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Econometric analysis of Fisher's equation
American Journal of Economics and Sociology, The, Jan, 2005 by Peter C.B. Phillips
I
Introduction
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SINCE IRVING FISHER (1896, 1930) formalized (1) the notion of a real rate of interest, the concept has played a significant role in the formulation of a wide range of economic models. These include individual agent decision making regarding investment, savings, and portfolio allocations, options pricing models in finance, and the modern theory of inflation targeting in macroeconomics, to name but a few. Naturally enough, in light of the role that the real rate plays in economic theory models, a good deal of attention has been devoted in the literature, especially in macroeconomics, to the measurement of the real rate and to the characterization of its temporal dependence properties. Prima facie, this task seems like a simple exercise in time series econometrics. However, the empirical analysis is complicated by two factors: (1) the apparent nonstationary behavior of the series involved, particularly interest rates but also sometimes inflation; and (2) the fact that the ex ante real rate of interest depends on inflation expectations and is therefore not directly measured. Perhaps because of these complicating factors, no consensus seems to have emerged about the time series properties of the real rate of interest, in spite of intensive empirical study. In particular, while economic theory models routinely assume that the real rate of interest is a constant, or fluctuates in a stationary way about a constant mean, the empirical work indicates that this is not so or at best holds only over short regimes.
Figure 1 shows monthly data for the ex post real interest rate in the United States over the period 1961:1-1985:12. The series is calculated by taking the U.S. 90-day treasury bill rate for the nominal interest rate and by using the U.S. monthly CPI (all commodities, with no adjustment for housing costs) to compute three-month inflation rates. The figure also shows subgroup means calculated over the subperiods 1961:1-1973:1, 1973:2-1982:1, and 1982:2-1985:12. The data cover the same period as that studied recently by Garcia and Perron (1994), who used regime shift methods to estimate the ex ante real rate over approximately these subperiods. These authors concluded that the ex ante real rate of interest was effectively constant but subject to occasional mean shifts over 1961-1985. They found two mean shifts over this time period and gave results very similar to those obtained by subgroup means that are displayed in Figure 1. For these data, at least, the conclusion does not seem unreasonable.
[FIGURE 1 OMITTED]
Figures 2 and 3 graph the ex post real interest rate series calculated in the same way over the longer periods 1961-1997 and 1934-1997. Over the 1961-1997 period the graphs shows subsample means for the additional two subperiods 1990-1993 and 1994-1997. Apparently, there is a need to allow for continuing regime shifts in the mean level if this approach to modeling the real rate of interest is to give acceptable results. For the longer period, an even larger number of mean shifts is needed to accommodate this approach, and the results seem much less satisfying--we merely illustrate what is involved in Figure 3. It is hard to conclude that one is doing any more than simply curve fitting in such exercises and then possibly providing ex post rationalizations for the mean shifts. Note, however, that the data in Figure 3 clearly do support the conclusion reached in Fama's (1975) influential study that the real rate has a constant mean over the period 1953-1971. Notwithstanding the apparent constancy over 1953-1971, so many mean shifts are needed to model the data this way that the Garcia-Perron conclusion that the ex ante real rate of interest fluctuates about a constant mean, subject to occasional mean shifts, seems much less reasonable over the longer period 1934-1997 than it does for 1961-1986. Indeed, over longer periods such as this, most formal tests (e.g., Rose 1988; Walsh 1987) support a conclusion that is the opposite of stationarity and favor unit root nonstationary. One of the goals of the present paper is to determine whether there are other hypotheses that are more reasonable than these two alternatives.
[FIGURES 2-3 OMITTED]
Another goal of the paper is to contribute some new methods to assist in the econometric analysis of data of this type. The methods given here furnish a new way of describing and characterizing data like interest rates and inflation that appear to have nonstationary elements. More specifically, the paper proposes a nonparametric spatial density estimate as a new descriptive tool for nonstationary time series. Many series like the interest rates shown in Figures 1-3 behave as if they have no fixed mean. The random wandering characteristic of these series is hard to describe quantitatively. What a spatial density does is provide useful quantitative information about the spatial location characteristics of a time series, in just the same way as a probability density can be used to characterize stationary time series. As will be shown, in contrast to a probability density, the spatial density is a random process. However, it turns out that we can still obtain consistent estimates of spatial densities using nonparametric techniques, like those of kernel methods, which have proved useful in studying iid and strictly stationary time series. We outline these new procedures here and provide an asymptotic theory that characterizes their large sample properties and facilitates inference.
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