Using the log-likelihood of a model, Eq. (39), we compare DSGE models: the model in section 2, the DSGE model without inflation idexation, and the DSGE model without habit formation. The log-likelihoods of models and the estimates of $ |\xi_{s}| $ are shown in Table 1. These results indicate that our model in section 2 is better than the other models. They also show that the inflation inertia and the habit persistence have crucial roles in empirical analysis based on DSGE models, and they are consistent with An and Schorfheide (2007), Hirose and Naganuma (2007), and related studies.

[Table 1 about here.]

## 5 Conclusion and Discussion

This paper proposes a new method to estimate parameters of dynamic general equilibrium models under a liquidity trap based on the Monte Carlo particle filter and a self-organizing state space model. This method is a natural extension of Yano (2009). Our method analyzes how stable structural parameters are. Adopting it creates the great advantage that the structural changes of parameters are detected naturally. The novel feature of our method is that we are able to estimate parameters of new Keynesian DSGE models under a liquidity trap (Krugman (1998)), because nonlinear, non-Gaussian, and non-stationary state space models allow model switching. Moreover, we estimate time-varying trends of macroeconomic data: real output, inflation rate, and real interest rate. To estimate trends of macroeconomic data, the Hodrick-Prescott filter, proposed by Hodrick and Prescott (1997), is often used. In recent years, the Baxter-King filter, proposed by Baxter and King (1999), and the Christiano-Fitzgerald filter, proposed by Christiano and Fitzgerald (2003) are also often used. Our method is an alternative to these filters, and it is a “structural” estimation of time-varying economic trends. We conclude that 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 empirical analysis, we estimate new Keynesian DSGE models under a liquidity trap using Japanese macroeconomic data, which include the “zero-interest-rate” period (1999-2006). The analysis shows that the growth rate of natural output declines in the late 1990s but becomes as high as about 2% in the mid-2000s. The target rate of inflation is too low in the 1990s and the 2000s, and it causes deflation in the Japanese economy. In the “zero-interest-rate” period, the impulse responses to technology shocks, aggregate demand shocks, and aggregate supply shocks are more volatile than the other period because the stabilizing effect of monetary policy is lost under the liquidity trap. These results are consistent with Section 4.2, Woodford (2003). Following Eggertsson and Woodford (2003), this problem can be solved by adopting the flexible 2% – 3% targeted rate of inflation based on GDP deflator.

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) $ ^{27} $ . In policy analysis of DSGE models, impulse response functions are often used. In our framework, the effectiveness of the traditional