early Greenspan era. An equilibrium real rate is negative from the early 2000s to the mid-2000s because the Fed cuts policy rates to prevent deflation. In our estimation, most structural variables are stable. However, The inverse of the elasticity of labor, $ \sigma_{L} $ , is varying from about 2.0 to about 3.5, and it is consistent that the estimates of $ \sigma_{L} $ , which are inferred by Smets and Wouters (2003), Levin et al. (2005), Del Negro et al. (2007), Smets and Wouters (2007), and related studies, have relatively large standard deviations.

Adopting the time-varying-parameter approach does not suggest that structural parameters are “time-varying.” 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. The TVP 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 appendix C, 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 the TVP approach creates the great advantage that the structural changes of parameters are detected naturally. Thus, our method is suitable to analyze how stable structural parameters are.

Our paper is closely related with Fernandez-Villaverde and Rubio-Ramirez (2007b) $ ^{7} $ . However, there exist several large differences between our paper and theirs. The first point is that they focus on the stabilities of parameters of the Taylor rule and several parameters, for example, Calvo pricing parameters. In contrast, our main concern is that natural rates, deep parameters, business cycles are simultaneously and consistently estimated using the TVP approach. The second point is that they use maximizing the likelihood of MCPF to estimate parameters, while, we adopt a self-organizing state space model for 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 third point is that we estimate a time-varying natural rate of real output, a time-varying inflation target, and a time-varying equilibrium real interest rate.

Canova and Sala (2009) point out that structural parameters, which is based on maximum likelihood estimation, of DSGE models is weakly identified. The main cause of the problem is that the curvature of the likelihood of a DSGE model and the population objective function may be small in certain regions of the parameter space. In our method, the search space of most time-varying parameters is possible to 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, for example, Del Negro et al. (2007) and Smets and Wouters (2007). Such kind of restrictions based on previous studies are “reasonable” and help to reduce the problem.

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.