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Structural Inference from Reduced Forms with Many Instruments

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Abstract

This paper develops exact finite sample and asymptotic distributions for structural equation tests based on partially restricted reduced form estimates. Particular attention is given to models with large numbers of instruments, wherein the use of partially restricted reduced form estimates is shown to be especially advantageous in statistical testing even in cases of uniformly weak instruments and reduced forms. Comparisons are made with methods based on unrestricted reduced forms, and numerical computations showing finite sample performance of the tests are reported. Some new results are obtained on inequalities between noncentral chi-squared distributions with different degrees of freedom that assist in analytic power comparisons.

Suggested Citation

  • Wayne Yuan Gao & Peter C.B. Phillips, 2016. "Structural Inference from Reduced Forms with Many Instruments," Cowles Foundation Discussion Papers 2062, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:2062
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    1. Phillips, Peter C B, 1985. "The Exact Distribution of LIML: II," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 26(1), pages 21-36, February.
    2. Donald W.K. Andrews & James H. Stock, 2005. "Inference with Weak Instruments," Cowles Foundation Discussion Papers 1530, Cowles Foundation for Research in Economics, Yale University.
    3. Chernozhukov, Victor & Hansen, Christian, 2008. "The reduced form: A simple approach to inference with weak instruments," Economics Letters, Elsevier, vol. 100(1), pages 68-71, July.
    4. Bekker, Paul & Kleibergen, Frank, 2003. "Finite-Sample Instrumental Variables Inference Using An Asymptotically Pivotal Statistic," Econometric Theory, Cambridge University Press, vol. 19(5), pages 744-753, October.
    5. Marcelo J. Moreira, 2003. "A Conditional Likelihood Ratio Test for Structural Models," Econometrica, Econometric Society, vol. 71(4), pages 1027-1048, July.
    6. Phillips, Peter C.B., 2006. "A Remark On Bimodality And Weak Instrumentation In Structural Equation Estimation," Econometric Theory, Cambridge University Press, vol. 22(5), pages 947-960, October.
    7. Phillips, P C B, 1980. "The Exact Distribution of Instrumental Variable Estimators in an Equation Containing n + 1 Endogenous Variables," Econometrica, Econometric Society, vol. 48(4), pages 861-878, May.
    8. Donald W. K. Andrews & Marcelo J. Moreira & James H. Stock, 2006. "Optimal Two-Sided Invariant Similar Tests for Instrumental Variables Regression," Econometrica, Econometric Society, vol. 74(3), pages 715-752, May.
    9. Andrews, Donald W.K. & Guggenberger, Patrik, 2017. "Asymptotic Size Of Kleibergen’S Lm And Conditional Lr Tests For Moment Condition Models," Econometric Theory, Cambridge University Press, vol. 33(5), pages 1046-1080, October.
    10. Peter C. B. Phillips, 2017. "Reduced forms and weak instrumentation," Econometric Reviews, Taylor & Francis Journals, vol. 36(6-9), pages 818-839, October.
    11. John C. Chao & Norman R. Swanson, 2005. "Consistent Estimation with a Large Number of Weak Instruments," Econometrica, Econometric Society, vol. 73(5), pages 1673-1692, September.
    12. Frank Kleibergen, 2002. "Pivotal Statistics for Testing Structural Parameters in Instrumental Variables Regression," Econometrica, Econometric Society, vol. 70(5), pages 1781-1803, September.
    13. Chen, Jeesen & Rubin, Herman, 1986. "Bounds for the difference between median and mean of gamma and poisson distributions," Statistics & Probability Letters, Elsevier, vol. 4(6), pages 281-283, October.
    14. Phillips, P.C.B., 1983. "Exact small sample theory in the simultaneous equations model," Handbook of Econometrics, in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 1, chapter 8, pages 449-516, Elsevier.
    15. Hansen, Christian & Hausman, Jerry & Newey, Whitney, 2008. "Estimation With Many Instrumental Variables," Journal of Business & Economic Statistics, American Statistical Association, vol. 26, pages 398-422.
    16. Phillips, P.C.B., 1989. "Partially Identified Econometric Models," Econometric Theory, Cambridge University Press, vol. 5(2), pages 181-240, August.
    17. Douglas Staiger & James H. Stock, 1997. "Instrumental Variables Regression with Weak Instruments," Econometrica, Econometric Society, vol. 65(3), pages 557-586, May.
    18. Knight, John L., 1977. "On the existence of moments of the partially restricted reduced-form estimators from a simultaneous-equation model," Journal of Econometrics, Elsevier, vol. 5(3), pages 315-321, May.
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    Cited by:

    1. Max-Sebastian Dovì & Anders Bredahl Kock & Sophocles Mavroeidis, 2024. "A Ridge-Regularized Jackknifed Anderson-Rubin Test," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 42(3), pages 1083-1094, July.
    2. Peter C. B. Phillips, 2022. "An Econometrician amongst Statisticians: T. W. Anderson," Cowles Foundation Discussion Papers 2333, Cowles Foundation for Research in Economics, Yale University.
    3. Christopher L. Skeels & Frank Windmeijer, 2018. "On the Stock–Yogo Tables," Econometrics, MDPI, vol. 6(4), pages 1-23, November.

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    More about this item

    Keywords

    Exact distributions; Partial identification; Partially restricted reduced form; Structural inference; Unidentified structure; Weak reduced form;
    All these keywords.

    JEL classification:

    • C23 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Models with Panel Data; Spatio-temporal Models
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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