International Journal of Forecasting (1992), 18(3), 439-454.
P. J. Brockwell1 and R. J. Hyndman2
- Statistics Department, Colorado State University, Fort Collins, CO 80523, USA.
- Statistics Department, University of Melbourne, Parkville, Vic. 3052, Australia.
Abstract: 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 2δ>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, …, 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 jn/n, j = 1, 2, …, with n large. This model is used to fit ‘narrow boundaryÙ CTAR models to both simulated and real data.
Keywords: Non-linear forecasting; Threshold autoregression; State-space representation; Maximum likelihood estimation; Continuous-time autoregression.