Weak identification in probit models with endogenous covariates

Abstract

Weak identification is a well-known issue in the context of linear structural models. However, for probit models with endogenous explanatory variables, this problem has been little explored. In this paper, we study by simulating the behavior of the usual z-test and the LR test in the presence of weak identification. We find that the usual asymptotic z-test exhibits large level distortions (over-rejections under the null hypothesis). The magnitude of the level distortions depends heavily on the parameter value tested. In contrast, asymptotic LR tests do not over-reject and appear to be robust to weak identification.

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Fig. 1
Fig. 2

Notes

  1. 1.

    A further exception is Magnusson (2007), who considered in an early version of his paper the probit model with endogenous covariates as an example and found medium level distortions. However, in later versions of the working paper and in the published version (Magnusson 2010) the probit example was deleted.

  2. 2.

    All simulations were done using R; see Core Team (2016). The GMM estimation was done using the package GMM, version 1.6-1. The case of iid observations can be implemented by the option vcov="iid"; see Chaussé (2010), p. 13. All R codes are available on request.

  3. 3.

    For the ML estimation, the exogenous variable \(x_i\) was drawn from an N(0, 16) distribution.

  4. 4.

    This result is similar to that of Magnusson (2007).

  5. 5.

    We also tried the option "CUE" for the continuous updating estimator. However, we always got the error message "node stack overflow".

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Correspondence to Joachim Wilde.

Additional information

The authors thank Leandro Magnusson and two anonymous referees for several useful comments, and Dietrich Trenkler and Sebastian Veldhuis for valuable assistance. This work was supported by the William Dow Chair in Political Economy (McGill University), the Bank of Canada (Research Fellowship), the Toulouse School of Economics (Pierre-de-Fermat Chair of excellence), the Universitad Carlos III de Madrid (Banco Santander de Madrid Chair of excellence), a Guggenheim Fellowship, a Konrad-Adenauer Fellowship (Alexander-von-Humboldt Foundation, Germany), the Natural Sciences and Engineering Research Council of Canada, the Social Sciences and Humanities Research Council of Canada, and the Fonds de recherche sur la société et la culture (Québec).

Appendix: Simulation results for model (5) with \(\beta _{11}=\pi _{21}= 0\)

Appendix: Simulation results for model (5) with \(\beta _{11}=\pi _{21}= 0\)

See Tables 7 and 8.

Table 7 Rejection frequencies of the z-test, H\(_0: \gamma _1 = 0\)
Table 8 Rejection frequencies of the z-test, H\(_0: \gamma _1 = 2\)

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Dufour, JM., Wilde, J. Weak identification in probit models with endogenous covariates. AStA Adv Stat Anal 102, 611–631 (2018). https://doi.org/10.1007/s10182-018-0325-8

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Keywords

  • Probit model
  • Weak identification
  • z-test

JEL Classification

  • C35