Establishing Conditions for the Functional Central Limit Theorem in Nonlinear and Semiparametric Time Series Processes

This paper considers methods of deriving sufficient conditions for the central limit theorem and functional central limit thorem to hold in a broad class of time series processes, including nonlinear processes and semiparametric linear processes. The common thread linking these results is the concept of near-epoch dependence on a mixing process, since powerful limit results are available under this limited-dependence property. The particular case of near-epoch dependence on an independent process provides a convenient framework for dealing with a range of nonlinear cases, including the bilinear, GARCH, and threshold autoregressive models. It is shown in particular that even SETAR processes with a unit root regime have short memory, under the right conditions. A simulation approach is also demonstrated, applicable to cases that are analytically intractable. A new FCLT is given for semiparametric linear processes, where the forcing processes are of the NED-on-mixing type, under conditions that are evidently close to necessary. ∗Email: [email protected]. Research supported by the ESRC under award L138251025. This paper is based on parts of a working paper circulated under the title “When is a Time Series I(0)?” I am grateful to Michael Jansson, Ron Gallant, an Associate Editor, and two anonymous referees for comments which have materially improved the paper. I retain all responsibility for errors.