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On the Existence and Application of Continuous-Time Threshold Autoregressions of Order Two

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

P. J. Brockwell  and
R. J. Williams
P. J. Brockwell*
Affiliation:
Royal Melbourne Institute of Technology
R. J. Williams*
Affiliation:
University of California, San Diego
*
∗ Postal address: Department of Statistics and Operations Research, Royal Melbourne Institute of Technology, GPO Box 2476V, Melbourne, Victoria 3001, Australia.
∗∗ Postal address: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093–0112, USA.
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Abstract

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.

Keywords

THRESHOLD AUTOREGRESSIONSTOCHASTIC DIFFERENTIAL EQUATIONDEGENERATE DIFFUSIONNON-LINEAR TIME SERIESCAMERON–MARTIN–GIRSANOV FORMULARANDOM TIME-CHANGEMARTINGALE PROBLEM

MSC classification

Primary: 60J60: Diffusion processes
Secondary: 60J35: Transition functions, generators and resolvents 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 1 , March 1997 , pp. 205 - 227
Copyright
Copyright © Applied Probability Trust 1997 

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Footnotes

Research supported in part by NSF Grant DMS 9504596 and GER 9023335.

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