Pj. Brockwell et Rj. Williams, ON THE EXISTENCE AND APPLICATION OF CONTINUOUS-TIME THRESHOLD AUTOREGRESSIONS OF ORDER-2, Advances in Applied Probability, 29(1), 1997, pp. 205-227
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-Gir
sanov formula and a random time-change are used to overcome the diffic
ulties associated with possible discontinuities and degeneracies in th
e 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 appro
ximating sequence is used to fit a CTAR(2) model to percentage relativ
e daily changes in the Australian All Ordinaries Index of share prices
by maximization of the 'Gaussian likelihood'. The advantages of non-l
inear relative to linear time series models are briefly discussed and
illustrated by means of the forecasting performance of the model fitte
d to the All Ordinaries Index.