# On continuous-time threshold autoregression

Peter J Brockwell and Rob J Hyndman
(1992) International Journal of Forecasting, 18(3), 439-454

DOI

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.