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Random coefficient autoregression, regime switching and long memory

Part of: Limit theorems Inference from stochastic processes Special processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Remigijus Leipus  and
Donatas Surgailis
Remigijus Leipus*
Affiliation:
Vilnius University and Institute of Mathematics and Informatics, Vilnius
Donatas Surgailis*
Affiliation:
Institute of Mathematics and Informatics, Vilnius
*
∗ Postal address: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Email address: remigijus.leipus@maf.vu.lt
∗∗ Postal address: Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania.
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Abstract

We discuss long-memory properties and the partial sums process of the AR(1) process {Xt, t ∈ 𝕫} with random coefficient {at, t ∈ 𝕫} taking independent values Aj ∈ [0,1] on consecutive intervals of a stationary renewal process with a power-law interrenewal distribution. In the case when the distribution of generic Aj has either an atom at the point a=1 or a beta-type probability density in a neighborhood of a=1, we show that the covariance function of {Xt} decays hyperbolically with exponent between 0 and 1, and that a suitably normalized partial sums process of {Xt} weakly converges to a stable Lévy process.

Keywords

Random coefficient autoregressionregime switchingrenewal processstable Lévy motionlong memory

MSC classification

Primary: 60K05: Renewal theory 62M10: Time series, auto-correlation, regression, etc.
Secondary: 60F05: Central limit and other weak theorems 60G52: Stable processes 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 3 , September 2003 , pp. 737 - 754
Copyright
Copyright © Applied Probability Trust 2003 

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References

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
Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Prob. Appl. 10, 323–331.Google Scholar
Davidson, J. and Sibbertsen, P. (2002). Generating schemes for long memory processes: regimes, aggregation and linearity. Preprint.Google Scholar
Diebold, F. X. and Inoue, A. (2001). Long memory and regime switching. J. Econometrics 105, 131–159.Google Scholar
Giraitis, L., Robinson, P. M. and Surgailis, D. (2000). A model for long memory conditional heteroscedasticity. Ann. Appl. Prob. 10, 1002–1024.Google Scholar
Gourieroux, C. and Jasiak, J. (2001). Memory and infrequent breaks. Econom. Lett. 70, 29–41.Google Scholar
Granger, C. W. J. (1980). Long memory relationships and the aggregation of dynamic models. J. Econometrics 14, 227–238.Google Scholar
Granger, C. W. J. and Hyung, N. (1999). Occasional structural breaks and long memory. Discussion paper 99-14, Department of Economics, University of California, San Diego.Google Scholar
Leipus, R. and Viano, M.-C. (2003). Long memory and stochastic trend. Statist. Prob. Lett. 61, 177–190.Google Scholar
Lewis, P. A. W. and Lawrence, A. J. (1981). A new autoregressive time series model in exponential variables (NEAR(1)). Adv. Appl. Prob. 13, 826–845.Google Scholar
Liu, M. (2000). Modeling long memory in stock market volatility. J. Econometrics 99, 139–171.Google Scholar
Mikosch, T., Resnick, S., Rootzén, H. and Stegeman, A. (2002). Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Prob. 12, 23–68.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
Parke, W. R. (1999). What is fractional integration? Rev. Econom. Statist. 81, 632–638.Google Scholar
Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York.Google Scholar
Pipiras, V., Taqqu, M. S. and Levy, J. B. (2002). Slow, fast and arbitrary growth conditions for the renewal reward processes when the renewals and the rewards are heavy-tailed. Preprint.Google Scholar
Pourahmadi, M. (1988). Stationarity of the solution of X_t = A_t X_t-1 + varepsilon_t and analysis of non-Gaussian dependent variables. J. Time Ser. Anal. 9, 225–239.Google Scholar
Robinson, P.M. (1978). Statistical inference for a random coefficient autoregressive model. Scand. J. Statist. 5, 163–168.Google Scholar
Surgailis, D. (2002a). Stable limits of empirical processes of moving averages with infinite variance. Stoch. Process. Appl. 100, 255–274.Google Scholar
Surgailis, D. (2002b). Stable limits of sums of bounded functions of long memory moving averages with finite variance. Preprint.Google Scholar
Taqqu, M. S. and Levy, J. B. (1986). Using renewal processes to generate long-range dependence and high variability. In Dependence in Probability and Statistics, eds Eberlein, E. and Taqqu, M. S., Birkhäuser, Boston, MA, pp. 73–89.Google Scholar
Tjøstheim, D., (1986). Some doubly stochastic time series models. J. Time Ser. Anal. 7, 51–72.Google Scholar
Vervaat, W. (1979). On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.Google Scholar