned}$$

around the mode $\hat{\alpha }_t$ (hence $\hat{\eta }_t$) such that

$$\begin{aligned} \ell _t \approx \widehat{\ell }_t + (\alpha _t-\hat{\alpha }_t)\widehat{\ell }_t' +\frac{1}{2}(\alpha _t-\hat{\alpha }_t)^2\widehat{\ell }''_t, \end{aligned}$$

where

$$\begin{aligned}&\ell '_t=\frac{\partial \ell _t}{\partial \alpha _t} =-\frac{1}{2}+\frac{\theta _1}{2}e^{-\frac{\theta _1\alpha _t}{2}}g_1(y_t)I(y_t<\psi _t)+\frac{\theta _2}{2}e^{-\frac{\theta _2\alpha _t}{2}}g_2(y_t)I(y_t\ge \psi _t),\\&\ell ''_t=\frac{\partial ^2\ell _t}{\partial \alpha _t^2} =-\frac{\theta _1^2}{4}e^{-\frac{\theta _1\alpha _t}{2}}g_1(y_t)I(y_t<\psi _t) - \frac{\theta _2^2}{4}e^{-\frac{\theta _2\alpha _t}{2}}g_2( y_t)I( y_t\ge \psi _t),\\&g_1(y_t)=e^{-\frac{\theta _1\mu }{2}}(1-p)\left| s_{\varvec{\theta }}( y_t-\psi _t)\right| ^{\theta _1}, \\&g_2(y_t)=e^{-\frac{\theta _2\mu }{2}}p\left| s_{\varvec{\theta }}(y_t-\psi _t)\right| ^{\theta _2}, \end{aligned}$$

$\widehat{\ell }_t=\ell _t|_{\alpha _t=\hat{\alpha }_t}$, $\widehat{\ell }'_t=\ell '_t|_{\alpha _t=\hat{\alpha }_t}$, and $\widehat{\ell }''_t=\ell ''_t|_{\alpha _t=\hat{\alpha }_t}$. How to obtain the mode $\hat{\alpha }_t$ is described below. Note that we do not use the mixture representation (11) to construct the block sampler for the SEPSV model. This is because if we use (11), the resulting $\ell ''_t$ is not strictly negative as required by Shephard and Pitt (1997).

The proposal distribution is formed as

$$\begin{aligned} q({\varvec{\eta }}|\Xi )\propto \prod _{t=s}^{s+m}\exp \left\{ -\frac{(\alpha _t^*-\alpha _t)^2}{2\sigma _t^{2*}}\right\} \prod _{t=s}^{s+m-1} f(\eta _t), \end{aligned}$$

(15)

where

$$\begin{aligned} \alpha ^*_t = \hat{\alpha }_t+\sigma _t^*\widehat{\ell }'_t,\quad \sigma ^*_t=-1/\widehat{\ell }''_t, \quad t=s,\dots ,s+m-1\quad \text {and} \quad t=T, \end{aligned}$$

and

$$\begin{aligned} \alpha ^*_t= \sigma _t^*\left( \widehat{\ell }'_t-\widehat{\ell }''_t\hat{\alpha }_t + \frac{\phi }{\tau ^2}\alpha _{t+1}\right) , \quad \sigma ^*_t = \left( \frac{\phi ^2}{\tau ^2}-\widehat{\ell }''_t\right) ^{-1},\quad t = s+m < T. \end{aligned}$$

Then consider the linear Gaussian state space model given by

$$\begin{aligned} \alpha ^*_t&= \alpha _t + \epsilon _t,\quad \epsilon _t\sim N(0, \sigma ^*_t),\quad t = s,\dots , s+m-1,\nonumber \\ \alpha _{t}&=\phi \alpha _{t-1} + \eta _t,\quad \eta _t\sim N(0,\tau ^2), \quad t = s+1,\dots ,s+m-1, \end{aligned}$$

(16)

with $\alpha _1\sim N(0, \tau ^2/(1-\phi ^2))$. To sample from (15), we first use the Kalman filter and disturbance smoother with respect to (16) to obtain the mode of the posterior distribution. Then the simulation smoother (de Jong and Shephard 1995; Durbin and Koopman 2002) with the accept–rejection (AR) algorithm is applied to sample values around the posterior mode. Finally, the candidate values are accepted based on the MH ratio.

Suppose the current state is ${\varvec{\eta }}$. The algorithm can be summarised as follows.

• 1. Initialise $(\hat{\eta }_s,\dots ,\hat{\eta }_{s+m-1})$ and $(\hat{\alpha }_s,\dots ,\hat{\alpha }_{s+m-1})$.

• 2. Find mode $(\hat{\alpha }_s,\dots ,\hat{\alpha }_{s+m-1})$ by repeating the following steps several times:

  • (a) Compute $(\alpha ^*_s,\dots ,\alpha ^*_{s+m-1})$ and $(\sigma ^{2*}_s,\dots ,\sigma ^{2*}_{s+m-1})$.

  • (b) Apply Kalman filter and disturbance smoother to the linear Gaussian state space model (16) and obtain the posterior mode $(\hat{\alpha }_s,\dots ,\hat{\alpha }_{s+m-1})$.

• 3. Given the posterior mode, compute $(\alpha ^*_s,\dots ,\alpha ^*_{s+m-1})$ and $(\sigma ^{2*}_s,\dots ,\sigma ^{2*}_{s+m-1})$.

• 4. (AR-step) Repeat the following until a candidate is accepted.

  • (a) Apply simulation smoother to (16) to draw a candidate $\tilde{{\varvec{\eta }}}=(\tilde{\eta }_s,\dots ,\tilde{\eta }_{s+m-1})$.

  • (b) Accept $\tilde{{\varvec{\eta }}}$ with probability

    $$\begin{aligned} \frac{\min \left\{ \pi (\tilde{{\varvec{\eta }}}|\Xi ),cq(\tilde{{\varvec{\eta }}}|\Xi )\right\} }{cq(\tilde{{\varvec{\eta }}}|\Xi )}, \end{aligned}$$

    where $c>0$ is a constant.

• 5. (MH-step) Accept $\tilde{{\varvec{\eta }}}$ with probability

  $$\begin{aligned} \min \left\{ 1,\frac{\pi (\tilde{{\varvec{\eta }}}|\Xi )\min \left\{ \pi ({\varvec{\eta }}|\Xi ),cq({\varvec{\eta }})\right\} }{\pi ({\varvec{\eta }}|\Xi )\min \left\{ \pi (\tilde{{\varvec{\eta }}}|\Xi ),cq(\tilde{{\varvec{\eta }}})\right\} } \right\} . \end{aligned}$$

To determine the blocks, we employ the stochastic knots (Shephard and Pitt 1997) based on $k_i=\text {int}\left[ T(i+U_i)/(K+2)\right]$ where $U_i\sim U(0,1)$. It is known that the stochastic knots approach can improve the efficiency of MCMC by randomly changing the conditioning sets over the iterations. The tuning parameter K is determined such that the size of block is neither too small nor too large (see, e.g., Omori and Watanabe 2008). When K is too small, the sampler would resemble the single move sampler using an MH algorithm and would be inefficient. Choosing K too large would lead to low acceptance rate and to inefficiency of the sampler as well. Therefore, we choose the value of K such that the number of points in a block is 30 on average. This choice of K works well in the numerical examples in this paper.

The sampling scheme for the rest, $\psi _0$, $\psi _1$, $\theta _1$, $\theta _2$, p, $\mu$, $\phi$, and $\tau ^2$, is the same as in the case of the single-move sampler with MH updates.

1.4 Sampling from the posterior predictive distribution

To compute one-day-ahead VaR and ES forecasts, the following steps are added after a sweep of an MCMC scheme described above. We generate daily return forecast utilising the mixture representation (11).

• 1. Generate $h_{t+1}$ from $N(\mu +\phi (h_t-\mu ), \tau ^2)$.

• 2. Generate u from U(0, 1).

  • if $\pi _1>u$

    • (a) Generate $u_1$ from $Ga(1+1/\theta _1,1)$.

    • (b) Generate $y_{t+1}$ from $U\left( \psi _0+\psi _1y_t-\frac{e^{h_{t+1}/2}}{s_{\varvec{\theta }}}\left( \frac{u_1}{1-p}\right) ^{1/\theta _1}, \psi _0+\psi _1y_t\right)$.

  • else

    • (a) Generate $u_2$ from $Ga(1+1/\theta _2,1)$.

    • (b) Generate $y_{t+1}$ from $U\left( \psi _0+\psi _1y_t, \psi _0+\psi _1y_t+\frac{e^{h_{t+1}/2}}{s_{\varvec{\theta }}}\left( \frac{u_2}{p}\right) ^{1/\theta _2}\right)$.