respectively. Notice that each stochastic disturbance $ \varepsilon_{t} $ is assumed to follow a time-varying volatility using the SV model mentioned in Section 2.

## 4 Method of Estimation

In this study, a hybrid MCMC (also referred to as Metropolis-within-Gibbs) was employed as an estimation method of the data-rich DSGE model following Boivin and Giannoni (2006), and Kryshko (2011). $ ^{10} $ The contribution of our study is to extend the data-rich DSGE to include SV shocks from i.i.d. shocks.

The benefit of employing a hybrid MCMC is not only to implement a sampling of the posterior of the model variables $ S_{t} $ but also to implement a sampling of the posterior of the structural shocks $ \varepsilon_{t} $ . Using the sampling of the structural shocks, we composed historical decompositions. The MH algorithm, which is a general estimation method used for the regular DSGE model, has disadvantages for policy analysis because it cannot generate posterior structural shocks and induces the impossibility of generating a credible interval of estimation in policy simulation.

The objects of estimation in the state-space model, (2.17), and (2.18) are the structural parameters $ \theta $ —the parameters in the measurement equations: $ \mathbf{\Gamma}(=\{\mathbf{\Lambda},\mathbf{\Psi},\mathbf{R}\}) $ , model variables $ S^{T}(=S_{1},S_{2},\cdots,S_{T}) $ , and stochastic volatilities $ \mathbf{H}^{T}(=h_{1},h_{2},\cdots h_{T}) $ . For convenience, let $ \log\sigma_{t} $ denote $ h_{t} $ , hereafter. Notice that estimating $ \theta,\Gamma,S^{T}H^{T} $ is necessary and sufficient for estimating our model because matrices $ \mathbf{G}(\theta),\mathbf{E}(\theta),\mathbf{Q}(\theta) $ in the transition equation (2.18) are a function of the structural parameters $ \theta $ .

General speaking, the Bayesian estimation of parameters $ \theta $ , $ \Gamma $ , $ H^{T} $ is implemented as described in the following steps.

Step I. We set priors of parameters $\theta$, $\Gamma$, $\boldsymbol{H}^{T}$, i.e. $p(\boldsymbol{\theta}, \boldsymbol{\Gamma}, \boldsymbol{H}^{T})$ where $p(\boldsymbol{\theta}, \boldsymbol{\Gamma}, \boldsymbol{H}^{T}) = p(\boldsymbol{\theta} | \boldsymbol{\Gamma} \boldsymbol{H}^{T}) p(\boldsymbol{\Gamma} | \boldsymbol{H}^{T}) p(\boldsymbol{H}^{T})$, since $\theta$, $\Gamma$, $\boldsymbol{H}^{T}$ are assumed to be independent.

Step II. Using Bayes theorem, posterior $p(\boldsymbol{\theta},\boldsymbol{\Gamma},\boldsymbol{H}^{T}|\boldsymbol{X}^{T})$ is derived from prior $p(\boldsymbol{\theta},\boldsymbol{\Gamma},\boldsymbol{H}^{T})$ and likelihood function $p(\boldsymbol{X}^{T}|\boldsymbol{\theta},\boldsymbol{\Gamma},\boldsymbol{H}^{T})$.

$$ p(\boldsymbol{\theta},\mathbf{\Gamma},\boldsymbol{H}^{T}|\boldsymbol{X}^{T})=\frac{p(\boldsymbol{X}^{T}|\boldsymbol{\theta},\mathbf{\Gamma},\boldsymbol{H}^{T}\;p(\boldsymbol{\theta},\mathbf{\Gamma},\boldsymbol{H}^{T})}{\int p(\boldsymbol{X}^{T}|\boldsymbol{\theta},\mathbf{\Gamma},\boldsymbol{H}^{T})p(\boldsymbol{\theta},\mathbf{\Gamma},\boldsymbol{H}^{T})d\boldsymbol{\theta}d\mathbf{\Gamma}d\boldsymbol{H}^{T}}. $$

Step III. We obtain representative values (mean, median, credible band etc.) of parameters $\theta$, $\Gamma$, $H^{T}$ from posterir $p(\theta, \Gamma, H^{T} | X^{T})$ using numerical technique.

However, it is troublesome to sample directly the joint posterior distribution $ p(\boldsymbol{\theta}, \boldsymbol{\Gamma}, \boldsymbol{H}^{T} | \boldsymbol{X}^{T}) $ of a state-space model (2.17), (2.18) in step II. Instead, using Gibbs sampling, we obtained the joint posterior $ p(\boldsymbol{\theta}, \boldsymbol{\Gamma}, \boldsymbol{H}^{T} | \boldsymbol{X}^{T}) $ from the conditional posterior $ \theta $ , $ \Gamma $ and $ H^{T} $ as described below,

$$ p(\boldsymbol{\theta}|\mathbf{\Gamma},\boldsymbol{H}^{T},\boldsymbol{X}^{T}),\qquad p(\mathbf{\Gamma}|\boldsymbol{\theta},\boldsymbol{H}^{T},\boldsymbol{X}^{T}),\qquad p(\boldsymbol{H}^{T}|\boldsymbol{\theta},\mathbf{\Gamma},\boldsymbol{X}^{T}) $$