Estimation and inference in regression discontinuity designs with asymmetric kernels

Abstract

We study the behaviour of the Wald estimator of causal effects in regression discontinuity design when local linear regression (LLR) methods are combined with an asymmetric gamma kernel. We show that the resulting statistic is no more complex to implement than existing methods, remains consistent at the usual non-parametric rate, and maintains an asymptotic normal distribution but, crucially, has bias and variance that do not depend on kernel-related constants. As a result, the new estimator is more efficient and yields more reliable inference. A limited Monte Carlo experiment is used to illustrate the efficiency gains. As a by product of the main discussion, we extend previous published work by establishing the asymptotic normality of the LLR estimator with a gamma kernel. Finally, the new method is used in a substantive application.

Keywords:

• regression discontinuity
• asymmetric kernels
• local linear regression

JEL Classification::

• C13
• C14
• C21

Notes

1. In contrast, the bias of the popular Nadaraya–Watson estimator disappears at a slower rate at the boundary and, therefore, larger sample sizes are required in this region in order to obtain a given level of accuracy [21].

2. The focus of this paper is on the gamma distribution. The working paper version of this article explores estimation of τ with the beta kernel. It turns out that the asymptotic distribution of this estimator is identical to the gamma-based estimator. However, the use of beta kernels requires the selection of two bandwidths: one to select which data are included in the estimation and another to allocate weights to these data. This makes this choice of kernel unattractive.

3. The estimators of m₋, p₊ and p₋ are defined similarly.

4. Note that, although the domain of the gamma kernel is [0, ∞), in practice the regions generated by the cut-off point in an RDD will rarely equal this interval. However, since the cut-off values are known to the researcher, it is always possible to map the different regions onto the interval [0, ∞).

5. Note, however, that the results in HTvK need less restrictive conditions, as their results only require an application of the dominated convergence theorem.

6. Further details about roll-call voting records and ADA's voting scores can be found in [12] or at ADA's website, http://www.adaction.org/.

7. An excellent and more detailed discussion is provided in [17].

8. The data set for this study is available at Enrico Moretti's website, http://emlab.berkeley.edu/moretti/.