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Time Series Regression With a Unit Root and Infinite-Variance Errors

Published online by Cambridge University Press:  11 February 2009

P.C.B. Phillips
P.C.B. Phillips
Affiliation:
Cowles Foundation for Research in EconomicsYale University
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Abstract

In [4] Chan and Tran give the limit theory for the least-squares coefficient in a random walk with i.i.d. (identically and independently distributed) errors that are in the domain of attraction of a stable law. This paper discusses their results and provides generalizations to the case of I(1) processes with weakly dependent errors whose distributions are in the domain of attraction of a stable law. General unit root tests are also studied. It is shown that the semiparametric corrections suggested by the author in other work [22] for the finite-variance case continue to work when the errors have infinite variance. Surprisingly, no modifications to the formulas given in [22] are required. The limit laws are expressed in terms of ratios of quadratic functional of a stable process rather than Brownian motion. The correction terms that eliminate nuisance parameter dependencies are random in the limit and involve multiple stochastic integrals that may be written in terms of the quadratic variation of the limiting stable process. Some extensions of these results to models with drifts and time trends are also indicated.

Type
Articles
Information
Econometric Theory , Volume 6 , Issue 1 , March 1990 , pp. 44 - 62
Copyright
Copyright © Cambridge University Press 1990

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References

1.Billingsley, P.Convergence of probability measures. New York: Wiley, 1968.Google Scholar
2.Breiman, L.On some limit theorems similar to the arc-sin law. Theory of Probability and its Applications 10 (1965): 323–331.CrossRefGoogle Scholar
3.Brockwell, P.J. & Davis, R.A.. Time series: theory and methods. New York: Springer, 1987.CrossRefGoogle Scholar
4.Chan, N.H. & Tran, L.T.. On the first order autoregressive process with infinite variance. Econometric Theory 5 (1989): 354–362.CrossRefGoogle Scholar
5.Cline, D.B.H. Estimation and linear prediction for regression, autoregression and ARMA with infinite variance data. Ph.D dissertation, Colorado State University, 1983.Google Scholar
6.Darling, D.A.A note on a limit theorem. Annals of Probability 3 (1975): 876–888.CrossRefGoogle Scholar
7.Davis, R. & Resnick, S.. Limit theory for moving averages of random variables with regularly varying tail probabilities. Annals of Probability 13 (1985): 179–195.CrossRefGoogle Scholar
8.Davis, R. & Resnick, S.Limit theory for the sample covariance and correlation functions of moving averages. Annals of Statistics 14 (1986): 533–558.CrossRefGoogle Scholar
9.Feller, W.An introduction to probability theory and its applications, Vol. 2 (2nd Ed.). New York: Wiley, 1971.Google Scholar
10.Gallant, A.R. & White, H.A unified theory of estimation and inference for nonlinear dynamic models. Oxford: Blackwell, 1988.Google Scholar
11.Hall, A.Testing for a unit root in the presence of moving average errors. Biometrika 76 (1989): 49–56.CrossRefGoogle Scholar
12.Ibragimov, I.A. & Linnik, Y.V.. Independent and stationary sequences of random variables. Groninger: Wolters-Noordhoff, 1971.Google Scholar
13.Ito, K.Lectures on stochastic processes. Bombay: Tata Institute, 1961.Google Scholar
14.Kanter, M.Linear sample spaces and stable processes. Journal of Functional Analysis 9 (1972): 441–459.CrossRefGoogle Scholar
15.Kopp, P.E.Martingales and stochastic integrals. Cambridge: Cambridge University Press, 1984.CrossRefGoogle Scholar
16.LePage, R., Woodroofe, M., & Zinn, J.. Convergence to a stable distribution via order statistics. Annals of Probability 9 (1981): 624–632.CrossRefGoogle Scholar
17.Logan, B., Mallows, C., Rice, S.O., & Shepp, L.. Limit distributions of self normalized sums. Annals of Probability 1 (1973): 788–809.CrossRefGoogle Scholar
18.McLeish, D.L.A maximal inequality and dependent strong laws. Annals of Probability 3 (1975): 829–839.CrossRefGoogle Scholar
19.Metivier, M.Semimartingales. New York: Walter de Gruyter, 1982.CrossRefGoogle Scholar
20.Park, J.Y. & Phillips, P.C.B.. Statistical inference in regressions with integrated processes: Part 1. Econometric Theory 4 (1988): 468–498.CrossRefGoogle Scholar
21.Park, J.Y. & Phillips, P.C.B.. Statistical inference in regressions with integrated processes: Part 2. Econometric Theory 5 (1989): 95–132.CrossRefGoogle Scholar
22.Phillips, P.C.B.Time series regression with a unit root. Econometrica 55 (1987): 277–301.CrossRefGoogle Scholar
23.Phillips, P.C.B. Multiple regression with integrated processes. In Prabhu, N.U. (ed.), Statistical Inference from Stochastic Processes, Contemporary mathematics, Vol. 80. Providence: American Mathematical Society, 1988.Google Scholar
24.Phillips, P.C.B. Spectral regression for cointegrated time series. In Barnett, W. (ed.), Nonparametric and Semiparametric Methods in Economics and Statistics. New York: Cambridge University Press, 1990 (forthcoming).Google Scholar
25.Phillips, P.C.B. & Hajivassiliou, V. Bimodal t-ratios. Cowles Foundation Discussion Paper No. 842, July 1987.Google Scholar
26.Phillips, P.C.B. & Hansen, B.. Statistical inference in instrumental variables regression with 1(1) processes. Cowles Foundation Discussion Paper No. 869, July 1988.Google Scholar
27.Resnick, S.I.Point processes regular variation and weak convergence. Advances in Applied Probability 18 (1986): 66–138.CrossRefGoogle Scholar