## 1 Introduction

In recent years, new Keynesian, dynamic stochastic general equilibrium models of monetary analysis have been rapidly developing. The early works of Kimball (1995), Roberts (1995), and Yun (1996) beget the subsequent many papers (see McCallum and Nelson (1999), Clarida et al. (1999), Gali (2002), and related studies which are referred therein) $ ^{2} $ . “Middle-size” new Keynesian models are developed by Christiano et al. (2005) and Smets and Wouters (2003), and their models are often adopted by practitioners in the government and the central bank. The fit performance of their models is discussed by Fout (2005), Trabandt (2006), and Del Negro et al. (2007).

Bayesian statistics are now becoming a standard tool to estimate these DSGE models. Smets and Wouters (2003), Onatski and Williams (2004), Levin et al. (2005), Del Negro et al. (2007), Smets and Wouters (2007), Hirose and Naganuma (2007), and many studies estimate parameters of new Keynesian DSGE models using Markov Chain Monte Carlo methods (MCMC). Fernandez-Villaverde and Rubio-Ramirez (2005) and Fernandez-Villaverde and Rubio-Ramirez (2007a) have shown that the Monte Carlo particle filter (MCPF) and maximizing likelihood can be successfully applied to estimate DSGE models $ ^{3} $ . An and Schorfheide (2007) is an excellent survey on this area, and see references cited therein $ ^{4} $ .

This paper proposes a new method to estimate parameters, natural rates, and business cycles of dynamic general equilibrium models simultaneously and consistently. It 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) $ ^{5} $ . 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. Adopting the TVP approach creates the great advantage that the structural changes of parameters are detected naturally. 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. Moreover, we estimate time-varying natural rates of macroeconomic data: natural output, a inflation rate, and a real interest rate $ ^{6} $ . To estimate natural rates of macroeconomic data, the Hodrick and Prescott (1997) filter, the Baxter and King (1999) filter, the Christiano and Fitzgerald (2003) filter, and other filtering algorithms are also often used in practice. Our method, however, is an alternative to these filters, and it is “structural” estimation of time-varying natural rates. Again, we would like to stress that our method simultaneously and consistently estimates natural rates, time-varying parameters, and business cycles.

In empirical analysis, we estimate a new Keynesian DSGE model using the US data from 1981:Q1 to 2007:Q4. 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. From the late Volcker era to the early Greenspan era, the reaction coefficient to inflation in the Taylor rule is increasing, and from the mid Greenspan era to the early Bernanke era it is stable around 4.0%. It also indicates that the behavior of the Fed had changed to realize mild inflation from the late Volcker era to the