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This paper analyses moment and near-epoch dependence properties for the general class of models in which the conditional variance is a linear function of squared lags of the process. It is shown how the properties of these processes depend independently on the sum and rate of convergence of the lag coefficients, the former controlling the existence of moments, and the latter the memory of the volatility process. Conditions are derived for existence of second and fourth moments, and also for the processes to be L 1 -and L 2 -near epoch dependent (NED), and also to be L 0 -approximable, in the absence of moments. The geometric convergence cases (GARCH and IGARCH) are compared with models having hyperbolic convergence rates, the FIGARCH, and a newly proposed generalization, the HYGARCH model. The latter model is applied to 10 daily dollar exchange rates for 1980-1996, with very similar results. When nested in the HYGARCH framework, the FIGARCH model appears as a valid simplification. However, when applied to data for Asian exchange rates over the 1997 crisis period, a distinctively different pattern emerges. The model exhibits remarkable parameter stability across the pre-and post-crisis periods.

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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.