On the existence and application of continuous-time threshold autoregressions of order two

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 that 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.