Hostname: page-component-cb9f654ff-9b74x Total loading time: 0 Render date: 2025-08-10T21:21:49.563Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Asymptotic Behavior of Least-SquaresEstimators in AR Time Series with Roots Near theUnit Circle

Published online by Cambridge University Press:  11 February 2009

P. Jeganathan
Get access Rights & Permissions [Opens in a new window]

Abstract

Some asymptotic properties of the least-squaresestimator of the parameters of an AR model of orderp, p ≥ 1, are studied when theroots of the characteristic polynomial of the givenAR model are on or near the unit circle.Specifically, the convergence in distribution isestablished and the corresponding limiting randomvariables are represented in terms of functionals ofsuitable Brownian motions.

Further, the preceding convergence in distribution isstrengthened to that of convergence uniformly overall Borel subsets. It is indicated that the methodemployed for this purpose has the potential of beingapplicable in the wider context of obtainingsuitable asymptotic expansions of the distributionsof leastsquares estimators.

Information

Type
Research Article
Information
Econometric Theory , Volume 7 , Issue 3 , September 1991 , pp. 269 - 306
Copyright
Copyright © Cambridge University Press 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

1 Ahtola, J.A. Studies in asymptotic and finite sample inference of non-stationary and nearly non-stationary autoregressive models. Unpublished Ph.D. thesis (1983). University of Wisconsin, Madison.Google Scholar
2. Bhattacharya, R.N. & Ranga Rao, R.. Normal Approximation and Asymptotic Expansions. New York: Wiley, 1976.Google Scholar
3. Billingsley, P. Convergence of Probability Measures. New York: Wiley, 1968.Google Scholar
4. Chan, N.H. & Wei, C.Z.. Limiting distributions of least squares estimate of unstable autoregressive processes. Annals of Statistics 16 (1988): 367–401.10.1214/aos/1176350711CrossRefGoogle Scholar
5. Chan, N.H. & Wei, C.Z.. Asymptotic inference of nearly nonstationary AR(1) process. Annals of Statistics 15 (1987): 1050–1063.10.1214/aos/1176350492CrossRefGoogle Scholar
6. Chan, N.H. On the parameter estimation for nearly nonstationary time series. Journal of American Statistical Association 83 (1988): 857–862.10.1080/01621459.1988.10478674CrossRefGoogle Scholar
7. Dickey, D.A. & Fuller, W.A.. Distribution of the estimators for autoregressive time series with a unit root. Journal of American Statistical Association 74 (1979): 427–431.Google Scholar
8. Fuller, W.A. Introduction to Statistical Time Series. New York: Wiley, 1976.Google Scholar
9. Götze, F. & Hipp, C.. Asymptotic expansions for sums of weakly dependent random vectors. Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete 64 (1983): 211–2?10.1007/BF01844607CrossRefGoogle Scholar
10. Hall, P. & Heyde, C.C.. Martingale Limit Theory and Its Application. New York: Academic Press, 1980.Google Scholar
11. Hasza, D.P. & Fuller, W.A.. Estimation for autoregressive processes with unit roots. Annals of Statistics 7 (1979): 1106–1120.10.1214/aos/1176344793CrossRefGoogle Scholar
12. Jeganathan, P. On the strong approximation of the distributions of estimators in linear stochastic models, I and II: stationary and explosive AR models. Annals of Statistics 16 (1988): 1283–1314.10.1214/aos/1176350962CrossRefGoogle Scholar
13. Jeganathan, P. Some aspects of asymptotic theory with applications to time series models. Technical Report, University of Michigan, Ann Arbor. (To appear in Econometric Theory.)Google Scholar
14. Lai, T.L. & Wei, C.Z.. A note on martingale difference equations satisfying the local Marcinkeiwicz-Zygmund condition. Bulletin of the Institute of Mathematics Academica Sinica 11 (1983): 1–13.Google Scholar
15. LeCam, L. Locally asymptotically normal families of distributions. University of California Publications in Statistics 3 (1960): 37–98.Google Scholar
16. LeCam, L. Asymptotic Methods in Statistical Decision Theory. New York: Springer-Verlag, 1986.Google Scholar
17. Nyblom, J. On the least squares estimators for the unstable AR(2) models. In Bulletin of International Statistical Institute Contributed Paper Volume, 45th Session (1985): 385–386.Google 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. Biometrica 74 (1987): 535–547.10.1093/biomet/74.3.535CrossRefGoogle Scholar
20. Phillips, P.C.B. Weak convergence to matrix stochastic integrals. Journal of Multivariate Analysis 24 (1988): 252–264.10.1016/0047-259X(88)90039-5CrossRefGoogle Scholar
21. Strasser, H. Martingale difference arrays and stochastic integrals. Probability Theory and Related Fields 72 (1986): 83–98.10.1007/BF00343897CrossRefGoogle Scholar
22. White, J.S. The limiting distribution of the serial correlation coefficient in the explosive case. Annals of Mathematical Statistics 29 (1958): 1188–1197.10.1214/aoms/1177706450CrossRefGoogle Scholar