Regression Discontinuity Suppose assignment into treated state depends on a continuously distributed observable variable

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1 Private school admission Regression Discontinuity Suppose assignment into treated state depends on a continuously distributed observable variable Observe higher test scores for private school students than public school students. Should everyone go to private school? Not Private School qualified Private School qualified T min Admissions test Regression Discontinuity Observe higher test scores for private school students than public school students. Should everyone go to private school? Private school admission What if we compare outcomes around the threshold? Not Private School qualified Private School qualified T min Admissions test 1

2 Possible model of regression discontinuity H: True Human Capital T: Test Score Θ: Random Error Expect Cov(H,T)>0 T = H + Θ If T > T min then qualify for private school Compare outcomes of similar H individuals around the threshold T min T<T min T>T min Range of common H H 2

3 Fuzzy Regression Discontinuity Y: test score D: private school τ: treatment effect X: admissions test c: T min Exact Regression Discontinuity G Economic Literature 48(2) (June 2010):

4 Continuity assumption Economic Literature 48(2) (June 2010): Features Regression Discontinuity valid if individuals cannot precisely control the assignment variable in the neighborhood of the threshold Imperfect manipulation of the assignment variable means that in the neighborhood, variation in the treatment is randomized as if it were a randomized experiment Control of outcomes at the threshold invalidate regression discontinuity 4

5 If discontinuous, then both X and treatment vary at threshold Characterization of randomized assignment as a regression discontinuity design 5

6 Role of imprecise control Y=outcome; D = treatment; X=assignment variable W = predetermined observable covariates that can be endogenous; If V is stochastic, then assignment is imprecise Two people with the same W and U will have a different observed Y only on random assignment into X due to V Conditional on W and U, X is continuous at threshold, so V is continuous at the threshold Role of imprecise control Y=outcome; D = treatment; X=assignment variable W = predetermined observable covariates that can be endogenous; Bayes theorem RHS continuous in x means LHS is continuous in x 6

7 Complete control means degenerate x Precise control means truncation at threshold Imprecise means continuity at threshold For fuzzy regression discontinuity, assignment is intent to treat effect Corrects for difference in expectation 1 If denominator = 1, assignment perfectly predicts treatment status 7

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9 Classic Regression setup Left-hand-side of cutoff Right-hand-side of cutoff Fuzzy discontinuity Intent to treat 9

10 Maimonides Rule: Class Size should never be larger than 40 School Classes Avg Class Enrollment int((e 1/40)+1) Size Average Class Size by School Enrollment School Enrollment Class Size Maimonides Rule in Action: Class Sizes in Israel Joshua D. Angrist and Jörn-Steffen Pischke Mostly harmless econometrics : an empiricist's companion. Princeton University Press

11 Class Size and Test Scores Joshua D. Angrist and Jörn-Steffen Pischke Mostly harmless econometrics : an empiricist's companion. Princeton University Press