Comments on "Econometric Analysis of Fisher's Equation" by Peter C. B. Phillips

American Journal of Economics and Sociology, The, Jan, 2005 by John Rust

I

Introductory Remarks

I AM HONORED to discuss Peter Phillips's fascinating contribution to this volume, although my remarks are tempered by the fact that this is not my area of expertise. Ostensibly this paper is an empirical analysis of the Fisher equation; however, I view its primary contribution as providing important new econometric tools for analyzing nonstationary time series. After summarizing its key theoretical contributions for the benefit of readers who are less familiar with continuous time stochastic processes, I offer my assessment of whether this elegant asymptotic approach yields new empirical insights into the Fisher equation. Overall, I think the paper leaves us with some important unresolved puzzles about the joint evolution of the nominal interest rate, the inflation rate, and the (ex post) real rate of interest. More heretically, I question whether the simple version of the Fisher equation provides a useful framework for thinking about the behavior of nominal interest rates, especially since the simplest versions of this equation ignore the tremendous advances in finance in the years since Fisher's death.

II

Theoretical Contributions

THE NEW ECONOMETRIC CONTRIBUTIONS include derivation of asymptotic distributions for the "spatial density" and its associated hazard function, and the exponent d in "fractionally integrated" time series processes. The latter result is especially important, since for the first time we have a unified framework for conducting inference in a class of potentially nonstationary long memory univariate time series models that obviates the need for overly restrictive parametric assumptions (e.g., Gaussian error terms).

The spatial density is simply a rescaled version of the well-known nonparametric kernel density estimator for IId data, and the hazard function is constructed from the spatial density by the standard formula. The novel result of this paper is Theorem III.A, which shows that an appropriately rescaled version of the kernel density estimator still converges with probability 1 when applied to data-generating processes (DGPs) that have a unit root. However rather than converging to a deterministic limit (the true marginal or stationary density of the DGP in the IID or ergodic cases, respectively), Phillips shows that when the DGP has a unit root the kernel density estimator converges in probability to a random limiting object knows as a local time, a concept originally introduced by the probabilist Paul Levy for the special case of Brownian motion back in the late 1930s. Levy called the local time L(r,s) mesure du voisinage, since it is (the density of) the measure of the time a Brownian path B(t) spends in the vicinity of the point s over the time interval [0,r]. The most straightforward way to define L(r,s) is in terms of the random occupation time that the Brownian path spends in the set A over the interval [0,r]:

(1) [GAMMA](r, A) [equivalent to] [[integral].sup.r.sub.0] I {B(s) [member of] A}ds.

The local time is just the density (Radon-Nikodym derivative) of the random measure [GAMMA](r, A):

(2) [GAMMA](r, A) = [integral]I{x [member of] A} L(r, x)dx.

The concept of local time can be generalized to the class of semi-martingales M(t), that is, processes that can be represented as M(t) = U(t) V(t), where U(t) is a (local) martingale and V(t) is a finite variation process (the reader is referred to Karatzas and Shreve 1991 for more details). Unfortunately, the definition of local time given in this paper may be difficult for some readers to grasp since it depends on the additional concept of quadratic variation [M] of the semi-martingale M defined, nonconstructively, as the unique increasing stochastic process obtained from the Doob-Meyer decomposition of the nonnegative (local) submartingale [U.sup.2](t), where U is the local martingale component of the semi-martingale M. The Doob-Meyer decomposition establishes that any nonnegative submartingale X(t) can be uniquely represented as the sum of a martingale Y(t) plus an increasing stochastic process Z(t). In the case of a Brownian motion, the quadratic variation turns out to be [B], = t since the Doob-Myer decomposition of the nonnegative super-martingale X(t) = [B.sup.2](t) is given by [B.sup.2](t) = ([B.sup.2](t) - t) t, where Y(t) = [B.sup.2](t) - t is a martingale, and Z(t) = t is the increasing process. Thus, in the Brownian motion case the definition of the local time [L.sub.M](r, s) reduces to

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the right-hand side of Equation (3) is clearly the same as the right-hand side of the definition given in Equation (5) of Phillips's paper, since dt = d[[B].sub.t] = d[M], in the case of a Brownian motion. Phillips provides some helpful intuition for local times via the inverse relation between local times and quadratic variation in Equation (6) of his paper, which he analogizes to the standard spectral density decomposition of long-run variance. This inverse relation of Equation (16) can be seen to be an immediate consequence of the original definition of local time as the density of occupation time in Equation (2) above since for the case A = (-[infinity], [infinity]) we can see from Equation (1) that [GAMMA](t, A) = t, so using the definition of local time in Equation (2) we have: