Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T05:30:52.255Z Has data issue: false hasContentIssue false
  • English
  • Français

A multiple-threshold AR(1) model

Published online by Cambridge University Press:  14 July 2016

K. S. Chan ,
Joseph D. Petruccelli ,
H. Tong  and
Samuel W. Woolford
K. S. Chan*
Affiliation:
Chinese University of Hong Kong
Joseph D. Petruccelli*
Affiliation:
Worcester Polytechnic Institute
H. Tong*
Affiliation:
Chinese University of Hong Kong
Samuel W. Woolford*
Affiliation:
Worcester Polytechnic Institute
*
∗Postal address: Department of Statistics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong.
∗∗Postal address: Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA.
∗Postal address: Department of Statistics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong.
∗∗Postal address: Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the model Zt = φ (0, k)+ φ(1, k)Zt–1 + at (k) whenever rk−1<Zt−1≦rk, 1≦k≦l, with r0 = –∞ and rl =∞. Here {φ (i, k); i = 0, 1; 1≦k≦l} is a sequence of real constants, not necessarily equal, and, for 1≦k≦l, {at(k), t≧1} is a sequence of i.i.d. random variables with mean 0 and with {at(k), t≧1} independent of {at(j), t≧1} for j ≠ k. Necessary and sufficient conditions on the constants {φ (i, k)} are given for the stationarity of the process. Least squares estimators of the model parameters are derived and, under mild regularity conditions, are shown to be strongly consistent and asymptotically normal.

Keywords

NON-LINEAR TIME SERIESSETAR MODELSAUTOREGRESSIVE MODELSMARKOV CHAINS
Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 2 , June 1985 , pp. 267 - 279
Copyright
Copyright © Applied Probability Trust 1985 

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

Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Orey, S. (1971) Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand Reinhold, New York.Google Scholar
Petruccelli, J. D. and Woolford, S. W. (1984) A threshold AR(1) model. J. Appl. Prob. 21, 270–286.CrossRefGoogle Scholar
Tong, H. (1983) Threshold Models in Non-Linear Time-Series Analysis. Lecture Notes in Statistics 21, Springer-Verlag, New York.CrossRefGoogle Scholar
Tong, H. and Lim, K. S. (1980) Threshold autoregression, limit cycles and cyclical data. J. R. Statist. Soc. B 42, 245–292.Google Scholar
Tweedie, R. L. (1975) Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space. Stoch. Proc. Appl. 3, 385–403.Google Scholar
Tweedie, R. L. (1976) Criteria for classifying general Markov chains. Adv. Appl. Prob. 8, 737–771.CrossRefGoogle Scholar
Tweedie, R. L. (1983) The existence of moments for stationary Markov chains. J. Appl. Prob. 20, 191–196.CrossRefGoogle Scholar