Overidentification tests for the exogeneity of instruments in discrete choice models

Endogeneity is often present in discrete choice models, precluding the consistent estimation of the model parameters. To correct for this problem, the researcher needs to gather exogenous instrumental variables, which should be independent of the error term of the model. This critical assumption can be tested using overidentification tests that rely on having more instruments than endogenous variables. For discrete choice models, instruments' exogeneity can be assessed using the Amemiya-Lee-Newey test, which relies in the estimation of an auxiliary GMM model build from reduced-form estimates. This paper proposes two alternative tests that are constructed as adaptations of the Refutability test and the Hausman test, into the discrete choice framework. The Refutability test consists in including instruments as additional variables in an auxiliary model that was corrected for endogeneity using all instruments available. The Hausman test is built from the comparison of the estimates attained using different subsets of instruments. Using a binary choice Monte Carlo experiment, the three tests are assessed in terms of size, power, and robustness to De Blander's condition for which all overidentification tests of this kind are blind. Results show that what is termed the modified Refutability test, which includes all instruments simultaneously, has smaller size distortion, larger power, and more robustness, compared to the state of the art Amemiya-Lee-Newey test and to the proposed Hausman test. Besides, the Amemiya-Lee-Newey and both versions of the Refutability test allow being agnostic about which instrument might be exogenous. The paper finishes highlighting the methodological and practical implications and limitations of these findings and suggesting future lines of research in this area.