As a consequence, our projection-based tests would be consistent against all alternative hy- potheses of the type of (3.2), avoiding the aforementioned problem. Second, and perhaps more importantly, even when one does not use NLS to estimate θ$_{0}$, we argue that the lack of power against alternatives collinear to the score function g ( X, θ$_{0}$ ) is not a main concern. As shown by Escanciano (2009 a ), every test based on ICM approach has trivial local power against these alternatives, and as a consequence, the global power of all ICM tests in the direction of the score function will also be low, see e.g. Strasser (1990). In fact, instead of considering this as a limitation one may consider this property as a feature of our tests: by acknowledging that such alternatives cannot be powerfully detected, our projection-based test statistics do not waste power in such directions, and therefore may have higher power against other, perhaps more important, alternatives; see Section 4 for an illustration. The above discussion raises the question: are there other classes of fixed alternative hy- potheses that our specification tests are not able to detect? As first pointed out by Shaikh et al. (2009), the answer is yes. Because our test statistics depend on X only through the propen- sity score q ( X, θ$_{0}$ ), our tests will have trivial power against the class of misspecified propensity scores, where in a set of positive probability, but where θ$_{0}$ now is the probability limit of ˆ θ$_{n}$ . In other words, tests based on (2.3) (or equivalently on (2.4)), will have trivial power against alternatives such that p ( X ) is different from q ( X, θ$_{0}$ ) but The leading case of such a class of alternatives is when the propensity score is correctly specified for a subvector of X , but not for the entire vector X , see e.g. Shaikh et al. (2009). Given the nonlinear nature of p ( X ), one may consider such a class of alternatives rather exceptional. However, they can still arise in practice under some particular circumstances as we will illustrate below. Consider the case where X = ( X$_{1}$, X$_{2}$ ) and that both covariates are relevant for the propen- sity score, i.e., p ( X ) = p ( X$_{1}$, X$_{2}$ ). Suppose that a researcher considers a Probit model for E [ p ( X ) | q ( X, θ$_{0}$ )] = q ( X, θ$_{0}$ ) a.s.. (3.3) E [ D − q ( X, θ$_{0}$ ) | q ( X, θ$_{0}$ )] = 0 a.s., E [ D − q ( X, θ$_{0}$ ) | X ] = p ( X ) − q ( X, θ$_{0}$ ) = 0 14