Local Average Treatment Effect

Definition

The Local Average Treatment Effect (LATE), introduced by Imbens and Angrist (1994), is the average causal effect of a binary treatment restricted to the subpopulation of compliers — units whose treatment status is shifted by the instrument. Unlike the Average Treatment Effect (ATE, population average) or ATT (average over the treated), LATE is instrument-specific: different instruments in the same data identify LATE for different complier subpopulations.

Key Ideas

Compliance types (Angrist-Imbens-Rubin 1996): compliers (treatment status responds to instrument), always-takers, never-takers; defiers excluded by monotonicity assumption.
Instrumental Variables (IV) = LATE: Under independence of instrument, exclusion restriction, and monotonicity, the Wald IV estimator identifies $\text{LATE} = E[Y(1)-Y(0)|\text{complier}]$ .
Fuzzy Regression Discontinuity (FRD) interpretation: The threshold $T^*$ acts as the instrument; LATE is further conditioned on $X=T^*$ — doubly local (see Fuzzy Regression Discontinuity).
Hausman test in LATE framework: The Hausman test $\tau^{OLS} = \tau^{IV}$ tests a weighted average of differences between compliance-type potential outcomes — unintuitive and weight-dependent. Angrist (2004) proposes the cleaner restriction $E[Y(1)|AT] - E[Y(1)|C] = E[Y(0)|NT] - E[Y(0)|C]$ .
External validity gap: $\text{LATE} \neq \text{ATE}$ unless potential outcomes are independent of compliance type given covariates. Bertanha-Imbens (2014) formalise this assumption and derive its testable implications in the FRD setting.

How It Works

With binary instrument $Z \in \{0,1\}$ and binary treatment $W$ :

$\text{LATE} = \frac{E[Y|Z=1] - E[Y|Z=0]}{E[W|Z=1] - E[W|Z=0]} = E[Y(1) - Y(0) \mid W(1) > W(0)]$

where $W(z)$ is the potential treatment when the instrument is set to $z$ . The denominator is the fraction of compliers in the population. The numerator is the Intent-to-Treat (ITT) effect.

Why It Matters

LATE provided a coherent interpretation of IV estimates under heterogeneous treatment effects — before Imbens-Angrist (1994), IV was understood primarily for constant-coefficient models. It clarifies that IV answers the question "what is the effect for those whom the instrument moves?" and thus what external validity requires: the complier subgroup must be representative of the population of interest. The framework is the canonical language for FRD analysis and is referenced in virtually all modern quasi-experimental applied microeconometrics.

Open Questions