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.We thank
Liudas Giraitis for providing the manuscript by Dalla, Giraitis, and
Hidalgo (2006). We are grateful to the two
referees and the editor for their detailed comments, which led to
substantial improvements. We also thank Michael Stein for helpful comments
on an earlier version. The work is supported in part by NSF grant
DMS-0478704.