A STATISTICAL TEST FOR THE STATIONARITY AND THE STABILITY OF EQUILIBRIUM

Introduction

One of the most important developments in econometrics in the 1980s is what has been conveniently summarized as the unit root. Concerning the least-squares estimator of an autoregressive parameter, say p, its asymptotic distribution when the true p is unity is different from that when | p| is less than unity, as shown by Dickey and Fuller (1979) and Phillips (1987). The point is important in applied econometrics because the finite sample distribution when |p| is less than but near unity resembles the asymptotic distribution for p = 1 more closely than the asymptotic distribution for |p| < 1. One implication is that the power in testing p= 1 against |p| < 1 by the least-squares estimator is bound to be low when the sample size is not very large. Nevertheless we are frequently compelled to discriminate between p= 1 and |p| < 1 by the economic and statistical problems.

However, as judged from the articles published or yet unpublished as of the time of the present writing, the latest research efforts are somewhat counteracting to the previous ones: (1) The Bayesian inference contains no such anomaly as found in the sampling approach (see Zellner, 1971, p. 187), but a number of problems arise on the prior. Sims (1988) points out that economic theories do not necessarily justify a sharp point prior placed on p = 1, and Wago and Tsurumi (1991) refer to the inference problems caused by such a prior. Phillips (1991) criticizes the flat prior as a representation of ignorance. (2) The wisdom of taking p= 1 for the null hypothesis has been questioned, and the stationarity for the null hypothesis has been investigated in Park (1990), Fukushige and Hatanaka (1989), Ogaki and Park (1989), Fisher and Park (1990), and Bierens (1990). Schotman and van Dijk (1990, 1991) treat the stationarity and the unit root symmetrically and derive a posterior odds ratio. (3) As regards Nelson and Plosser (1982), which revealed the importance of the unit root in economic data, Schmidt and Phillips (1992), Choi (1990), and Haldrup (1990) find drawbacks in the method used. Each proposes a revised method within the framework of sampling approach, and Choi (1990) in particular argues that his revision reverses the conclusion of Nelson and Plosser. From the standpoint of the Bayesian inference DeJong and Whiteman (1991) also reverse the conclusion of Nelson and Plosser (1982). It seems that the Box-Jenkins modeling of trends should be considered with a grain of salt.