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Testing Exogeneity of Multinomial Regressors in Count Data Models: Does Two-stage Residual Inclusion Work?

  • Andrea Geraci , Daniele Fabbri and Chiara Monfardini EMAIL logo


We study a simple exogeneity test in count data models with possibly endogenous multinomial treatment. The test is based on Two Stage Residual Inclusion (2SRI), an estimation method which has been proved to be consistent for a general class of nonlinear parametric models. Results from a broad set of simulation experiments provide novel evidence on important features of this approach. We find differences in the finite sample performance of various likelihood-based tests, analyze their robustness to misspecification arising from neglected over-dispersion or from incorrect specification of the first stage model, and uncover that standardizing the variance of the first stage residuals leads to better results. An original application to testing the endogeneity status of insurance in a model of healthcare demand corroborates our Monte Carlo findings.

Correction note

[Correction added after online publication 8 November 2016: For consistency, qij was changed to qij* on p. 4, line 35 and p. 5, line 10. Also, the formatting of qi was corrected from italics to bold in eqs. 9 and 10.]

Appendix 1

Table A1:

Summary Statistics of Dependent Variables Generated in Monte Carlo Study.

Multinomial treatment dummies
Count variable – yi

Summary statistics are computed on the 5000 observations of the first replication of the experiment.

Appendix 2

Figure A1: Empirical Power Plot of Wald Tests using Raw and Standardized Residuals.
Figure A1:

Empirical Power Plot of Wald Tests using Raw and Standardized Residuals.

Figure A2: Empirical Power Plot of LR and LM Tests using Raw and Standardized Residuals.
Figure A2:

Empirical Power Plot of LR and LM Tests using Raw and Standardized Residuals.

Figure A3: True Latent Factors against Estimated Residuals.
Figure A3:

True Latent Factors against Estimated Residuals.

Appendix 3

Table A2:

NB2 Estimator with Correctly Specified Residuals: Rejection Frequencies of Exogeneity Tests.

Nom. sizeRaw residualsStandardized residuals
Emp. sizeEmp. powerEmp. sizeEmp. power
Wald test (Murphy Topel correction)
Wald test (no correction)
Likelihood ratio test
Lagrange multiplier test

No of replications of the Monte Carlo experiment (R)=5.000; Saple size for each replication (N)=5.000. Raw residuals and Standardized residuals are computed after estimation of the first stage equations using, respectively:q^ij=(dijq^ij),for j=0, 1, 2 and q^ij=p^ij1/2(1p^ij)1/2(dijp^ij), for j=0, 1, 2.


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The online version of this article (DOI: offers supplementary material, available to authorized users.

Published Online: 2016-11-8

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