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BARTLETT CORRECTION IN THE STABLE AR(1) MODEL WITH INTERCEPT AND TREND

Published online by Cambridge University Press:  01 June 2009

Noud P.A. van Giersbergen
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Abstract

Bartlett corrections are derived for testing hypotheses about the autoregressive parameter ρ in the stable (a) AR(1) model, (b) AR(1) model with intercept, (c) AR(1) model with intercept and linear trend. The correction is found explicitly as a function of ρ. In the models with deterministic terms, the correction factor is asymmetric in ρ. Furthermore, the Bartlett correction is monotonically increasing in ρ and tends to infinity when ρ approaches the stability boundary of + 1. Simulation results indicate that the Bartlett corrections are useful in controlling the size of the likelihood ratio statistic in small samples, although these corrections are not the ultimate panacea.

Type
Brief Report
Information
Econometric Theory , Volume 25 , Issue 3 , June 2009 , pp. 857 - 872
Copyright
Copyright © Cambridge University Press 2009

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Footnotes

The author thanks a co-editor for encouraging remarks and two referees for critical comments that have led to a complete revision of this paper. A third referee is appreciated for drawing attention to the impact of the starting value. Furthermore, helpful comments from Peter Boswijk and participants of the ESEM 2004 meeting (Madrid, Spain) and the UvA-Econometrics seminar (Amsterdam, The Netherlands) are gratefully acknowledged. All errors remain my responsibility.

References

REFERENCES

Andrews, D.W.K. (1993) Exactly median-unbiased estimation of first-order autoregressive/ unit-root models. Econometrica 61, 139–165.10.2307/2951781CrossRefGoogle Scholar
Bartlett, M.S. (1937) Properties of sufficiency and statistical tests. Proceedings of the Royal Society of London, Series A 160, 268–282.Google Scholar
Bravo, F. (1999) A correction factor for unit root test statistics. Econometric Theory 15, 218–227.10.1017/S0266466699152046CrossRefGoogle Scholar
Chen, W.W. & Deo, R.S. (2006) A Smooth Transition to the Unit Root Distribution via the Chi-Square Distribution with Interval Estimation for Nearly Integrated Autoregressive Processes. Working paper (MPRA 1215) presented at the ESEM 2007 (Budapest).Google Scholar
Cribari-Neto, F. & Cordeiro, G. (1996) On Bartlett and Bartlett-type corrections. Econometric Reviews 15, 339–367.10.1080/07474939608800361CrossRefGoogle Scholar
Evans, G.B.A. & Savin, N.E. (1981) Testing for unit roots: 1. Econometrica 49, 753–779.10.2307/1911521CrossRefGoogle Scholar
Hayakawa, T. (1977) The likelihood ratio criterion and the asymptotic expansion of its distribution (see Annals of the Institute of Statistical Mathematics, 39, 681 for correction). Annals of the Institute of Statistical Mathematics 29, 359–378.10.1007/BF02532797CrossRefGoogle Scholar
Jensen, J.L. & Wood, A.T.A. (1997) On the non-existence of a Bartlett correction for unit root tests. Statistics and Probability Letters 35, 181–187.10.1016/S0167-7152(97)00012-6CrossRefGoogle Scholar
Johansen, S. (2004) Asymptotic and finite sample properties of the Dickey-Fuller test. In Welfe, A. (ed.), New Directions in Macromodelling, vol. 269, pp. 49–68. Elsevier Science.10.1016/S0573-8555(04)69003-1CrossRefGoogle Scholar
Kiviet, J.F. & Dufour, J.-M. (1997) Exact tests in single equation autoregressive distributed lag models. Journal of Econometrics 80, 325–353.10.1016/S0304-4076(97)00048-1CrossRefGoogle Scholar
Lagos, B.M. & Morettin, P.A. (2004) Improvement of the likelihood ratio test statistic in ARMA models. Journal of Time Series Analysis 25, 83–101.10.1111/j.1467-9892.2004.00338.xCrossRefGoogle Scholar
Larsson, R. (1998) Bartlett corrections for unit root test statistics. Journal of Time Series Analysis 19, 425–438.10.1111/1467-9892.00101CrossRefGoogle Scholar
Lawley, D.N. (1956) A general method for approximating to the distribution of likelihood ratio criteria. Biometrika 43, 295–303.10.1093/biomet/43.3-4.295CrossRefGoogle Scholar
Nielsen, B. (1997) Bartlett correction of the unit root test in autoregressive models. Biometrika 84, 500–504.10.1093/biomet/84.2.500CrossRefGoogle Scholar
Nielsen, B. (1999) The likelihood-ratio test for rank in bivariate canonical correlation analysis. Biometrika 86, 279–288.10.1093/biomet/86.2.279CrossRefGoogle Scholar
Omtzigt, P.H. (2003) Notes on the Bartlett Correction of AR(1) processes. Manuscript, European University Institute, Florence.Google Scholar
Phillips, P.C.B. (1977) Approximations to some finite sample distributions associated with a first order stochastic difference equation. Econometrica 45, 463–486.10.2307/1911222CrossRefGoogle Scholar
Sul, D., Phillips, P.C.B. & Choi, C.-Y. (2005) Prewhitening bias in HAC estimation. Oxford Bulletin of Economics and Statistics 67, 517–546.10.1111/j.1468-0084.2005.00130.xCrossRefGoogle Scholar
Taniguchi, M. (1988) Asymptotic expansions of the distribution of some test statistics for Gaussian ARMA processes. Journal of Multivariate Analysis 27, 494–511.10.1016/0047-259X(88)90144-3CrossRefGoogle Scholar
Taniguchi, M. (1991) Higher Order Asymptotic Theory for Time Series Analysis. Springer-Verlag.10.1007/978-1-4612-3154-7CrossRefGoogle Scholar
van Giersbergen, N.P.A. (2004) Bartlett Correction in the Stable AR(1) Model with Intercept and Trend. UvA-Econometrics, Working paper 2004/07.Google Scholar
Wolfram, S. (1991) Mathematica: A System for Doing Mathematics by Computer. Addison-Wesley.Google Scholar