Continuous time threshold autoregressive models


Peter J Brockwell, Rob J Hyndman, Gary K Grunwald


16 February 1991

Publication details

Statistica Sinica 1, 401-410




The importance of non-linear models in time series analysis has been recognized increasingly over the past ten years. A number of discrete time non-linear processes have been introduced and found valuable for the modelling of observed series. Among these processes are the discrete time threshold models, discussed extensively in the book of Tong (1983). The purpose of this paper is to define a continuous time analogue of the threshold AR(p) process and to discuss some of its properties. For the continuous time threshold AR(1) process (henceforth denoted CTAR(1)) we derive the stationary distribution (under appropriate assumptions) and consider problems of prediction and inference. The techniques developed apply equally well both to regularly and to irregularly spaced observations.

Keywords: non-linear model, stationary distribution, prediction, Gaussian likelihood.