where $ \boldsymbol{v}_{t}=\left(\epsilon_{Y,t}^{v},\epsilon_{\pi,t}^{v},\epsilon_{i,t}^{v}\right)^{T}\sim N(\mathbf{0},\mathbf{\Sigma}_{v,t}) $ with $ \mathbf{\Sigma}_{v,t}=diag((\sigma_{Y,t}^{v})^{2},(\sigma_{\pi,t}^{v})^{2},(\sigma_{i,t}^{v})^{2})^{16} $ . The growth rate of real output, $ YGR_{t} $ , is a log difference of real GDP per capita, rate of inflation, $ INF_{t} $ , is the a log difference of GDP deflator, and nominal interest rate, $ INT_{t} $ , is the uncollateralized overnight call rate. Real growth, inflation, and interest rate are annualized. The symbols, $ Y^{s} $ , $ \pi^{s} $ , and $ r^{s} $ are the trend of real output, the target rate of inflation, and the trend of real interest rates, respectively.

In our method, we estimate the parameters of Eq. (16)-(24) using the TVP approach. Thus, we define the vector of time-varying parameters as follows.

$$ \tilde{\theta}_{t}=[h_{t},\sigma_{C,t},\xi_{p,t},\sigma_{L,t},\xi_{Z,t},\rho_{i,t},\phi_{Y,t},\phi_{\pi,t},\sigma_{Y,t},\sigma_{\pi,t},\sigma_{i,t},\sigma_{Z,t},Y_{t}^{s},\pi_{t}^{s},r_{t}^{s},\sigma_{Y,t}^{v},\sigma_{\pi,t}^{v},\sigma_{i,t}^{v}], $$

where t is a time index. Note that we suppose that $ \beta $ is given. Structural linear rational expectations models are redefined by

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

Reduced linear rational expectations models are also redefined by

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

In 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 often used in state space modeling to estimate invariant 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 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 the liquidity trap (Krugman (1998)) because NNNSS allows model switching.

## 3 Estimation Method

To estimate a state vector $ x_{t} $ and a time-varying-parameter vector, $ \pmb{\theta}_{t} $ , we adopt the Monte Carlo Particle Filter (MCPF), proposed by Kitagawa (1996) and Gordon et al. (1993), and a self-organizing state space model, proposed by Kitagawa (1998).

### 3.1 Nonlinear, Non-Gaussian, and Non-stationary State Space Model

In this subsection, we describe a nonlinear, non-Gaussian, and non-stationary state space model and a self-organizing state space model (MCPF is described in the next subsection).

A nonlinear, non-Gaussian, and non-stationary state space model for the time series $ Y_{t}, t=\{1,2,\cdots,T\} $ is defined as follows.

$$ \boldsymbol{x}_{t}=f_{t}(\boldsymbol{x}_{t-1},\boldsymbol{v}_{t},\boldsymbol{\xi}_{s}), $$

$$ \mathbf{Y}_{t}=h_{t}(\boldsymbol{x}_{t},\boldsymbol{\epsilon}_{t},\boldsymbol{\xi}_{o}), $$