In the regression discontinuity design (RDD), it is common practice to assess the credibility of the design by testing the continuity of the density of the running variable at the cut-off, e.g., McCrary (2008). In this paper we propose a new test for continuity of a density at a point based on the so-called g-order statistics, and study its properties under a novel asymptotic framework. The asymptotic frameworkis intended to approximate a small sample phenomenon: even though the total number n of observations may be large , the number of effective observations local to the cut-off is often small. Thus, while traditional asymptotic analysis in RDD requires a growing number of observations local to the cut-off as n grows, our framework allows for the number q of observations local to the cut-off to be fixed as n grows. The new test is easy to implement, asymptotically valid under weaker conditions than those used by competing methods,  exhibits finite sample validity under stronger conditions than those needed for its asymptotic validity , and has favorable power properties against certain alternatives . In a simulation study, we find that the new test controls size remarkably well across designs . We finally apply our test to the design in Lee (2008), a well-known application of the RDD to study incumbency advantage.
In the regression discontinuity design (RDD), it is common practice to assess the credibility of the design by testing the continuity of the density of the running variable at the cut-off, e.g., McCrary (2008). In this paper we propose an approximate sign test for continuity of a density at a point based on the so-called g-order statistics, and study its properties under two complementary asymptotic frameworks. In the first asymptotic framework , the number q of observations local to the cut-off is fixed as the sample size n diverges to infinity, while in the second framework q diverges to infinity slowly as n diverges to infinity. Under both of these frameworks, we show that the test we propose is asymptotically valid in the sense that it has limiting rejection probability under the null hypothesis not exceeding the nominal level. More importantly, the test is easy to implement, asymptotically valid under weaker conditions than those used by competing methods, and exhibits finite sample validity under stronger conditions than those needed for its asymptotic validity . In a simulation study, we find that the approximate sign test provides good control of the rejection probability under the null hypothesis while remaining competitive under the alternative hypothesis . We finally apply our test to the design in Lee (2008), a well-known application of the RDD to study incumbency advantage.
R
a new
an approximate sign
R
a novel asymptotic framework. The asymptotic frameworkis intended to approximate a small sample phenomenon: even though the total number n of observations may be large
two complementary asymptotic frameworks. In the first asymptotic framework
R
of effective
q of
R
often small. Thus, while traditional asymptotic analysis in RDD requires a growing number of observations local to the cut-off as n grows, our framework allows for the number q of observations local to
fixed as the sample size n diverges to infinity, while in
R
cut-off to be fixed as n grows. The new test is
second framework q diverges to infinity slowly as n diverges to infinity. Under both of these frameworks, we show that the test we propose is asymptotically valid in the sense that it has limiting rejection probability under the null hypothesis not exceeding the nominal level. More importantly, the test is
A
and
D
, and has favorable power properties against certain alternatives
R
new test controls size remarkably well across designs
approximate sign test provides good control of the rejection probability under the null hypothesis while remaining competitive under the alternative hypothesis