Step 4: Draw the coefficients $ \Phi_{i}^{(g+1)}=((\phi_{i},\rho_{i},\omega_{i})) $ of stochastic volatility processess.

Having generated a sample $ H^{T,(g+1)} $ , we sample the elements of vector $ \Phi_{i}^{(g+1)} $ from the density

$$ p(\Phi_{i}\mid\sigma_{i,t}^{*},d_{i,t}u_{i,t})\propto p(\sigma_{i,t}^{*}\mid d_{it},u_{i,t}\Phi_{i})p(\Phi_{i}). $$

The density $ p(\sigma_{i,t}^{*} \mid d_{it}, u_{i,t} \Phi_{i}) $ is found from the output of Kalman filter recursion applied to the state space model (A.1) and (A.2). For the sampling we rely on the Metropolis-Hasting algorithm with a proposal density based on random walk such as

$$ \mathbf{\theta}^{(proposal)}=\mathbf{\theta}^{(g-1)}+\mathbf{u}_{t},\quad\mathbf{u}_{t}\sim N(\mathbf{0},c\mathbf{\Sigma}), $$

where c is an adjustment constant.

### A.2 Simulation Smoother of Model Variable

Step 3.1 of algorithm of data-rich DSGE described in Section 4 employs simulation smoother (de Jong and Shephard, 1995) which generate sampling of model variables $ S_{t} $ from conditional posterior distribution, $ p(\mathbf{S}^{T}|\mathbf{\Gamma}^{(g-1)},\boldsymbol{\theta},\mathbf{X}^{T}) $ . $ ^{20} $ On the other hand, Boivin and Giannoni (2006), and Kryshko (2011) employ smoothing method proposed by Carter and Kohn (1994). But their method does not apply only to sample positive definite matrix as variance covariance matrix of state variables so that their method discontinue on the way of sampling in MCMC pointed out by Chib (2001, p.3614). As a result, Kryshko (2011) transforms to ad hoc variance covariance matrix of state variables. To avoid this problem, our algorithm employs simulation smoother instead of Carter and Kohn's (1994) algorithm. Accordingly, our algorithm accomplishes generalization of estimating data-rich DSGE model.

To simplify representation of algorithm of simulation smoother, we rewrite state space model of $ (2.17) $ and $ (2.18) $ described in Section 2.1 into $ (A.8) $ , and $ (A.9) $ as below.

$$ \tilde{\boldsymbol{X}}_{t}=\tilde{\boldsymbol{\Lambda}}\tilde{\boldsymbol{S}}_{t}+\boldsymbol{\nu}_{t},\qquad\boldsymbol{\nu}_{t}\sim N(\mathbf{0},\mathbf{R}), $$

$$ \tilde{\boldsymbol{S}}_{t}=\tilde{\mathbf{G}}\tilde{\boldsymbol{S}}_{t-1}+\tilde{\mathbf{E}}\boldsymbol{\varepsilon}_{t},\qquad\boldsymbol{\varepsilon}_{t}\sim N(\mathbf{0},\mathbf{Q}(\boldsymbol{\theta})), $$

The following four steps are conducted to generate a new draw of model variables.

Step 1: Kalman filter for state space model is implemented.

Kalman filter is represented as

$$ \begin{array}{r l r}{\eta_{t}}&{=}&{\tilde{\boldsymbol{X}}_{t}-\tilde{\boldsymbol{\Lambda}}\tilde{\boldsymbol{S}}_{t|t},\qquad\mathbf{F}_{t}=\tilde{\boldsymbol{\Lambda}}\tilde{\mathbf{P}}_{t|t}\tilde{\boldsymbol{\Lambda}}^{\prime}+\mathbf{R},\qquad\mathbf{K}_{t}=\tilde{\mathbf{G}}\tilde{\mathbf{P}}_{t|t}\tilde{\boldsymbol{\Lambda}}^{\prime}\mathbf{F}_{t}^{-1},}\end{array} $$

$$ \mathbf{L}_{t}=\tilde{\mathbf{G}}-\mathbf{K}_{t}\tilde{\mathbf{\Lambda}},\qquad\tilde{S}_{t+1|t+1}=\tilde{\mathbf{G}}\tilde{S}_{t|t}+\mathbf{K}_{t}\boldsymbol{\eta}_{t},\qquad\tilde{\mathbf{P}}_{t+1|t+1}=\tilde{\mathbf{G}}\tilde{\mathbf{P}}_{t|t}\mathbf{L}_{t}^{\prime}+\tilde{\mathbf{E}}\mathbf{Q}(\boldsymbol{\theta})\tilde{\mathbf{E}}^{\prime}, $$