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

Johann Caro-Burnett

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.

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

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.
Paper Structure (17 sections, 16 theorems, 95 equations, 2 figures, 7 tables)

This paper contains 17 sections, 16 theorems, 95 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Illustrative Design: Two Parallel Instruments
  3. Estimation and Inference
  4. General Framework
  5. Applications
  6. Example 1: Cash Transfers and Long-Term Effects on Education
  7. Example 2: Credit Market in South Africa
  8. Example 3: Persistence in Voter Turnout
  9. Simulations
  10. Conclusions
  11. Appendix
  12. Summary of all results with proofs
  13. Step 1: Residualizing with respect to $(x_i^{\Sigma},x_i^{\Pi})$.
  14. Step 2: Design cells.
  15. Step 3: Conditional covariance with $d_i$.
  16. ...and 2 more sections

Key Result

Lemma 1

Under Assumption assumption:exclu,

Figures (2)

  • Figure 1: Thresholds for treatment take-up for potential compliers and potential defiers.
  • Figure 2: Sampling distributions of standard IV using $(z_1,z_2)$ and DIIV across four environments. Panels vary by which behavioral type is more responsive to the instruments, implemented through larger absolute coefficients $\kappa_{\theta}(m_j)$, and by the variance of the latent noise term $\eta_i$. Vertical dashed lines indicate $\tau_F = 2$ and $\tau_C = 3$. Reported probabilities $p_\theta^j$ correspond to type-$\theta$-specific shifts in treatment take-up induced by instrument $j$, and $\lambda$ denotes the DIIV weight on $\tau_C$ in the convex combination $\lambda \tau_C + (1-\lambda)\tau_F$.

Theorems & Definitions (34)

  • Lemma 1
  • proof
  • Definition
  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 24 more