Single equation endogenous binary response models

This paper studies single equation models for binary outcomes incorporating instrumental variable restrictions. The models are incomplete in the sense that they place no restriction on the way in which values of endogenous variables are generated. The models are set, not point, identifying. The paper explores the nature of set identification in single equation IV models in which the binary outcome is determined by a threshold crossing condition. There is special attention to models which require the threshold crossing function to be a monotone function of a linear index involving observable endogenous and exogenous explanatory variables. Identified sets can be large unless instrumental variables have substantial predictive power. A generic feature of the identified sets is that they are not connected when instruments are weak. The results suggest that the strong point identifying power of triangular "control function" models - restricted versions of the IV models considered here - is fragile, the wide expanses of the IV model's identified set awaiting in the event of failure of the triangular model's restrictions

Instrumental Values

This paper studies the identification of partial differences of nonseparable structural functions. The paper considers triangular structures with no more stochastic unobservables than observable outcomes, that exhibit a degree of monotonicity with respect to variation in certain stochastic unobservables. It is shown that, the existence of a set of instrumental values of covariates, over which the stochastic unobservables exhibit local quantile invariance and over which a local order condition holds, defines a model which identifies certain partial differences of structural functions. This result is useful when covariates exhibit discrete variation. The paper also considers the identification of partial derivatives in smooth structures when covariates exhibit continuous variation

Instrumental variable models for discrete outcomes

Single equation instrumental variable models for discrete outcomes are shown to be set not point identifying for the structural functions that deliver the values of the discrete outcome. Identified sets are derived for a general nonparametric model and sharp set identification is demonstrated. Point identification is typically not achieved by imposing parametric restrictions. The extent of an identified set varies with the strength and support of instruments and typically shrinks as the support of a discrete outcome grows. The paper extends the analysis of structural quantile functions with endogenous arguments to cases in which there are discrete outcomes