This paper studies the problem of specification testing in partially identified models defined by a finite number of moment equalities and inequalities (i.e. (in)equalities). Under the null hypothesis, there is at least one parameter value that simultaneously satisfies 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 confidence sets for the parameters of interest are empty or not, referred to as Test BP.
We propose two new specification 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 confidence set construction. Test RS requires a separate procedure to compute, but has the best power. The separate procedure is computationally easier than confidence set construction in typical cases.