Real wage forecast error:

$$ \eta_{t}^{w}=\hat{w}_{t}-E_{t-1}\hat{w}_{t}, $$

Equity premium forecast error:

$$ \eta_{t}^{q}=\hat{q}_{t}-E_{t-1}\hat{q}_{t}, $$

Investment forecast error:

$$ \eta_{t}^{i n v}=\widehat{i n v}_{t}-E_{t-1}\widehat{i n v}_{t}, $$

Consumption forecast error:

$$ \eta_{t}^{c}=\hat{c}_{t}-E_{t-1}\hat{c}_{t}, $$

Rental rate forecast error:

$$ \eta_{t}^{r k}=\hat{r}_{t}^{k}-E_{t-1}\hat{r}_{t}^{k}. $$

### 3.5 Log-linearized model system

From Eq. (3.1) through Eq. (3.21), the system of the log-linearized model neighborhood the steady state are integrated as follows:

$$ \mathbf{\Gamma}_{0}\bar{\boldsymbol{S}}_{t}=\mathbf{\Gamma}_{1}\bar{\boldsymbol{S}}_{t-1}+\mathbf{\Psi}\boldsymbol{\varepsilon}_{t}+\mathbf{\Pi}\boldsymbol{\eta}_{t}, $$

where $ \bar{S}_{t} $ is a vector of endogenous variables: $ \bar{S}_{t} = [\hat{y}_{t}, \hat{\pi}_{t}, \hat{w}_{t}, \hat{k}_{t}, \widehat{inv}_{t}, \hat{c}_{t}, \hat{R}_{t}, \hat{r}_{t}^{k}, \hat{l}_{t}, E_{t}\hat{\pi}_{t+1}, E_{t}\hat{w}_{t+1}, E_{t}\hat{q}_{t+1}, E_{t}\widehat{inv}_{t+1}, E_{t}\hat{c}_{t+1}, E_{t}\hat{r}_{t+1}^{k}, u_{t}^{c}, u_{t}^{inv}, u_{t}^{L}, u_{t}^{a}, u_{t}^{g}]' $ , $ \varepsilon_{t} $ is a vector of exogenous shocks: $ \varepsilon_{t} = [\varepsilon_{t}^{c}, \varepsilon_{t}^{inv}, \varepsilon_{t}^{q}, \varepsilon_{t}^{L}, \varepsilon_{t}^{w}, \varepsilon_{t}^{a}, \varepsilon_{t}^{p}, \varepsilon_{t}^{g}, \varepsilon_{t}^{m}]' $ . $ \eta_{t} $ is a vector of forecast errors: $ \eta_{t} = [\eta_{t}^{\pi}, \eta_{t}^{w}, \eta_{t}^{q}, \eta_{t}^{inv}, \eta_{t}^{c}, \eta_{t}^{rk}]' $ . $ \Gamma_{0} $ , $ \Gamma_{1} $ , $ \Psi $ , and $ \Pi $ are the matrices of parameters. Employing Sims' (2002) method to Eq. (3.22), we can derive the rational expectation equilibrium as Eq. (2.8).

## 4 Estimation method

In this study, a hybrid MCMC (also referred to as the Metropolis-within-Gibbs algorithm) is employed as an estimation method for a data-rich DSGE model following Boivin and Giannoni (2006) and Kryshko (2011). The MCMC algorithm for estimating DSGE models was developed by Schorfheide (2000). Our method extends Schorfheide's (2000) algorithm to form a hybrid MCMC algorithm. The algorithm of the hybrid MCMC is described in Gamerman and Lopes (2006, see Chapter 6). We also adopt a sped-up algorithm for sampling state variables developed by Jung-backer and Koopman (2008), but find that it drops the accuracy of estimates of state variables, so we omit the algorithm from the MCMC process. Adopting a simulation smoother (as explained in the Appendix) to sample state variables is the difference between our study and theirs, and is the advantage of our study over theirs.

The benefit of employing a hybrid MCMC is not only that it enables the sampling of the posterior of state variables $ \bar{S}_{t} $ , but the sampling of the posterior of structural shocks $ \varepsilon_{t} $ . Using the sampling of structural shocks, we compose historical decompositions, described in Appendix A3. On the other hand, the Metropolis-Hastings (MH) algorithm – which is a general estimation method used for the regular DSGE model – has disadvantages for policy analysis because it cannot generate