Specification tests for partially identified models defined by moment inequalities

This paper studies the problem of specifi cation testing in partially identi fied models defi ned by a fi nite number of moment equalities and inequalities (i.e. (in)equalities). Under the null hypothesis, there is at least one parameter value that simultaneously satis fies all of the moment (in)equalities whereas under the alternative hypothesis there is no such parameter value. This problem has not been directly addressed in the literature (except in particular cases), although several papers have suggested a test based on checking whether con fidence sets for the parameters of interest are empty or not, referred to as Test BP. We propose two new speci fication tests, denoted Tests RS and RC, that achieve uniform asymptotic size control and dominate Test BP in terms of power in any finite sample and in the asymptotic limit. Test RC is particularly convenient to implement because it requires little additional work beyond the con fidence set construction. Test RS requires a separate procedure to compute, but has the best power. The separate procedure is computationally easier than confi dence set construction in typical cases.