Answer-first: Instrumental variables identify the Local Average Treatment Effect (LATE) because the instrument shifts treatment for a specific subgroup—compliers—and only for them. Under relevance, exclusion, and monotonicity, the IV estimator isolates the causal effect for units whose treatment status is changed by the instrument.
Warm intro (and where to find the “why” pages)
If you’re staring at an IV or LATE question at 11:47pm, feeling stuck, behind, or low‑key panicking because the “compliers” logic feels abstract, you’re not alone. LATE questions look simple—“IV identifies the effect for compliers”—but under exam pressure, students freeze when asked to explain why IV only recovers a local effect and what assumptions make that effect credible.
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Answer first
IV identifies the Local Average Treatment Effect because the instrument affects treatment for some units (compliers) but not others (always-takers, never-takers, defiers). Under monotonicity, exclusion, and relevance, the IV estimator isolates the causal effect for the subgroup whose treatment status is actually moved by the instrument.
Problem setup
Let \(Z_i\) be an instrument, \(D_i\) the treatment, and \(Y_i\) the outcome. The first stage is:
\[ D_i = \pi_0 + \pi_1 Z_i + v_i, \]
and the structural equation is:
\[ Y_i = \beta D_i + u_i. \]
The IV estimator is:
\[ \hat{\beta}_{IV} = \frac{\mathbb{E}[Y_i \mid Z_i=1] – \mathbb{E}[Y_i \mid Z_i=0]}{\mathbb{E}[D_i \mid Z_i=1] – \mathbb{E}[D_i \mid Z_i=0]}. \]
Step-by-step solution (WordPress-safe MathJax)
Step 1: Decompose the population into compliance types
Each unit belongs to one of four groups:
- Compliers: take treatment if encouraged by the instrument.
- Always-takers: take treatment regardless of the instrument.
- Never-takers: never take treatment.
- Defiers: do the opposite of the instrument (ruled out by monotonicity).
Step 2: Understand what the numerator captures
The reduced form:
\[ \mathbb{E}[Y_i \mid Z_i=1] – \mathbb{E}[Y_i \mid Z_i=0] \]
reflects the instrument’s effect on outcomes. Only compliers change treatment status when \(Z\) changes, so only compliers contribute to this difference.
Step 3: Understand what the denominator captures
The first stage:
\[ \mathbb{E}[D_i \mid Z_i=1] – \mathbb{E}[D_i \mid Z_i=0] \]
measures how much the instrument shifts treatment. Again, only compliers contribute.
Step 4: Divide the two effects
Because both numerator and denominator isolate compliers, the ratio identifies:
\[ \beta_{LATE} = \mathbb{E}[Y_i(1) – Y_i(0) \mid \text{compliers}]. \]
Step 5: State the identifying assumptions clearly
IV identifies LATE under three conditions:
- Relevance: the instrument affects treatment.
- Exclusion: the instrument affects the outcome only through treatment.
- Monotonicity: no defiers.
These assumptions ensure that the IV estimator isolates the causal effect for compliers.
Intuition
IV works by using a natural or policy-induced “nudge” that pushes some people into treatment. Only those who respond to the nudge (compliers) provide identifying variation. The IV estimator is literally the effect of the nudge on outcomes divided by the effect of the nudge on treatment.
That ratio is the causal effect for compliers—the Local Average Treatment Effect.
Common exam mistakes
- Forgetting to state monotonicity. This is the #1 grading criterion.
- Claiming IV identifies the ATE. It identifies LATE unless strong assumptions hold.
- Ignoring compliance types. Always-takers and never-takers do not identify anything.
- Misinterpreting exclusion. The instrument must affect the outcome only through treatment.
- Using weak instruments. Weak first stages cause bias and instability.