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 (2.12), and (2.13) are structural parameters $ \theta $ , parameters in measurement equations $ \Gamma(=\{\Lambda,\Psi,\mathbf{R}\}) $ , and state variables $ \bar{\pmb{S}}^{T}(=\bar{\pmb{S}}_{1},\bar{\pmb{S}}_{2},\cdots,\bar{\pmb{S}}_{T}) $ . Note that to estimate $ \theta $ , $ \Gamma $ , $ S^{T} $ is necessary and sufficient for estimating our model because the parameters in the transition equation $ \mathbf{G}(\boldsymbol{\theta}),\mathbf{H}(\boldsymbol{\theta}),\mathbf{Q}(\boldsymbol{\theta}) $ are a function of the structural parameters $ \theta $ .

Generally speaking, a Bayesian estimation of parameters $ \theta $ , $ \Gamma $ is implemented as the following steps.

Step I. We set the priors of parameters $ \theta $ , $ \Gamma $ , i.e., $ p(\theta, \text{and}\Gamma) $ where $ p(\theta, \Gamma) = p(\theta|\Gamma)p(\Gamma) $ , as $ \theta $ , $ \Gamma $ are assumed to be independent.

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

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

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

However, it directly sampling the joint posterior distribution $ p(\boldsymbol{\theta}, \boldsymbol{\Gamma} | \boldsymbol{X}^{T}) $ of a state space model (2.12), (2.13) in step II is problematic. Instead, we obtain the joint posterior $ p(\boldsymbol{\theta}, \boldsymbol{\Gamma} | \boldsymbol{X}^{T}) $ from the conditional posterior $ \theta $ and $ \Gamma $ as below, using Gibbs sampling.

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

In addition, as parameter $ \Gamma $ is a dependent state variable $ \bar{S}_{t} $ , we have to separate two conditional posterior $ p(\bar{\pmb{S}}^{T}|\pmb{\Gamma},\pmb{\theta},\pmb{X}^{T}) $ and $ p(\pmb{\Gamma}|\bar{\pmb{S}}^{T},\pmb{\theta},\pmb{X}^{T}) $ from the above conditional posterior $ p(\pmb{\Gamma}|\pmb{\theta},\pmb{X}^{T}) $ and insert $ \bar{S}_{t} $ into posterior. We also adopt a simulation smoother for sampling from $ p(\bar{\pmb{S}}^{T}|\pmb{\Gamma},\pmb{\theta},\pmb{X}^{T}) $ as a data augmentation method, Gibbs sampling for sampling from $ p(\pmb{\Gamma}|\bar{\pmb{S}}^{T},\pmb{\theta},\pmb{X}^{T}) $ , and the MH algorithm for sampling from $ p(\pmb{\theta}|\pmb{\Gamma},\pmb{X}^{T}) $ , respectively. In this way, different algorithms are employed for different parameters in the hybrid MCMC. In sum, we show five steps in a hybrid MCMC to estimate a data-rich DSGE model as follows. $ ^{2} $

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

Step 2. Solve the DSGE model numerically at $ \boldsymbol{\theta}^{(g-1)} $ based on Sims' (2002) method and obtain $ \mathbf{G}(\boldsymbol{\theta}^{(g-1)}) $ , $ \mathbf{H}(\boldsymbol{\theta}^{(g-1)}) $ , and $ \mathbf{Q}(\boldsymbol{\theta}^{(g-1)}) $ in Eq. (3.5).

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