Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-05T09:31:27.327Z Has data issue: false hasContentIssue false
  • English
  • Français

Aggregation of a random-coefficient ar(1) process with infinite variance and idiosyncratic innovations

Part of: Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Donata Puplinskaitė  and
Donatas Surgailis
Donata Puplinskaitė*
Affiliation:
Vilnius University and Institute of Mathematics and Informatics, Vilnius
Donatas Surgailis*
Affiliation:
Institute of Mathematics and Informatics, Vilnius
*
∗ Postal address: Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania. Email address: [email protected]
∗∗ Postal address: Institute of Mathematics and Informatics, Akademijos 4, LT-08663 Vilnius, Lithuania. Email address: [email protected]
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.

Contemporaneous aggregation of N independent copies of a random-coefficient AR(1) process with random coefficient a ∈ (−1, 1) and independent and identically distributed innovations belonging to the domain of attraction of an α-stable law (0 < α < 2) is discussed. We show that, under the normalization N1/α, the limit aggregate exists, in the sense of weak convergence of finite-dimensional distributions, and is a mixed stable moving average as studied in Surgailis, Rosiński, Mandrekar and Cambanis (1993). We focus on the case where the slope coefficient a has probability density vanishing regularly at a = 1 with exponent b ∈ (0, α − 1) for α ∈ (1, 2). We show that in this case, the limit aggregate {X̅t} exhibits long memory. In particular, for {X̅t}, we investigate the decay of the codifference, the limit of partial sums, and the long-range dependence (sample Allen variance) property of Heyde and Yang (1997).

Keywords

Aggregationrandom-coefficient AR(1) processinfinite variancemixed stable moving averagecodifferenceself-similar processlong memory

MSC classification

Primary: 62M10: Time series, auto-correlation, regression, etc.
Secondary: 60G18: Self-similar processes 60G52: Stable processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 2 , June 2010 , pp. 509 - 527
Copyright
Copyright © Applied Probability Trust 2010 

References

Astrauskas, A. (1983). Limit theorems for sums of linearly generated random variables. Lithuanian Math. J. 23, 127–134.CrossRefGoogle Scholar
Astrauskas, A., Levy, J. B. and Taqqu, M. S. (1991). The asymptotic dependence structure of the linear fractional Lévy motion. Lithuanian Math. J. 31, 1–19.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
Celov, D., Leipus, R. and Philippe, A. (2007). Time series aggregation, disaggregation, and long memory. Lithuanian Math. J. 47, 379–393.CrossRefGoogle Scholar
Cioczek-Georges, R. and Mandelbrot, B. B. (1995). Stable fractal sums of pulses: the general case. Preprint. Available at http://yale.academia.edu/BenoitMandelbrot/Papers/21711.Google Scholar
Cioczek-Georges, R., Mandelbrot, B. B., Samorodnitsky, G. and Taqqu, M. S. (1995). Stable fractal sums of pulses: the cylindrical case. Bernoulli 1, 201–216.CrossRefGoogle Scholar
Cox, D. R. (1984). Long-range dependence: a review. In Statistics: An Appraisal, eds David, H. A. and David, H. T. Iowa State University Press, Iowa, pp. 55–74.Google Scholar
Ding, Z. and Granger, C. W. J. (1996). Modeling volatility persistence of speculative returns: a new approach. J. Econometrics 73, 185–215.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Giraitis, L., Leipus, R. and Surgailis, D. (2010). Aggregation of random coefficient GLARCH(1,1) process. Econometric Theory 26, 406–425.CrossRefGoogle Scholar
Gonçalves, E. and Gouriéroux, C. (1988). Aggrégation de processus autorégressifs d'ordre 1. Ann. Econom. Statist. 12, 127–149.Google Scholar
Granger, C. W. J. (1980). Long memory relationship and the aggregation of dynamic models. J. Econometrics 14, 227–238.CrossRefGoogle Scholar
Heyde, C. C. and Yang, Y. (1997). On defining long-range dependence. J. Appl. Prob. 34, 939–944.CrossRefGoogle Scholar
Ibragimov, I. A. and Linnik, Y. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.Google Scholar
Kazakevičius, V., Leipus, R. and Viano, M.-C. (2004). Stability of random coefficient ARCH models and aggregation schemes. J. Econometrics 120, 139–158.CrossRefGoogle Scholar
Koul, H. L. and Surgailis, D. (2001). Asymptotics of empirical processes of long memory moving averages with infinite variance. Stoch. Process. Appl. 91, 309–336.CrossRefGoogle Scholar
Lamperti, J. (1962). Semi-stable stochastic processes. Trans. Amer. Math. Soc. 104, 62–78.CrossRefGoogle Scholar
Leipus, R. and Viano, M.-C. (2002). Aggregation in ARCH models. Lithuanian Math. J. 42, 54–70.CrossRefGoogle Scholar
Loève, M. (1963). Probability Theory, 3rd edn. Van Nostrand, Princeton, NJ.Google Scholar
Mikosch, T. (2003). Modelling dependence and tails of financial time series. In Extreme Values in Finance, Telecommunications and the Environment, eds Finkenstädt, B. and Rootzén, H., Chapman and Hall, New York, pp. 185–286.Google Scholar
Mikosch., T. and Samorodnitsky, G. (2000). Ruin probability with claims modeled by a stationary ergodic stable process. Ann. Prob. 28, 1814–1851.CrossRefGoogle Scholar
Oppenheim, G. and Viano, M.-C. (2004). Aggregation of random parameters Ornstein-Uhlenbeck or AR processes: some convergence results. J. Time Ser. Anal. 25, 335–350.CrossRefGoogle Scholar
Puplinskaitė, D. and Surgailis, D. (2009). Aggregation of random-coefficient AR(1) process with infinite variance and common innovations. Lithuanian Math. J. 49, 446–463.CrossRefGoogle Scholar
Robinson, P. M. (1978). Statistical inference for a random coefficient autoregressive model. Scand. J. Statist. 5, 163–168.Google Scholar
Rosiński, J. (1995). On the structure of stationary stable processes. Ann. Prob. 23, 1163–1187.CrossRefGoogle Scholar
Samorodnitsky, G. (2004). Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes. Ann. Prob. 32, 1438–1468.CrossRefGoogle Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.Google Scholar
Surgailis, D. (1979). On the Markov property of a class of linear infinitely divisible fields. Z. Wahrscheinlichkeitsth 49, 293–311.CrossRefGoogle Scholar
Surgailis, D., Rosiński, J., Mandrekar, V. and Cambanis, S. (1992). Stable generalized moving averages. Preprint, University of North Carolina.Google Scholar
Surgailis, D., Rosiński, J., Mandrekar, V. and Cambanis, S. (1993). Stable mixed moving averages. Prob. Theory Relat. Fields 97, 543–558.CrossRefGoogle Scholar
Zaffaroni, P. (2004). Contemporaneous aggregation of linear dynamic models in large economies. J. Econometrics 120, 75–102.CrossRefGoogle Scholar
Zaffaroni, P. (2007a). Aggregation and memory of models of changing volatility. J. Econometrics 136, 237–249.CrossRefGoogle Scholar
Zaffaroni, P. (2007b). Contemporaneous aggregation of GARCH processes. J. Time Ser. Anal. 28, 521–544.CrossRefGoogle Scholar