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.