Sharp Testable Implications of Encouragement Designs

TL;DR

This paper derives sharp, testable inequalities for encouragement designs with multi-valued treatments and instruments, enabling empirical testing of key assumptions in causal inference models.

Contribution

It introduces a novel set of sharp inequalities for encouragement designs and proposes tests to empirically verify these restrictions in observed data.

Findings

Some restrictions are violated in empirical data

Identifies substitution patterns causing violations

Provides a constructive method for testing assumptions

Abstract

This paper studies a potential outcome model with a continuous or discrete outcome, a discrete multi-valued treatment, and a discrete multi-valued instrument. We derive sharp, closed-form testable implications for a class of restrictions on potential treatments where each value of the instrument encourages towards at most one unique treatment choice; such restrictions serve as the key identifying assumption in several prominent recent empirical papers. Borrowing the terminology used in randomized experiments, we call such a setting an encouragement design. The testable implications are inequalities in terms of the conditional distributions of choices and the outcome given the instrument. Through a novel constructive argument, we show these inequalities are sharp in the sense that any distribution of the observed data that satisfies these inequalities is compatible with this class of restrictions on potential treatments. Based on these inequalities, we propose tests of the restrictions. In an empirical application, we show some of these restrictions are violated and pinpoint the substitution pattern that leads to the violation.