NONLINEAR TRANSFORMATIONS OF INTEGRATED TIME SERIES:A RECONSIDERATION

Abstract

Abstract. In this paper I reconsider two of the questions raised by Granger and Hallman (Nonlinear transformations of integrated time series. J. Time Ser. Anal. 12 (1991), 207–24):(i) If Xt is I(1) and Zt=h(Xt), is Zt also I(1)? (ii) Can Xt and h(Xt) be cointegrated? The distinction between I(1) and I(0) processes is replaced by the distinction between long memory and short memory processes, where for short memory I mean strong mixing. By exploiting the fact that random walks (with positive trend component) are martingales (submartingales) and are also first-order Markov, I show that (a) unbounded convex (concave) and strictly monotonic transformations of random walks are always long memory processes, (b) polynomial, strictly convex (concave) transformations of random walks display a unit root component, but the first differences of such transformations need not be short memory, and (c) Xt and h(Xt), with h an unbounded convex (concave) or strictly monotonic function, can never be cointegrated.