terior of structural shocks $ \varepsilon_{t} $ . Using the sampling of structural shocks, we compose historical decompositions, described in the next subsection. On the other hand, 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 (2.7), and (2.8) 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 parameters in transition equation $ \mathbf{G}(\theta) $ , $ \mathbf{H}(\theta) $ , $ \mathbf{Q}(\theta) $ is a function of structural parameters $ \theta $ .

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

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

Step II. Using Bayes theorem, posterior $ p(\boldsymbol{\theta}, \boldsymbol{\Gamma} | \boldsymbol{X}^{T}) $ is derived from prior $ p(\boldsymbol{\theta}, \boldsymbol{\Gamma}) $ and 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 posterir $ p(\pmb{\theta}, \pmb{\Gamma} | \pmb{X}^{T}) $ using numerical technique.

However, it is troublesome to sample directly joint posterior distribution $ p(\boldsymbol{\theta}, \boldsymbol{\Gamma} | \boldsymbol{X}^{T}) $ of a state space model (2.7), (2.8) in step II. Instead, we obtain the joint posterior $ p(\boldsymbol{\theta}, \boldsymbol{\Gamma} | \boldsymbol{X}^{T}) $ from conditional posterior $ \theta $ and $ \Gamma $ as below, using Gibbs sampling.

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

In addition, since parameter $ \Gamma $ is dependent state variable $ S_{t} $ , we have to separate two conditional posterior $ p(\mathbf{S}^{T}|\mathbf{\Gamma},\theta,\mathbf{X}^{T}) $ and $ p(\mathbf{\Gamma}|\mathbf{S}^{T},\theta,\mathbf{X}^{T}) $ from above conditional posterior $ p(\mathbf{\Gamma}|\theta,\mathbf{X}^{T}) $ and insert $ S_{t} $ into posterior. We also adopt a simulation smoother for sampling from $ p(\mathbf{S}^{T}|\mathbf{\Gamma},\theta,\mathbf{X}^{T}) $ as a data augmentation method, Gibbs sampling for sampling from $ p(\mathbf{\Gamma}|\mathbf{S}^{T},\theta,\mathbf{X}^{T}) $ , and MH algorithm for sampling from $ p(\theta|\mathbf{\Gamma},\mathbf{X}^{T}) $ , respectively. In this way, different algorithms