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GENERALIZED EMPIRICAL LIKELIHOOD ESTIMATORS AND TESTS UNDER PARTIAL, WEAK, AND STRONG IDENTIFICATION

Published online by Cambridge University Press:  19 July 2005

Patrik Guggenberger  and
Richard J. Smith
Patrik Guggenberger
Affiliation:
UCLA
Richard J. Smith
Affiliation:
cemmap, UCL and IFS and University of Warwick
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Abstract

The purpose of this paper is to describe the performance of generalized empirical likelihood (GEL) methods for time series instrumental variable models specified by nonlinear moment restrictions as in Stock and Wright (2000, Econometrica 68, 1055–1096) when identification may be weak. The paper makes two main contributions. First, we show that all GEL estimators are first-order equivalent under weak identification. The GEL estimator under weak identification is inconsistent and has a nonstandard asymptotic distribution. Second, the paper proposes new GEL test statistics, which have chi-square asymptotic null distributions independent of the strength or weakness of identification. Consequently, unlike those for Wald and likelihood ratio statistics, the size of tests formed from these statistics is not distorted by the strength or weakness of identification. Modified versions of the statistics are presented for tests of hypotheses on parameter subvectors when the parameters not under test are strongly identified. Monte Carlo results for the linear instrumental variable regression model suggest that tests based on these statistics have very good size properties even in the presence of conditional heteroskedasticity. The tests have competitive power properties, especially for thick-tailed or asymmetric error distributions.This paper is a revision of Guggenberger's job market paper “Generalized Empirical Likelihood Tests under Partial, Weak, and Strong Identification.” We are thankful to the editor, P.C.B. Phillips, and three referees for very helpful suggestions on an earlier version of this paper. Guggenberger gratefully acknowledges the continuous help and support of his adviser, Donald Andrews, who played a prominent role in the formulation of this paper. He thanks Peter Phillips and Joseph Altonji for their extremely valuable comments. We also thank Vadim Marner for help with the simulation section and John Chao, Guido Imbens, Michael Jansson, Frank Kleibergen, Marcelo Moreira, Jonathan Wright, and Motohiro Yogo for helpful comments. Aspects of this research have been presented at the 2003 Econometric Society European Meetings; York Econometrics Workshop 2004; Seminaire Malinvaud; CREST-INSEE; and seminars at Albany, Alicante, Austin (Texas), Brown, Chicago, Chicago GSB, Harvard/MIT, Irvine, ISEG/Universidade Tecnica de Lisboa, Konstanz, Laval, Madison (Wisconsin), Mannheim, Maryland, NYU, Penn, Penn State, Pittsburgh, Princeton, Rice, Riverside, Rochester, San Diego, Texas A&M, UCLA, USC, and Yale. We thank all the seminar participants. Guggenberger and Smith received financial support through a Carl Arvid Anderson Prize Fellowship and a 2002 Leverhulme Major Research Fellowship, respectively.

Type
Research Article
Information
Econometric Theory , Volume 21 , Issue 4 , August 2005 , pp. 667 - 709
Copyright
© 2005 Cambridge University Press

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References

REFERENCES

Anderson, T.W. & H. Rubin (1949) Estimators of the parameters of a single equation in a complete set of stochastic equations. Annals of Mathematical Statistics 21, 570–582.Google Scholar
Andrews, D.W.K. (1994) Empirical process methods in econometrics. In R. Engle & D. McFadden (eds.), Handbook of Econometrics, vol. 4, 2247–2294. North-Holland.
Brown, B.W. & W.K. Newey (1998) Efficient semiparametric estimation of expectations. Econometrica 66, 453–464.Google Scholar
Caner, M. (2003) Exponential Tilting with Weak Instruments: Estimation and Testing. Working paper, University of Pittsburgh.
Dufour, J. (1997) Some impossibility theorems in econometrics with applications to structural and dynamic models. Econometrica 65, 1365–1387.Google Scholar
Guggenberger, P. (2003) Econometric essays on generalized empirical likelihood, long-memory time series, and volatility. Ph.D. thesis, Yale University.
Guggenberger, P. & R.J. Smith (2003) Generalized Empirical Likelihood Tests in Time Series Models with Potential Identification Failure. Working paper, UCLA and University of Warwick.
Guggenberger, P. & M. Wolf (2004) Subsampling Tests of Parameter Hypotheses and Overidentifying Restrictions with Possible Failure of Identification. Working paper, UCLA.
Hansen, L.P. (1982) Large sample properties of generalized method of moment estimators. Econometrica 50, 1029–1054.Google Scholar
Hansen, L.P., J. Heaton, & A. Yaron (1996) Finite-sample properties of some alternative GMM estimators. Journal of Business & Economic Statistics 14, 262–280.Google Scholar
Imbens, G. (1997) One-step estimators for over-identified generalized method of moments models. Review of Economic Studies 64, 359–383.Google Scholar
Imbens, G. (2002) Generalized method of moments and empirical likelihood. Journal of Business & Economic Statistics 20, 493–506.Google Scholar
Imbens, G., R.H. Spady, & P. Johnson (1998) Information theoretic approaches to inference in moment condition models. Econometrica 66, 333–357.Google Scholar
Kitamura, Y. (1997) Empirical likelihood methods with weakly dependent processes. Annals of Statistics 25, 2084–2102.Google Scholar
Kitamura, Y. & M. Stutzer (1997) An information-theoretic alternative to generalized method of moments estimation. Econometrica 65, 861–874.Google Scholar
Kleibergen, F. (2001) Testing parameters in GMM without assuming that they are identified. Econometrica, forthcoming.Google Scholar
Kleibergen, F. (2002a) Pivotal statistics for testing structural parameters in instrumental variables regression. Econometrica 70, 1781–1805.Google Scholar
Kleibergen, F. (2002b) Two Independent Pivotal Statistics That Test Location and Misspecification and Add-Up to the Anderson-Rubin Statistic. Working paper, Brown University.
Moreira, M.J. (2003) A conditional likelihood ratio test for structural models. Econometrica 71, 1027–1048.Google Scholar
Nelson, C.R. & R. Startz (1990) Some further results on the exact small sample properties of the instrumental variable estimator. Econometrica 58, 967–976.Google Scholar
Newey, W.K. (1985) Generalized method of moments specification testing. Journal of Econometrics 29, 229–256.Google Scholar
Newey, W.K. & R.J. Smith (2004) Higher order properties of GMM and generalized empirical likelihood estimators. Econometrica 72, 219–255.Google Scholar
Newey, W.K. & K.D. West (1987) Hypothesis testing with efficient method of moments estimation. International Economic Review 28, 777–787.Google Scholar
Otsu, T. (2003) Generalized Empirical Likelihood Inference under Weak Identification. Working paper, University of Wisconsin.
Owen, A. (1988) Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75, 237–249.Google Scholar
Owen, A. (1990) Empirical likelihood ratio confidence regions. Annals of Statistics 18, 90–120.Google Scholar
Pakes, A. & D. Pollard (1989) Simulation and the asymptotics of optimization estimators. Econometrica 57, 1027–1057.Google Scholar
Phillips, P.C.B. (1984) The exact distribution of LIML: I. International Economic Review 25, 249–261.Google Scholar
Phillips, P.C.B. (1989) Partially identified econometric models. Econometric Theory 5, 181–240.Google Scholar
Qin, J. & J. Lawless (1994) Empirical likelihood and general estimating equations. Annals of Statistics 22, 300–325.Google Scholar
Smith, R.J. (1997) Alternative semi-parametric likelihood approaches to generalized method of moments estimation. Economic Journal 107, 503–519.Google Scholar
Smith, R.J. (2001) GEL Criteria for Moment Condition Models. Working paper, University of Bristol. Revised version CWP 19/04, cemmap, IFS and UCL. http://cemmap.ifs.org.uk/wps/cwp0419.pdf.
Staiger, D. & J.H. Stock (1997) Instrumental variables regression with weak instruments. Econometrica 65, 557–586.Google Scholar
Startz, R., E. Zivot, & C.R. Nelson (2004) Improved inference in weakly identified instrumental variables regression. In Frontiers of Analysis and Applied Research: Essays in Honor of Peter C.B. Phillips. Cambridge University Press.
Stock, J.H. & J.H. Wright (2000) GMM with weak identification. Econometrica 68, 1055–1096.Google Scholar
Stock, J.H., J.H. Wright, & M. Yogo (2002) A survey of weak instruments and weak identification in generalized method of moments. Journal of Business & Economic Statistics 20, 518–529.Google Scholar
van der Vaart, A.W. & J.A. Wellner (1996) Weak Convergence and Empirical Processes. Springer.
Wooldridge, J. (2002) Econometric Analysis of Cross Section and Panel Data. MIT Press.