Local Average Treatment Effects

Summary

When treatment effects are heterogeneous, IV estimates the average causal effect on compliers — individuals whose treatment status is changed by the instrument. This is the LATE, introduced by Imbens and Angrist (1994).

The Four Types

With binary instrument $z_i$ and binary treatment $D_i$:

Type	$D_i$ when $z_i=0$	$D_i$ when $z_i=1$	Behavior
Compliers	0	1	Follow the instrument
Always-takers	1	1	Always treated
Never-takers	0	0	Never treated
Defiers	1	0	Do the opposite

The LATE Theorem

Under monotonicity (no defiers) and exclusion restriction:

$$\frac{E[Y_i|z_i=1]-E[Y_i|z_i=0]}{E[D_i|z_i=1]-E[D_i|z_i=0]}=E[Y_{1i}-Y_{0i}|\text{complier}]$$

The Wald/IV estimand is the average treatment effect on compliers.

Why LATE Matters

• Different instruments identify different complier populations → different LATEs
• Example: twins and sex-composition instruments for family size give different estimates because they affect different women
• LATE ≠ ATE in general — the policy-relevant parameter depends on context

Characterizing Compliers

You can’t identify individual compliers, but you can describe them statistically:

• Compliance rate: $P(\text{complier})=E[D_i|z_i=1]-E[D_i|z_i=0]$
• Complier characteristics: compute $P(\text{complier}|X=x)/P(\text{complier})$ using Bayes’ rule

See Also

• Instrumental Variables
• Regression Discontinuity Designs — fuzzy RD estimates LATE at the cutoff
• Differences-in-Differences — parallel-trends DiD is also a complier-flavored estimand under heterogeneous effects
• The Selection Problem — LATE is fundamentally a solution to the selection problem for non-compliant units
• Mostly Harmless Econometrics - Overview
• Hierarchical Models — Bayesian partial pooling as an alternative framework for treatment effect heterogeneity
• Regression and the CEF — the CEF provides the population target that LATE identifies in the complier subpopulation
• Bayesian Difference in Differences — DiD treatment effects under heterogeneous compliance connect to the LATE framework