Local Whittle Estimation of Fractional Integration for Nonlinear Processes

We study asymptotic properties of the local Whittle estimator of the long memory parameter for a wide class of fractionally integrated nonlinear time series models+ In particular, we solve the conjecture posed by Phillips and Shimotsu ~2004, Annals of Statistics 32, 656–692! for Type I processes under our framework, which requires a global smoothness condition on the spectral density of the short memory component+ The formulation allows the widely used fractional autoregressive integrated moving average ~FARIMA! models with generalized autoregressive conditionally heteroskedastic ~GARCH! innovations of various forms, and our asymptotic results provide a theoretical justification of the findings in simulations that the local Whittle estimator is robust to conditional heteroskedasticity+ Additionally, our conditions are easily verifiable and are satisfied for many nonlinear time series models+