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THE FINITE-SAMPLE DISTRIBUTION OF POST-MODEL-SELECTION ESTIMATORS AND UNIFORM VERSUS NONUNIFORM APPROXIMATIONS

Published online by Cambridge University Press:  08 January 2003

Hannes Leeb  and
Benedikt M. Pötscher
Hannes Leeb
Affiliation:
University of Vienna
Benedikt M. Pötscher
Affiliation:
University of Vienna
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Abstract

In Pötscher (1991, Econometric Theory 7, 163–185) the asymptotic distribution of a post-model-selection estimator, both unconditional and conditional on selecting a correct model (minimal or not), has been derived. Limitations of these results are (i) that they do not provide information on the distribution of the post-model-selection estimator conditional on selecting an incorrect model and (ii) that the quality of this asymptotic approximation to the finite-sample distribution is not uniform with respect to the underlying parameters. In the present paper we first obtain the unconditional and also the conditional finite-sample distribution of the post-model-selection estimator, which turn out to be complicated and difficult to interpret. Second, we obtain approximations to the finite-sample distributions that are as simple and easy to interpret as the asymptotic distributions obtained in Pötscher (1991) but at the same time are close to the finite-sample distributions uniformly with respect to the underlying parameters. As a by-product, we also obtain the asymptotic distribution conditional on selecting an incorrect model.We thank the co-editor Richard Smith and the two referees for helpful comments on a previous version of this paper. Hannes Leeb's research was supported by the Austrian Science Foundation (FWF), project P13868-MAT.

Type
Research Article
Information
Econometric Theory , Volume 19 , Issue 1 , February 2003 , pp. 100 - 142
Copyright
© 2003 Cambridge University Press

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References

REFERENCES

Billingsley, P. & F. Topsoe (1967) Uniformity in weak convergence. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 7, 1–16.Google Scholar
Chandra, T.K. (1989) Multidimensional Polya's theorem. Bulletin of the Calcutta Mathematical Society 81, 227–231.Google Scholar
Elsinger, H. (1994) “Pre-Test” Schätzer und Modellselektion im linearen Regressionsmodell. Master's thesis, Sozial- und Wirtschaftswissenschaftliche Fakultät der Universität Wien.
Feller, W. (1966) An Introduction to Probability Theory and Its Applications, vol. 1, 2nd ed. New York: Wiley.
Gaenssler, P. & W. Stute (1977) Wahrscheinlichkeitstheorie. Berlin: Springer.
Giles, D.E.A. & V. Srivastava (1993) The exact distribution of a least squares regression coefficient estimator after a preliminary t-test. Statistics and Probability Letters 16, 59–64.Google Scholar
Giles, J.A. & D.E.A. Giles (1993) Pre-test estimation and testing in econometrics: Recent developments. Journal of Economic Surveys 7, 145–197.Google Scholar
Gradstejn, I.S. & I.M. Ryzik (1985) Table of Integrals, Series, and Products, corrected and enlarged ed. New York: Academic Press.
Judge, G.G. & M.E. Bock (1978) The Statistical Implications of Pre-Test and Stein-Rule Estimators in Econometrics. Amsterdam: North-Holland.
Judge, G.G. & T.A. Yancey (1986) Improved Methods of Inference in Econometrics. Amsterdam: North-Holland.
Kabaila, P. (1995) The effect of model selection on confidence regions and prediction regions. Econometric Theory 11, 537–549.Google Scholar
Leeb, H. & B.M. Pötscher (2000) The Finite-Sample Distribution of Post-Model-Selection Estimators, and Uniform versus Non-uniform Approximations. Technical Report, Department of Statistics, University of Vienna.
Leeb, H. & B.M. Pötscher (2001) Can One Estimate the Distribution of Post-Model-Selection Estimators? Manuscript, Department of Statistics, University of Vienna.
Magnus, J.R. (1999) The traditional pre-test estimator. Theory of Probability and Its Applications 44, 293–308.Google Scholar
Pötscher, B.M. (1991) Effects of model selection on inference. Econometric Theory 7, 163–185.Google Scholar
Pötscher, B.M. (1995) Comment on “The effect of model selection on confidence regions and prediction regions.” Econometric Theory 11, 550–559.Google Scholar
Pötscher, B.M. & A.J. Novak (1998) The distribution of estimators after model selection: Large and small sample results. Journal of Statistical Computation and Simulation 60, 19–56.Google Scholar
Sen, P.K. (1979) Asymptotic properties of maximum likelihood estimators based on conditional specification. Annals of Statistics 7, 1019–1033.Google Scholar