Regression Discontinuity Designs Slides

# Regression Discontinuity Designs Slides - WNE#3 Whats New...

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“What’s New in Econometrics” Lecture 3 Regression Discontinuity Designs Guido Imbens NBER Summer Institute, 2007

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Outline 1. Introduction 2. Basics 3. Graphical Analyses 4. Local Linear Regression 5. Choosing the Bandwidth 6. Variance Estimation 7. Specification Checks 1
1. Introduction A Regression Discontinuity (RD) Design is a powerful and widely applicable identification strategy. Often access to, or incentives for participation in, a service or program is assigned based on transparent rules with crite- ria based on clear cutoff values, rather than on discretion of administrators. Comparisons of individuals that are similar but on different sides of the cutoff point can be credible estimates of causal effects for a specific subpopulation. Good for internal validity, not much external validity. 2

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2. Basics Two potential outcomes Y i (0) and Y i (1), causal effect Y i (1) Y i (0), binary treatment indicator W i , covariate X i , and the observed outcome equal to: Y i = Y i ( W i ) = Y i (0) if W i = 0 , Y i (1) if W i = 1 . (1) At X i = c incentives to participate change. Two cases, Sharp Regression Discontinuity : W i = 1 { X i c } . (SRD) and Fuzzy Regression Discontinuity Design : lim x c Pr( W i = 1 | X i = x ) = lim x c Pr( W i = 1 | X i = x ) , (FRD) 3
Sharp Regression Discontinuity Example (Lee, 2007) What is effect of incumbency on election outcomes? (More specifically, what is the probability of a Democrat winning the next election given that the last election was won by a Demo- crat?) Compare election outcomes in cases where previous election was very close. 4

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SRD Key assumption: E [ Y (0) | X = x ] and E [ Y (1) | X = x ] are continuous in x . Under this assumption, τ SRD = lim x c E [ Y i | X i = x ] lim x c E [ Y i | X i = x ] . (FRD estimand) The estimand is the difference of two regression functions at a point. Extrapolation is unavoidable. 5
Fuzzy Regression Discontinuity Examples (VanderKlaauw, 2002) What is effect of financial aid offer on acceptance of college admission. College admissions oﬃce puts applicants in a few categories based on numerical score. Financial aid offer is highly correlated with category. Compare individuals close to cutoff score. 6

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FRD What do we look at in the FRD case: ratio of discontinuities in regression function of outcome and treatment: τ FRD = lim x c E [ Y i | X i = x ] lim x c E [ Y i | X i = x ] lim x c E [ W i | X i = x ] lim x c E [ W i | X i = x ] . (FRD Estimand) 7
Interpretation of FRD (Hahn, Todd, VanderKlaauw) Let W i ( x ) be potential treatment status given cutoff point x , for x in some small neigborhood around c (which requires that the cutoff point is at least in principle manipulable) W i ( x ) is non-increasing in x at x = c . A complier is a unit such that lim x X i W i ( x ) = 0 , and lim x X i W i ( x ) = 1 . Then lim x c E [ Y i | X i = x ] lim x c E [ Y i | X i = x ] lim x c E [ W i | X i = x ] lim x c E [ W i | X i = x ] = E [ Y i (1) Y i (0) | unit i is a complier and X i = c ] .

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