Misspecification in Moment Inequality Models: Back to Moment Equalities?

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Specification tests for partially identified models defined by moment inequalities (2013, background): “…are typically related to functional forms or distributional assumptions.3 If these assumptions are not supported by the data, and so the model is misspecified, the resulting statistical inferences are usually invalid (see, e.g., Ponomareva and Tamer, 2011; Bugni, Canay, and Guggenberger, 2012).”
Nonparametric Inference Based on Conditional Moment Inequalities (2014, background): “1) with 0 = ( 0; 0(z0)); Wi = (Y u i ; Y l i ; Xi; Zi); m1(Wi; 0) = Y u i X 0 i 0 0(z0); and m2(Wi; 0) = X 0 i 0 + 0(z0) Y l i : Example 3 is a partially-linear version of the interval-outcome regression model considered in Manski and Tamer (2002) and widely discussed in the moment inequality litera”
Sensitivity to Missing Data Assumptions: Theory and an Evaluation of the U.S. Wage Structure (2010, methodology): “6 in van der Vaart and Wellner (1996) it follows that the class F is Donsker.”