Misspecification in moment inequality models: back to moment equalities?

Abstract

Summary Consider the linear model E[y|x] = x ′ β where one is interested in learning about β given data on y and x and when y is interval measured, i.e., we observe ([y0,y1],x) such that P(y 2 [y0,y1]) = 1. Moment inequality procedures use the implication E[y0|x] � x ′ β � E[y1|x]. As compared to least squares in the classical regression model, estimates obtained using an objective function based on these moment inequalities do not provide a clear approximation to the underlying unobserved conditional mean function. Most importantly, under misspecification, it is not unusual that no parameter β satisfies the previous inequalities for all values of x, and hence minima of an objective function based on these moment inequalities are typically tight. We construct set estimates for β in the linear model that have a clear interpretation when the model is misspecified. These sets are based on moment equality models. We illustrate these sets and compare them to estimates obtained using moment inequality based methods. In addition to the linear model with interval outcomes we also analyze the binary missing data model with a monotone instrument assumption (MIV), we find there that when this assumption is misspecified, bounds can still be non-empty, and can differ from parameters obtained via maximum likelihood. We also examine a bivariate discrete game with multiple equilibria. In sum, misspecification in moment inequality models is of a different flavor than in moment equality models, and so care should be taken with the 1) interpretation of the estimates and 2) the size of the ‘identified set.’