Hostname: page-component-7dd5485656-s6l46 Total loading time: 0 Render date: 2025-10-31T23:19:01.362Z Has data issue: false hasContentIssue false
  • English
  • Français

On some nonstationary, nonlinear random processes and their stationary approximations

Part of: Parametric inference Inference from stochastic processes

Published online by Cambridge University Press:  08 September 2016

Suhasini Subba Rao
Suhasini Subba Rao*
Affiliation:
Universität Heidelberg
*
∗ Current address: Department of Statistics, Texas A&M University, 3143 TAMU, College Station, TX 77843-3143, USA. Email address: suhasini@stat.tamu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper our object is to show that a certain class of nonstationary random processes can locally be approximated by stationary processes. The class of processes we are considering includes the time-varying autoregressive conditional heteroscedastic and generalised autoregressive conditional heteroscedastic processes, amongst others. The measure of deviation from stationarity can be expressed as a function of a derivative random process. This derivative process inherits many properties common to stationary processes. We also show that the derivative processes obtained here have alpha-mixing properties.

Keywords

Derivative processlocal stationaritynonlinearitynonstationaritystate space model

MSC classification

Primary: 62M10: Time series, auto-correlation, regression, etc.
Secondary: 62F10: Point estimation

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 4 , December 2006 , pp. 1155 - 1172
Copyright
Copyright © Applied Probability Trust 2006 

References

Basrak, B., Davis, R. A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95–115.CrossRefGoogle Scholar
Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedasticity. J. Econometrics 31, 307–327.CrossRefGoogle Scholar
Bougerol, P. and Picard, N. (1992a). Stationarity of GARCH processes and of some nonnegative time series. J. Econometrics 52, 115–127.CrossRefGoogle Scholar
Bougerol, P. and Picard, N. (1992b). Strict stationarity of generalized autoregressive processes. Ann. Prob. 20, 1714–1730.CrossRefGoogle Scholar
Brandt, A. (1986). The stochastic equation Y n+1=A n Y n B n with stationary coefficients. Adv. Appl. Prob. 18, 211–220.Google Scholar
Cramér, H. (1961). On some classes of nonstationary stochastic processes. In Proc. 4th Berkeley Symp. Math. Statist. Prob., Vol. II, ed. Neyman, J., University of California Press, Berkeley, pp. 57–78.Google Scholar
Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 16, 1–37.Google Scholar
Dahlhaus, R. and Subba Rao, S. (2006). Statistical inference for time-varying ARCH processes. Ann. Statist. 34, 1075–1114.Google Scholar
Davidson, J. (1994). Stochastic Limit Theory. Clarendon Press, New York.Google Scholar
Engle, R. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007.CrossRefGoogle Scholar
Kesten, H. and Spitzer, F. (1984). Convergence in distribution for products of random matrices. Z. Wahrscheinlichkeitsth. 67, 363–386.Google Scholar
Moulines, E., Priouret, P. and Roueff, F. (2005). On recursive estimation for time varying autoregressive processes. Ann. Statist. 33, 2610–2654.Google Scholar
Nicholls, D. F. and Quinn, B. G. (1982). Random Coefficient Autoregressive Models: an Introduction (Lecture Notes Statist. 11). Springer, New York.Google Scholar
Priestley, M. B. (1965). Evolutionary spectra and non-stationary processes. J. R. Statist. Soc. B 27, 204–237.Google Scholar
Straumann, D. and Mikosch, T. (2006). Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equations approach. To appear in Ann. Statist. Google Scholar
Subba Rao, T. (1977). On the estimation of bilinear time series models. Bull. Internat. Statist. Inst. 41, 413–417.Google Scholar
Terdik, G. (1999). Bilinear Stochastic Models and Related Problems of Nonlinear Time Series Analysis. A Frequency Domain Approach (Lecture Notes Statist. 142). Springer, New York.Google Scholar
Tweedie, R. L. (1983). Criteria for rates of convergence of Markov chains, with application to queueing and storage theory. In Probability, Statistics and Analysis, eds Kingman, J. and Reuter, G., Cambridge University Press, pp. 260–277.CrossRefGoogle Scholar