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Finite-Sample Instrumental Variables Inference using an Asymptotically Pivotal Statistic


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  • Paul A. Bekker

    (University of Groningen)

  • Frank Kleibergen

    (University of Amsterdam)


The paper considers the K-statistic, Kleibergen’s (2000) adaptation ofthe Anderson-Rubin (AR) statistic in instrumental variables regression.Compared to the AR-statistic this K-statistic shows improvedasymptotic efficiency in terms of degrees of freedom in overidentifiedmodels and yet it shares, asymptotically, the pivotal property of theAR statistic. That is, asymptotically it has a chi-square distributionwhether or not the model is identified. This pivotal property is veryrelevant for size distortions in finite-sample tests. Whereas Kleibergen(2000) focuses especially on the asymptotic behavior of the statistic,the present paper concentrates on finite-sample properties in a Gaussianframework. In that case the AR statistic has an F-distribution.However, the K-statistic is not exactly pivotal. Its finite-sample distributionis affected by nuisance parameters. Here we consider the twoextreme cases, which provide tight bounds for the exact distribution.The first case amounts to perfect identification—which is similar tothe asymptotic case—where the statistic has an F-distribution. Inthe other extreme case there is total underidentification. For the lattercase we show how to compute the exact distribution. Thus weprovide tight bounds for exact confidence sets based on the efficientK-statistic. Asymptotically the two bounds converge, except whenthere is a large number of redundant instruments. This paper has resulted in a publication in Econometric Theory , 2003, 19, 744-53.

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Bibliographic Info

Paper provided by Tinbergen Institute in its series Tinbergen Institute Discussion Papers with number 01-055/4.

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Date of creation: 13 Jun 2001
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Handle: RePEc:dgr:uvatin:20010055

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  1. Charles R. Nelson & Richard Startz, 1988. "Some Further Results on the Exact Small Sample Properties of the Instrumental Variable Estimator," NBER Technical Working Papers, National Bureau of Economic Research, Inc 0068, National Bureau of Economic Research, Inc.
  2. Zivot, E & Startz, R & Nelson, C-R, 1997. "Valid Confidence Intervals and Inference in the Presence of Weak Instruments," Working Papers, University of Washington, Department of Economics 97-17, University of Washington, Department of Economics.
  3. Douglas Staiger & James H. Stock, 1994. "Instrumental Variables Regression with Weak Instruments," NBER Technical Working Papers, National Bureau of Economic Research, Inc 0151, National Bureau of Economic Research, Inc.
  4. Frank Kleibergen, 2002. "Pivotal Statistics for Testing Structural Parameters in Instrumental Variables Regression," Econometrica, Econometric Society, Econometric Society, vol. 70(5), pages 1781-1803, September.
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Cited by:
  1. Whitney Newey & Frank Windmeijer, 2005. "GMM with many weak moment conditions," CeMMAP working papers, Centre for Microdata Methods and Practice, Institute for Fiscal Studies CWP18/05, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  2. Bekker, Paul A. & Lawford, Steve, 2008. "Symmetry-based inference in an instrumental variable setting," Journal of Econometrics, Elsevier, Elsevier, vol. 142(1), pages 28-49, January.
  3. Dufour, Jean-Marie & Taamouti, Mohamed, 2007. "Further results on projection-based inference in IV regressions with weak, collinear or missing instruments," Journal of Econometrics, Elsevier, Elsevier, vol. 139(1), pages 133-153, July.
  4. D. S. Poskitt & C. L. Skeels, 2005. "Small Concentration Asymptotics and Instrumental Variables Inference," Monash Econometrics and Business Statistics Working Papers, Monash University, Department of Econometrics and Business Statistics 4/05, Monash University, Department of Econometrics and Business Statistics.
  5. Frank Kleibergen, 2004. "Expansions of GMM statistics that indicate their properties under weak and/or many instruments and the bootstrap," Econometric Society 2004 North American Summer Meetings, Econometric Society 408, Econometric Society.


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