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On the First-Order Autoregressive Process with Infinite Variance
Published online by Cambridge University Press: 18 October 2010
Abstract
For a first-order autoregressive process Yt = βYt−1 + ∈t where the ∈t'S are i.i.d. and belong to the domain of attraction of a stable law, the strong consistency of the ordinary least-squares estimator bn of β is obtained for β = 1, and the limiting distribution of bn is established as a functional of a Lévy process. Generalizations to seasonal difference models are also considered. These results are useful in testing for the presence of unit roots when the ∈t'S are heavy-tailed.
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REFERENCES
1.Chan, N.H. & Wei, C.Z.. Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics 15 (1987): 1050–1063.CrossRefGoogle Scholar
2.Chatterji, S.D.An Lp convergence theorem. Annals of Mathematical Statistics 40 (1969): 1068–1070.CrossRefGoogle Scholar
3.Davis, R.A. & Resnick, S.I.. Limit theory for the sample covariance and correlation functions of moving averages. Annals of Statistics 14 (1986): 533–558.CrossRefGoogle Scholar
4.Dickey, D.A., Haza, D.P., & Fuller, W.A.. Testing for unit roots in seasonal time series. Journal of the American Statistical Association 79 (1984): 355–367.CrossRefGoogle Scholar
5.DuMouchel, W.H.Estimating the stable index α in order to measure tail thickness. Annals of Statistics 11 (1983): 1019–1031.CrossRefGoogle Scholar
6.Fama, E., The behavior of stock market prices. Journal of Business 38 (1965): 34–105.CrossRefGoogle Scholar
7.Feller, W.An introduction to probability theory and its applications, Vol. 2. 2nd Edition. New York: Wiley, 1976.Google Scholar
9.Hannan, E.J. & Kanter, M.. Autoregressive processes with infinite variance. Journal of Applied Probability 14 (1977): 411–415.CrossRefGoogle Scholar
10.Jacod, J.Calcul Stochastique et Problemes de Martingales. Lectures Notes in Mathematics, No. 714. Berlin and New York: Springer-Verlag, 1979.CrossRefGoogle Scholar
11.Jain, N.C.A Donsker-Varadhan type of invariance principle. Z. Wahrscheinlichkeitstheorie View. Gebiete. 59(1982): 117–138.CrossRefGoogle Scholar
12.Knight, K., Rate of convergence of centered estimates of autoregressive parameters for infinite variance regressions. Journal of Time Series Analysis 8 (1987): 51–60.CrossRefGoogle Scholar
13.Kopp, P.E.Martingales and stochastic integrals. Cambridge: Cambridge University Press, 1984.CrossRefGoogle Scholar
14.Lai, T.L. & Wei, C.Z.. Least-squares estimates in stochastic regression models with applications to identification and control of dynamical systems. Annals of Statistics 10 (1982): 154–166.CrossRefGoogle Scholar
15.Mandelbrot, B., Long-run linearity, locally Gaussian process, H-spectra, and infinite variances. International Economic Review 10(1969): 82–111.CrossRefGoogle Scholar
16.McCullough, H., Continuous-time processes with stable increments. Journal of Business 51 (1978): 601–619.CrossRefGoogle Scholar
17.Millar, P.W.Path behavior of processes with stationary independent increments. Z. Wahrscheinlichkeitstheorie View. Gebiete. 17 (1971): 53–73.CrossRefGoogle Scholar
18.Monroe, I.On the γ-variation of processes with stationary independent increments. Annals of Mathematical Statistics 43 (1972): 1213–1220.CrossRefGoogle Scholar
19.Phillips, P.C.B.Towards a unified asymptotic theory for autoregression. Biometrika 74 (1987): 535–547.CrossRefGoogle Scholar
21.Resnick, S.I.Point processes, regular variation, and weak convergence. Advances in Applied Probability 18 (1986): 66–138.CrossRefGoogle Scholar
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