Abstract
This paper investigates the asymptotics of the log likelihood ratio test for a unit root in an autoregressive (AR) process of general order. The main result is that the expectation and variance (in fact, all moments) of the test statistic may, to the order of T-1, where T is the number of observations, be approximated by the expectation and variance of the corresponding test in an AR(1) process. This result has obvious implications for the asymptotics of unit root tests for panels. An explicit formula for the approximation error of a test in an AR(2) process is also given.
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References
Abadir, K. M. (1995). The limiting distribution of the t ratio under a unit root, Econometric Theory, 11, 775–793.
Anderson, T. W. (1971). The Statistical Analysis of Time Series, Wiley, New York.
Chung, K. L. (1974). A Course in Probability Theory, 2nd ed., Academic Press, London.
Evans, G. B. A. and Savin, N. E. (1981). The calculation of the limiting distribution of the least squares estimator of the parameter in a random walk model, Ann. Statist., 9, 1114–1118.
Im, K. S., Pesaran, M. H. and Shin, Y. (1996). Testing for unit roots in heterogeneous panels,. Unpublished manuscript, University of Cambridge.
Johansen, S. (1995). Likelihood-based Inference in Cointegrated Vector Autoregressive Models, Oxford University Press, Oxford.
Larsson, R. (1994). Bartlett corrections for unit root test statistics, Preprint no. 2, Institute of Mathematical Statistics, University of Copenhagen.
Larsson, R. (1995a). The asymptotic distributions of some test statistics in near-integrated AR processes, Econometric Theory, 11, 306–330.
Larsson, R. (1995b). Small sample corrections for unit root test statistics, Report no. 13, Department of Mathematics, Uppsala University.
Larsson, R. (1997). On the asymptotic expectations of some unit root tests in a first order autoregressive process in the presence of trend, Ann. Inst. Statist. Math., 49, 585–599.
Mann, H. B. and Wald, A. (1943). On stochastic limit and order relationships, Ann. Math. Statist., 14, 217–226.
Nielsen, B. (1995). Bartlett correction of the unit root test in autoregressive models, Discussion Paper no. 98, Nuffield College.
Nielsen, B. (1996). Moments of unit root tests in autoregressive models with drifts, Unpublished manuscript, Institute of Mathematical Statistics, University of Copenhagen.
Phillips, P. C. B. (1987). Time series regression with a unit root, Econometrica, 55, 277–301.
Rao, M. M. (1978). Asymptotic distribution of an estimator of the boundary parameter of an unstable process, Ann. Statist., 6, 185–190.
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Larsson, R. The Order of the Error Term for Moments of the Log Likelihood Ratio Unit Root Test in an Autoregressive Process. Annals of the Institute of Statistical Mathematics 50, 29–48 (1998). https://doi.org/10.1023/A:1003445229753
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DOI: https://doi.org/10.1023/A:1003445229753