Econometrics & Causal Inference
Wald Instrumental Variables Estimator and Complier LATE Formula
Correct imperfect randomized encouragement non-compliance using the Angrist-Imbens-Rubin Wald IV ratio LATE = (E[Y|Z=1] - E[Y|Z=0]) / (E[D|Z=1] - E[D|Z=0]).
Scale Intention-to-Treat (ITT) by exact first-stage complier share without selection bias
When a randomized encouragement trial sends a training voucher (Z = 1) vs no voucher (Z = 0), some encouraged users ignore training (never-takers) while some unencouraged users enroll anyway (always-takers). Comparing actual trainees D = 1 against non-trainees D = 0 ("As-Treated") reintroduces severe self-selection bias! Under monotonicity and exclusion, Angrist, Imbens, and Rubin (1996) proved that dividing the Intention-to-Treat outcome effect ITT_Y = E[Y|Z=1] - E[Y|Z=0] by the first-stage uptake jump pi_C = E[D|Z=1] - E[D|Z=0] recovers the exact Local Average Treatment Effect (LATE / CACE) on compliers!
Check by hand: Encouraged arm Z = 1 has uptake p_D1 = 0.65 (65% trained, B2) and mean earnings Y_1 = 54.20 ($k, C2); Control arm Z = 0 has always-taker uptake p_D0 = 0.15 (15% trained, B3) and mean earnings Y_0 = 50.20 ($k, C3). First-stage complier share is pi_C = 0.65 - 0.15 = 0.50 (50% compliers, 15% always-takers, 35% never-takers). Intention-to-Treat is ITT_Y = 54.20 - 50.20 = +4.00, so Complier LATE = 4.00 / 0.50 = +8.0000 ($8,000 causal lift per complier)!
| Encouragement Design Quantity | Formula Expression | Sample Output |
|---|---|---|
| Intention-to-Treat Effect ITT_Y | E[Y | Z = 1] - E[Y | Z = 0] | +4.0000 (Diluted by non-compliers) |
| First-Stage Complier Share pi_C | E[D | Z = 1] - E[D | Z = 0] | 0.5000 (50% compliers) |
| Always-Taker & Never-Taker Shares | p_D0 and 1 - p_D1 | 0.1500 and 0.3500 (Sum = 1.0000) |
| Wald IV Complier Effect (LATE) | ITT_Y / pi_C = (C2 - C3) / (B2 - B3) | +8.0000 |
First-Stage Complier Share & Weak-Instrument Guard Formula
Verify monotonicity (p_D1 > p_D0) and guard weak first stages (pi_C < 0.05), where tiny denominator noise explodes IV variance.
=IF(OR(B2 < 0, B2 > 1, B3 < 0, B3 > 1, B2 - B3 < 0.05), "Weak or Defier First Stage", ROUND(B2 - B3, 4))Wald Instrumental-Variables LATE / CACE Formula
Divide ITT_Y by pi_C. Verify the full-compliance special case: set B2 = 1.00 and B3 = 0.00 (100% compliance in both arms) — pi_C becomes 1.0000 and Wald LATE equals ITT_Y = 4.0000 identically!
=LET(pi_c, B2 - B3, itt_y, C2 - C3, IF(OR(B2 < 0, B2 > 1, B3 < 0, B3 > 1, pi_c < 0.05), "Weak or Defier First Stage", ROUND(itt_y / pi_c, 4)))Try the example
Use the original values or change them to test your understanding.