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THE IMPACT OF A HAUSMAN PRETEST ON THE ASYMPTOTIC SIZE OF A HYPOTHESIS TEST

Published online by Cambridge University Press:  18 August 2009

Patrik Guggenberger
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Abstract

This paper investigates the asymptotic size properties of a two-stage test in the linear instrumental variables model when in the first stage a Hausman (1978) specification test is used as a pretest of exogeneity of a regressor. In the second stage, a simple hypothesis about a component of the structural parameter vector is tested, using a t-statistic that is based on either the ordinary least squares (OLS) or the two-stage least squares estimator (2SLS), depending on the outcome of the Hausman pretest. The asymptotic size of the two-stage test is derived in a model where weak instruments are ruled out by imposing a positive lower bound on the strength of the instruments. The asymptotic size equals 1 for empirically relevant choices of the parameter space. The size distortion is caused by a discontinuity of the asymptotic distribution of the test statistic in the correlation parameter between the structural and reduced form error terms. The Hausman pretest does not have sufficient power against correlations that are local to zero while the OLS-based t-statistic takes on large values for such nonzero correlations. Instead of using the two-stage procedure, the recommendation then is to use a t-statistic based on the 2SLS estimator or, if weak instruments are a concern, the conditional likelihood ratio test by Moreira (2003).

Type
Research Article
Information
Econometric Theory , Volume 26 , Issue 2 , April 2010 , pp. 369 - 382
Copyright
Copyright © Cambridge University Press 2009

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Footnotes

I would like to thank the Sloan Foundation for a 2009 fellowship and the NSF for support under grant number SES-0748922, co-editor Richard Smith, seminar participants, Don Andrews, Badi Baltagi, Ivan Canay, Gary Chamberlain, Victor Chernozhukov, Phoebus Dhrymes, Ivan Fernandez-Val, Raffaella Giacomini, William Greene, Jinyong Hahn, Bruce Hansen, Jerry Hausman, Guido Imbens, Dale Jorgenson, Anna Mikusheva, Marcelo Moreira, Ulrich Müller, Whitney Newey, Pierre Perron, Jack Porter, Zhongjun Qu, John Rust, Jim Stock, and Michael Wolf for comments, and the Economics Department at Harvard and the Cowles Foundation at Yale for their hospitality.

References

REFERENCES

Anderson, T.W. & Rubin, H. (1949) Estimation of the parameters of a single equation in a complete set of stochastic equations. Annals of Mathematical Statistics 20, 46–63.10.1214/aoms/1177730090CrossRefGoogle Scholar
Andrews, D.W.K. & Guggenberger, P. (2009a) Hybrid and size-corrected subsampling methods. Econometrica 77, 721–762.Google Scholar
Andrews, D.W.K. & Guggenberger, P. (2009b) Incorrect asymptotic size of subsampling procedures based on post-consistent model selection estimators. Journal of Econometrics, forthcoming.10.1016/j.jeconom.2009.02.001Google Scholar
Andrews, D.W.K. & Guggenberger, P. (2009c) Validity of subsampling and “plug-in asymptotic” inference for parameters defined by moment inequalities. Econometric Theory 25, 669–709.10.1017/S0266466608090257CrossRefGoogle Scholar
Andrews, D.W.K. & Guggenberger, P. (2010a) Asymptotic size and a problem with subsampling and with the m out of n bootstrap. Econometric Theory 26, forthcoming.10.1017/S0266466609100051Google Scholar
Andrews, D.W.K. & Guggenberger, P. (2010b) Applications of subsampling, hybrid and size-correction methods. Journal of Econometrics, forthcoming.10.1016/j.jeconom.2010.01.002Google Scholar
Andrews, D.W.K., Moreira, M.J., & Stock, J.H. (2006) Optimal invariant similar tests for instrumental variables regression. Econometrica 74, 715–752.10.1111/j.1468-0262.2006.00680.xCrossRefGoogle Scholar
Angrist, J. & Krueger, A. (1991) Does compulsory school attendance affect schoolings and earnings? Quarterly Journal of Economics 106, 979–1014.10.2307/2937954Google Scholar
Dhrymes, P.J. (2008) Tests for Endogeneity, Instrument Suitability, the Hausman Test and Weak Instruments. Working paper, Columbia University.Google Scholar
Guggenberger, P. (2007) The Impact of a Hausman Pretest on the Size of a Hypothesis Test: The Panel Data Case. Working paper, UCLA.10.2139/ssrn.1126333Google Scholar
Guggenberger, P. (2008) The Impact of a Hausman Pretest on the Size of Hypothesis Tests. Cowles Foundation Discussion paper 1651.10.2139/ssrn.1126333Google Scholar
Guggenberger, P. & Smith, R.J. (2005) Generalized empirical likelihood estimators and tests under partial, weak and strong identification. Econometric Theory 21, 667–709.10.1017/S0266466605050371Google Scholar
Hahn, J., Ham, J., & Moon, H.R. (2007) The Hausman Test and Weak Instruments. Working paper, UCLA.Google Scholar
Hahn, J. & Hausman, J.A. (2002) A new specification test for the validity of instrumental variables. Econometrica 70, 163–189.10.1111/1468-0262.00272Google Scholar
Hall, A.R., Rudebusch, G.D., & Wilcox, D.W. (1996) Judging instrument relevance in instrumental variables estimation. International Economic Review 37, 283–298.10.2307/2527324Google Scholar
Hansen, C., Hausman, J.A., & Newey, W. (2004) Estimation with many instrumental variables. Journal of Business and Economic Statistics, forthcoming.Google Scholar
Hausman, J.A. (1978) Specification tests in econometrics. Econometrica 46, 1251–1271.10.2307/1913827Google Scholar
Hausman, J.A., Stock, J.H., & Yogo, M. (2005) Asymptotic properties of the Hahn-Hausman test for weak instruments. Economics Letters 89, 332–342.10.1016/j.econlet.2005.06.007Google Scholar
Judge, G.G. & Bock, M.E. (1978) The Statistical Implications of Pre-Test and Stein Rule Estimators in Econometrics. North-Holland.Google Scholar
Kabaila, P. (1995) The effect of model selection on confidence regions and prediction regions. Econometric Theory 11, 537–549.10.1017/S0266466600009403Google Scholar
Kleibergen, F. (2002) Pivotal statistics for testing structural parameters in instrumental variables regression. Econometrica 70, 1781–1805.10.1111/1468-0262.00353CrossRefGoogle Scholar
Leeb, H. & Pötscher, B.M. (2005) Model selection and inference: Facts and fiction. Econometric Theory 21, 21–59.10.1017/S0266466605050036CrossRefGoogle Scholar
Leeb, H. & Pötscher, B.M. (2009) Model selection. In Andersen, T.G., Davis, R.A., Kreib, J.-P., & Mikosch, T. (eds.), Handbook of Financial Time Series, pp. 889–925. Springer-Verlag.10.1007/978-3-540-71297-8_39Google Scholar
Moreira, M.J. (2003) A conditional likelihood ratio test for structural models. Econometrica 71, 1027–1048.10.1111/1468-0262.00438CrossRefGoogle Scholar
Moreira, M.J. (2009) Tests with correct size when instruments can be arbitrarily weak. Journal of Econometrics, forthcoming.10.1016/j.jeconom.2009.01.012Google Scholar
Pötscher, B.M. (1991) Effects of model selection on inference. Econometric Theory 7, 163–185.10.1017/S0266466600004382Google Scholar
Staiger, D. & Stock, J.H. (1997) Instrumental variables regression with weak instruments. Econometrica 65, 557–586.10.2307/2171753CrossRefGoogle Scholar
Wong, K. (1997) Effects on inference of pretesting the exogeneity of a regressor. Economic Letters 56, 267–271.10.1016/S0165-1765(97)00172-9Google Scholar