# On continuous-time threshold autoregression

The use of non-linear models in time series analysis has expanded rapidly in the last ten years, with the development of several useful classes of discrete-time non-linear models. One family of processes which has been found valuable is the class of self-exciting threshold autoregressive (SETAR) models discussed extensively in the books of Tong (1983, 1990). In this paper we consider problems of modelling and forecasting with continuous-time threshold autoregressive (CTAR) processes. Techniques for analyzing such models have been proposed by Tong and Yeung (1991) and Brockwell, Hyndman and Grunwald (1991). In this paper we define a CTAR(p) process X(t) with boundary width 2\delta>0 as the first component of a p-dimensional Markov process X(t), defined by a stochastic differential equation. We are primarily concerned with the problems of model-fitting and forecasting when observations are available at times 1, 2, \dots, N; however, the techniques considered apply equally well to irregularly spaced observations. For practical computations with CTAR processes we approximate the process X(t) by a linearly interpolated discrete-time Markov process whose transitions occur at times j/n, j = 1, 2, \dots with n large. This model is used to fit ‘narrow boundary’ CTAR models to both simulated and real data.