On the Two Frameworks for Regression Discontinuity Design:Which One Should We Use?
Regression discontinuity(RD)design is one of the most popular quasi-experimental approaches to causal inference in economics.There are two major frameworks for RD design,which differ in their assumptions,choice of bandwidth,and inference.The continuity-based framework assumes that the conditional expectations of potential out-comes are continuous at the cutoff.The local average treatment effect at the cutoff is identified and nonparametric local polynomial regression can be used for estimation and inference.The local randomization framework assumes that the run-ning variable is as good as randomly assigned in a small window around the cutoff.The average treatment effect in this window is identified,and estimation and inference can be implemented using methods from analysis of experiments.An important question is how to choose between the two RD frameworks.The continuity framework appeared ear-lier in literature and is widely used in empirical studies.The local randomization framework is a latecomer,currently used mostly as an alternative approach or a robustness check.However,this paper views the local randomization framework as a promising framework,which is expected to play an increasingly important role in the future for the following reasons.Firstly,although the continuity assumption is weaker than the local randomization assumption,the continuity frame-work implicitly assumes that the running variable is exogenous within the chosen optimal bandwidth while implementing local polynomial regression.However,the optimal bandwidth is usually chosen to minimize mean squared error(MSE),with no regard of guaranteeing the exogeneity of the running variable.Empirical researchers often tacitly view RD as local randomized experiments and take this exogeneity condition for granted.The local randomization framework chooses the bandwidth by a series of covariate balance tests,often resulting in a much smaller window such that the exogeneity can be more easily satisfied.However,a consequence of choosing a small window is that the effective observations within this window may be small,and one may have to rely on an exact finite-sample Fisherian approach for inference.Secondly,the continuity framework assumes that the running variable is continuous with a positive density at the cut-off,which is not applicable with a discrete running variable,unless additional assumptions are introduced.The local ran-domization framework is applicable with a continuous or discrete running variable.Thirdly,the external validity of the continuity framework is weak since it can only identify the local average treat-ment effect at the cutoff.The local randomization framework enjoys a stronger external validity since it can identify the av-erage treatment effect in a window around the cutoff,although the window is usually narrow.In summary,the continuity framework and the local randomization framework have respective advantages and disad-vantages,but the latter is expected to become more important in practice.Furthermore,this paper compares their relative performance via Monte Carlo simulations.The results of simulations show that if the running variable is exogenous in the wider MSE-optimal bandwidth,then the estimators of the continuity framework are more efficient.However,if the run-ning variable is only exogenous in the narrower window chosen under the local randomization framework,the estimators of the continuity framework are inconsistent,while the estimator of the local randomization framework remains consis-tent.Additionally,this paper demonstrates the detailed implementation of the two RD frameworks in Stata using the clas-sic example of U.S.Senate elections(Cattaneo et al.,2015).A major disadvantage of the local randomization framework is that it often chooses a very narrow window.This typi-cally results in a significant loss of observations such that the statistical inference has to rely on the exact finite-sample Fisherian approach.Since the effective sample size under the local randomization framework may be small and subject to the influence of outliers,this paper proposes leave-one-out estimation as a robustness check.
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