Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series processes

Abstract

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.

Testing for instability in covariance structures

“…As far as serial dependence is concerned, the requirement that y t be NED is typical in nonlinear time series analysis (see [17]) and it implies that y t is a mixingale [13]. Many of the DGPs considered in the literature generate NED series -examples include GARCH, bilinear and threshold models (see [14]). Part (ii) illustrates the trade-off between the memory of y t (i.e., its NED size α), and its largest existing moment: as α (the memory of y t ) approaches 1, r has to increase.…”

“…However, such difference is crucial: by virtue of the weighing scheme proposed in (13), we are able to detect the presence of breaks closer to either end of the sample than afforded by (14). More specific comments on the power properties of tests based on (13) versus tests based on (14) are in the remarks to Theorem 4; here we point out that the price to pay is that we are not able to study the limiting distribution of the supremum of (13) using the IP shown in Theorem 1, but conversely the SIP is needed.…”

Estimation and inference in unstable nonlinear least squares models

“…Sufficient conditions for geometric ergodicity can be found in Chan and Tong (1986), Davidson (2002), and require in general that, for all i = 1, . .…”

Functional central limit theorems for ARCH(∞) models

“…If ξ = 1,then this convergence implies the CLT. There are numerous literatures considering the CLT and FCLT for various GARCH family models (Berkes et al, 2008;Billingsley, 1968;Davidson, 2002;De Jong and Davidson, 2000;Herrndorf, 1984;Lee, 2014aLee, , 2014b.…”