When treatment effects are heterogeneous (different people benefit differently), IV estimates the Local Average Treatment Effect (LATE) — the causal effect specifically for compliers, the subpopulation whose treatment status is changed by the instrument.

Why Heterogeneity Matters

Constant effects (y_1i − y_0i = ρ for all i) is unrealistic. Different people benefit differently from treatment. This raises two concerns:

• Internal validity: What exactly is IV estimating?
• External validity: Do the results generalize to other populations?

Setup: Generalized Potential Outcomes

Define potential outcomes indexed by both treatment and instrument:

4.4.1 The LATE Theorem (Imbens & Angrist, 1994)

Four Assumptions

Assumption	Formal Statement	Intuition
A1: Independence	{yi(d,z), d1i, d0i} ⊥ zi	Instrument is as good as randomly assigned
A2: Exclusion	yi(d, 0) = yi(d, 1) for d = 0, 1	Instrument affects outcome only through treatment
A3: First stage	E[d1i − d0i] ≠ 0	Instrument affects treatment on average
A4: Monotonicity	d1i ≥ d0i for all i (or vice versa)	No one is pushed away from treatment by the instrument

Proof Sketch

Why Monotonicity?

Without monotonicity, some people are "defiers" (d_1i < d_0i). The reduced form becomes:

Positive effects could be canceled by defiers, making the reduced form misleading. Monotonicity rules out this possibility.

4.4.2 The Compliant Subpopulation

The instrument partitions the population into three groups:

Group	Definition	Draft Lottery Example
Compliers	d1i = 1, d0i = 0	Served because of draft eligibility
Always-takers	d1i = d0i = 1	Volunteered regardless
Never-takers	d1i = d0i = 0	Exempted / deferred regardless

Special Cases: LATE = ATT or LATE = Effect on Non-treated

Scenario	Example	Why
No always-takers: E[d|z=0]=0	JTPA training experiment	Treated = compliers only → LATE = ATT
No never-takers: d1i=1 for all i	Twins instrument, Minneapolis DV experiment	Non-treated = compliers only → LATE = E[y₁−y₀|d=0]

4.4.3 IV in Randomized Trials (Bloom 1984)

In a randomized trial with one-sided non-compliance (some offered treatment decline, but no control subject gets treatment), the IV estimand is the effect of treatment on the treated.

Example: JTPA Training Experiment

	By Training Status (OLS)	By Assignment (ITT)	IV Estimate (ATT)
Men	$3,970	$1,117	$1,825
Women	$2,133	$1,243	$1,942

OLS (by actual training) overstates the effect due to selection. ITT understates it because only 60% complied. IV = ITT ÷ 0.6 gives the causal effect on compliers = ATT.

4.4.4 Counting and Characterizing Compliers

Size of Complier Group

Proportion of Treated Who Are Compliers

Complier Characteristics

Although individual compliers can't be identified, the distribution of characteristics can be described:

Example: Complier Characteristics for Twins vs. Same-Sex Instruments

Characteristic	Sample Mean	Twins Ratio	Same-Sex Ratio
Age ≥ 30 at first birth	0.003	1.39	1.00
College graduate	0.132	1.14	0.70

Twins compliers are older and more educated; same-sex compliers are less educated. This helps explain why twins IV gives smaller labor supply effects (labor supply consequences of childbearing decline with education).

4.5.1 Multiple Instruments

With two instruments z_1i and z_2i, each having its own complier group, 2SLS produces:

where ρ_j is the LATE using instrument j alone, and λ depends on the relative strength of each instrument in the first stage. Instruments with a stronger first stage get more weight.

4.5.2 Covariates in the Heterogeneous-Effects Model

When the instrument is only valid conditional on covariates X_i (e.g., draft eligibility conditional on year of birth):

Saturate and Weight Theorem (Angrist & Imbens 1995)

With a fully saturated first stage (separate effect of z for each value of X) and saturated covariates in the second stage, 2SLS estimates:

A weighted average of covariate-specific LATEs, with weights proportional to the variance of first-stage fitted values at each X value. Covariate values where the instrument creates more variation in treatment get more weight.

Abadie's Kappa Weighting (Abadie 2003)

2SLS approximates the causal response function for compliers: E[y_i | d_i, X_i, complier]. The kappa-weighting function:

"finds" compliers by down-weighting always-takers (d=1, z=0) and never-takers (d=0, z=1). With a linear model for P(z=1|X), Abadie's estimator equals 2SLS.

4.5.3 Average Causal Response with Variable Treatment Intensity

When treatment is multi-valued (e.g., years of schooling s ∈ {0, 1, …, S}), the Wald estimand becomes:

Key insight: The weight at each point s is proportional to the shift in the CDF of treatment at that point, which can be estimated from data:

Application: Compulsory Schooling Laws

Acemoglu & Angrist (2000) show that child labor and compulsory attendance laws shift the schooling distribution mainly in grades 8–12, with no effect on post-secondary schooling. Therefore, IV estimates using these instruments capture returns to schooling in the high-school range, not the college range.

Continuous Treatment: Average Derivative

When treatment is continuous (e.g., price), the IV estimand is a weighted average derivative:

Example: Angrist, Graddy & Imbens (2000) estimate the demand for fish at Fulton Fish Market using weather instruments. Stormy weather drives up prices, and IV recovers the demand elasticity averaged over the range of storm-induced price shifts.

Research question: Does having a third child causally reduce female labor supply?

The Identification Problem

Simple OLS comparison of mothers with 2 vs. 3+ children confounds causation with selection: women who have more children may have inherently stronger family-orientation preferences, leading to both more children and less labor supply.

Two Instruments for a Third Child

Among mothers with ≥2 children, Angrist & Evans use two sources of exogenous variation:

	Twins at second birth	Same-sex (first two children)
Logic	Twins mechanically create ≥3 children	Parents prefer mixed-sex sibship → more likely to try for a third
First stage	0.625 (very strong)	0.067 (modest)
Validity	Twin births are essentially random	Child sex composition is random

Results

Outcome	OLS	Twins IV	Same-sex IV
Employment	−0.167	−0.083	−0.135
Weeks worked	−8.05	−3.83	−6.23

Why Estimates Differ: Different Compliers

Each instrument identifies effects for a different complier subpopulation:

Mapping to ATE / ATT / ITT / LATE

Estimand	Definition	In This Study
ATE	E[Y(1)−Y(0)] for entire population	Effect of 3rd child on all mothers with 2 children — not directly observed
ATT	E[Y(1)−Y(0) | D=1] for treated	Effect on mothers who actually had a 3rd child — OLS (−0.167) tries but is biased by selection
ITT	E[Y|Z=1]−E[Y|Z=0] by assignment	Effect of being assigned twins/same-sex — reduced form, always unbiased
LATE	E[Y(1)−Y(0) | compliers]	Twins: −0.083 | Same-sex: −0.135 — different compliers give different LATEs

Mathematical Relationships

Size Relationships

Relationship	Condition
|ITT| < |LATE|	Always (compliance rate < 1)
ATT ≥ ATE (typically)	High-benefit individuals self-select into treatment
LATE = ATT	No always-takers (Bloom 1984)
LATE₁ ≠ LATE₂	Different IVs → different compliers (Angrist & Evans)
LATE = ATE	Homogeneous treatment effects (constant effect)

Method → Estimand Mapping

Method	Estimates	Generalizability
RCT (full compliance)	ATE	Broad
RCT (non-compliance) + IV	LATE	Compliers only
DID / Matching / PSM	ATT	Groups similar to treated
RDD	LATE at cutoff	Near cutoff only

Concept	Key Point
LATE	IV = E[y₁−y₀ | compliers], not ATE or ATT in general
Four Assumptions	Independence, Exclusion, First stage, Monotonicity
Monotonicity	No defiers; all affected people are pushed in the same direction
Complier size	= First stage; characteristics via first-stage ratio across subgroups
Bloom's Result	One-sided non-compliance → LATE = ATT (e.g., JTPA)
Multiple instruments	2SLS = weighted average of instrument-specific LATEs
Covariates	2SLS = weighted average of covariate-specific LATEs
ACR Theorem	Multi-valued treatment → weighted average of unit causal effects along the response function