Abstract
Estimation in a first order autoregressive process with trend isconsidered. Integral expressions for the asymptotic bias of the estimatorunder a unit root and for the expectation of the limit distribution of thelog likelihood ratio test for a unit root are given, and evaluatednumerically.
Similar content being viewed by others
References
Abadir, K. M. (1993). OLS bias in a nonstationary autoregression, Econom. Theory, 9, 81–93.
Andrews, D. W. K. (1993). Exactly median-unbiased estimation of first order autoregressive/unit root models, Econometrica, 61, 139–165.
Billingsley, P. (1968). Convergence of Probability Measures, Wiley, New York.
Evans, G. B. A. and Savin, N. E. (1981). Testing for unit roots: 1, Econometrica, 49, 753–779.
Evans, G. B. A. and Savin, N. E. (1984). Testing for unit roots: 2, Econometrica, 52, 1241–1269.
Jacobson, T. and Larsson, R. (1997). Numerical aspects of a likelihood ratio test statistic for cointegrating rank, to appear in Computational Statistics and Data Analysis.
Johansen, S. (1988). Statistical analysis of cointegration vectors, J. Econom. Dynamics Control, 12, 231–254.
Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in gaussian vector autoregressive models, Econometrica, 59, 1551–1580.
Johansen, S. and Juselius, K. (1990). Maximum likelihood estimation and inference on cointegration with application to the demand for money, Oxford Bulletin of Economics and Statistics, 52, 169–210.
Larsson, R. (1994). Bartlett corrections for unit root test statistics, Preprint No. 2, Institute of Mathematical Statistics, University of Copenhagen.
Larsson, R. (1995). Small sample corrections for unit root test statistics, Report No. 13, Department of Mathematics, Uppsala University, Sweden.
Le Breton, A. and Pham, D. T. (1989). On the bias of the least squares estimator for the first order autoregressive process, Ann. Inst. Statist. Math., 41, 555–563.
Liptser, R. S. and Shiryayev, A. N. (1978). Statistics of Random Processes I & II, Springer, New York.
Magnus, J. R. (1978). The moments of products of quadratic forms in normal variables, Statistica Nederlandica, 32, 201–210.
Mikulski, P. W. and Monsour, M. J. (1994). Moments of the limiting distribution for the boundary case in the first order autoregressive process, Amer. J. Math. Management Sci., 14, 327–347.
Nielsen, B. (1995). Bartlett correction of the unit root test in autoregressive models, Discussion Paper No. 98, Nuffield College, U.K.
Pham, D. T. (1990). Approximate distribution of parameter estimates for first-order autoregressive models, J. Time Ser. Anal., 13, 147–170.
Phillips, P. C. B. and Durlauf, S. N. (1986). Multiple time series regression with integrated processes, Rev. Econom. Stud., LIII, 473–495.
Phillips, P. C. B. and Perron, P. (1988). Testing for a unit root in time series regression, Biometrika, 75, 335–346.
Author information
Authors and Affiliations
About this article
Cite this article
Larsson, R. On the Asymptotic Expectations of Some Unit Root Tests in a First Order Autoregressive Process in the Presence of Trend. Annals of the Institute of Statistical Mathematics 49, 585–599 (1997). https://doi.org/10.1023/A:1003131215117
Issue Date:
DOI: https://doi.org/10.1023/A:1003131215117