Silviase/jgov_v1.4_full_ann huggingface dataset:
filter, the Christiano-Fitzgerald filter, and other filtering algorithms are also often used in practice. Our method is an alternative to these filters, and it is “structural” estimation of time-varying economic trends. In empirical analysis, we estimate new Keynesian DSGE models under a liquidity trap using Japanese macroeconomic data, which includes the “pre-zero-interest-rate” period (1980-1998), the “zero-interest-rate” period (1999-2006), and the “post-zero-interest-rate” period (2007). One restriction on our method, however, exists. We assume that the timings of when the economy is trapped in a liquidity trap and its subsequent escape are given. In other words, these timings are exogenous.

In most previous papers on DSGE models, structural parameters of them are assumed to be “deep (invariant).” Our method, however, analyzes how stable structural parameters are. The time-varying-parameter approach is practically often used in state space modeling to estimate parameters, for example, Kitagawa (1998) and Liu and West (2001). Even if we assume the random walk priors, which are described in section 3, it does not indicate that the deep parameters of DSGE models are “time-varying.” Our framework is just a practical one to estimate deep parameters. Adopting it creates the great advantage that the structural changes of parameters are detected naturally. Thus, it is suitable to analyze how stable structural parameters are. The second advantage of our method is that we are able to estimate new Keynesian DSGE models under a liquidity trap (Krugman (1998)) because NNNSS allows model switching.

Our paper is closely related with Fernandez-Villaverde and Rubio-Ramirez (2007b) $ ^{11} $ . However, there exist several large differences between our paper and theirs. The first point is that they focus on the stabilities of “structural” parameters of the Taylor rule. In contrast, we estimate any parameters using the TVP approach. The second point is that they use MCPF to estimate the second-order approximation of DSGE models, whereas, we focus on the nonlinearity of the Taylor rule of the economy under a liquidity trap. The third point is that they use maximizing the likelihood of MCPF to estimate parameters, while, we adopt a self-organizing state space model for parameter estimation. Yano (2008a) reports that the variances of the estimates of a self-organizing state space model are smaller than the ones of the maximizing-likelihood approach. The fourth point is that we estimate a time-varying trend of real output, a time-varying inflation target, and a time-varying equilibrium real interest rate.

This paper is structured as follows. In section 2, we describe a new Keynesian DSGE model. In section 3, we explain our method based on the Monte Carlo particle filter and a self-organizing state space model. In section 4, we show the results of our empirical analysis. In section 5, we describe conclusions and discussions.
## 2 The Model

### 2.1 Households

In the economy, there is a continuum of households indexed by $ j \in (0,1) $ . The households consume and provide labor. The utility of the household j is given by

$$ E_{0}^{j}\sum_{t=0}^{\infty}\beta^{t}\Big[\frac{1}{1-\sigma_{C}}\big(C_{j,t}-h C_{j,t-1}\big)^{1-\sigma_{c}}-\frac{1}{1+\sigma_{L}}L_{j,t}^{1+\sigma_{L}}\Big], $$

where $ E_{0}^{j} $ is the expectation operator, conditional on household j's information at time 0, $ C_{j,t} $ is household j's consumption, $ L_{j,t} $ is household j's labor hours, t is a time index, and h, $ \chi $ , and $ \eta $ are constants. The constraint condition of the household j is given by

$$ C_{j,t}+\frac{B_{j,t}}{P_{t}}\leq W_{t}L_{j,t}+(1+i_{t-1})\frac{B_{j,t-1}}{P_{t}}+\Pi_{j,t}, $$

where $ B_{j,t} $ is household j's domestic bonds, $ W_{t} $ is the average real wage, $ i_{t} $ is the short-term nominal interest rate, and $ \Pi_{j,t} $ is the profit of the firm j. In addition to Eq. (2), we assume the households are subject to the no-Ponzi condition.

$$ \lim_{T\to\infty}E_{0}[(\prod_{t=0}^{T}\frac{1}{1+i_{t}})B_{j,T}]=0. $$

### 2.2 Final Good Sector

In the final good sector, a single final good is produced by a perfectly competitive, representative firm. The final good is produced using a continuum of intermediate good, $ Y_{j,t} $ , indexed by $ j \in (0,1) $ . The final good, $ Y_{t} $ , is produced using the aggregate technology.

$$ Y_{t}=\left[\int_{0}^{1}\left(Y_{j,t}\right)^{\frac{1}{1+\lambda_{p}}}d j\right]^{1+\lambda_{p}}, $$

where $ Y_{j,t} $ is the quantity of intermediate good j, $ \lambda_{p} $ is a parameter. The demand curve for $ Y_{j,t} $ is given by

$$ Y_{j,t}=\left(\frac{P_{j,t}}{P_{t}}\right)^{-\frac{1+\lambda_{p}}{\lambda_{p}}}Y_{t}, $$

where $ P_{j,t} $ is the price of intermediate good j and $ P_{t} $ is the aggregate price of the final good. The aggregate price is given by

$$ P_{t}=\left[\int_{0}^{1}\left(P_{j,t}\right)^{-\frac{1}{\lambda_{p}}}d j\right]^{-\lambda_{p}}. $$

### 2.3 Intermediate Goods Firms

In the intermediate goods sector, monopolistic competitive domestic firms produce intermediate goods which is indexed by $ j \in (0,1) $ . The firm j's production function is given by

$$ Y_{j,t}=Z_{t}L_{j,t}, $$

$ L_{j,t} $ is labor to produce the jth intermediate good. The technology level, $ Z_{t} $ , is given by

$$ \log Z_{t}=\left(1-\xi_{Z}\right)\log\bar{Z}+\xi_{Z}\log Z_{t-1}+\epsilon_{Z,t}, $$
where $ \epsilon_{Z,t}\sim N(0,\sigma_{Z,t}^{2}) $ and $ \bar{Z} $ and $ \xi_{Z} $ are constants. The firms' real marginal costs are given by

$$ MC_{t}=\frac{W_{t}}{Z_{t}}. $$

In the sticky prices model, proposed by Calvo (1983), a fraction $ 1 - \xi_{p} $ of all firms re-optimize their nominal prices while the remaining $ \xi_{p} $ fraction of all firms do not re-optimize their nominal prices. Following Christiano et al. (2005), firms that cannot re-optimize their price index to lagged inflation are as follows.

$$ P_{j,t}=\pi_{t-1}P_{j,t-1}, $$

where $ \pi_{t}=P_{t}/P_{t-1} $ . We call this price setting “lagged inflation indexation.” The firm j chooses $ P_{j,t} $ to maximize

$$ E_{t}\sum_{l=0}^{\infty}\big(\beta\xi_{p}\big)^{l}\Big[\frac{P_{j,t}}{P_{t+l}}X_{t l}-M C_{t+1}\Big]Y_{j,t+l}, $$

$$ subject to Y_{j,t}=\left(\frac{P_{j,t}}{P_{t}}\right)^{-\frac{1+\lambda_{p}}{\lambda_{p}}}Y_{t}, $$

where $X_{tl}$ is

$$ X_{tl}=\left\{\begin{array}{l}\pi_{t}\times\pi_{t+1}\times\cdots\times\pi_{t+l-1}for l\geq1\\ 0for l=0.\end{array}\right. $$

The aggregate price index of sticky prices and inflation indexation is obtained by

$$ P_{t}=\left[\left(1-\xi_{p}(\tilde{P}_{t})^{\frac{1}{1-\lambda_{p}}}+\xi_{p}(\pi_{t-1}P_{t-1})^{\frac{1}{1-\lambda_{p}}}\right]^{1-\lambda_{p}}\right.. $$

### 2.4 Monetary Policy

The monetary authority is assumed to determine the nominal interest rate according to the nonlinear Taylor rule (the Taylor rule with non-negativity constraint on the short-term nominal interest rate.)

$$ i_{t}=\operatorname*{m a x}\bigl[r_{0},i_{t-1}^{\rho_{i}}\bigl(Y_{t}^{\phi_{Y}}\pi_{t}^{\phi_{\pi}}\bigr)^{(1-\rho_{i})}e^{\epsilon_{t}^{i}}\bigr], $$

where $ r_{0} \geq 0 $ is the lower bound of the nominal interest rate, $ \rho_{i} $ , $ \phi_{Y} $ , and $ \phi_{\pi} $ are constants, and $ \epsilon_{i,t} \sim N(0, \sigma_{i,t}^{2}) $ . In ordinary cases, $ r_{0} $ is zero or nearly equal to zero.

### 2.5 Market Clearing

In the economy, we ignore capital and investment. Thus, in the final market equilibrium, the final good production is equivalent to the households' demand for consumption.

$$ Y_{t}=C_{t}, $$

where $ C_{t}=\left[\int_{0}^{1}\left(C_{j,t}\right)^{\frac{1}{1+\lambda_{p}}}dj\right]^{1+\lambda_{p}} $ and $ Y_{j,t}=C_{j,t} $

### 2.6 Linearized Model

We linearize the model described above around the non-stochastic steady state. The linearized model consists of the hybrid new IS curve (HNISC), the hybrid new Keynesian Phillips curve (HNKPC), the
nonlinear Taylor rule (NTR), $ ^{12} $ and several equations. HNISC is obtained as follows $ ^{13} $ .

$$ \hat{Y}_{t}=\frac{h}{1+h}\hat{Y}_{t-1}+\frac{1}{1+h}E_{t}\hat{Y}_{t+1}-\frac{1}{\sigma_{C}(1+h)}E_{t}\big[\hat{i}_{t}-\hat{\pi}_{t+1}\big]+\epsilon_{Y,t}. $$

HNKPC is obtained as follows.

$$ \hat{\pi}_{t}=\frac{1}{1+\beta}\hat{\pi}_{t-1}+\frac{\beta}{1+\beta}E_{t}\hat{\pi}_{t+1}+\frac{(1-\xi_{p})(1-\beta\xi_{p})}{\xi_{p}(1+\beta)}(\hat{W}_{t}-\hat{Z}_{t})+\epsilon_{\pi,t}. $$

The other equation are

$$ \hat{W}_{t}=\sigma_{C}(\hat{Y}_{t}-h\hat{Y}_{t-1})+\sigma_{L}\hat{L}_{t}, $$

$$ \hat{Y}_{t}=\hat{Z}_{t}+\hat{L}_{t}, $$

and

$$ \hat{Z}_{t}=\xi_{Z}\hat{Z}_{t-1}+\epsilon_{Z,t}, $$

where $ \epsilon_{Y,t}\sim N(0,\sigma_{Y,t}^{2}) $ , $ \epsilon_{\pi,t}\sim N(0,\sigma_{\pi,t}^{2}) $ , and $ \epsilon_{Z,t}\sim N(0,\sigma_{Z,t}^{2}) $ . The linearized NTR is given by

$$ \hat{i}_{t}=\left\{\begin{array}{l}0,\quad(if a liquidity trap),\\\rho_{i}\hat{i}_{t-1}+(1-\rho_{i})(\phi_{Y}\hat{Y}_{t}+\phi_{\pi}\hat{\pi}_{t})+\epsilon_{t}^{i},\quad(if not a liquidity trap).\end{array}\right. $$

### 2.7 State Space Model

Structural linear rational expectations models are given by

$$ \mathbf{\Gamma}_{0}\boldsymbol{x}_{t}=\mathbf{\Gamma}_{1}\boldsymbol{x}_{t-1}+\mathbf{\Psi}\boldsymbol{z}_{t}+\mathbf{\Pi}\boldsymbol{\eta}_{t}+\boldsymbol{C},t=1,\ldots,T, $$

where $ \pmb{x}_{t}=\left[E_{t}\hat{Y}_{t+1},E_{t}\hat{\pi}_{t+1},\hat{Y}_{t},\hat{\pi}_{t},\hat{i}_{t},\hat{W}_{t},\hat{L}_{t},\hat{Z}_{t}\right]^{t},\quad\pmb{z}_{t}=\left(\epsilon_{Y,t},\epsilon_{\pi,t},\epsilon_{i,t},\epsilon_{z,t}\right)^{T}\sim N(\mathbf{0},\mathbf{\Sigma}_{t}) $ with $ \mathbf{\Sigma}_{t}=diag((\sigma_{Y,t})^{2},(\sigma_{\pi,t})^{2},(\sigma_{i,t})^{2},(\sigma_{z,t})^{2}) $ , $ \Pi=0 $ , and C=0. Sims (2002) proposes the solution of linear rational expectations models using QZ decomposition $ ^{14} $ . Following Sims (2002), reduced linear rational expectations models are obtained by

$$ \boldsymbol{x}_{t}=\boldsymbol{\Theta}_{1}\boldsymbol{x}_{t-1}+\boldsymbol{\Theta}_{0}\boldsymbol{z}_{t}+\boldsymbol{\Theta}_{c}, $$

where $ \Theta_{1} $ , $ \Theta_{c} $ , and $ \Theta_{0} $ are described in Sims (2002) $ ^{15} $ . The measurement equation of the model is

$$ \begin{aligned}\begin{bmatrix}{{{YGR_{t}}}} \\{{{INFL_{t}}}} \\{{{INT_{t}}}}\end{bmatrix}=\begin{bmatrix}{{{Y^{s}}}} \\{{{\pi^{s}}}} \\{{{r^{s}+\pi^{s}}}}\end{bmatrix}+\begin{bmatrix}{{{0}}}&{{{0}}}&{{{4}}}&{{{0}}}&{{{0}}}&{{{0}}}&{{{0}}}&{{{0}}} \\{{{0}}}&{{{0}}}&{{{0}}}&{{{4}}}&{{{0}}}&{{{0}}}&{{{0}}}&{{{0}}} \\{{{0}}}&{{{0}}}&{{{0}}}&{{{0}}}&{{{4}}}&{{{0}}}&{{{0}}}&{{{0}}}\end{bmatrix}\begin{bmatrix}{{{E_{t}\hat{Y}_{t+1}}}} \\{{{E_{t}\hat{\pi}_{t+1}}}} \\{{{\hat{Y}_{t}}}} \\{{{\hat{\pi}_{t}}}} \\{{{\hat{i}_{t}}}} \\{{{\hat{W}_{t}}}} \\{{{\hat{L}_{t}}}} \\{{{\hat{Z}_{t}}}}\end{bmatrix}+\boldsymbol{v}_{t},\end{aligned} $$
where

$$ F_{t}(\boldsymbol{z}_{t-1},\boldsymbol{v}_{t},\boldsymbol{\xi}_{s})=\begin{bmatrix}f_{t}(\boldsymbol{x}_{t-1},\boldsymbol{v}_{t})\\ \theta\end{bmatrix} $$

and

$$ H_{t}(\boldsymbol{z}_{t},\boldsymbol{\epsilon}_{t},\boldsymbol{\xi}_{o})=h_{t}(\boldsymbol{x}_{t})+\boldsymbol{\epsilon}_{t}. $$

This NNNSS is called a self-organizing state space (SOSS) model. Moreover, Kitagawa (1998) suggests to adopt a time-varying parameter approach if we can use a “symmetric” random walk model, $ \tilde{\pmb{\theta}}_{t} = \tilde{\pmb{\theta}}_{t-1} + \pmb{v}_{2,t} $ , with $ \pmb{v}_{2,t} $ a white noise sequence distributed with a density function $ p_{2}(v_{2,t}) $ . The nonlinear function $ F_{t} $ is then defined by

$$ F_{t}(\boldsymbol{z}_{t-1},\boldsymbol{v}_{t},\boldsymbol{\xi}_{s})=\begin{bmatrix}f_{t}(\boldsymbol{x}_{t-1},\boldsymbol{v}_{1,t})\\ \tilde{\boldsymbol{\theta}}_{t-1}+\boldsymbol{v}_{2,t}\end{bmatrix}, $$

where the system noise vector is $ \boldsymbol{v}_{t} = (\boldsymbol{v}_{1,t}, \boldsymbol{v}_{2,t})^{t} $ . In our method we combine time-varying parameters, $ \tilde{\pmb{\theta}}_{t-1} $ , and invariant parameters, $ \pmb{\theta} $ . An augmented vector of time-varying/invariant parameters is defined $ \boldsymbol{\Theta}_{t-1} = (\tilde{\pmb{\theta}}_{t}, \pmb{\theta})^{t} $ . The augmented state vector is redefined as follows.

$$ \boldsymbol{z}_{t}=\left[\boldsymbol{x}_{t}\atop\boldsymbol{\Theta}_{t}\right]. $$

Thus, our nonlinear, non-Gaussian, and non-stationary function $F_{t}$ is redefined by

$$ F_{t}(\boldsymbol{z}_{t-1},\boldsymbol{v}_{t},\boldsymbol{\xi}_{s})=\left[\begin{aligned}f_{t}(\boldsymbol{x}_{t-1},\boldsymbol{v}_{1,t})\\ \tilde{\boldsymbol{\theta}}_{t-1}+\boldsymbol{v}_{2,t}\\ \boldsymbol{\theta}\end{aligned}\right]. $$

Note that the measurement equation is the second equation of (32). In our method, we stress that states, time-varying parameters, and invariant parameters are estimated simultaneously. Therefore, our problem is how to estimate $ z_{t} $ .

### 3.2 Monte Carlo Particle Filter

The Monte Carlo particle filter is a variant of sequential Monte Carlo algorithms. In MCPF, expectations of a posterior distribution are approximated using “particles” that have weights.

$$ E[p(\boldsymbol{z}_{t}|\mathbf{Y}_{1:t})]\simeq\frac{1}{\sum_{m=1}^{M}w_{t}^{m}}\sum_{m=1}^{M}w_{t}^{m}\delta(\boldsymbol{z}_{t}-\boldsymbol{z}_{t}^{m}), $$

where $ w_{t}^{m} $ is the weight of a particle $ z_{t}^{m} $ , M is the number of particles, and $ \delta $ is the Dirac's delta function $ ^{21} $ . Weights $ w_{t}^{m} $ $ m=\{1,2,\cdots,M\} $ are defined as follows.

$$ w_{t}^{m}=r(\psi(\pmb{y}_{t},\pmb{z}_{t}^{m}))\Big|\frac{\partial\psi}{\partial\pmb{y}_{t}}\Big|, $$

$$ \delta(x)=0,\mathrm{if}x\neq0, $$

$$ \int_{\infty}^{\infty}\delta(x)d x=1. $$
where $ \psi $ is the inverse function of the function $ h^{22} $ . The right hand side of Eq. (36) is the likelihood function of an NNNSS model. In the standard algorithm of MCPF, the particles $ x_{t}^{m} $ are resampled with sampling probabilities proportional to $ w_{t}^{1},\cdots,w_{t}^{M} $ . Resampling algorithms are discussed in Kitagawa (1996). After resampling, we have $ w_{t}^{m}=1/M $ . Therefore, Eq. (35) is rewritten as

$$ E[p(\boldsymbol{z}_{t}|\boldsymbol{Y}_{1:t})]\simeq\frac{1}{M}\sum_{m=1}^{M}\delta(\boldsymbol{z}_{t}-\hat{\boldsymbol{z}}_{t}^{m}), $$

where $ \hat{z}_{t}^{m} $ are particles after resampling. Particles $ x_{t}^{m}\;m=\{1,2,\cdots,M\} $ are sampled from a system equation:

$$ \boldsymbol{z}_{t}^{m}\sim p(\boldsymbol{z}_{t}|\boldsymbol{z}_{t-1}^{m},\boldsymbol{\xi}_{s}). $$

Kitagawa (1996) shows that the log-likelihood of $\theta$ is approximated by

$$ l(\boldsymbol{\theta})\simeq\sum_{t=1}^{T}\log(\sum_{m=1}^{M}w_{t}^{m})-T\log M, $$

where T is the number of observations. Using Eq. (39), we can compare the fits of DSGE models. In self-organizing state space modeling, the augmented state vector is estimated using MCPF. Thus, states and parameters are estimated simultaneously without maximizing the log-likelihood of Eq. (32) because the parameter vector $ \theta $ in Eq. (32) is approximated by particles and it is estimated as the state vector in Eq. (31) $ ^{23} $ .

On a self-organizing state space model, however, Hürseler and Künsch (2001) points out a problem: determination of initial distributions of parameters for a self-organizing state space model. The estimated parameters of a self-organizing state space model comprise a subset of the initial distributions of parameters. We must know the posterior distributions of parameters to estimate parameters adequately. However, the posterior distributions of the parameters are generally unknown. Parameter estimation fails if we do not know their appropriate initial distributions. Yano (2008a) proposes a method to seek initial distributions of parameters for a self-organizing state space model using the simplex Nelder-Mead algorithm to solve the problem. In this paper, we use uniform distributions for initial distributions of time-varying parameters because most time-varying parameters are restricted to be more than zero and less than unity.

### 3.3 Time-varying Parameters

In this paper, we assume the “symmetric” random walk prior (the Litterman prior) to estimate time-varying parameters (see Doan et al. (1984)). The random walk prior is given by

$$ \tilde{\boldsymbol{\theta}}_{t}=\tilde{\boldsymbol{\theta}}_{t-1}+\boldsymbol{v}_{2,t}, $$

where $ \boldsymbol{v}_{2,t}\sim q(\boldsymbol{v}_{2,t}|\boldsymbol{\Sigma}_{\boldsymbol{\xi}_{s}}) $ , $ q(\boldsymbol{v}_{2,t}|\boldsymbol{\Sigma}_{\boldsymbol{\xi}_{s}}) $ is a Gaussian distribution, and $ \Sigma_{\xi_{s}} $ is a diagonal matrix. In general, the diagonal components, $ \{\xi_{1,s},\xi_{2,s},\cdots,\xi_{L,s}\} $ , of $ \Sigma_{\xi_{s}} $ are different. In this paper, however, to reduce computational complexity, we assume as follows.

$$ \xi_{1,s}=\xi_{2,s}=\cdots=\xi_{L,s}=|\xi_{s}|. $$
The time evolution of a coefficient is then given by

$$ \tilde{\theta}_{i,t}=\tilde{\theta}_{i,t-1}+|\xi_{s}|v_{2,i,t}, $$

where $ v_{2,i,t} \sim N(0,1) $ . Note that $ \sigma_{L,t} $ , $ \xi_{Z,t} $ , $ \phi_{Y,t} $ , $ \phi_{\pi,t} $ , $ \sigma_{Y,t} $ , $ \sigma_{\pi,t} $ , $ \sigma_{i,t} $ , $ \sigma_{Z,t} $ , $ \sigma_{Y,t}^{v} $ , $ \sigma_{\pi,t}^{v} $ , $ \sigma_{i,t}^{v} $ are restricted to be positive and $ h_{t} $ , $ \sigma_{C,t} $ , $ \xi_{p,t} $ , and $ \rho_{i,t} $ are restricted to be more than zero and less than unity. The particles that violate these restrictions are numerically discarded before resampling.

### 3.4 Algorithm

In our method, we adopt not a smoothing algorithm but a filtering algorithm because the rational expectations hypothesis is consistent with the latter. If we use a smoothing algorithm to estimate time-varying parameters, the estimates of them include the information at times $ t + 1 $ , $ t + 2 $ , $ \cdots $ which is not known at time t. Our method to estimate time-varying parameters of DSGE models is summarized as follows:

1. In time t, generate $ z_{t} $ based on the results at time t - 1.

2. Using particles, the linear rational expectations system is solved to obtain the state transition equation Eq. (23).

3. If a particle implies indeterminacy (or non-existence of a stable rational expectations solution), then the weight of the particle, $ w_{t}^{m} $ , is set to zero.

4. If a unique stable solution exists, then the weight of a particle is calculated using Eq. (36).

5. Resampling particles with sampling probabilities proportional to $ w_{t}^{1}, \cdots, w_{t}^{M} $

6. Replace t with $ t + 1 $ .

7. Go to 1.

## 4 Empirical Analysis

We use data from 1981:Q1 up to 2007:Q4 $ ^{24} $ . We assume the Japanese economy was trapped in a liquidity trap (the non-negativity constraint on nominal short-term interest rates) at 1999:Q1. Moreover, we suppose the economy escapes from the trap at 2006:Q4 because the quantitative-easing policy and the zero-interest-rate policy of the BOJ are ended at 2006:Q1 and 2006:Q3, respectively. The simulation settings used in empirical analysis are described in appendix B.

Figure 1 shows the annualized estimates of $ Y_{t}^{s} $ , $ \pi_{t}^{s} $ , and $ r_{t}^{s~25} $ . The black lines in all figures are means of particles, and the green and red lines are 95% confidence intervals, which are calculated using 100 bootstrap samples of particles. From the mid-1980s to the early 1990, $ Y_{t}^{s} $ is from about 2% to 5%, and the periods are called the “bubble economy.” From the mid-1990s to the early 2000s, $ Y_{t}^{s} $ is relatively small,
Figure 4 shows the estimates of time-varying parameters of NTR. The inertia term, $ \rho_{i} $ , is from 0.6 to 0.7 in most periods. It indicates that the BOJ makes the nominal short-term interest rate smooth. The coefficient of the output gap, $ \phi_{Y} $ , is from 1.0 to 1.6 in most periods, and it shows that the BOJ's reaction of output gap is stable from 1985 to present. The coefficient of the inflation rate, $ \phi_{\pi} $ , increases from 0.5 to 3.0, and this result shows that the BOJ is increasing the focus on the inflation rate from 1985 to 2007.

[Figure 4 about here.]

The stochastic volatilities, $ \sigma_{Y,t} $ , $ \sigma_{\pi,t} $ , $ \sigma_{i,t} $ , $ \sigma_{Z,t} $ , are shown in Figure 5. The stochastic volatilities, $ \sigma_{Y,t}^{v} $ , $ \sigma_{\pi,t}^{v} $ , $ \sigma_{i,t}^{v} $ , are shown in Figure 6.

[Figure 5 about here.]

[Figure 6 about here.]

In conventional DSGE modeling with invariant parameters, impulse response functions are often used for policy analysis. IRFs show the responses of variables to a shock, for example, a monetary shock, a technology shock, or a demand shock. In our method, however, IRFs are responses of variables, which are expectational conditional on information known at time t, to a shock because we estimate time-varying parameters of DSGE and a parameter at time t may change at time $ t+1 $ . Expectational impulse response functions to a contractionary monetary shock at 1985:Q1, 1990:Q1, 1995:Q1, and 2007:Q4 are shown in Figure 7, 8, 9, and 10, respectively. These figures show that the responses of inflation to a monetary shock are different. In 1985:Q1 a contractionary monetary shock caused a positive inflation rate, and it is inconsistent with monetary theory. This odd response is originated from the small value of $ \phi_{\pi} $ , which indicates that the BOJ understates the inflation rate in the 1980s. The increasing of $ \phi_{\pi} $ makes the responses of inflation normalized from 1990s to 2007. The responses of $ \hat{Y}_{t} $ , $ \hat{w}_{t} $ , and $ \hat{L}_{t} $ are consistent with monetary theory.

[Figure 7 about here.]

[Figure 8 about here.]

[Figure 9 about here.]

[Figure 10 about here.]

EIRFs to technology shock at 1985:Q1, 1990:Q1, 1995:Q1, 2000:Q1, and 2005:Q1, are shown in Figure 11, 12, 13, 14, and 15, respectively. The responses in figures in the "pre-zero-interest-rate" period are more volatile than ones in figures in the "zero-interest-rate" period because the stabilizing effect of monetary policy is lost under the liquidity trap.

[Figure 11 about here.]

[Figure 12 about here.]