Toggling the Defiers to Relax Monotonicity: The Difference-in-Instrumental-Variables Estimand

TL;DR

This paper proposes the DIIV estimand, which uses two instruments with opposing effects to identify causal effects without assuming monotonicity, broadening the applicability of IV methods.

Contribution

It introduces a novel DIIV estimand that relaxes the monotonicity assumption by leveraging two instruments with opposing compliance patterns.

Findings

DIIV recovers a convex combination of effects on compliers and defiers.

When monotonicity holds, DIIV aligns with standard IV estimates.

The method is simple to implement with standard two-stage least squares.

Abstract

Standard instrumental variables (IV) methods identify a Local Average Treatment Effect under monotonicity, which rules out defiers. In many empirical environments, however, distinct instruments may induce heterogeneous and even opposing behavioral responses. This paper introduces the Difference-in-Instrumental-Variables (DIIV) estimand, which exploits two instruments with opposing compliance patterns to recover a point-identified and behaviorally interpretable causal effect without imposing monotonicity. The estimand yields a convex combination of the marginal treatment effects on compliers and defiers, with weights reflecting differential shifts in treatment take-up across instruments. When monotonicity holds, DIIV coincides with the standard IV estimand. The approach can be implemented using simple linear transformations and standard two-stage least squares procedures. Applications using replication data illustrate its applicability in practice.