Simultaneous equations models for discrete outcomes: coherence, completeness, and identification

This paper studies simultaneous equations models for two or more discrete outcomes. These
models may be incoherent, delivering no values of the outcomes at certain values of the latent
variables and covariates, and they may be incomplete, delivering more than one value of the
outcomes at certain values of the covariates and latent variates. We revisit previous approaches
to the problems of incompleteness and incoherence in such models, and we propose a new
approach for dealing with these. For each approach, we use random set theory to characterize
sharp identification regions for the marginal distribution of latent variables and the structural
function relating outcomes to covariates, illustrating the relative identifying power and tradeoffs
of the different approaches. We show that these identified sets are characterized by systems of
conditional moment equalities and inequalities, and we provide a generically applicable algorithm
for constructing these. We demonstrate these results for the simultaneous equations model for
binary outcomes studied in for example Heckman (1978) and Tamer (2003) and the triangular
model with a discrete endogenous variable studied in Chesher (2005) and Jun, Pinkse, and Xu
(2011) as illustrative examples.