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A Continuous Time Approximation to the Stationary First-Order Autoregressive Model

Published online by Cambridge University Press:  11 February 2009

Pierre Perron
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Abstract

We consider the least-squares estimator in a strictly stationary first-order autoregression without an estimated intercept. We study its continuous time asymptotic distribution based on an asymptotic framework where the sampling interval converges to zero as the sample size increases. We derive a momentgenerating function which permits the calculation of percentage points and moments of this asymptotic distribution and assess the adequacy of the approximation to the finite sample distribution. In general, the approximation is excellent for values of the autoregressive parameter near one. We also consider the behavior of the power function of tests based on the normalized leastsquares estimator. Interesting nonmonotonic properties are uncovered. This analysis extends the study of Perron [15] and helps to provide explanations for the finite sample results established by Nankervis and Savin [13].

Type
Research Article
Information
Econometric Theory , Volume 7 , Issue 2 , June 1991 , pp. 236 - 252
Copyright
Copyright © Cambridge University Press 1991

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References

REFERENCES

1. Arnold, L. Stochastic Differential Equations: Theory and Applications. New York: John Wiley, 1974.Google Scholar
2. Basmann, R.L., Richardson, D.H. & Rohr, R.J.. Finite sample distributions associated with stochastic difference equations-some experimental evidence. Econometrica 42 (1974): 825–839.10.2307/1913791CrossRefGoogle Scholar
3. Bobkoski, M.J. Hypothesis testing in nonstationary time series. Ph.D. Thesis, University of Wisconsin, Madison, WI, 1983.Google Scholar
4. Cavanagh, C. Roots local to unity. Mimeo, Harvard University, 1986.Google Scholar
5. Chan, N.H. The parameter inference for nearly nonstationary time series. Journal of the American Statistical Association 83 (1988): 857–862.10.1080/01621459.1988.10478674CrossRefGoogle Scholar
6. Chan, N.H. & Wei, C.Z.. Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics 15 (1987): 1050–1063.10.1214/aos/1176350492CrossRefGoogle Scholar
7. Evans, G.B.A. & Savin, N.E.. Testing for unit roots: 1. Econometrica 49 (1981): 753–779.10.2307/1911521CrossRefGoogle Scholar
8. Evans, G.B.A. & Savin, N.E.. Testing for unit roots: 2. Econometrica 52 (1984): 1241–1269.10.2307/1910998CrossRefGoogle Scholar
9. Gurland, J. Inversion formulae for the distribution of ratios. Annals of Mathematical Statistics 19 (1948): 228–237.10.1214/aoms/1177730247CrossRefGoogle Scholar
10. Mann, H.B. & Wald, A.. On the statistical treatment of linear stochastic difference equations. Econometrica 11 (1943): 173–220.10.2307/1905674CrossRefGoogle Scholar
11. Mehta, J.S. & Swamy, P.A.V.B.. The existence of moments of some simple, bayes estimators of coefficients in a simultaneous equation model. Journal of Econometrics 7 (1978): 1–14.10.1016/0304-4076(78)90002-7CrossRefGoogle Scholar
12. Nabeya, S. & Tanaka, K.. A general approach to the limiting distribution for estimators in time series regression with nonstable autoregressive errors. Econometrica 58 (1990): 145–163.10.2307/2938338CrossRefGoogle Scholar
13. Nankervis, J. & Savin, N.E.. The exact moments of the least-squares estimator for the autoregressive model: corrections and extensions. Journal of Econometrics 37 (1988): 381–388.10.1016/0304-4076(88)90012-7CrossRefGoogle Scholar
14. Nankervis, J. & Savin, N.E.. The exact moments of the least-squares estimator for the autoregressive model: corrections and extensions. Working Paper Series No. 86–23 (Department of Economics, University of Iowa, Iowa City, IA), 1988.CrossRefGoogle Scholar
15. Perron, P. A continuous-time approximation to the unstable first-order autoregressive model: the case without an intercept. Econometrica 59 (1991): 211–236.10.2307/2938247CrossRefGoogle Scholar
16. Perron, P. The calculation of the limiting distribution of the least-squares estimator in a near-integrated model. Econometric Theory 5 (1989): 241–255.10.1017/S026646660001241XCrossRefGoogle Scholar
17. Phillips, P.C.B. Approximations to some finite sample distributions associated with a first-order stochastic difference equation. Econometrica 45 (1977): 463–485.10.2307/1911222CrossRefGoogle Scholar
18. Phillips, P.C.B. Time series regression with a unit root. Econometrica 55 (1987): 277–301.10.2307/1913237CrossRefGoogle Scholar
19. Phillips, P.C.B. Towards a unified asymptotic theory for autoregression. Biometrika 74 (1988): 535–547.10.1093/biomet/74.3.535CrossRefGoogle Scholar
20. Press, W.H., Flannery, B.P., Teukolsky, S.A. & Vetterling, W.T.. Numerical Recipes: The Art of Scientific Computing. Cambridge: Cambridge University Press, 1986.Google Scholar
21. Rubin, H. Consistency of maximum likelihood estimator in the explosive case. In Koopmans, T.C. (ed.), Statistical Inference in Dynamic Economic Models. New York: Wiley, 1950.Google Scholar
22. Sawa, T. The exact moments of the least-squares estimator for the autoregressive model. Journal of Econometrics 8 (1978): 159–172.10.1016/0304-4076(78)90025-8CrossRefGoogle Scholar