We consider inference in regression discontinuity designs when the running variable only takes a moderate number of distinct values. In particular, we study the common practice of using confidence intervals (CIs) based on standard errors that are clustered by the running variable as a means to make inference robust to model misspecification (Lee and Card, 2008). We derive theoretical results and present simulation and empirical evidence showing that these CIs do not guard against model misspecification, and that they have poor coverage properties. We therefore recommend against using these CIs in practice. We instead propose two alternative CIs with guaranteed coverage properties under easily interpretable restrictions on the conditional expectation function. * We thank Joshua Angrist, Tim Armstrong, Guido Imbens, Philip Oreopoulos, and Miguel Urquiola and seminar participants at Columbia University, Villanova University, and the 2017 SOLE Annual Meeting for helpful comments and discussions.

a b s t r a c tThis paper proposes a fully nonparametric procedure to evaluate the effect of a counterfactual change in the distribution of some covariates on the unconditional distribution of an outcome variable of interest. In contrast to other methods, we do not restrict attention to the effect on the mean. In particular, our method can be used to conduct inference on the change of the distribution function as a whole, its moments and quantiles, inequality measures such as the Lorenz curve or Gini coefficient, and to test for stochastic dominance. The practical applicability of our procedure is illustrated via a simulation study and an empirical example.

We analyze the statistical properties of nonparametric regression estimators using covariates which are not directly observable, but have be estimated from data in a preliminary step. These so-called generated covariates appear in numerous applications, including two-stage nonparametric regression, estimation of simultaneous equation models or censored regression models. Yet so far there seems to be no general theory for their impact on the final estimator's statistical properties. Our paper provides such results. We derive a stochastic expansion that characterizes the influence of the generation step on the final estimator, and use it to derive rates of consistency and asymptotic distributions accounting for the presence of generated covariates.Comment: Published in at http://dx.doi.org/10.1214/12-AOS995 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

We propose a specification test for a wide range of parametric models for conditional distribution function of an outcome variable given a vector of covariates.The test is based on the Cramer-von Mises distance between an unrestricted estimate of the joint distribution function of the data, and an restricted estimate that imposes the structure implied by the model. The procedure is straightforward to implement, is consistent against fixed alternatives, has non-trivial power against local deviations from the null hypothesis of order n −1/2 , and does not require the choice of smoothing parameters. We also provide an empirical application using data on wages in the US.JEL Classification: C12, C14, C31, C52, J31 . Financial support by Deutsche Forschungsgemeinschaft (SFB 823) is gratefully acknowledged. We would like to thank Blaise Melly for helpful comments. The usual disclaimer applies.

The key assumption in regression discontinuity analysis is that the distribution of potential outcomes varies smoothly with the running variable around the cutoff. In many empirical contexts, however, this assumption is not credible; and the running variable is said to be manipulated in this case. In this paper, we show that while causal effects are not point identified under manipulation, they remain partially identified under a general model that covers a wide range of empirical patterns. We derive sharp bounds on causal parameters for both sharp and fuzzy designs under our general model, and show how additional structure can be used to further narrow the bounds. We use our methods to study the disincentive effect of unemployment insurance on (formal) reemployment in Brazil, and show that our bounds remain informative, despite the fact that manipulation has a sizable effect on our estimates of causal parameters.