Statistica Sinica 17(2007), 43-62

STABILITY OF CYCLIC THRESHOLD AND

THRESHOLD-LIKE AUTOREGRESSIVE TIME

SERIES MODELS

Thomas R. Boucher and Daren B. H. Cline

Plymouth State University and Texas A&M University

Abstract: We investigate the stability, in terms of $V$-uniform ergodicity or transience, of cyclic threshold autoregressive time series models. These models cycle through one of a number of collections of subregions of the state space when the process is large. Our results can be applied in cases where the model has multiple cycles and/or affine thresholds. The bounds on the parameter space are sharper than those in previous results, and are easily verified. We extend these results to autoregressive nonlinear time series that can be approximated well by a threshold model (threshold-like).

Key words and phrases: Ergodicity, Markov chain, nonlinear autoregressive time series, nonlinear time series, threshold autoregressive time series.