ON THE EXISTENCE AND APPLICATION OF CONTINUOUS-TIME THRESHOLD AUTOREGRESSIONS OF ORDER TWO

P. J. BROCKWELL and R. J. WILLIAMS


A continuous-time threshold autoregressive process of order two (CTAR(2)) is constructed as the first component of the unique (in law) weak solution of a stochastic differential equation. The Cameron-Martin-Girsanov formula and a random time-change are used to overcome the difficulties associated with possible discontinuities and degeneracies in the coefficients of the stochastic differential equation. A sequence of approximating processes which are well-suited to numerical calculations is shown to converge in distribution to a solution of this equation, provided the initial state vector has finite second moments. The approximating sequence is used to fit a CTAR(2) model to percentage relative daily changes in the Australian All Ordinaries Index of share prices by maximization of the "Gaussian likelihood". The advantages of non-linear relative to linear time series models are briefly discussed and illustrated by means of the forecasting performance of the model fitted to the All Ordinaries Index.

In Advances in Applied Probability, 29 (1997), 205-227.