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A weak instrument F-test in linear IV models with multiple endogenous variables

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  • Sanderson, Eleanor
  • Windmeijer, Frank

Abstract

We consider testing for weak instruments in a model with multiple endogenous variables. Unlike Stock and Yogo (2005), who considered a weak instruments problem where the rank of the matrix of reduced form parameters is near zero, here we consider a weak instruments problem of a near rank reduction of one in the matrix of reduced form parameters. For example, in a two-variable model, we consider weak instrument asymptotics of the form π1=δπ2+c/n where π1 and π2 are the parameters in the two reduced-form equations, c is a vector of constants and n is the sample size. We investigate the use of a conditional first-stage F-statistic along the lines of the proposal by Angrist and Pischke (2009) and show that, unless δ=0, the variance in the denominator of their F-statistic needs to be adjusted in order to get a correct asymptotic distribution when testing the hypothesis H0:π1=δπ2. We show that a corrected conditional F-statistic is equivalent to the Cragg and Donald (1993) minimum eigenvalue rank test statistic, and is informative about the maximum total relative bias of the 2SLS estimator and the Wald tests size distortions. When δ=0 in the two-variable model, or when there are more than two endogenous variables, further information over and above the Cragg–Donald statistic can be obtained about the nature of the weak instrument problem by computing the conditional first-stage F-statistics.

Suggested Citation

  • Sanderson, Eleanor & Windmeijer, Frank, 2016. "A weak instrument F-test in linear IV models with multiple endogenous variables," Journal of Econometrics, Elsevier, vol. 190(2), pages 212-221.
  • Handle: RePEc:eee:econom:v:190:y:2016:i:2:p:212-221
    DOI: 10.1016/j.jeconom.2015.06.004
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    1. Kleibergen, Frank & Paap, Richard, 2006. "Generalized reduced rank tests using the singular value decomposition," Journal of Econometrics, Elsevier, vol. 133(1), pages 97-126, July.
    2. Rothenberg, Thomas J., 1984. "Approximating the distributions of econometric estimators and test statistics," Handbook of Econometrics, in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 2, chapter 15, pages 881-935, Elsevier.
    3. Andrews,Donald W. K. & Stock,James H. (ed.), 2005. "Identification and Inference for Econometric Models," Cambridge Books, Cambridge University Press, number 9780521844413.
    4. Joshua D. Angrist & Jörn-Steffen Pischke, 2009. "Mostly Harmless Econometrics: An Empiricist's Companion," Economics Books, Princeton University Press, edition 1, number 8769.
    5. Douglas Staiger & James H. Stock, 1997. "Instrumental Variables Regression with Weak Instruments," Econometrica, Econometric Society, vol. 65(3), pages 557-586, May.
    6. José Luis Montiel Olea & Carolin Pflueger, 2013. "A Robust Test for Weak Instruments," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 31(3), pages 358-369, July.
    7. Cragg, John G. & Donald, Stephen G., 1993. "Testing Identifiability and Specification in Instrumental Variable Models," Econometric Theory, Cambridge University Press, vol. 9(2), pages 222-240, April.
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    More about this item

    Keywords

    Weak instruments; Multiple endogenous variables; F-test;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C36 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Instrumental Variables (IV) Estimation

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