Rocío Titiunik

Robust Nonparametric Confidence Intervals for Regression‐Discontinuity Designs

In the regression‐discontinuity (RD) design, units are assigned to treatment based on whether their value of an observed covariate exceeds a known cutoff. In this design, local polynomial estimators are now routinely employed to construct confidence intervals for treatment effects. The performance of these confidence intervals in applications, however, may be seriously hampered by their sensitivity to the specific bandwidth employed. Available bandwidth selectors typically yield a “large” bandwidth, leading to data‐driven confidence intervals that may be biased, with empirical coverage well below their nominal target. We propose new theory‐based, more robust confidence interval estimators for average treatment effects at the cutoff in sharp RD, sharp kink RD, fuzzy RD, and fuzzy kink RD designs. Our proposed confidence intervals are constructed using a bias‐corrected RD estimator together with a novel standard error estimator. For practical implementation, we discuss mean squared error optimal bandwidths, which are by construction not valid for conventional confidence intervals but are valid with our robust approach, and consistent standard error estimators based on our new variance formulas. In a special case of practical interest, our procedure amounts to running a quadratic instead of a linear local regression. More generally, our results give a formal justification to simple inference procedures based on increasing the order of the local polynomial estimator employed. We find in a simulation study that our confidence intervals exhibit close‐to‐correct empirical coverage and good empirical interval length on average, remarkably improving upon the alternatives available in the literature. All results are readily available in R and STATA using our companion software packages described in Calonico, Cattaneo, and Titiunik (2014d, 2014b).

When Natural Experiments Are Neither Natural nor Experiments

Natural experiments help to overcome some of the obstacles researchers face when making causal inferences in the social sciences. However, even when natural interventions are randomly assigned, some of the treatment–control comparisons made available by natural experiments may not be valid. We offer a framework for clarifying the issues involved, which are subtle and often overlooked. We illustrate our framework by examining four different natural experiments used in the literature. In each case, random assignment of the intervention is not sufficient to provide an unbiased estimate of the causal effect. Additional assumptions are required that are problematic. For some examples, we propose alternative research designs that avoid these conceptual difficulties.

Optimal Data-Driven Regression Discontinuity Plots

Exploratory data analysis plays a central role in applied statistics and econometrics. In the popular regression-discontinuity (RD) design, the use of graphical analysis has been strongly advocated because it provides both easy presentation and transparent validation of the design. RD plots are nowadays widely used in applications, despite its formal properties being unknown: these plots are typically presented employing ad hoc choices of tuning parameters, which makes these procedures less automatic and more subjective. In this article, we formally study the most common RD plot based on an evenly spaced binning of the data, and propose several (optimal) data-driven choices for the number of bins depending on the goal of the researcher. These RD plots are constructed either to approximate the underlying unknown regression functions without imposing smoothness in the estimator, or to approximate the underlying variability of the raw data while smoothing out the otherwise uninformative scatterplot of the data. In addition, we introduce an alternative RD plot based on quantile spaced binning, study its formal properties, and propose similar (optimal) data-driven choices for the number of bins. The main proposed data-driven selectors employ spacings estimators, which are simple and easy to implement in applications because they do not require additional choices of tuning parameters. Altogether, our results offer an array of alternative RD plots that are objective and automatic when implemented, providing a reliable benchmark for graphical analysis in RD designs. We illustrate the performance of our automatic RD plots using several empirical examples and a Monte Carlo study. All results are readily available in R and STATA using the software packages described in Calonico, Cattaneo, and Titiunik. Supplementary materials for this article are available online.

Randomization Inference in the Regression Discontinuity Design: An Application to Party Advantages in the U.S. Senate

Abstract In the Regression Discontinuity (RD) design, units are assigned a treatment based on whether their value of an observed covariate is above or below a fixed cutoff. Under the assumption that the distribution of potential confounders changes continuously around the cutoff, the discontinuous jump in the probability of treatment assignment can be used to identify the treatment effect. Although a recent strand of the RD literature advocates interpreting this design as a local randomized experiment, the standard approach to estimation and inference is based solely on continuity assumptions that do not justify this interpretation. In this article, we provide precise conditions in a randomization inference context under which this interpretation is directly justified and develop exact finite-sample inference procedures based on them. Our randomization inference framework is motivated by the observation that only a few observations might be available close enough to the threshold where local randomization is plausible, and hence standard large-sample procedures may be suspect. Our proposed methodology is intended as a complement and a robustness check to standard RD inference approaches. We illustrate our framework with a study of two measures of party-level advantage in U.S. Senate elections, where the number of close races is small and our framework is well suited for the empirical analysis.

Interpreting Regression Discontinuity Designs with Multiple Cutoffs

We consider a regression discontinuity (RD) design where the treatment is received if a score is above a cutoff, but the cutoff may vary for each unit in the sample instead of being equal for all units. This multi-cutoff regression discontinuity design is very common in empirical work, and researchers often normalize the score variable and use the zero cutoff on the normalized score for all observations to estimate a pooled RD treatment effect. We formally derive the form that this pooled parameter takes and discuss its interpretation under different assumptions. We show that this normalizing-and-pooling strategy so commonly employed in practice may not fully exploit all the information available in a multi-cutoff RD setup. We illustrate our methodological results with three empirical examples based on vote shares, population, and test scores.

Regression Discontinuity Designs Using Covariates

We study regression discontinuity designs when covariates are included in the estimation. We examine local polynomial estimators that include discrete or continuous covariates in an additive separable way, but without imposing any parametric restrictions on the underlying population regression functions. We recommend a covariate-adjustment approach that retains consistency under intuitive conditions and characterize the potential for estimation and inference improvements. We also present new covariate-adjusted mean-squared error expansions and robust bias-corrected inference procedures, with heteroskedasticity-consistent and cluster-robust standard errors. We provide an empirical illustration and an extensive simulation study. All methods are implemented in R and Stata software packages.

The Incumbency Curse: Weak Parties, Term Limits, and Unfulfilled Accountability

We study how representation works in a context where accountability to voters is restricted because of term limits and accountability to parties is limited because of party weakness. Analyzing all Brazilian mayoral elections between 1996 and 2012 using a regression discontinuity design, we show that becoming the incumbent party results in large subsequent electoral losses. We theorize that the presence of term limits, combined with political parties to which politicians are only weakly attached, affects the incentives and behavior of individual politicians in such a way that their parties’ suffer systematic losses. A descriptive analysis of an original dataset on the career paths of Brazilian mayors suggests that our assumptions are an accurate description of Brazil’s political context, and we find support for three central empirical implications of our theoretical explanation. Moreover, based on an analysis of additional data from Mexico, Peru, Chile, Costa Rica, and Colombia, we show that the negative effects found in Brazil also exist in other democracies.

On interpreting the regression discontinuity design as a local experiment

We discuss the two most popular frameworks for identification, estimation and inference in regression discontinuity (RD) designs: the continuity-based framework, where the conditional expectations of the potential outcomes are assumed to be continuous functions of the score at the cutoff, and the local randomization framework, where the treatment assignment is assumed to be as good as randomized in a neighborhood around the cutoff. Using various examples, we show that (i) assuming random assignment of the RD running variable in a neighborhood of the cutoff implies neither that the potential outcomes and the treatment are statistically independent, nor that the potential outcomes are unrelated to the running variable in this neighborhood; and (ii) assuming local independence between the potential outcomes and the treatment does not imply the exclusion restriction that the score affects the outcomes only through the treatment indicator. Our discussion highlights key distinctions between “locally randomized” RD designs and real experiments, including that statistical independence and random assignment are conceptually different in RD contexts, and that the RD treatment assignment rule places no restrictions on how the score and potential outcomes are related. Our findings imply that the methods for RD estimation, inference, and falsification used in practice will necessarily be different (both in formal properties and in interpretation) according to which of the two frameworks is invoked.

An Overview of Geographically Discontinuous Treatment Assignments with an Application to Children’s Health Insurance ☆

Abstract We study research designs where a binary treatment changes discontinuously at the border between administrative units such as states, counties, or municipalities, creating a treated and a control area. This type of geographically discontinuous treatment assignment can be analyzed in a standard regression discontinuity (RD) framework if the exact geographic location of each unit in the dataset is known. Such data, however, is often unavailable due to privacy considerations or measurement limitations. In the absence of geo-referenced individual-level data, two scenarios can arise depending on what kind of geographic information is available. If researchers have information about each observation’s location within aggregate but small geographic units, a modified RD framework can be applied, where the running variable is treated as discrete instead of continuous. If researchers lack this type of information and instead only have access to the location of units within coarse aggregate geographic units that are too large to be considered in an RD framework, the available coarse geographic information can be used to create a band or buffer around the border, only including in the analysis observations that fall within this band. We characterize each scenario, and also discuss several methodological challenges that are common to all research designs based on geographically discontinuous treatment assignments. We illustrate these issues with an original geographic application that studies the effect of introducing copayments for the use of the Children’s Health Insurance Program in the United States, focusing on the border between Illinois and Wisconsin.

Redistricting and the Personal Vote: When Natural Experiments are Neither Natural nor Experiments

Natural experiments are increasingly prominent in the social sciences. However, natural experiments often have more in common with traditional observational studies than with randomized experiments. We illustrate our argument by examining the case of using redistricting to estimate the personal vote. Strikingly, even if voters were redistricted randomly, previous uses of redistricting would not identify the causal effect of interest. We also find that the redistricting process is sufficiently nonrandom as to require significant covariate adjustment to overcome confounding. To overcome these difficulties, we propose a new design for estimating the personal vote and the partisan incumbency advantage that relies on the implementation of multiple redistricting plans. Analyzing data from U.S. House elections in California and Texas, we find that there is a large partisan incumbency advantage in both states but that the effect of the personal vote is zero in Texas and small in California.