Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T19:25:08.712Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximations and boundary conditions for continuous-time threshold autoregressive processes

Part of: Stochastic analysis Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Rob J. Hyndman
Rob J. Hyndman*
Affiliation:
The University of Melbourne
*
∗Postal address: Department of Statistics, University of Melbourne, Parkville, V1C 3052, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

Continuous-time threshold autoregressive (CTAR) processes have been developed in the past few years for modelling non-linear time series observed at irregular intervals. Several approximating processes are given here which are useful for simulation and inference. Each of the approximating processes implicitly defines conditions on the thresholds, thus providing greater understanding of the way in which boundary conditions arise.

Keywords

CONTINUOUS-TIME AUTOREGRESSIONTHRESHOLD AUTOREGRESSIONNON-LINEAR STOCHASTIC DIFFERENTIAL EQUATIONSUNEQUALLY SPACED TIME SERIES

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 62M10: Time series, auto-correlation, regression, etc.
Type
Research Article
Information
Journal of Applied Probability , Volume 31 , Issue 4 , December 1994 , pp. 1103 - 1109
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Atkinson, J. D. and Caughey, T. K. (1968) Spectral density of piecewise linear first order systems excited by white noise. J. Non-Linear Mechanics 3, 137–156.Google Scholar
Brockwell, P. J. and Hyndman, R. J. (1992) On continuous-time threshold autoregression. Int. J. Forecasting 8, 157–173.Google Scholar
Brockwell, P. J., Hyndman, R. J. and Grunwald, G. K. (1991) Continuous time threshold autoregressive models. Statistica Sinica 1, 401–410.Google Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
Hyndman, R. J. (1992a) Continuous time threshold autoregressive models. Unpublished PhD thesis, University of Melbourne.Google Scholar
Hyndman, R. J. (1992b) Forecast regions for non-linear and non-normal time series models. Research Report No. 7, Dept. of Statistics, University of Melbourne.Google Scholar
Khazen, E. M. (1961) Estimating the density of the probability distribution for random processes in systems with non-linearities of piecewise-linear type. Theory Prob. Appl. 6, 214–220.Google Scholar
Kloeden, P. E. and Platen, E. (1992) Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Berlin.Google Scholar
Kurtz, T. G. (1975) Semigroups of conditioned shifts and approximation of Markov processes. Ann. Prob. 3, 618–642.Google Scholar