Abstract

A violation of the instrument exogeneity condition makes the instrumental variable approach cause more harm than good: the estimator is less efficient than the OLS estimator, while the bias could be even higher than that of the OLS. Usually, in the economic literature, the exogeneity of instruments is justified using economic-theoretical arguments that could be subjective and are not testable from data. The Sargan test [Sargan, 1958] is based on the non-testable assumption that the just-identifying instruments satisfy the exogeneity condition. The approach in [Kiviet, 2020] is based on non-testable expert opinions about the admissible range for the extent of the correlation of an endogenous variable with the error term. The method in [Kitagawa, 2015] can be used only for binary treatments and discrete instruments. In this work, we model the joint distribution of the error term of the OLS model, the instrumental variables, and the error term for the reduced-form equation of the endogenous regressor by a Gaussian copula. We show that exogeneity of instrumental variables is equivalent to the exogeneity of their standard normal transformations with the same CDF value. Then, we establish a Wald test for the exogeneity of the instrumental variables. We also show that this method can be used to test the exogeneity of a regressor. We demonstrate the performance of our test using simulation studies. Our simulations show that if the instruments are actually endogenous, our test rejects the exogeneity hypothesis approximately 93% of the time at the 5% significance level. Conversely, when instruments are truly exogenous, it dismisses the exogeneity assumption less than 30% of the time on average for data with 200 observations and less than 2% of the time for data with 1000 observations. In particular, our test remains robust even when the error term deviates from a normal distribution. We also present the performance of our test for the TSLS approach in [Angrist and Krueger, 1991], showing that the interaction between the quarter of birth and year of birth are exogenous from the error term. Our test also provides strong evidence for the endogeneity of the length of education while showing that this variable becomes exogenous after adding the demographic variables. Our results demonstrate our test’s effectiveness, offering significant value to applied econometricians.

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• Keyword(s):
• machine learning 23
• operations 143

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