Rejoinder

estimates. However, such an approach has a number of shortcomings. Most importantly, this simple comparison does not provide a direct test of the validity of CIA. Perhaps CIA does not hold but various biases cancel off and we are just lucky to find that the matching and RD estimates are similar around the cutoff point. In that setting, we cannot generalize beyond the cutoff since we do not know whether CIA holds, and we have not gained much relative to RD in terms of external validity. The main contribution of the Angrist and Rokkanen article is to provide a simple and clever test of CIA in the context of an RD design. The idea is that in an RD design, if outcomes on each side of the cutoff are a flat function of the running variable, we can consistently estimate the treatment effect without controlling for the running variable. A useful analogy here is that if the running variable was purely random, the RD design would be just like a randomized experiment where the running variable plays the role of the randomizer used to select subjects into the treatment and control groups. One can then do a simple means comparison without having to control for the randomizer. Angrist and Rokkanen implement this idea in practice using a simple omitted variable argument. If the running variable is uncorrelated with outcomes conditional on covariates and treatment status (the flat function discussed above), we can estimate the treatment effect without controlling for the running variable (the omitted variable). This suggests a test of CIA based on this simple regression. If the running variable is not significant, we can go ahead and rely on CIA to estimate the treatment effect away from the cutoff. Angrist and Rokkanen show that the assumption holds in practice in an empirical example on selective schools in Boston where a rich set of covariates is available. More generally, the proposed approach will most likely “work” when rich covariates predict well the running variable. This is the case in the example studied in the article where admission is based on a test score, and rich information about a variety of baseline test scores is available. A possible explanation for why the approach works here is that baseline scores do an excellent job predicting underlying scholastic aptitude. Conditional on this information, all that is left in the admission test score is noise on the day of the test (e.g., student has a cold, etc.). While this is an ideal setting for CIA to hold, it also means that the admission process is badly designed, since all the admission test does is to “noise up” good information that would be available in the baseline covariates. Noising up things is obviously good from an evaluation perspective, but it is not clear that there are that many settings where we can think of the running variable as the sum of a predicted component based on baseline covariates plus a pure noise term. For instance, think of a rule for providing individuals with job training based on current earnings plus a score based on an interview. Lagged values of earnings could do a good job predicting the running variable (current earnings plus test scores), but the innovation in earnings relative to past values would for sure not be a purely random component. It could, for instance, be related to a recent job loss. Because of a mix of persistence and mean reversion (Ashenfelter’s dip) we would then expect future earnings (the outcome variable) to be correlated with the running variable even after conditioning for past earnings. The test of CIA proposed by Angrist and Rokkanen would of course pick this up and tell us we cannot do matching in this case. The main point here is that while lagged values and other baseline covariates may often do an excellent job predicting the running variables, for CIA to hold we need the unpredicted part of the running variable to be “pure noise,” something that may not happen that often in practice. Experience will tell whether or not we can “get away from the cutoff” in the large number of applied problems in which RD designs have proven useful. This point aside, Angrist and Pekkanen’s proposal for “going away” from the cutoff is simple, intuitive, and promising, and will likely become a standard tool used by applied researchers working with RD designs.