Testing for Instrument Independence in the Selection Model

We develop a specification test for the independent instrument assumption in the sample selection model. We test the emptiness of the identification region of Manski (2003): the set of outcome distributions that are compatible with data and the restriction of statistical independence between the instrument and outcome. The size of the identification region is characterized by a scalar parameter, the integrated envelope, and in particular the identification region is empty if and only if the integrated envelope exceeds one. Since the empty identification region implies a violation of the exclusion restriction, we obtain a nonparametric specification test for the instrument exclusion restriction by developing a testing procedure for whether the integrated envelope exceeds one. This test procedure has a non-pivotal asymptotic distribution and it is well-known that in this case the standard nonparametric bootstrap is not valid to obtain the critical values. We therefore develop a modified bootstrap procedure and show its validity. Monte Carlo simulations examine the finite sample performance of this bootstrap procedure. We use the procedure to test the independence of the instrument used by Blundell et al. (2003).