Silviase/jgov_v1.4_full_ann huggingface dataset:
### 2.5 Monetary Policy

The monetary authority is assumed to determine the nominal interest rate according to the Taylor rule, proposed by Taylor (1993).

$$ i_{t}=(i_{t-1})^{\rho_{i}}(Y_{t}^{\phi_{Y}}\pi_{t}^{\phi_{\pi}})^{1-\rho_{i}} $$

where $ \phi_{Y} $ and $ \phi_{\pi} $ are constants and $ \epsilon_{i,t} \sim N(0, \sigma_{i,t}^{2}) $ .

### 2.6 Market Clearing

In the final market equilibrium, the final good production is equivalent to the households' demand for consumption, investment, and the expenditure of the government.

$$ Y_{t}=C_{t}+I_{t}+G_{t}, $$

where $ Y_{t}=\int_{0}^{1}Y_{j,t}dj $ , $ C_{t}=\int_{0}^{1}C_{j,t}dj $ , $ I_{t}=\int_{0}^{1}I_{j,t}dj $ , and $ G_{t} $ is a government expenditure.

### 2.7 Linearized Model

We linearize the model described above around the non-stochastic steady state. The linearized model consists of the new IS curve (NISC), the new Keynesian Phillips curve (NKPC), the Taylor rule (TR), and several equations. NISC is obtained as follows $ ^{8} $ .

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

NKPC is obtained as follows.

$$ \hat{\pi}_{t}=\frac{1}{1+\beta}\hat{\pi}_{t-1}+\frac{\beta}{1+\beta}E_{t}\hat{\pi}_{t+1}+\frac{(1-\xi_{p})(1-\beta\xi_{p})}{\xi_{p}(1+\beta)}\big[(1-\alpha)\hat{W}_{t}+\alpha\hat{r}_{t}^{K}-\hat{Z}_{t}\big]+\epsilon_{\pi,t}. $$

The other equations are

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

$$ \hat{L}_{t}=-\hat{W}_{t}+\hat{r}_{t}^{K}+\hat{K}_{t}, $$

$$ \hat{I}_{t}=\frac{1}{1+\beta}\hat{I}_{t-1}+\frac{\beta}{1+\beta}\hat{I}_{t+1}+\frac{\nu}{1+\beta}\hat{Q}_{t}+\epsilon_{I,t}, $$

$$ \hat{Q}_{t}=-E_{t}[\hat{i}_{t}-\hat{\pi}_{t+1}]+\frac{1-\delta}{1-\delta+\bar{r}^{K}}E_{t}\hat{Q}_{t+1}+\frac{\bar{r}^{K}}{1-\delta+\bar{r}^{K}}E_{t}\hat{r}_{t+1}^{K}+\epsilon_{Q,t}, $$

$$ \hat{K}_{t}=(1-\delta)\hat{K}_{t-1}+\delta\hat{I}_{t-1}, $$

$$ \hat{Y}_{t}=\Psi_{C}\hat{C}_{t}+\Psi_{I}\hat{I}_{t}+\Psi_{G}\hat{G}_{t}, $$

$$ \hat{Y}_{t}=\hat{Z}_{t}+\alpha\hat{K}_{t}+(1-\alpha)\hat{L}_{t}, $$

$$ \hat{G}_{t}=\rho_{G}\hat{G}_{t-1}+\epsilon_{G,t}, $$
In our method, we estimate the parameters of Eq. (18)-(32) using the TVP approach, which is explained in section 3. Thus, we define the vector of time-varying parameters as follows.

$$ \begin{align*}\tilde{\theta}_{t}=&[h_{t},\xi_{p,t},\sigma_{L,t},\nu_{t},\xi_{Z,t},\rho_{i,t},\phi_{Y,t},\phi_{\pi,t},\rho_{G,t},\sigma_{C,t},\sigma_{\pi,t},\sigma_{I,t},\sigma_{Q,t},\sigma_{i,t},\sigma_{G,t},\sigma_{Z,t},\\&Y_{t}^{s},I_{t}^{s},\pi_{t}^{s},L_{t}^{s},r_{t}^{s},\sigma_{Y,t}^{v},\sigma_{C,t}^{v},\sigma_{\pi,t}^{v},\sigma_{I,t}^{v},\sigma_{W,t}^{v},\sigma_{L,t}^{v},\sigma_{i,t}^{v}].\end{align*} $$

Note that we calibrate seven parameters: $ \beta $ , $ \alpha $ , $ \delta $ , $ \bar{r}^{K} = 1/\beta - 1 + \delta $ , $ \Psi_{C,t} $ , $ \Psi_{I,t} $ , and $ \Psi_{G,t} $ (see section 3.1.) Reduced linear rational expectations models are also redefined by

$$ \boldsymbol{x}_{t}=\boldsymbol{\Theta}_{1,t}\boldsymbol{x}_{t-1}+\boldsymbol{\epsilon}_{1,t}, $$

where $ \epsilon_{1,t}=\Theta_{0,t}z_{t} $ and $ \epsilon_{1,t}^{\prime}=\Theta_{0,t}^{\prime}z_{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 appendix C, it does not indicate that the deep parameters are “time-varying.” Our framework is just a practical one to estimate deep parameters. Adopting our framework 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 in the liquidity trap (Krugman (1998)) because NNNSS, which is described in section 3, allows model switching.

### 2.9 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. (31).

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 the particle is calculated using Eq. (C7).

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.
## 3 Empirical Analysis

We use data from 1981:Q1 up to 2007:Q4 $ ^{13} $ . The simulation settings used in empirical analysis are described in appendix B.

### 3.1 Preliminary Setting

Following Smets and Wouters (2007), we calibrate four parameters: $ \beta = 0.99 $ , $ \alpha = 0.3 $ , $ \delta = 0.06 $ , $ \bar{r}^{K} = 1/\beta - 1 + \delta $ , $ \Psi_{C,t} = 0.6 $ , $ \Psi_{I,t} = 0.2 $ , and $ \Psi_{G,t} = 0.2 $ . For preliminary setting for our method, we estimate our DSGE model using MCMC $ ^{14} $ . In Table 1, the estimates of MCMC are shown. For our method, we determine

Table 1: Preliminary Parameter Estimation Based on MCMC

prior density	posterior mean	confidence interval
h	beta	0.5619	0.4709	0.6687
$ \xi_{P} $	beta	0.8092	0.7604	0.8568
$ \sigma_{L} $	norm	0.3471	0.2451	0.4336
$ \nu $	norm	0.1185	0.0618	0.1832
$ \xi_{Z} $	beta	0.8839	0.8396	0.9328
$ \rho_{G} $	beta	0.6585	0.5252	0.7818
$ \rho_{i} $	norm	0.4370	0.3543	0.5313
$ \phi_{Y} $	norm	0.0200	0.0024	0.0777
$ \phi_{\pi} $	norm	1.0196	1.0044	1.0370

the prior distributions of time-varying parameters based on Table 1. The other simulation settings are described in appendix B.

In Table 2, we show the search space of TVPs to reduce the weakly identification problem, which is pointed out by Canova and Sala (2009). In our method, the space of some time-varying parameters can be strictly limited in a “reasonable” region because the MCP filter is a simulation-based algorithm. For example, the search space of the time-varying reaction parameter to inflation in the Taylor rule is limited from 1.0 to 3.0 because the range is consistent with the Taylor principle and the results of previous studies (Del Negro et al. (2007) and Smets and Wouters (2007)). If the particles are outside of the range, they are just discarded in simulation. Such kind of restrictions based on previous studies are “reasonable” and help to reduce the problem.

Figure 1 shows the annualized estimates of $ Y_{t}^{s} $ , $ \pi_{t}^{s} $ , and $ r_{t}^{s} $ $ {}^{15} $ . 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. To estimate natural rates of macroeconomic data, the Hodrick and Prescott (1997) filter is often used. In recent years, the Baxter and King (1999) filter and the Christiano and Fitzgerald (2003) filter are also often used. Our method, however, is an alternative to these filters, and it is “structural” estimation of time-varying natural rates $ {}^{16} $ . From the mid-1980s to the mid-2000s, $ Y_{t}^{s} $ is varying between about 2.0% and 4.0%. It suggests
Figure 1: Time-varying natural rates and targets
Figure 2: Investment-specific Technology Change
Figure 3: Endogenous variables
Figure 4: Time-varying parameters
Table 3: Estimates of previous studies

Mean (90% distribution intervals)	LOWW	DSSW	SW	HN
$ \sigma_{C} $	2.19 (1.68-2.74)	-	1.38 (1.16-1.59)	1.75 (1.51-1.97)
h	0.29 (0.20-0.38)	0.81 (0.77-0.85)	0.71 (0.64-0.78)	0.59 (0.39-0.79)
$ \sigma_{L} $	1.49 (0.95-2.12)	2.09 (0.95-3.19)	1.83 (0.91-2.78)	0.79 (0.46-1.11)
$ \xi_{p} $	0.83 (0.81-0.86)	0.83 (0.79-0.87)	0.66 (0.56-0.74)	0.82 (0.78-0.86)

in most periods. It indicates that the Fed makes the change of nominal short-term interest rates smooth. The coefficient of the output gap, $ \phi_{Y} $ , is from 1.8 to 2.1 in most periods. In the 1990s, $ \phi_{Y} $ , is relatively small rather than in the late 1980s, and it shows the change of monetary policy of the Fed. From 1985 to 1990, the coefficient of the inflation rate, $ \phi_{\pi} $ , increases from 2.5 to 4.0, and this result shows that the Fed is increasing the focus on the inflation rate from 1985 to the early 1990. From the mid-1990 to present, $ \phi_{\pi} $ is around 4.0 and very stable. It also indicates that the behavior of the Fed has changed to realize mild inflation and made it stable from the mid-Greenspan era to the early Bernanke era.

The stochastic volatilities, $ \sigma_{Y,t} $ , $ \sigma_{\pi,t} $ , $ \sigma_{i,t} $ , $ \sigma_{Z,t} $ , are shown in Figure 6. The stochastic volatilities, $ \sigma_{Y,t}^{v} $ , $ \sigma_{\pi,t}^{v} $ , $ \sigma_{i,t}^{v} $ , are shown in Figure 7.

In practice, the Hodrick and Prescott (1997) filter and the Baxter and King (1999) filter are often used to estimate the natural output of the US economy. Our method is an alternative to these filters, and it is “structural” estimation of time-varying economic trends. In Figure 8, we compare our annualized estimates of output gap with estimates of the HP filter and the CF filter. In the upper panel of Figure 8, we show our estimate (the black line) and the estimate of the HP filter (the blue line). In the lower panel of Figure 8, we show our estimate (the black line) and the estimate of the CF filter (the green line). The result of the HP filter is not consistent with our result. However, the one of the CF filter coincides with our one.

## 4 Conclusion and Discussion

This paper proposes a new method to estimate parameters, natural rates, and business cycles of dynamic stochastic general equilibrium models simultaneously and consistently. The method is based on the Monte Carlo particle filter, proposed by Kitagawa (1996) and Gordon et al. (1993), and a self-organizing state space model, proposed by Kitagawa (1998). In our method, we estimate the parameters and the natural rates using the time-varying-parameter approach, which is often used to infer invariant parameters practically. In most previous papers on DSGE models, structural parameters of them are assumed to be “deep (invariant).” However, our method analyzes how stable structural parameters are. Our method is based on Bayesian statistics and nonlinear, non-Gaussian, and non-stationary state space modeling to estimate unknown parameters and states. Adopting the TVP approach
Figure 5: Time-varying parameters of the Taylor rule
Figure 6: Stochastic Volatilities in the System Equation
Figure 7: Stochastic Volatilities in the Measurement Equation
Figure 8: Output gap: Comparing filtering methods
creates the great advantage that the structural changes of parameters are detected naturally. Moreover, we estimate time-varying natural rates of macroeconomic data: real output, inflation rate, and real interest rate. Our method is an alternative to detrending methods based on the Hodrick-Prescott filter, the Baxter-King filter, the Christiano-Fitzgerald filter, and other filtering algorithms.

In empirical analysis, we estimate a new Keynesian DSGE model using the US data. The analysis shows that the average of the growth of natural output is 3.02 and the average of inflation target is 2.46 from 1985 to 2007. It also indicates that the coefficients of the Taylor rule are stable and the Fed focus on the stability of inflation in the Greenspan era. An equilibrium real interest rate is negative from the early 2000s to the mid-2000s because the Fed cut rates to prevent deflation. Our estimate of output gap relatively coincides with the estimates, which are calculated by the HP/CF filters, although our method is totally different from the filters.

In a new study, we are estimating new Keynesian, small open economy DSGE models, new Keynesian DSGE models with liquidity-constraint households, Christiano et al. (2005), and second-order approximation of DSGE models. Furthermore, our method can be easily extended to estimate state-dependent-pricing models with random menu costs, proposed by Dotsey et al. (1999) $ ^{17} $ . In policy analysis of DSGE models, impulse response functions are often used. In our framework, the effectiveness of the traditional way is ambiguous because parameters in DSGE models are time-varying. If we calculate impulse response function at time t, the results of them may be meaningless because the parameters may have changed at time $ t + 1 $ . Canova and Gambetti (2006) proposes the use of generalized impulse response functions in time-varying structural vector autoregressions. However, in time-varying analysis of DSGE models, it is an open question. Additionally, the cause of investment-specific growth is also an open question.

## Appendix A Data Source

We use quarterly macroeconomic data on the US economy from 1981:Q1 to 2007:Q4.

- 3-Month Treasury Bill (Secondary Market Rate, Averages of Business Days), averaged over three months: St. Louis Fed, Economic Data - FRED,

• Seasonally-adjusted Real Gross Domestic Product: St. Louis Fed, Economic Data - FRED

• Seasonally-adjusted GDP deflator: St. Louis Fed, Economic Data - FRED

• Civilian Noninstitutional Population, averaged over three months: St. Louis Fed, Economic Data - FRED

• Seasonally-adjusted Real Private Nonresidential Fixed Investment: St. Louis Fed, Economic Data - FRED

• Seasonally-adjusted Average Weekly Hours: BLS
• Seasonally-adjusted Average Hourly Earnings of Production Workers: BLS

## Appendix B Simulation Setting

The discount factor, $ \beta $ , is calibrated to be 0.99. We use uniform distributions for initial prior distributions of states, time-varying parameters, and parameters: $ \mathit{uniform}(-1,1) $ for states, $ \mathit{uniform}(0,1) $ for time-varying parameters, and $ \mathit{uniform}(0,0.2) $ for parameters. The number of particle is 10,000 at time t. Thus, we generate 270,000 random variables at time t. In Eq. (30), we set C to zero.

## Appendix C Estimation Method

To estimate a state vector $ x_{t} $ and a time-varying-parameter vector, $ \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).

### Appendix C.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.

$$ \begin{aligned}\boldsymbol{x}_{t}&=f_{t}(\boldsymbol{x}_{t-1},\boldsymbol{\epsilon}_{1,t},\boldsymbol{\xi}_{s}),\\\boldsymbol{Y}_{t}&=h_{t}(\boldsymbol{x}_{t},\boldsymbol{v}_{t},\boldsymbol{\xi}_{o}),\end{aligned} $$

where $ x_{t} $ is an unknown $ n_{x} \times 1 $ state vector, $ \epsilon_{1,t} $ is $ n_{\epsilon} \times 1 $ system noise vector with a density function $ q(\mathbf{\epsilon}_{1}|\cdot)^{18} $ , $ v_{t} $ is $ n_{v} \times 1 $ observation noise vector with a density function $ r(\mathbf{v}|\cdot) $ . The function $ f_{t}: R^{n_{x}} \times R^{\epsilon_{v}} \to R^{n_{x}} $ is a possibly nonlinear time-varying function and the function $ h_{t}: R^{n_{x}} \times R^{n_{v}} \to R^{n_{y}} $ is a possibly nonlinear time-varying function. The first equation of (C1) is called a system equation and the second equation of (C1) is called an observation equation. We would like to emphasize the functions, $ f_{t} $ and $ h_{t} $ , are possibly time dependent. A system equation depends on a possibly unknown $ n_{s} \times 1 $ parameter vector, $ \xi_{s} $ , and an observation equation depends on a possibly unknown $ n_{o} \times 1 $ parameter vector, $ \xi_{o} $ . This NNSS specifies the two following conditional density functions.

$$ \begin{aligned}&p(\boldsymbol{x}_{t}|\boldsymbol{x}_{t-1},\boldsymbol{\xi}_{s}),\\&p(\boldsymbol{Y}_{t}|\boldsymbol{x}_{t},\boldsymbol{\xi}_{o}).\\ \end{aligned} $$

We define a parameter vector $ \theta $ as follows.

$$ \boldsymbol{\theta}=\left[\begin{aligned}\xi_{s}\\ \xi_{o}\end{aligned}\right]. $$

We denote that $ \theta_{j} $ , $ (1 \leq j \leq J) $ is the jth element of $ \theta $ and $ J(=n_{s}+n_{o}) $ is the number of elements of $ \theta $ . This type of state space model (C1) contains a broad class of linear, nonlinear, Gaussian, or non-Gaussian time series
models. In state space modeling, estimating the state space vector $ x_{t} $ is the most important problem. For the linear Gaussian state space model, the Kalman filter, which is proposed by Kalman (1960), is the most popular algorithm to estimate the state vector $ x_{t} $ . For nonlinear or non-Gaussian state space models, there are many algorithms. For example, the extended Kalman filter (Jazwinski (1970)) is the most popular algorithm; other examples are the Gaussian-sum filter (Alspach and Sorenson (1972)), the dynamic generalized model (West et al. (1985)), and the non-Gaussian filter and smoother (Kitagawa (1987)). In recent years, MCPF for NNNSS has been a popular algorithm because it is easily applicable to various time series models $ ^{19} $ .

In econometric analysis, generally, we don’t know the parameter vector $ \theta $ . In our framework, the unknown parameter vectors are $ \xi_{o} $ and $ \xi_{s} $ $ ^{20} $ . In traditional parameter estimation, maximizing the log-likelihood function of $ \theta $ is often used. The log-likelihood of $ \theta $ in MCPF is proposed by Kitagawa (1996). However, MCPF is problematic to estimate the parameter vector $ \theta $ because the likelihood of the filter contains errors from the Monte Carlo method. Thus, you cannot use nonlinear optimizing algorithm like Newton’s method $ ^{21} $ . To solve the problem, Kitagawa (1998) proposes a self-organizing state space model. In Kitagawa (1998), an augmented state vector is defined as follows.

$$ \boldsymbol{z}_{t}=\begin{bmatrix}\boldsymbol{x}_{t}\\ \boldsymbol{\Theta}_{t}\end{bmatrix}, $$

where $ \boldsymbol{\Theta}_{t} = (\tilde{\boldsymbol{\theta}}_{t}, \boldsymbol{\theta})^{t} $ , $ \tilde{\theta}_{t} $ is a vector of time-varying parameters, and $ \theta $ is a vector of invariant parameters. Note that $ \tilde{\theta}_{t} = \tilde{\theta}_{t-1} + \epsilon_{2,t} $ , with $ \epsilon_{2,t} $ a white noise sequence distributed with a density function $ p_{2}(\epsilon_{2,t} | \Sigma_{\xi_{s}}) $ . An augmented system equation and an augmented measurement equation are defined as

$$ \begin{aligned}&\boldsymbol{z}_{t}=F_{t}(\boldsymbol{z}_{t-1},\boldsymbol{\epsilon}_{t},\boldsymbol{\xi}_{s}),\\&\boldsymbol{Y}_{t}=H_{t}(\boldsymbol{z}_{t},\boldsymbol{v}_{t},\boldsymbol{\xi}_{o}),\\ \end{aligned} $$

where

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

and

$$ H_{t}(z_{t},\boldsymbol{v}_{t},\boldsymbol{\xi}_{o})=h_{t}(\boldsymbol{x}_{t},\boldsymbol{v}_{t},\boldsymbol{\xi}_{o}) $$

where $ \boldsymbol{\epsilon}_{t} = (\epsilon_{1,t}, \epsilon_{2,t})^{t} $ . This NNNSS is called a self-organizing state space (SOSS) model. 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} $ .
### Appendix C.2 Monte Carlo Particle Filter

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

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

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 $ ^{22} $ . Weights $ w_{t}^{m} $ $ m=\{1,2,\cdots,M\} $ are defined as follows.

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

where $ \psi $ is the inverse function of the function $ h^{23} $ . The right hand side of Eq. (C7) 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. (C6) 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. (C10), 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. (C5) because the parameter vector $ \theta $ in Eq. (C5) is approximated by particles and it is estimated as the state vector in Eq. (C4) $ ^{24} $ .

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

$$ \begin{aligned}&\delta(x)=0,\mathrm{if}x\neq0,\\&\int_{\infty}^{\infty}\delta(x)dx=1.\\ \end{aligned} $$