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This chapter studies the large sample properties of a subclassification-based estimator of the dose–response function under ignorability. Employing standard regularity conditions, it is shown that the estimator is root-n consistent, asymptotically linear, and semiparametric efficient in large samples. A consistent estimator of the standard-error is also developed under the same assumptions. In a Monte Carlo experiment, we investigate the finite sample performance of this simple and intuitive estimator and compare it to others commonly employed in the literature.

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.

“The Elephant in the Corner: A Cautionary Tale About Measurement Error in Treatment Effects Models” by Daniel L. Millimet discusses the current use of the unobserved-outcome framework to estimate population-averaged treatment effects, and it exposes the sensitivity of these estimators to assumption of no measurement error. The Monte Carlo simulation evidence in this chapter indicates that “nonclassical measurement error in the covariates, mean-reverting measurement error in the outcome, and simultaneous measurement errors in the outcome, treatment assignment, and covariates have a dramatic, adverse effect on the performance of the various estimators even with relatively small and infrequent errors” (Millimet article, p. 1–39). To some extent, all the estimators analyzed by Millimet are based on weak functional form assumptions and use semiparametric or nonparametric methods. Millimet's results indicate the need for measurement error models be they parametric or nonparametric models, see Schennach (2007), Hu and Schennach (2008), and Matzkin (2007) for some recent research in nonparametric approaches. Chapter 7 develops a Bayesian estimator that can handle some of the measurement errors discussed in this chapter.