How can Instrumental Variables Identify the Local Average Treatment Effect (LATE)

What Is the Local Average Treatment Effect?

Formally:

\[ LATE = E[Y_1 – Y_0 \mid \text{Compliers}] \]

IV identifies this effect — not the ATE — because only compliers respond to the instrument.

Why Instrumental Variables Identify LATE

1. The instrument affects treatment but not outcomes directly

This is the exclusion restriction.

2. Individuals differ in how they respond to the instrument

The population contains:

• Compliers — take treatment when encouraged
• Never‑takers — never take treatment
• Always‑takers — always take treatment
• Defiers — do the opposite of encouragement

3. Monotonicity rules out defiers

This ensures the instrument moves treatment in only one direction.

4. Only compliers change treatment status

Thus, the instrument only reveals information about compliers.

5. The Wald estimator isolates the complier effect

\[ \frac{E[Y \mid Z=1] – E[Y \mid Z=0]}{E[D \mid Z=1] – E[D \mid Z=0]} \] equals the average treatment effect for compliers.

How LATE Arises from the IV Estimator (Step by Step)

Step 1: Decompose the first stage

\[ E[D \mid Z=1] – E[D \mid Z=0] = P(\text{Compliers}) \]

Step 2: Decompose the reduced form

\[ E[Y \mid Z=1] – E[Y \mid Z=0] \] captures the treatment effect only for compliers.

Step 3: Divide reduced form by first stage

\[ \frac{\text{Reduced Form}}{\text{First Stage}} = LATE \]

Step 4: Interpret the result

IV identifies the causal effect for the subgroup whose treatment is moved by the instrument.

Numerical Example

Suppose:

• Reduced form: \(E[Y \mid Z=1] – E[Y \mid Z=0] = 2\)
• First stage: \(E[D \mid Z=1] – E[D \mid Z=0] = 0.5\)

Then:

\[ LATE = \frac{2}{0.5} = 4 \]

The treatment increases the outcome by 4 units for compliers.