Applied Microeconometrics Chapter 8 Regression Discontinuity (RD)

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1 1 / 26 Applied Microeconometrics Chapter 8 Regression Discontinuity (RD) Romuald Méango and Michele Battisti LMU, SoSe 2016

2 Overview What is it about? What are its assumptions? What are the main applications? (MHE Chapter 6, Imbens and Wooldridge Lecture 3) Sharp RD Designs A classical example of Sharp RDD: Lee (Journal of Econometrics 2008) Fuzzy RDD Designs (i.e. IV) Some Tips for RD: charts and tests 2 / 26

3 Regression Discontinuity 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 criteria 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: a local RCT. Good for internal validity, not much external validity. 3 / 26

4 Sharp RDD Regression discontinuity research designs exploit the fact that some rules can provide good quasi-experiments when you compare people (or cities, firms, countries,...) who are just affected by the rule with people who are just not affected by the rule Sharp RD: treatment is a deterministic function of a covariate X Fuzzy RD: exploits discontinuities in the probability of treatment conditional on a covariate X (the discontinuity is then used as an IV). Examples? RD captures the causal effect by distinguishing the nonlinear and discontinuous function, 1(Xi X 0 ) from the smooth function f (X i ). 4 / 26

5 Sharp Regression Discontinuity 5 / 26

6 Sharp RD In Sharp RD designs we exploit that treatment status is a deterministic and discontinuous function of a covariate x i. D i = 1 if x i x 0 and D i = 0 if x i < x 0, where x 0 is a known threshold or cutoff. Once we know x i we know D i. There is no value of x i at which we observe both treatment and control observations. The method relies on extrapolation across covariate values. For this reason we cannot be agnostic about regression functional form in RD 6 / 26

7 Just a little bit more math Linear Case Suppose that in addition to the assignment mechanism above, potential outcomes can be described by a linear, constant effects model: E[Y 0i X i ] = α + βx i (1) Y 1i = Y 0i + ρ (2) We can write this as a regression equation Y i = α + βx i + ρd i + η i (3) Here, D i is not only correlated with X i but it is a deterministic function of X i. 7 / 26

8 Sharp RD Identifying assumption Key identifying assumption: E[Y 0i X i ] and E[Y 1i X i ] are continuos in X i around X 0. In other words, all other unobserved determinants of Y are continuously related to the running variable X. This allows us to use average outcomes of units just below the cutoff as a valid counterfactual for units right above the cutoff. This assumption cannot be directly tested. But there are some tests which give suggestive evidence whether the assumption is satisfied 8 / 26

9 Sharp RD: Nonlinear case Sometimes the trend relation E[Y 0i x i ] is nonlinear Suppose the nonlinear relationship is E[Y 0i x i ] = f (X i ) for some reasonably smooth function f (X i ). In this case we can construct RD estimates by fitting: Y i = f (x i ) + ρd i + η i (4) There are 2 ways of approximating f (x i ): 1. Use a nonparametric kernel method 2. Use a pth order polynomial: i.e. estimate: Y i = α + β 1 x i + β 2 x 2 i β p x p i + ρd i + η i (5) 9 / 26

10 Different Polynomials on the two sides of the Discontinuity We can generalize the function f (x i ) by allowing the x i terms to differ on both sides of the threshold by including them both individually and interacting them with D i. Lee and Lemieux (2010): allowing different functions on both sides of the discontinuity should be the main results in an RD paper (as otherwise we use values from both sides of the cutoff the estimate the function on each side). 10 / 26

11 Different Polynomials on the two sides of the Discontinuity 11 / 26 In that case: 2 p E[Y 0i X i ] = α + β 01 Xi + β 02 X i β 0p X i (6) 2 p E[Y 1i X i ] = α + ρ + β 11 Xi + β 12 X i β 1p X i (7) where X i = X i X 0. Centering at X 0 ensures that the treatment effect at X i = X 0 is the coefficient on D i in a regression model with interaction terms (because you do not have to add values of the D i interacted with X to get the treatment effect at X 0 ).

12 12 / 26 Different Polynomials on the two sides of the Discontinuity To derive a regression model that can be used to estimate the causal effect we use the fact that D i is a deterministic function of X i : E[Y i X i ] = E[Y 0i X i ] + (E[Y 1i X i ] E[Y 0i X i ])D i (8) The regression model which you estimate is then: Y i = α + β 01 x i + β 02 x 2 i +ρd i + β 1D i x i + β 2D i x 2 i β 0p x p i (9) βpd i x p i + η i (10) where β 1 = β 11 β 01, β p = β 1p β 0p The treatment effect at X 0 is ρ. The treatment effect at X i X 0 = c > 0 is ρ + β 1 c + β 2 c β pc p

13 Example of Sharp RD: Lee (2008) Lee (2008) uses a sharp RD design to estimate the probability that the incumbent wins an election. A large political science literature suggests that incumbents may use privileges and resources of office to gain an advantage over potential challengers. An OLS regression of incumbency status on election success is likely to be biased because of unobserved differences. Incumbents have already won an election so they may just be better. In words, Lee s identifying assumption is that... What are possible examples violating it? 13 / 26

14 Example of Sharp RD: Lee (2008) Lee analyzes the incumbency effect using Democratic incumbents for US congressional elections. He analyzes the probability of winning the election in year t+1 by comparing candidates who just won compared to candidates who just lost the election in year t. 14 / 26

15 Lee (2008) 15 / 26

16 Validity of RD Estimates The validity of RD estimates depends crucially on the assumption that the polynomials provide an adequate representation of E[Y 0i X i ]. If not, what looks like a jump may simply be a non-linearity in f (X i ) that the polynomials have not accounted for. 16 / 26

17 Fuzzy RD (IV) Fuzzy RD exploits discontinuities in the probability of treatment conditional on a covariate. The discontinuity becomes an instrumental variable for treatment status. Example: effect of class size on student test score (Angrist and Lavy 1999). D i is no longer deterministically related to crossing a threshold but there is a jump in the probability of treatment at X 0 : P[D i = 1 X i ] = g 1 (X i ) if x i x o P[D i = 1 X i ] = g 0 (X i ) if x i < x o where g 1 (X i ) and g 0 (X i ) can be anything as long as they differ at x / 26

18 Fuzzy RD (IV) The relationship between the probability of treatment and Xi can be written as: P[D i = 1 X i ] = g 0 (X i ) + [g 1 (X i ) g 0 (X i )]T i (11) where T i = 1(X i X 0 ) One can use both T i and interaction terms as instruments for D 1 As in the sharp RD case one can allow the smooth function to be different on both sides of the discontinuity. First Stage: estimate D i as a function of T 1 and interaction terms. Second Stage: same as equation for sharp RD above, with fitted values 18 / 26

19 Fuzzy RD: What are we estimating Same assumptions as in the standard IV framework. As with other binary IVs one then estimates LATE: the average treatment effect of the compliers. In RD the compliers are those whose treatment status changes as we move the value of xi from just the left of x 0 to just to the right of x / 26

20 Graphical Analysis (Lee and Lemieux 2010) Outcomes by forcing variable (X i ) 20 / 26

21 Graphical Analysis (Lee and Lemieux 2010) Covariates by forcing variable (X i ) 21 / 26

22 Graphical Analysis (Lee and Lemieux 2010) Density by forcing variable (X i ) 22 / 26

23 Some useful Tests in RD Testing the continuity of the density of X (not required but worrisome if failed) Use covariate as outcomes (placebo) Jumps at non-discountinuity points (placebo) 23 / 26

24 24 / 26 On Friday One presentation and discussion in class (Theresa presents, Osman will act as the discussant): Isen, Adam Do Local Government Fiscal Spillovers Exist? Evidence from Counties, Municipalities, and School Districts. Journal of Public Economics

25 25 / 26 Feedback Link for anonymous feedback:

26 26 / 26 Bibliography See our Reading List.