In addition, since parameter $ \Gamma $ is dependent on model variable $ S_{t} $ , we have to separate the two conditional posterior $ p(S^{T}|\Gamma,\theta,H^{T},X^{T}) $ and $ p(\Gamma|S^{T},\theta,H^{T},X^{T}) $ from the above conditional posterior $ p(\Gamma|\theta,H^{T},X^{T}) $ and insert $ S_{t} $ into the posterior. We also adopted a forward-backward recursion for sampling from $ p(S^{T}|\Gamma,\theta,H^{T},X^{T}) $ and from $ p(H^{T}|\Gamma,\theta,S^{T},X^{T}) $ as a data augmentation method, Gibbs sampling for sampling from $ p(\Gamma|S^{T},\theta,H^{T},X^{T}) $ , and from the MH algorithm for sampling from $ p(\theta|\Gamma,H^{T},X^{T}) $ , respectively. In this way, different algorithms were employed for different parameters in a hybrid MCMC. In sum, we show six steps of the hybrid MCMC for estimating a data-rich DSGE model as follows. $ ^{11} $

Step 1. Specify initial values of parameters $ \theta^{(0)} $ , $ \Gamma^{(0)} $ , and $ H^{T(0)} $ . And set iteration index g = 1.

Step 2. Solve the DSGE model numerically at $ \theta^{(g-1)} $ based on Sims' (2002) method and obtain matrices $ \mathbf{G}(\theta^{(g-1)}) $ , $ \mathbf{E}(\theta^{(g-1)}) $ , and $ \mathbf{Q}(\theta^{(g-1)}) $ in equation (2.18).

Step 3. Draw $ \Gamma^{(g)} $ from $ p(\Gamma \mid \theta^{(g-1)}, H^{T(g-1)}, X^{T}) $ .

(3.1) Generate model variables $ S_{t}^{(g)} $ and structural shocks $ \varepsilon_{t}^{(g)} $ from $ p(S^{T}, \varepsilon^{T} \mid \Gamma^{(g-1)}, \theta^{(g-1)}, H^{T(g-1)}, X^{T}) $ using simulation smoother by de Jong and Shephard (1995).

(3.2) Generate parameters $ \mathbf{\Gamma}^{(g)} $ from $ p(\mathbf{\Gamma} \mid \mathbf{S}^{T(g)}, \boldsymbol{\theta}^{(g-1)}, \mathbf{H}^{T(g-1)}, \mathbf{X}^{T}) $ based on the sampled draw $ \bar{\mathbf{S}}^{T(g)} $ using Gibbs sampling by Chib and Greenberg(1994).

Step 4. Draw $ H^{T(g-1)} $ from $ p(H^{T} \mid \theta^{(g-1)}, \Gamma^{(g)}, \varepsilon^{T,(g)}, X^{T}) $ .

(4.1) Generate stochastic volatility $ H^{T(g)} $ from $ p(H^{T} \mid \Gamma^{(g)}, \theta^{(g-1)}, \varepsilon^{T,(g)}, u^{T(g-1)}, \Phi^{(g-1)}, X^{T}) $ , using a draw of $ \varepsilon^{T,(g)} $ at Step 3.1, and the forward-backward recursion by Cater and Kohn (1994).

(4.2) Generate the indicators of the mixture approximation $ u^{T(g)} $ using discrete density proposed by Omori et al. (2007).

(4.3) Generate the coefficients $ \Phi^{(g)} $ of stochastic volatility process using Metropolis step.

Step 5. Draw deep parameters $ \theta^{(g)} $ from $ p(\theta \mid \Gamma^{(g)}, H^{t(g)}, X^{T}) $ using Metropolis step:

(5.1) Sample from proposal density $ p(\theta|\theta^{(g-1)}) $ and, using the sampled draw $ \theta^{\mathrm{(proposal)}} $ , calculate the acceptance probability q as follows.

$$ q=\min\left[\frac{p(\boldsymbol{\theta}^{(\mathrm{p r o p o s a l})}\mid\boldsymbol{\Gamma}^{(g)},\boldsymbol{H}^{t(g)},\boldsymbol{X}^{T})p(\boldsymbol{\theta}^{(g-1)}\mid\boldsymbol{\theta}^{(\mathrm{p r o p o s a l})})}{p(\boldsymbol{\theta}^{(g-1)}\mid\boldsymbol{\Gamma}^{(g)},\boldsymbol{H}^{t(g)},\boldsymbol{X}^{T})p(\boldsymbol{\theta}^{(\mathrm{p r o p o s a l})}\mid\boldsymbol{\theta}^{(g-1)})},1\right]. $$

(5.2) Accept $ \theta^{\mathrm{(proposal)}} $ with probability q and reject it with probability 1 - q. Set $ \theta^{(g)} = \theta^{\mathrm{(proposal)}} $ when accepted and $ \theta^{(g)} = \theta^{(g-1)} $ when rejected.

Step 6. Set iteration index $ q = q + 1 $ and return to Step 2 up to q = G.

The algorithm of sampling stochastic volatilities $ H^{T} $ in Step 4 is explained in Appendix A1. And a simulation smoother in Step 3.1 and of sampling parameters $ \Gamma $ in Step 3.2 are in Appendix A2 and A3, respectively. $ ^{12} $