Properties of nonlinear transformations of fractionally integrated processes

This paper shows that the properties of nonlinear transformations of a fractionally integrated process strongly depend on whether the initial series is stationary or not. Transforming a stationary Gaussian I(d) process with d > 0 leads to a long-memory process with the same or a smaller long-memory parameter depending on the Hermite rank of the transformation. Any nonlinear transformation of an antipersistent Gaussian I(d) process is 1(0)). For non-stationary I(d) processes, every polynomial transformation is non-stationary and exhibits a stochastic trend in mean and in variance. In particular, the square of a non-stationary Gaussian I(d) process still has long memory with parameter d, whereas the square of a stationary Gaussian I(d) process shows less dependence than the initial process. Simulation results for other transformations are also discussed. (C) 2002 Elsevier Science B.V. All rights reserved.