This note shows that the local average treatment effect (LATE) assumptions of Angrist and Imbens are weaker than imposing an ordered, discrete-choice selection model if one imposes the standard assumption of constant thresholds in the latter. However, the note extends results of Vytlacil to show that the LATE assumptions are equivalent to an ordered, discrete-choice selection model if one allows for random thresholds in the latter. A nonparametric representation result for ordered, discrete-choice models is produced as a by-product of these results.

This paper examines the properties of instrumental variables (IV) applied to models with essential heterogeneity, that is, models where responses to interventions are heterogeneous and agents adopt treatments (participate in programs) with at least partial knowledge of their idiosyncratic response. We analyze two-outcome and multiple-outcome models, including ordered and unordered choice models. We allow for transition-specific and general instruments. We generalize previous analyses by developing weights for treatment effects for general instruments. We develop a simple test for the presence of essential heterogeneity. We note the asymmetry of the model of essential heterogeneity: outcomes of choices are heterogeneous in a general way; choices are not. When both choices and outcomes are permitted to be symmetrically heterogeneous, the method of IV breaks down for estimating treatment parameters.