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Relative to the randomized controlled trial (RCT), the basic regression discontinuity (RD) design suffers from lower statistical power and lesser ability to generalize causal estimates away from the treatment eligibility cutoff. This chapter seeks to mitigate these limitations by adding an untreated outcome comparison function that is measured along all or most of the assignment variable. When added to the usual treated and untreated outcomes observed in the basic RD, a comparative RD (CRD) design results. One version of CRD adds a pretest measure of the study outcome (CRD-Pre); another adds posttest outcomes from a nonequivalent comparison group (CRD-CG). We describe how these designs can be used to identify unbiased causal effects away from the cutoff under the assumption that a common, stable functional form describes how untreated outcomes vary with the assignment variable, both in the basic RD and in the added outcomes data (pretests or a comparison group’s posttest). We then create the two CRD designs using data from the National Head Start Impact Study, a large-scale RCT. For both designs, we find that all untreated outcome functions are parallel, which lends support to CRD’s identifying assumptions. Our results also indicate that CRD-Pre and CRD-CG both yield impact estimates at the cutoff that have a similarly small bias as, but are more precise than, the basic RD’s impact estimates. In addition, both CRD designs produce estimates of impacts away from the cutoff that have relatively little bias compared to estimates of the same parameter from the RCT design. This common finding appears to be driven by two different mechanisms. In this instance of CRD-CG, potential untreated outcomes were likely independent of the assignment variable from the start. This was not the case with CRD-Pre. However, fitting a model using the observed pretests and untreated posttests to account for the initial dependence generated an accurate prediction of the missing counterfactual. The result was an unbiased causal estimate away from the cutoff, conditional on this successful prediction of the untreated outcomes of the treated.

Regression discontinuity (RD) models are commonly used to nonparametrically identify and estimate a local average treatment effect. Dong and Lewbel (2015) show how a derivative of this effect, called treatment effect derivative (TED) can be estimated. We argue here that TED should be employed in most RD applications, as a way to assess the stability and hence external validity of RD estimates. Closely related to TED, we define the complier probability derivative (CPD). Just as TED measures stability of the treatment effect, the CPD measures stability of the complier population in fuzzy designs. TED and CPD are numerically trivial to estimate. We provide relevant Stata code, and apply it to some real datasets.

This study applies asymmetric rather than conventional symmetric analysis to advance theory in occupational psychology. The study applies systematic case-based analyses to model complex relations among conditions (i.e., configurations of high and low scores for variables) in terms of set memberships of managers. The study uses Boolean algebra to identify configurations (i.e., recipes) reflecting complex conditions sufficient for the occurrence of outcomes of interest (e.g., high versus low financial job stress, job strain, and job satisfaction). The study applies complexity theory tenets to offer a nuanced perspective concerning the occurrence of contrarian cases – for example, in identifying different cases (e.g., managers) with high membership scores in a variable (e.g., core self-evaluation) who have low job satisfaction scores and when different cases with low membership scores in the same variable have high job satisfaction. In a large-scale empirical study of managers (n = 928) in four (contextual) segments of the farm industry in New Zealand, this study tests the fit and predictive validities of set membership configurations for simple and complex antecedent conditions that indicate high/low core self-evaluations, job stress, and high/low job satisfaction. The findings support the conclusion that complexity theory in combination with configural analysis offers useful insights for explaining nuances in the causes and outcomes to high stress as well as low stress among farm managers. Some findings support and some are contrary to symmetric relationship findings (i.e., highly significant correlations that support main effect hypotheses).

This chapter analyzes a geographic quasi-experiment embedded in a cluster-randomized experiment in Honduras. In the experiment, average treatment effects of conditional cash transfers on school enrollment and child labor were large – especially in the poorest experimental blocks – and could be generalized to a policy-relevant population given the original sample selection criteria. In contrast, the geographic quasi-experiment yielded point estimates that, for two of three dependent variables, were attenuated. A judicious policy analyst without access to the experimental results might have provided misleading advice based on the magnitude of point estimates. We assessed two main explanations for the difference in point estimates, related to external and internal validity.

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.

The purpose of this paper is to estimate the causal effect of vocational high school (VHS) education on employment likelihood relative to general high school (GHS) education in Turkey using Census data.

To address non-random selection into high school types, the authors collect construction dates of the VHSs at the town level and use various measures of VHS availability in the town by the age of 13 as instrumental variables.

The first-stage estimates suggest that the availability of VHS does not affect the overall high school graduation rates, but generates a substitution from GHS to VHS. The OLS estimates yield the result that individuals with a VHS degree are around 5 percentage points more likely to be employed compared to those with a GHS degree. When the authors use measures of VHS availability as instruments, they still find positive and statistically significant effect of VHS degree on employment likelihood relative to GHS degree. However, once they include town-level controls or town fixed effects, IV estimates get much smaller and become statistically insignificant.

The authorsconclude that, although VHS construction generates a substitution from GHS to VHS education, this substitution is not transformed into increased employment rates in a statistically significant way.

This chapter develops a novel bootstrap procedure to obtain robust bias-corrected confidence intervals in regression discontinuity (RD) designs. The procedure uses a wild bootstrap from a second-order local polynomial to estimate the bias of the local linear RD estimator; the bias is then subtracted from the original estimator. The bias-corrected estimator is then bootstrapped itself to generate valid confidence intervals (CIs). The CIs generated by this procedure are valid under conditions similar to Calonico, Cattaneo, and Titiunik’s (2014) analytical correction – that is, when the bias of the naive RD estimator would otherwise prevent valid inference. This chapter also provides simulation evidence that our method is as accurate as the analytical corrections and we demonstrate its use through a reanalysis of Ludwig and Miller’s (2007) Head Start dataset.

This chapter elaborates on the usefulness of embracing complexity theory, modeling outcomes rather than directionality, and modeling complex rather than simple outcomes in strategic management. Complexity theory includes the tenet that most antecedent conditions are neither sufficient nor necessary for the occurrence of a specific outcome. Identifying a firm by individual antecedents (i.e., noninnovative vs. highly innovative, small vs. large size in sales or number of employees, or serving local vs. international markets) provides shallow information in modeling specific outcomes (e.g., high sales growth or high profitability) – even if directional analyses (e.g., regression analysis, including structural equation modeling) indicate that the independent (main) effects of the individual antecedents relate to outcomes directionally – because firm (case) anomalies almost always occur to main effects. Examples: a number of highly innovative firms have low sales while others have high sales and a number of noninnovative firms have low sales while others have high sales. Breaking-away from the current dominant logic of directionality testing – null hypothesis significance testing (NHST) – to embrace somewhat precise outcome testing (SPOT) is necessary for extracting highly useful information about the causes of anomalies – associations opposite to expected and “statistically significant” main effects. The study of anomalies extends to identifying the occurrences of four-corner strategy outcomes: firms doing well in favorable circumstances, firms doing badly in favorable circumstances, firms doing well in unfavorable circumstances, and firms doing badly in unfavorable circumstances. Models of four-corner strategy outcomes advance strategic management beyond the current dominant logic of directional modeling of single outcomes.

Identification in a regression discontinuity (RD) design hinges on the discontinuity in the probability of treatment when a covariate (assignment variable) exceeds a known threshold. If the assignment variable is measured with error, however, the discontinuity in the relationship between the probability of treatment and the observed mismeasured assignment variable may disappear. Therefore, the presence of measurement error in the assignment variable poses a challenge to treatment effect identification. This chapter provides sufficient conditions to identify the RD treatment effect using the mismeasured assignment variable, the treatment status and the outcome variable. We prove identification separately for discrete and continuous assignment variables and study the properties of various estimation procedures. We illustrate the proposed methods in an empirical application, where we estimate Medicaid takeup and its crowdout effect on private health insurance coverage.

While there exist many surveys on the use stochastic frontier analysis (SFA), many important issues and techniques in SFA were not well elaborated in the previous surveys, namely, regular models, copula modeling, nonparametric estimation by Grenander’s method of sieves, empirical likelihood and causality issues in SFA using regression discontinuity design (RDD) (sharp and fuzzy RDD). The purpose of this paper is to encourage more research in these directions.

A literature survey.

While there are many useful applications of SFA to econometrics, there are also many important open problems.

This is the first survey of SFA in econometrics that emphasizes important issues and techniques such as copulas.