Nonlinear Transformations of Integrated Time Series:a Reconsideration

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 X t is I(1) and Z t = h ( X t ), is Z t also I(1)? (ii) Can X t and h ( X t ) 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) X t and h ( X t ), with h an unbounded convex (concave) or strictly monotonic function, can never be cointegrated.