## 5 Preliminary Settings and Data Description

### 5.1 Specifications of Four Alternative Cases

This study considered four alternative cases based on the number of observation variables (11 vs. 40 observable variables) and on the specification of the volatilities of the structural shocks (constant vs. time-varying volatility) as summarized in Table 1. The reason why four cases are considered is that we wanted to verify whether data-rich information decomposes the measurement errors and the model variables from more robust data, and whether relaxation of specifying the volatilities depicts a rapid change of the more detailed shocks in the data-rich approach. The first case (referred to as Case A) dealt with one of the standard DSGE models that used 11 observable variables in the measurement equation (2.11) and the structural shocks with i.i.d. Normal distribution in the transition equation (2.12). In Case A, each observable variable was connected with its specified model variable by one-to-one matching. The second case (Case B) was extended to the data-rich approach with i.i.d shocks, including 40 observable variables, which indicate more or less four observable variables corresponding to one specified model variable. The third case (Case C) extends to SV shocks from Case A. And the fourth case (Case D) extends to the data-rich approach with SV shocks from Case B. Nishiyama et al. (2011) have already studied the comparison of a standard DSGE approach (Case A) and a data-rich approach (Case B). This study focused on the remaining two cases (C and D) with SV shock, using Case A as the reference model.

## [Insert Table 1 around here]

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 for converging effectively. First, the posterior mode of structural parameters $ \theta $ in a regular DSGE model without measurement errors is derived from numerical calculation and set as initial values $ \theta^{(0)} $ . Second, a implementing simulation smoother of state variables $ S_{t} $ using $ \theta^{(0)} $ , we get initial value $ S_{t}^{(0)} $ . Finally, initial values $ \Gamma^{(0)} $ of measurement equations are obtained by OLS using $ S_{t}^{(0)} $ and $ X^{T} $ .

Next, on generating structural parameters $ \theta $ from proposal density in Step 5.1, we adopt a random walk MH algorithm following previous works. Proposal density $ \theta^{\mathrm{(proposal)}} $ is represented as

$$ \mathbf{\theta}^{(p r o p o s a l)}=\mathbf{\theta}^{(g-1)}+\mathbf{u}_{t},\quad\mathbf{u}_{t}\sim N(\mathbf{0},c\mathbf{\Sigma}), $$

where, $ \Sigma $ is variance covariance matrix of random walk process, and c is the adjustment coefficient. The matrix $ \Sigma $ is the Hessian $ (-l''^{-1}(\hat{\boldsymbol{\theta}}) $ of log posterior distribution $ (l(\boldsymbol{\theta})=\ln p(\boldsymbol{\theta}|\boldsymbol{\Gamma},\boldsymbol{X}^{T}) $ when obtaining initial value $ \boldsymbol{\theta}^{(0)} $ . And the case of sampling in MH algorithm,

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

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

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

In this equation, acceptance rate q is not dependent on proposal density $ p(\boldsymbol{\theta}|\boldsymbol{\theta}^{(g-1)}) $ . As a result, it is the advantage of a random walk MH algorithm that we do not need to adopt a proposal density close to posterior density. However, when proposal value $ \boldsymbol{\theta}^{\mathrm{(proposal)}} $ departs from the previous sample $ \boldsymbol{\theta}^{(g-1)} $ , acceptance rate q becomes small and efficiency of MCMC worsens. To avoid this, the adjustment coefficient c should be small, but doing so narrows the range of sampling space of $ \boldsymbol{\theta}^{\mathrm{(proposal)}} $ . Roberts et al. (1997) and Neal and Roberts (2008) reported that the optimal acceptance rate q of a random walk MH algorithm is around 25%. Accordingly, adjustment coefficient c of this study is set so that the acceptance rate is close to around 25%.