Table 2: Time-varying Parameter Search Space (TVPs' time indices are abbreviated)

Parameter	Search Space
$ Y^{s} $	(0.0, 5)
$ \phi_{\pi} $	(1.0, 3)
$ \phi_{Y} $	(0, 2)
$ h, \xi_{p}, \xi_{z}, \rho_{G}, \rho_{i} $	(0, 1)
$ \sigma_{L}, \nu $	positive
$ \sigma_{C}, \sigma_{\pi}, \sigma_{I}, \sigma_{Q}, \sigma_{i}, \sigma_{G}, \sigma_{Z} $	positive
$ \sigma_{Y}^{v}, \sigma_{C}^{v}, \sigma_{\pi}^{v}, \sigma_{I}^{v}, \sigma_{W}^{v}, \sigma_{L}^{v}, \sigma_{i}^{v} $	positive
other parameters	no limitation

that the average of the growth of natural output is 3.02 in the long run. From the mid-1980s to the mid-2000s, $ \pi_{t}^{s} $ is varying between about 1.5% and 3.5%. The average of $ \pi_{t}^{s} $ is about 2.46 from 1985 to 2007. It points that the inflation target rate of the Fed is about 2.5% in the past 20 years. The $ r_{t}^{s} $ is an estimate of equilibrium real interest rate. This result shows that an equilibrium real interest rate is negative from the early 2000s to the mid-2000s because the Fed cuts short term nominal interest rates to flight against deflation in the early 2000s.

Figure 2 shows the Investment-Specific Growth (ISG). It indicates that ISG of the US economy slowed down in the late 1980s, and recovers from the early 1990s to the early 2000s. The behavior of ISG coincides with the booming economy in the 1990s, and is probably related with investment-specific technology changes (IST), proposed by Greenwood et al. (1997). While IST has a microeconomic foundation, our ISG is practically introduced to make the estimation of $ Y^{s} $ stable because the growth rate of investment is different from the growth rate of natural output. Although our result suggests that ISG has an important role in the long-run growth, the cause of it is still an open question.

Figure 3 shows the estimates of the endogenous variables. The expectational output gap, $ E_{t}\hat{Y}_{t} $ , and the output gap, $ \hat{Y}_{t} $ , have fundamentally similar shapes. They indicate that the serious recessions happen in 1991. From 1992 to the late 1990s, the output gap is keeping positive. This result shows the booming economy in the Clinton era. From the early 1990s to the late 1990s, the inflation expectation, $ E_{t}\hat{\pi}_{t} $ , and the inflation rate, $ \hat{\pi}_{t} $ , are negative because of disinflation of the US economy. The interest rate, $ \hat{i}_{t} $ , shows the deviation from the equilibrium real interest rate, and it presents the fact that the Fed cuts short term nominal interest rates in recessions and hikes the rates in the economic booms. The symbol, $ \hat{Z}_{t} $ , shows the negative technology shocks that happened in the early 1990s, and it corresponds to the recessions in 1991.

Fig. 4 shows the estimates of time-varying parameters. These estimates indicate that some “structural” parameters are time-varying. The results indicate that habit persistence, h, the Calvo parameter, $ \xi_{p} $ , and the coefficient of AR(1) technology process, $ \xi_{z} $ , are relatively stable. The parameter, $ \sigma_{C} $ , is about 1 in most periods, and it indicates that the utility function of consumption is nearly equal to the log-type utility function. The inverse of the elasticity of labor, $ \sigma_{L} $ is varying from about 2.0 to about 3.5. In table 3, the estimates of $ \sigma_{L} $ , which are inferred by Levin et al. (2005) (LOWW), Del Negro et al. (2007) (DSSW), Smets and Wouters (2007) (SW), and Hirose and Naganuma (2007) (HN) using the US data, are shown, and they have relatively large standard deviations. These results of previous studies are caused by the fluctuation of time-varying $ \sigma_{L} $ .

Figure 5 shows the estimates of time-varying parameters of TR. The inertia term, $ \rho_{i} $ , is from 0.55 to 0.65.