$$ h_{i,t}=\mu+\phi(h_{i,t-1}-\mu)+\nu_{i,t},\nu_{i,t}\sim\mathbf{i.i.d.}\mathbf{N}(0,\omega_{i}^{2}) $$

where $ \tilde{\varepsilon}_{i,t} = \log[(\varepsilon_{i,t})^2 + \bar{c}] $ ; $ \bar{c} $ is the offset constant (set to 0.001); $ h_{it} = \log \sigma_{i,t} $ and $ z_{it}^*} = \log (z_{it}^2) $ . M is the number of structural shocks. Since the squared shocks $ \varepsilon_{i,t}^2 $ is very small, an offset constant is used to make the estimation procedure more robust. Eqs.(A.4) and (A.5) are linear, but non-Gaussian state space form, because $ z_{it}^* $ are distributed as a log $ \chi^2(1) $ . In order to transform the system in a Gaussian state space form, a mixture of normals approximation of the log $ \chi^2(1) $ distribution is used, as described in Kim et al. (1998) and Omori et al. (2007). A draw of $ z_{i,t}^* $ is implemented from the mixture normal distribution given as

$$ f(z_{i,t}^{*})=\sum_{k=1}^{K}q_{k}f_{N}(z_{i,t}^{*}\mid u_{i,t}=k),i=1,\cdots\cdots,M, $$

where $ u_{i,t} $ is the indicator variable selecting which member of the mixture of normals has to be used at period t for shock i. And $ q_{k} $ is the probability of $ u_{i,k}=k $ ; $ q_{k}=\operatorname*{Pr}(u_{i,t}=k) $ , and $ f_{N}(\cdot) $ denotes the probability density function of normal distribution. Omori et al (2007) select a mixture of ten normal densities (K=10) with component probabilities $ q_{k} $ , means $ m_{k} $ , and variances $ v_{j}^{2} $ , for $ k=1,2,\cdots,10 $ , chosen to match a number of moments of the log $ \chi^{2}(1) $ distribution. The constant $ \{q_{k}, m_{k}, v_{k}^{2}\} $ are reported as Table blow.

## Table of Selection Probability Function $ (q_{k}, m_{k}, v_{k}^{2}, a_{k}, b_{k}) $

k	q_{k}	m_{k}	v_{k}^{2}	a_{k}	b_{k}
1	0.00609	1.92677	0.11265	1.01418	0.50710
2	0.04775	1.34744	0.17788	1.02248	0.51124
3	0.13057	0.73504	0.26768	1.03403	0.51701
4	0.20674	0.02266	0.40611	1.05207	0.52604
5	0.22715	-0.85173	0.62699	1.08153	0.54076
6	0.18842	-1.97278	0.98583	1.13114	0.56557
7	0.12047	-3.46788	1.57469	1.21754	0.60877
8	0.05591	-5.55246	2.54498	1.37454	0.68728
9	0.01575	-8.68384	4.16591	1.68327	0.84163
10	0.00115	-14.65000	7.33342	2.50097	1.25049

Using generator of the mixture normal distribution above, the system has an approximate linear and Gaussian state space form. Therefore, a new draw of the stochastic volatilities $ \mathbf{H}^{T(g+1)} $ can be obtained recursively with standard Gibbs sampler for state space form using the algorithm of Carter and Kohn (1994).

## Step 3: Draw the indicators of the mixture approximation $ u^{T(g+1)} $

In the case of SV with leverage effect, we need to modify the indicator $ u_{i,t} $ for the mixture normal described in Step 2, compared with Justiniano and Primiceri (2008). We follow the algorithm proposed by Omori et al. (2007), and obtain a new draw of indicators $ u_{i,t} $ which is generated conditional on $ \varepsilon_{i,t}^{(g+1)} $ , $ \boldsymbol{H}^{T,(g+1)} $ by independently sampling each from the discrete density defined by

$$ \pi(u_{i,t}=k\mid\varepsilon_{i,t},h_{it},\Phi)\propto\pi(u_{i,t}=k\mid\sigma_{i,t}^{*},d_{it},h_{it},\Phi)\propto\pi(u_{it}=k\mid z_{i,t}^{*},\eta_{i,t},d_{i,t},\Phi) $$

$$ \propto q_{k}v_{k}^{-1}\mathbf{e x p}\left\{-\frac{(z_{i,t}^{*}-m_{k})^{2}}{2v_{k}^{2}}-\frac{[\eta_{i,t}-d_{i,t}\rho_{i}\omega_{i}\mathbf{e x p}\left(m_{k}/2\right)\{a_{k}+b_{k}\left(z_{i,t}^{*}-m_{k}\right)\}]^{2}}{\omega_{i}^{2}(1-\rho_{i}^{2})}\right\} $$