Local Average Treatment Effect

Definition

The Local Average Treatment Effect (LATE) is the average causal effect of a treatment for the subpopulation of compliers — units whose treatment status is actually switched by the instrument. When a binary instrument Z moves a unit from $D=0$ to $D=1$, that unit is a complier, and $\text{LATE} = E[Y(1)-Y(0) \mid D(1)-D(0)=1]$. LATE is local in the sense that it is specific to the complier population defined by a particular instrument; a different instrument with different compliers identifies a different LATE, even in the same setting.

Key Ideas

• Complier subpopulation: For any binary instrument, the population partitions into four types based on the pair $(D(0), D(1))$: compliers ($0 \to 1$), never-takers ($0 \to 0$), always-takers ($1 \to 1$), and defiers ($1 \to 0$). Under monotonicity (no defiers), LATE is the only subgroup for which the instrument has identifying power.
• LATE is not ATE: LATE $\neq$ the population average treatment effect (ATE) unless treatment effects are homogeneous. Compliers may have above-average or below-average treatment effects relative to the full population, so external validity is never guaranteed by instrument validity alone.
• Estimation: LATE is identified as the ratio of the reduced-form effect (Z on Y) to the first-stage effect (Z on D). Equivalently, $\text{LATE} = \text{Wald estimator} = (\bar{Y}_{Z=1} - \bar{Y}_{Z=0}) / (\bar{D}_{Z=1} - \bar{D}_{Z=0})$.
• Compliers are unobservable: We cannot identify individual compliers in the data — only their share (= first-stage coefficient) and their average outcome under each treatment value. We know the proportion of compliers but not which units they are.
• Strong instruments → larger complier share → better external validity: A first stage of $50\%$ means half the population are compliers; a first stage of $5\%$ means the LATE applies to a tiny, potentially unrepresentative slice.
• Essential heterogeneity (Heckman and Vytlacil): When agents select into treatment based on private knowledge of their own returns, instrumental variables (IV) identifies the marginal treatment effect (MTE) at the threshold of the instrument, which may differ substantially from both average treatment effect (ATE) and average treatment effect on the treated (ATT) depending on the sign of selection.
• ATT vs. LATE (Heckman 1996): The mean treatment effect on the treated — $E[Y_1-Y_0 \mid D=1]$ — identifies causal effects for an observable subpopulation (the treated), whereas LATE identifies effects for an unobservable one (compliers). Heckman argues ATT requires only mean independence conditions (weaker than Angrist, Imbens, and Rubin (AIR)'s full independence, and without monotonicity), making it the preferable evaluation parameter when the policy question concerns those actually treated. LATE is instrument-dependent and changes as the instrument changes; ATT is a fixed, substantively meaningful parameter.
• Heckman's A-2' assumes always-takers = compliers (AIR Rejoinder 1996): AIR formally show that Heckman's assumption (A-2') — given random assignment, the exclusion restriction, and monotonicity — implies $E[Y(1)-Y(0) \mid \text{always-taker}] = E[Y(1)-Y(0) \mid \text{complier}]$. Identifying ATT requires assuming that the average effect for volunteers equals that for draftees — precisely the strong assumption AIR avoided. Heckman's "weaker" identification conditions smuggle this equality in without making it explicit.
• One-sided noncompliance: LATE = ATT (AIR Rejoinder 1996): In randomized eligibility designs where $D(0)=0$ for all units (the control group cannot access treatment), there are no always-takers and monotonicity holds by construction. In this special case LATE equals ATT exactly. Heckman (1995) himself acknowledged that such designs "can be placed in an instrumental variables framework" and identify $E[\Delta \mid D=1,X]$ — which is LATE.
• ATE vs. LATE (Robins and Greenland 1996): The average treatment effect in the full population — $E[Y(1)-Y(0)]$ — is often of greater public health interest than LATE (compliers) or intent-to-treat (ITT). Under AIR's assumptions alone, ATE is not point-identified but can be bounded (Robins-Manski bounds). ATE is identifiable without monotonicity under Robins' structural nested mean model (SNMM): if within-arm average treatment effects are the same for treated and untreated subjects, ATE = the IV estimand. In bioequivalence trials (two active treatments), LATE is not identified even under monotonicity, and ITT is uninformative — only ATE methods apply.

The AIR (1996) Five Assumptions

For the IV estimand to equal LATE, all five must hold (Angrist, Imbens, Rubin 1996):

1. Stable Unit Treatment Value Assumption (SUTVA): Each unit's potential outcomes depend only on its own treatment, not others' (no spillovers, no multiple treatment versions).
2. Ignorable assignment: The instrument Z is randomly assigned (or conditionally ignorably assigned).
3. Exclusion restriction: Z affects Y only through D — holding D fixed, changing Z has no direct effect on Y. Untestable from the data; must be justified substantively.
4. Instrument relevance: $E[D(1)-D(0)] \neq 0$ — the instrument moves treatment rates on average. Directly testable (the first stage).
5. Monotonicity: $D(1) \geq D(0)$ for all units — no defiers. Rules out units who take treatment when assigned to control but not when assigned to treatment.

Sensitivity of LATE to Assumption Violations

• Exclusion restriction violation: $\text{Bias} = E[\text{direct } Z \to Y \text{ for noncompliers}] \times (\text{noncomplier odds})$. Scales inversely with instrument strength — weak instruments amplify exclusion restriction violations.
• Monotonicity violation: Bias involves the defier share weighted by the difference in treatment effects between compliers and defiers. Again, weaker instruments make this worse.

How LATE Appears in This Wiki

Every IV result in this wiki is a LATE. Key examples:

• Maestas, Mullen, and Strand (2013): Disability Determination Services (DDS) examiner allowance-rate IV identifies LATE for marginal applicants near the award threshold ($\approx 23\%$ of applicants); causal work reduction $\approx 28$ pp.
• French and Song (2014): Administrative Law Judge (ALJ) lottery IV at appeals stage identifies a different LATE (appeals-stage compliers); causal work reduction $\approx 26$ pp. Convergence with Maestas, Mullen, and Strand (2013) despite different instruments and complier populations is the strongest available evidence.
• Deshpande and Li (2019): Field office closing IV; compliers are applicants who would have applied from the closed office had it remained open.
• Chen (2012): State-level Disability Insurance (DI) policy changes as IV; compliers are spousal households at the DI eligibility margin.
• Angrist (2001): Immigration quota legislation as IV for immigrant sex ratios; compliers are ethnic-year cells whose sex ratio shifted due to the quota.
• Angrist et al. (2010): KIPP Lynn admissions lottery as IV for charter school attendance; LATE for lottery compliers = $+0.35\sigma$/year math, $+0.12\sigma$/year reading (concentrated in Limited English Proficiency (LEP)/special education (SPED) subgroup).

LATE as a Weighted Average of MTE (Heckman 2008)

Heckman and Vytlacil (1999, 2005) show that LATE is a particular weighted average of the Marginal Treatment Effect (MTE), with weights determined by the instrument's propensity score distribution. Specifically, LATE from instrument Z identifies $\int \text{MTE}(u) \cdot h_{IV}(u)\, du$, where $h_{IV}(u)$ is the instrument-specific weight proportional to the density of propensity scores in the interval moved by the instrument. This explains why different instruments yield different LATEs even in the same setting: each instrument moves a different segment of the propensity score distribution and thus weights the MTE differently. A practical implication: LATE for a narrow instrument (e.g., a lottery moving $10\%$ from treatment to non-treatment) applies only to the compliers near that margin, and extrapolating to the full population requires either a constant-MTE assumption or recovering the full MTE curve.

P1 adequacy of LATE: For P1 policy questions (evaluating whether a past program worked for those enrolled), LATE is sufficient when the instrument mimics the program's assignment mechanism — the complier weights then match the program population. For P2 and P3 questions (forecasting effects in new environments, evaluating new policies), LATE weights do not match the relevant population, and re-weighting requires structural assumptions about MTE heterogeneity.

Open Questions

• Can LATE estimates from different instruments with different complier populations be formally combined into a population ATE, and under what assumptions?
• When instrument strength is low (weak instruments), even small exclusion restriction violations can dominate the estimate — how should applied researchers communicate this fragility?
• The Marginal Treatment Effect (MTE) framework (Heckman and Vytlacil) offers a unifying estimand — can it be connected to the discrete compliance typology in practice when instruments are discrete?

Related

• Instrumental Variables
• Marginal Treatment Effect
• Rubin Causal Model
• Causal Effects of DI Receipt
• DI Application Costs and Take-Up
• Joshua Angrist
• Guido Imbens
• Donald Rubin
• James Heckman
• James Robins
• Sander Greenland
• Paul Rosenbaum

Sources

• Angrist Imbens and Rubin 1996 — Identification of Causal Effects Using Instrumental Variables
• Heckman 1996 — Identification of Causal Effects Using Instrumental Variables Comment
• Robins and Greenland 1996 — Identification of Causal Effects Using Instrumental Variables Comment
• Angrist Imbens and Rubin 1996 — Identification of Causal Effects Using Instrumental Variables Rejoinder
• Heckman 2008 — Econometric Causality
• Maestas Mullen and Strand 2013 — Does Disability Insurance Receipt Discourage Work
• French and Song 2014 — The Effect of Disability Insurance Receipt on Labor Supply
• Deshpande and Li 2019 — Who Is Screened Out Application Costs and the Targeting of Disability Programs
• Angrist et al 2010 — Inputs and Impacts in Charter Schools KIPP Lynn
• Brand and Simon-Thomas 2012 — Causal Effect Heterogeneity
• Card 1999 — The Causal Effect of Education on Earnings
• Moffitt 2005 — Remarks on the Analysis of Causal Relationships in Population Research
• Van der Klaauw 2014 — From Micro Data to Causality Forty Years of Empirical Labor Economics