Unit Roots in Macroeconomic Time Series: Some Critical Issues

Abstract

An enormous amount of analytical literature has recently appeared on the topic of in macroeconomic time series. Indeed, tests L for the presence of unit roots and techniques for dealing with them have together comprised one of the most active areas, over the past decade, in the entire field of macroeconomics. The issues at hand have involved substantive questions about the nature of macroeconomic growth and fluctuations in developed economies and technical questions about model formulation and estimation in systems that include unit-root variables. The present paper attempts to describe several of the main issues and to evaluate alternative positions. It does not pretend to be a comprehensive survey of the literature or to provide an even-handed treatment of issues, however(1) Instead, it attempts to develop a convincing perspective on the topic, one that is consistent with the views of many active researchers in the area but that may nevertheless be somewhat idiosyncratic. The exposition that is presented below is designed to be predominantly nontechnical in nature. Indeed, it takes a rather old-fashioned approach econometric issues and uses recently developed concepts only sparingly. It does, however, rely extensively on national conventions involving the time series operator, L. Under these conventions the symbol L may be manipulated as if it were an algebraic symbol while its effect, when applied to a time series variable, is to shift the variable's date back in time by one period. Thus Lx sub t = X sub t-1 while bLLx sub t = bL sup 2 x sub t = bx sub t-2, etc. In addition, the notation alpha(L) will denote a polynomial expression in the lag operator as follows: alpha(L) = alpha sub 0 + alpha sub 1 L + alpha sub 2 L sup 2 +alpha sub 3L sup 3 + .... Therefore, alpha(L)x sub t = alpha sub 0 x sub t +alpha sub 1 x sub t-1 + alpha sub 2 x sub t-2 +.... Using this notation, then, a distributed-lag regression relation of Yt on current and lagged values of x sub t could be written as y sub t = alpha(L)x sub t + xi sub t, with xi sub t a stochastic disturbance term. Furthermore, polynomials in L, which are often restricted to have only a finite number of terms, may be multiplied as in the following example:(2) if alpha(L) = alpha sub 0 + alpha sub 1 L + alpha sub 2 L sup 2 and beta(L) = beta sub 0 + beta sub 1 L, then alpha(L)beta(L) = BETA sub 0 ALPHA sub 0 + BETA sub 0 ALPHA sub 1 L + beta sub 0alpha sub 2 l sup 2 + alpha sub 0 beta sub 1 L + alpha sub 1 beta sub 1 L sup 2 + alpha sub 2 beta sub 1 L sup 3 . Finally, division by a lag polynomial means that the implied inverse, (y-l(L), is a polynomial such that alpha sup -1 (L)alpha(L) = 1. Thus alpha(L)beta sup -1 (L) yields a polynomial gamma(L) such that beta(L)gamma(L) = alpha(L). It should be mentioned that the first coefficient of a lag polynomial, such as alpha sub 0 , is often normalized so as to equal one. A brief outline of our discussion is as follows. In Section 1, the distinction between trend stationarity and difference stationarity of time series is introduced. That distinction is then related to the root concept in Section 2, which is primarily devoted to a description of attempts by researchers to determine whether the time series of real GNP values for the United States is difference or trend stationary (i.e., does or does not have an autoregressive unit root). Two approaches, involving different strategies for the specification of maintained and tested hypotheses, are discussed. Then in Section 3 a third approach, which presumes that the real GNP series is a sum of trend-stationary and difference-stationary components, is considered. From the discussion in Sections 2 and 3 together, it is concluded that the relevant question is not whether the GNP series is difference stationary, but what is the relative contribution of the two components. It is also concluded that an accurate answer is not obtainable with the amount of data available. …