Abstract
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 
Appendix 1
Summary Statistics of Dependent Variables Generated in Monte Carlo Study.
| dgp1 | dgp2 | dgp3 | ||||
|---|---|---|---|---|---|---|
| Endog. | Exog. | Endog. | Exog. | Endog. | Exog. | |
| % | % | % | % | % | % | |
| Multinomial treatment dummies | ||||||
| d0i | 24.63 | 24.63 | 24.68 | 24.68 | 21.94 | 21.94 |
| d1i | 34.18 | 34.18 | 34.38 | 34.38 | 34.12 | 34.12 |
| d2i | 41.19 | 41.19 | 40.94 | 40.94 | 43.94 | 43.94 |
| Count variable – yi | ||||||
| Mean | 7.622 | 5.224 | 1.021 | 0.678 | 5.756 | 5.058 |
| Variance | 456.329 | 50.768 | 11.704 | 1.087 | 82.778 | 47.227 |
| Value | % | % | % | % | % | % |
| 0 | 22.30 | 20.48 | 55.60 | 58.28 | 21.84 | 20.90 |
| 1 | 14.72 | 14.86 | 24.04 | 25.80 | 16.26 | 15.22 |
| 2 | 10.98 | 11.16 | 10.54 | 9.88 | 10.38 | 11.28 |
| 3 | 8.16 | 9.00 | 3.80 | 3.72 | 8.80 | 9.60 |
| 4 | 6.28 | 7.28 | 2.32 | 1.32 | 5.74 | 7.60 |
| 5 | 4.84 | 5.60 | 1.20 | 0.64 | 5.32 | 5.64 |
| 6 | 4.20 | 4.54 | 0.60 | 0.14 | 4.30 | 4.52 |
| 7 | 3.66 | 4.18 | 0.40 | 0.12 | 3.60 | 3.32 |
| 8 | 2.68 | 3.14 | 0.38 | 0.04 | 3.48 | 3.02 |
| 9 | 2.36 | 2.78 | 0.30 | 0.04 | 2.26 | 2.64 |
| 10 | 1.94 | 2.32 | 0.18 | 0.02 | 2.00 | 2.12 |
| >10 | 17.88 | 14.46 | 0.64 | 0 | 16.02 | 14.14 |
Summary statistics are computed on the 5000 observations of the first replication of the experiment.
Appendix 2
Empirical Power Plot of Wald Tests using Raw and Standardized Residuals.
Empirical Power Plot of LR and LM Tests using Raw and Standardized Residuals.
True Latent Factors against Estimated Residuals.
Appendix 3
NB2 Estimator with Correctly Specified Residuals: Rejection Frequencies of Exogeneity Tests.
| Nom. size | Raw residuals | Standardized residuals | ||
|---|---|---|---|---|
| Emp. size | Emp. power | Emp. size | Emp. power | |
| Wald test (Murphy Topel correction) | ||||
| 0.01 | 0.0068 | 0.7760 | 0.0084 | 0.7344 |
| 0.05 | 0.0468 | 0.9172 | 0.0468 | 0.8942 |
| 0.10 | 0.0978 | 0.9564 | 0.0922 | 0.9410 |
| Wald test (no correction) | ||||
| 0.01 | 0.0102 | 0.7874 | 0.0110 | 0.7476 |
| 0.05 | 0.0536 | 0.9204 | 0.0486 | 0.8980 |
| 0.10 | 0.1066 | 0.9570 | 0.0954 | 0.9432 |
| Likelihood ratio test | ||||
| 0.01 | 0.0104 | 0.7876 | 0.0108 | 0.7452 |
| 0.05 | 0.0532 | 0.9200 | 0.0490 | 0.8972 |
| 0.10 | 0.1062 | 0.9572 | 0.0968 | 0.9426 |
| Lagrange multiplier test | ||||
| 0.01 | 0.0110 | 0.7902 | 0.0116 | 0.7482 |
| 0.05 | 0.0542 | 0.9204 | 0.0520 | 0.8990 |
| 0.10 | 0.1084 | 0.9572 | 0.1018 | 0.9434 |
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:
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The online version of this article (DOI: https://doi.org/10.1515/jem-2014-0019) offers supplementary material, available to authorized users.
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