Regression Discontinuity STP Advanced Econometrics Lecture Douglas G. Steigerwald UC Santa Barbara March 2006 D. Steigerwald (Institute) Regression Discontinuity 03/06 1 / 11
Intuition Reference: Mostly Harmless Econometrics Chapter 6 Rule leads to identi cation Are National Merit Scholars more likely to go to graduate school? Rule: Award Scholarship if (PSAT) score c Compare: students with score just below c to students with score just above c D. Steigerwald (Institute) Regression Discontinuity 03/06 2 / 11
Sharp Discontinuity De nition treatment for observation i 1 if xi x = 0 0 if x i < x 0 x 0 known threshold deterministic function of x i Example: discontinuous function of x i = 1 if i received scholarship x i score for i x 0 threshold (rule) for awarding scholarship D. Steigerwald (Institute) Regression Discontinuity 03/06 3 / 11
Constant Mean Model Outcome - attending graduate school Treatment - receiving scholarship Y 0i Y 1i outcome without treatment outcome with treatment To measure impact: compare means for two groups E [Y 0i ] = α E [Y 1i ] = α + ρ H 0 : ρ = 0 Ȳ 1 Ȳ 0 H 1 : ρ > 0 Issue: no individual is in both groups if higher scores lead to higher rates of graduate school attendance, then could mistakenly conclude that scholarships matter (draw linear function of scores and take averages on each side of threshold) D. Steigerwald (Institute) Regression Discontinuity 03/06 4 / 11
Linear Model Higher scores likely increase graduate school attendance, regardless of scholarship To measure impact: compare conditional means for two groups E [Y 0i jx i ] = α + βx i E [Y 1i jx i ] = α + ρ + βx i estimate signi cance of ρ in Y i = α + βx i + ρ + u i Again: misspeci cation of conditional mean could be mistaken for discontinuity (draw nonlinear function of scores and t linear function with jump at x 0 ) D. Steigerwald (Institute) Regression Discontinuity 03/06 5 / 11
Local Nonparametrics Two key issues in parametric models of regression discontinuity 1 requires correct speci cation of conditional mean to identify discontinuity 2 assumes treatment e ect is constant for all treated individuals Both issues can be addressed with local nonparametric estimators local use only data within δ of x 0 nonparametric kernel density estimator or local linear estimator D. Steigerwald (Institute) Regression Discontinuity 03/06 6 / 11
Robustness Check: Local Estimators Key idea: as data window shrinks toward x 0, fewer polynomial terms are needed Example 1 Use all data: estimate signi cance of ρ in Y i = α + β 1 x i + β 2 x 2 i + β 3 x 3 i + ρ + u i 2 Drop 25% of data furthest from x 0 : estimate Y i = α + β 1 x i + β 2 x 2 i + ρ + u i 3 Drop 50% of data furthest from x 0 : estimate Y i = α + β 1 x i + ρ + u i estimates of ρ should be stable across speci cations D. Steigerwald (Institute) Regression Discontinuity 03/06 7 / 11
Robustness Check: Other Factors Example: Does incumbency increase re-election? Alternative argument: likely to be re-elected sample selection, good politicians are more Outcome - Probability of re-election (victory in period t + 1) x i = v D v R v D vote share for Democrats Treatment - Winning election in period t: x i > 0 Graph 1 - Outcome shows a sharp jump at 0 Issue: could be these districts simply favor Democrats Graph 2 - Number of past Dem victories, increases in x i but shows no jump at 0 Second issue: look for bunching near 0, indicating endogenous reaction to threshold D. Steigerwald (Institute) Regression Discontinuity 03/06 8 / 11
Fuzzy Discontinuity De nition treatment for observation i g1 (x P ( = 1jx i ) = i ) if x i x 0 g 0 (x i ) if x i < x 0 x 0 known threshold not a deterministic function of x i Example: not a discontinuous function of x i = 1 if i received scholarship x i score for i x 0 threshold for scores, but other factors (family income) are also used to award scholarship D. Steigerwald (Institute) Regression Discontinuity 03/06 9 / 11
Inconsistency If is not a deterministic function of x i, then the standard RD estimator is inconsistent Standard RD: estimate signi cance of ρ in Y i = α + βx i + ρ + u i family income is in the error and is correlated with proposed instrument T i = 1 (x i x 0 ) correlated with uncorrelated with u i if scores and family income are uncorrelated D. Steigerwald (Institute) Regression Discontinuity 03/06 10 / 11
2SLS First Stage = γ 0 + γ 1 x i + γ 2 T i + v i yields ˆ Second Stage Y i = α + βx i + ρ ˆ + u i D. Steigerwald (Institute) Regression Discontinuity 03/06 11 / 11