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SECOND ORDER EXPANSION OF THE T-STATISTIC IN AR(1) MODELS

Published online by Cambridge University Press:  15 September 2014

Anna Mikusheva
Anna Mikusheva*
Affiliation:
MIT
*
*Address correspondence to Anna Mikusheva, Department of Economics, MIT, E17-210, 77 Massachusetts Avenue, Cambridge, MA 02139; e-mail: [email protected].
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Abstract

The purpose of this paper is to differentiate between several asymptotically valid methods for confidence set construction for the autoregressive coefficient in AR(1) models. We show that the nonparametric grid bootstrap procedure suggested by Hansen (1999, Review of Economics and Statistics 81, 594–607) achieves a second order refinement in the local-to-unity asymptotic approach when compared with a modified version of Stock’s (1991, Journal of Monetary Economics 28, 435–459) and Andrews’ (1993, Econometrica 61, 139–165) grid testing procedures. We establish a second order expansion of the t-statistic in an AR(1) model in the local-to-unity asymptotic approach, which differs drastically from the usual Edgeworth-type expansions by approximating the statistic around a nonstandard and nonpivotal limit.

Type
ARTICLES
Information
Econometric Theory , Volume 31 , Issue 3 , June 2015 , pp. 426 - 448
Copyright
Copyright © Cambridge University Press 2014 

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Footnotes

Financial support from the Castle-Krob Career Development Chair and Sloan Research Fellowship is gratefully acknowledged. I am grateful to Jim Stock, Marcelo Moreira, Gary Chamberlain, Peter Phillips, Denis Chetverikov, and three anonymous referees for helpful comments.

References

REFERENCES

Andrews, D.W.K. (1993) Exactly median-unbiased estimation of first order autoregressive/unit root models. Econometrica 61(1), 139–165.CrossRefGoogle Scholar
Andrews, D.W.K. & Chen, H.-Y. (1994) Approximately median-unbiased estimation of autoregressive models. Journal of Business & Economic Statistics 12(2), 187–204.Google Scholar
Andrews, D.W.K. & Guggenberger, P. (2009) Hybrid and size-corrected subsample methods. Econometrica 77(3), 721–762.Google Scholar
Andrews, D.W.K. & Guggenberger, P. (2010) Asymptotic size and a problem with subsampling and with the m out of n bootstrap. Econometric Theory 26, 426–468.CrossRefGoogle Scholar
Basawa, I.V., Mallik, A.K., McCormick, W.P., Reeves, J.H., & Taylor, R.L. (1991) Bootstrapping unstable first-order autoregressive processes. Annals of Statistics 19(2), 1098–1101.CrossRefGoogle Scholar
Bose, A. (1988) Edgeworth correction by bootstrap in autoregressions. Annals of Statistics 16(4), 1709–1722.Google Scholar
Breiman, L. (1992) Probability. Addison-Wesley Series in Statistics.CrossRefGoogle Scholar
Hansen, B.E. (1999) The grid bootstrap and the autoregressive model. Review of Economics and Statistics 81(4), 594–607.CrossRefGoogle Scholar
Kurtz, T.G. & Protter, P. (1991) Weak limit theorems for stochastic integrals and stochastic differential equations. Annals of Probability 19(3), 1035–1070.CrossRefGoogle Scholar
Mikusheva, A. (2007) Uniform inference in autoregressive models. Econometrica 75(5), 1411–1452.CrossRefGoogle Scholar
Obloj, J. (2004) The Skorokhod embedding problem and its offspring. Probability Surveys 1, 321–392.CrossRefGoogle Scholar
Onatski, A. & Uhlig, H. (2012) Unit roots in white noise. Econometric Theory 28(3), 485–508.CrossRefGoogle Scholar
Park, J.Y. (2003) Bootstrap unit root tests. Econometrica 71(6), 1845–1895.CrossRefGoogle Scholar
Park, J.Y. (2006) A bootstrap theory for weakly integrated processes. Journal of Econometrics 133(2), 639–672.CrossRefGoogle Scholar
Phillips, P.C.B. (1987a) Asymptotic expansions in nonstationary vector autoregressions. Econometric Theory 3(1), 45–68.CrossRefGoogle Scholar
Phillips, P.C.B. (1987b) Time series regression with a unit root. Econometrica 55(2), 277–301.CrossRefGoogle Scholar
Phillips, P.C.B. (1987c) Toward a unified asymptotic theory for autoregression. Biometrika 74(3), 535–547.CrossRefGoogle Scholar
Phillips, P.C.B. (1991) Bayesian routes and unit roots: De Rebus Prioribus Semper Est Disputandum. Journal of Applied Econometrics 6, 435–473.CrossRefGoogle Scholar
Phillips, P.C.B. (2012) On confidence intervals for autoregressive roots and predictive regression. Unpublished manuscript.CrossRefGoogle Scholar
Skorokhod, A.V. (1965) Studies in the Theory of Random Processes. Addison-Wesley.Google Scholar
Stock, J. (1991) Confidence intervals for the largest autoregressive root in US macroeconomic time series. Journal of Monetary Economics 28, 435–459.CrossRefGoogle Scholar