Overidentification test in a nonparametric treatment model with unobserved heterogeneity

We provide an overidentification test for a nonparametric treatment model where individuals are allowed to select into treatment based on unobserved gains. Our test can be used to test the validity of instruments in a framework with essential heterogeneity (Imbens and Angrist 1994). The essential ingredient is to assume that a binary and a continuous instrument are available. The testable restriction is closely related to the overidentification of the Marginal Treatment Effect. We suggest a test statistic and characterize its asymptotic distribution and behavior under local alternatives. In simulations, we investigate the validity and finite sample performance of an easy-to-implement wild bootstrap procedure. Finally, we illustrate the applicability of our method by studying two instruments from the literature on teenage pregnancies. This research is motivated by the observation that in the presence of essential heterogeneity classical GMM overidentification tests can not be used to test for the validity of instruments (Heckman and Schmierer 2010). The test complements existing tests by considering for the first time the subpopulation of compliers. Our approach can be interpreted as a test of index sufficiency and is related to the test of the validity of the matching approach in Heckman et. al 1996,1998. Conditional on covariates the propensity score aggregates all information that the instruments provide about observed outcomes given that the model is correctly specified. The estimated propensity score enters our test statistic as a generated regressor. We quantify the effect of the first stage estimation error and find that in order to have good power against local alternatives we have to reduce the bias from estimating the first stage nonparametrically, e.g., by fitting a higher order local polynomial. For the second stage no bias reduction is necessary. Previous literature (Ying Ying Lee 2013) establishes the validity of a multiplier bootstrap to conduct inference in a treatment model with nonparametrically estimated regressors. Our simulations illustrate that a much easier to implement na ve wild bootstrap procedure can have good properties. In our application we consider two instruments that have been used in the analysis of the effect of teenage child bearing on high-school graduation. For the binary instrument we use teenage miscarriage and for the continuous instrument we use age of first menstrual period. If teenage girls select into pregnancy based on some unobserved heterogeneity that is correlated with their likelihood of graduation miscarriage does not constitute a valid instrument. Our test confirms this line of argument by rejecting that the treatment model is correctly specified.