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Sensitivity Analysis for Instrumental Variables Under Joint Relaxations of Monotonicity and Independence

Pedro Picchetti

Abstract

In this paper I develop a breakdown frontier approach to assess the sensitivity of Local Average Treatment Effects (LATE) estimates to violations of monotonicity and independence of the instrument. I parametrize violations of independence using the concept of $c$-dependence from Masten & Poirier (2018) and allow for the share of defiers to be greater than zero but smaller than the share of compliers. I derive identified sets for the LATE and the Average Treatment Effect (ATE) in which the bounds are functions of these two sensitivity parameters. Using these bounds, I derive the breakdown frontier for the LATE, which is the weakest set of assumptions such that a conclusion regarding the LATE holds. I derive consistent sample analogue estimators for the breakdown frontiers and provide a valid bootstrap procedure for inference. Monte Carlo simulations show the desirable finite-sample properties of the estimators and an empirical application shows that the conclusions regarding the effect of family size on unemployment from Angrist & Evans (1998) are highly sensitive to violations of independence and monotonicity.

Sensitivity Analysis for Instrumental Variables Under Joint Relaxations of Monotonicity and Independence

Abstract

In this paper I develop a breakdown frontier approach to assess the sensitivity of Local Average Treatment Effects (LATE) estimates to violations of monotonicity and independence of the instrument. I parametrize violations of independence using the concept of -dependence from Masten & Poirier (2018) and allow for the share of defiers to be greater than zero but smaller than the share of compliers. I derive identified sets for the LATE and the Average Treatment Effect (ATE) in which the bounds are functions of these two sensitivity parameters. Using these bounds, I derive the breakdown frontier for the LATE, which is the weakest set of assumptions such that a conclusion regarding the LATE holds. I derive consistent sample analogue estimators for the breakdown frontiers and provide a valid bootstrap procedure for inference. Monte Carlo simulations show the desirable finite-sample properties of the estimators and an empirical application shows that the conclusions regarding the effect of family size on unemployment from Angrist & Evans (1998) are highly sensitive to violations of independence and monotonicity.

Paper Structure

This paper contains 14 sections, 16 theorems, 168 equations, 5 figures.

Table of Contents

  1. Introduction
  2. General Framework
  3. Partial Identification of Potential Probabilities
  4. Partial Identification of Treatment Effects
  5. Partial Identification of $LATE_{co}$
  6. Breakdown Frontier
  7. Numerical Illustration
  8. Partial Identification of the ATE
  9. Estimation and Inference
  10. LATE and Breakdown Frontier
  11. Bootstrap Inference
  12. Monte Carlo Simulations
  13. Empirical Application
  14. Conclusion

Key Result

Proposition 1

Suppose Assumptions 1-3 hold. Then the sharp identified set for $\mathbb{P}\left(Y(D(z))=y,D(z)=d|X=x\right)$ is where

Figures (5)

  • Figure 1: Identified Sets for the LATE
  • Figure 2: Breakdown Frontiers
  • Figure 3: Sampling Distribution of the Breakdown Frontier Estimator
  • Figure 4: Finite-Sample Bias of the Breakdown Frontier Estimator
  • Figure 5: Breakdown Frontier for the Effect of Family Size on Unemployment

Theorems & Definitions (16)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Corollary 1
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • ...and 6 more