(3.1) Generate unobserved state variables $ \bar{\pmb{S}}_{t}^{(g)} $ from $ p(\bar{\pmb{S}}^{T} \mid \pmb{\Gamma}^{(g-1)}, \pmb{\theta}^{(g-1)}, \pmb{X}^{T}) $ using the simulation smoother of de Jong and Shephard (1995).

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

Step 4. Draw deep parameters $ \boldsymbol{\theta}^{(g)} $ from $ p(\boldsymbol{\theta} \mid \boldsymbol{\Gamma}^{(g)}, \boldsymbol{X}^{T}) $ using the Metropolis step:

(4.1) Sample from proposal density $ p(\boldsymbol{\theta}|\boldsymbol{\theta}^{(g-1)}) $ and, using the sampled draw $ \boldsymbol{\theta}^{(\text{proposal})} $ , calculate the acceptance probability q as in the following sub-step (4.2).

$$ q=\mathbf{m i n}\left[\frac{p(\pmb{\theta}^{(\mathrm{p r o p o s a l})}\mid\pmb{\Gamma}^{(g)},\pmb{X}^{T})p(\pmb{\theta}^{(g-1)}|\pmb{\theta}^{(\mathrm{p r o p o s a l})})}{p(\pmb{\theta}^{(g-1)}\mid\pmb{\Gamma}^{(g)},\pmb{X}^{T})p(\pmb{\theta}^{(\mathrm{p r o p o s a l})}|\pmb{\theta}^{(g-1)})},1\right]. $$

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

Step 5. Set the iteration index $ q = q + 1 $ and return to Step 2 up to q = G.

The algorithms of the simulation smoother in sub-step 3.1 and of the sampling parameters $ \Gamma $ in sub-step 3.2 are explained in detail in Appendices A1 and A2. Here, we supplement Steps 1 and 4.

On setting the initial values of parameters $ \boldsymbol{\theta}^{(0)} $ and $ \Gamma^{(0)} $ in Step 1, it is known that arbitrary initial values are eventually converged in MCMC. However, we require a huge number of iterations in MCMC simulation to converge to target posterior distributions in the case of a considerable number of parameters. Accordingly, following Boivin and Giannoni (2006) we set initial values as below to ensure effective convergence. First, the posterior mode of the structural parameters $ \theta $ in a regular DSGE model without measurement errors is derived from numerical calculation and set as initial values $ \boldsymbol{\theta}^{(0)} $ . Second, by implementing a simulation smoother of state variables $ \bar{S}_{t} $ using $ \boldsymbol{\theta}^{(0)} $ , we obtain an initial value $ \bar{\boldsymbol{S}}^{(0)} $ . Finally, the initial values $ \Gamma^{(0)} $ of measurement equations are obtained through ordinary least square (OLS) analysis using $ \bar{S}_{t}^{(0)} $ and $ X^{T} $ .

Next, on generating the structural parameters $ \theta $ from the proposal density in sub-step 4.1, we adopt a random walk MH algorithm following previous studies. A new draw from the proposal density, $ \theta^{\mathrm{(proposal)}} $ , is represented as

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

where $ \Sigma $ is variance covariance matrix of the random walk process, and parameter c is the adjustment coefficient of covariance in the proposal distribution. The matrix $ \Sigma $ is the Hessian $ (-l^{\prime\prime-1}(\hat{\boldsymbol{\theta}})) $ of the log posterior distribution $ (l(\boldsymbol{\theta}) = \ln p(\boldsymbol{\theta}|\boldsymbol{\Gamma}, \boldsymbol{X}^{T})) $ when obtaining the initial value $ \boldsymbol{\theta}^{(0)} $ . In the case of the sampling in MH algorithm,

$$ p(\pmb{\theta}^{(g-1)}|\pmb{\theta}^{(\mathrm{p r o p o s a l})})=p(\pmb{\theta}^{(\mathrm{p r o p o s a l})}|\pmb{\theta}^{(g-1)}) $$

is held in a stationary state of the target distribution, so that acceptance rate q is reduced to the following equation:

$$ q=\mathbf{m i n}\left[\frac{f(\pmb{\theta}^{(\mathrm{p r o p o s a l})})}{f(\pmb{\theta}^{(g-1)})},1\right], $$