Properties of Nonlinear Transformations of Fractionally Integrated Processes
October 2002
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This paper shows that the properties of nonlinear transformations of a fractionally integrated process depend strongly 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 I(0). For non-stationary I(d) processes, every integer power transformation is non-stationary and exhibits a deterministic 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.
- process
- transformation
- series
- long-memory
- long-memory parameter
- parameter
- autocorrelation
- time series
- proposition
- gaussian
- hermite
- function
- moment
- xt ~ i
- result
- section
- hermite rank
- yt z t
- trend
- table
