[Paper Review] Heterogeneous Treatment Effects with Mismeasured Endogenous Treatment

Takuya Ura|arXiv (Cornell University)|Nov 13, 2015

Advanced Causal Inference Techniques43 references8 citations

TL;DR

This paper develops a nonparametric identification framework for local average treatment effects (LATE) when the treatment is both endogenous and mismeasured, using a binary instrumental variable. It derives sharp bounds on LATE under exclusion restriction and deterministic monotonicity, showing the Wald estimand is an upper bound but not the sharp bound, and provides uniformly valid confidence intervals for inference.

ABSTRACT

This paper studies the identifying power of an instrumental variable in the nonparametric heterogeneous treatment effect framework when a binary treatment is mismeasured and endogenous. Using a binary instrumental variable, I characterize the sharp identified set for the local average treatment effect under the exclusion restriction of an instrument and the deterministic monotonicity of the true treatment in the instrument. Even allowing for general measurement error (e.g., the measurement error is endogenous), it is still possible to obtain finite bounds on the local average treatment effect. Notably, the Wald estimand is an upper bound on the local average treatment effect, but it is not the sharp bound in general. I also provide a confidence interval for the local average treatment effect with uniformly asymptotically valid size control. Furthermore, I demonstrate that the identification strategy of this paper offers a new use of repeated measurements for tightening the identified set.

Motivation & Objective

• To address the joint challenges of endogeneity and measurement error in treatment effect estimation, particularly in nonparametric heterogeneous treatment effect models.
• To extend the classical instrumental variable framework to settings where the true treatment is mismeasured but the instrument is valid under exclusion and monotonicity.
• To characterize the sharp identified set for the local average treatment effect (LATE) when the true treatment is unobserved and subject to general (possibly endogenous) measurement error.
• To provide uniformly asymptotically valid confidence intervals for LATE under partial identification.
• To demonstrate how repeated measurements can tighten the identified set, offering a new application of repeated data in measurement error models.

Proposed method

• Uses a binary instrumental variable under the exclusion restriction (instrument affects outcome only through true treatment) and deterministic monotonicity (instrument increases true treatment status).
• Characterizes the sharp identified set for LATE as a function of the intent-to-treat effect and the total variation distance between the joint distribution of outcome and measured treatment under different instrument values.
• Derives bounds on LATE that depend on the sign and magnitude of the intent-to-treat effect and the total variation distance, with the Wald estimand serving as an upper bound.
• Applies the framework to data distributions where the true treatment is latent and only the mismeasured treatment is observed, using conditional moment restrictions.
• Uses the compliers-defiers-for-marginals condition to link the observed data distribution to the latent complier subpopulation.
• Constructs a uniformly valid confidence interval for LATE by inverting a test based on the identified set, ensuring asymptotic size control.

Experimental results

Research questions

• RQ1 What is the sharp identified set for the local average treatment effect when the treatment is both endogenous and mismeasured?
• RQ2 How does the Wald estimand relate to the true LATE under mismeasurement, and is it sharp?
• RQ3 Can repeated measurements of the treatment be used to tighten the identified set for LATE under mismeasurement and endogeneity?
• RQ4 Under what conditions is the local average treatment effect point-identified, and when is it only partially identified?
• RQ5 How can uniformly valid confidence intervals be constructed for LATE in the presence of mismeasured endogenous treatments?

Key findings

• The sharp identified set for LATE is bounded by the intent-to-treat effect and the total variation distance between the outcome and measured treatment distributions under different instrument values.
• The Wald estimand is an upper bound on the true LATE but is not the sharp bound in general, especially when measurement error is endogenous.
• When the intent-to-treat effect is positive, the identified set is [ΔE[Y|Z], ΔE[Y|Z]/TV(Y,T)]; when negative, it is [ΔE[Y|Z]/TV(Y,T), ΔE[Y|Z]]; and when zero, the identified set is {0}.
• The size of the compliers (P(C_V)) is bounded by the total variation distance, with TV(Y,T) ≤ ΔE[T|Z] ≤ 1, which constrains the sharp identified set.
• The method allows for uniformly valid confidence intervals for LATE, ensuring correct asymptotic size control under partial identification.
• Repeated measurements of the treatment can be used to tighten the identified set, offering a new methodological application of repeated data in measurement error models.