ON CONTINUOUS-TIME THRESHOLD ARMA PROCESSES

A recent paper of Brockwell and Hyndman (Internat. J. Forecasting (1992)) considers the use of continuous-time threshold autoregressive (CTAR) processes in the modelling and forecasting of time series data. As in the linear case, the continuous-time model is particularly advantageous for dealing with irregularly spaced data. In this paper we consider an analogous continuous-time threshold ARMA (p, q) process with 0 less-than-or-equal-to q < p, expressing the process in terms of an underlying p-dimensional diffusion process. Recursions are derived for the likelihood of observations {y(t1),...,y(t(n))} in terms of the transition probabilities of the diffusion process. In the case when the underlying white noise and moving average coefficients are constant, the characteristic function of the transition probability distribution of the underlying diffusion process is expressed, via the Cameron-Martin-Girsanov formula, as an explicit functional of standard Brownian motion. Approximate numerical techniques for computing Gaussian likelihoods are examined and applied to the modelling of the Canadian Lynx Data.