Testing monotonicity of conditional treatment effects under regression discontinuity designs

Abstract

Summary Researchers are often interested in the relationship between treatment effects and observed individual heterogeneity. This paper proposes the first nonparametric monotonicity test for conditional treatment effects under the popular regression discontinuity framework. The proposed test examines whether the average treatment effect or the local average treatment effect has a monotonic relationship with some of the observed individual characteristics. We show consistency and asymptotic uniform size control of the proposed test, and apply the test to study treatment effect heterogeneity in two empirical settings.

Regression Discontinuity Designs

“…The continuity-based identification idea for average sharp RD treatment effects has been extended in several directions. For example, Frandsen, Frolich, and Melly (2012) study quantile treatment effects, Xu (2017) investigates treatment effects using generalized nonlinear models for limited dependent outcome variables, and Hsu and Shen (2019) focus on treatment effects conditional on pre-intervention covariates. All extensions and generalizations retain the two basic sharp RD features: (i) treatment assignment is binary and applies to all units in a crosssectional random sample, and (ii) there is a maximum discontinuity in the treatment assignment rule at the cutoff, that is,…”

“…Cellini, Ferreira, and Rothstein (2010) consider a different type of multi-dimensionality induced by a dynamic context in which the RD design occurs in multiple periods for the same units and the score is re-drawn in every period so that a unit may be assigned to treatment in one period but control in future periods; see also Hsu and Shen (2021a) for an econometric analysis of a dynamic RD design within the continuity-based framework. Lv, Sun, Lu, and Li (2019) consider the generalization of the RD design to survival data settings, where the treatment is assigned at most once per unit and the outcome of interest is the units' survival time in a particular state, which may be censored.…”

“…near the cutoff is a reasonable assumption Hsu and Shen (2021b). also consider the role of monotonicity in RD designs.…”

Covariate Adjustment in Regression Discontinuity Designs

“…A different continuity-based approach to examine RD treatment effect heterogeneity on different subpopulations is proposed by Hsu and Shen (2019). The authors impose stronger continuity conditions than in the standard RD design, requiring (in the sharp RD case) continuity of the expectation of the potential outcomes conditional on both the running variable and the additional covariates, and continuity of the conditional distribution of the additional covariates given the running variable.…”

“…Under regularity conditions, the asymptotic distribution of the local polynomial estimators of the moment conditions can then be used to derive the distribution of test statistics under the null hypothesis. In a follow-up paper, Hsu and Shen (2021) employ similar methods to develop a monotonicity test to assess whether a conditional local average treatment effect in a sharp RD design or a conditional local average treatment effect for compliers in a fuzzy RD design has a monotonic relationship with an observed pre-determined covariate-that is, the null hypothesis is that the conditional local average treatment effect, seen as a function of a covariate z, is non-decreasing in z.…”