Regression Discontinuity Designs (RDD) Flashcards
idea of RDD
exploit particularities in laws and institutions, usually thresholds
- treatment occurs just above (below) threshold
- control group are observations just below (above) threshold
identification requirement: RDD
discontinuity (‘jump’) in the mapping Z → D at some threshold z0 (necessary requirement)
identifying assumption:
continuity of Z |D → Y (in potential outcomes) and of other variables (Z → X ) around threshold z0
⇒ ‘identifying requirement’ and ‘identifying assumption’ are necessary and sufficient, respectively, for identification of causality
Two types of RDD
- ‘sharp’ or ‘clean’ RDD
- ‘fuzzy’ RDD
clean RDD estimation
- D is deterministically related by Z
- i.e. all subjects above (below) threshold z0 are treated …
- … while all subjects below (above) threshold z0 are
Implications:
- comparison of observations just above and just below threshold yields estimate of
causal effect - several complications in model specification and estimation (see below)
fuzzy RDD estimation and implication
- D is probabilistically related by Z
- i.e. treatment probability is higher for subjects above (below) threshold z0 than for those just below (above)
Omplications
- being above (zi ≥ z0) or below (zi < z0)threshold is used as instrument for actual treatment status di
- estimation analogue to intention-to-treat design with Wald or 2SLS-IV estimation
- ‘double local’ interpretation of estimated effect
a. local in the LATE sense (applies only to compliers)
b. local because only estimated at threshold
Formalization of RDD
1) zi≥z0 di= 0 if zi<z0></z0> E[Y0|Z] = α+βz (1)
E[Y1|Z] = E[Y0|Z] + δ (2)
Y1 = Y0+δ (3)
where δ is the causal effect of interest, assumed to be locally constant.
Note that model 1 describes observed outcomes for all z < z0 and potential outcomes for all z ≥ z0. In turn, 3 describes observed outcomes for all z ≥ z0 and potential outcomes for all z < z0.
Because for every i either Y 1 or Y 0 is observed, one can write the regression Yi =α+βzi +δdi +εi,
where di is an ‘above threshold’ dummy variable. Inclusion of other controls (+X′γ)
possible but ideally matters little (continuity assumption).
Main problems with RDD estimation
P1 observations from treatment group influence prediction of control group outcomes, and vice versa
P2.a observations far from z0 change yˆ close to z0 which can lead to bias
P2.b complex functional forms necessary to adequately model the data, which often
creates bias near threshold
P3 there may be very few observations close to the threshold
SOlution of Estimation Problems for RDD
P1 estimate separate regressions on either side of the threshold
P2 narrow z-range of observations considered (so called ‘bandwidth’) to reduce bias P3 avoid too small bandwidths
⇒ there is an unavoidable trade-off between bias and precision
Testing RDD Assumptions
- in-depth knowledge of laws and institutions necessary
- check whether there is an actual jump in d; is it sharp or fuzzy?- check whether d and y do not display jumps at other points of z where there is no threshold. - estimate ‘effects’ using preferred RD specification at many placebo thresholds. - rule of thumb: should only be able to reject the null of ‘no jump’ in 5% of cases - continuity harder to test
- assumption cannot be verified (because potential outcomes by definition unobserved)
- but can be falsified because assumption is usually likely violated if
(a) y or d are very ‘jumpy’ along z,
(b) there is sorting around z0 (e.g. due to manipulated population numbers),
(c) or if there are other changes at the same threshold
→ check whether one of them applies
Checks to falsify continuity assumption
(a) check if there are discrete jumps in y or d at values of z where there are no thresholds (placebo tests)
(b) check for smooth distribution of other variables x around z0 (works for observables)
(b) check for smooth density around threshold, e.g. by visual inspection and McCrary (2007) test (also speaks to unobservables)
(c) in-depth knowledge of legal and institutional environment necessary
Advantages of RDD
Advantages
- often only way for causal identification of certain effects
- particularly high internal validity if continuity holds
- assumptions fairly well and transparently testable (e.g. compare to IVs)
- graphical display of results often intuitive
Disadvantages of RDD
- similar to any natural experiment
- opportunities for RDD cannot be created, just discovered
- method influences what is been studied
- local identification only → external validity
(how far away from threshold is δ still constant?) - other pitfalls to be aware of when using population threshold RDDs → seminar next week
Basic idea of non-paramtric identification?
- data divided into many slices or ‘bins’
- many local means estimated
- kernel-weighting within slices possible
