The practical importance of inference with robustness against large bandwidths for causal effects in regression discontinuity and kink designs is widely recognized. Existing robust methods cover many cases, but do not handle uniform inference for CDF and quantile processes in fuzzy designs, despite its use in the recent literature in empirical microeconomics. In this light, this paper extends the literature by developing a unified framework of inference with robustness against large bandwidths that applies to uniform inference for quantile treatment effects in fuzzy designs, as well as all the other cases of sharp/fuzzy mean/quantile regression discontinuity/kink designs. We present Monte Carlo simulation studies and an empirical application for evaluations of the Oklahoma pre-K program.
The literature on regression kink designs develops identification results for average effects of continuous treatments (Card, Lee, Pei, and Weber, 2015), average effects of binary treatments (Dong, 2018), and quantile-wise effects of continuous treatments (Chiang and Sasaki, 2019), but there has been no identification result for quantile-wise effects of binary treatments to date. In this paper, we fill this void in the literature by providing an identification of quantile treatment effects in regression kink designs with binary treatment variables. For completeness, we also develop large sampl…
In this paper, we study the estimation and inference of the quantile treatment effect under covariate-adaptive randomization. We propose two estimation methods: (1) the simple quantile regression and (2) the inverse propensity score weighted quantile regression. For the two estimators, we derive their asymptotic distributions uniformly over a compact set of quantile indexes, and show that, when the treatment assignment rule does not achieve strong balance, the inverse propensity score weighted estimator has a smaller asymptotic variance than the simple quantile regression estimator. For the in…
This paper considers the instrumental variable quantile regression model (Chernozhukov and Hansen, 2005, 2013) with a binary endogenous treatment. It offers two identification results when the treatment status is not directly observed. The first result is that, remarkably, the reduced-form quantile regression of the outcome variable on the instrumental variable provides a lower bound on the structural quantile treatment effect under the stochastic monotonicity condition (Small and Tan, 2007; DiNardo and Lee, 2011). This result is relevant, not only when the treatment variable is subject to mis…
Identification in a regression discontinuity (RD) research design hinges on the discontinuity in the probability of treatment when a covariate (assignment variable) exceeds a known threshold. When 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 direct challenge to treatment effect identification. This paper provides sufficient conditions to identify the RD treatment eff…
Propensity score matching has been a long-standing tradition for handling confounding in causal inference, however requiring stringent model assumptions. In this article, we propose double score matching(DSM) for general causal estimands utilizing two balancing scores including the propensity score and prognostic score. To gain the protection of possible model misspecification, we posit multiple candidate models for each score. We show that the de-biasing DSM estimator achieves the multiple robustness property in that it is consistent for the true causal estimand if any model of the propensity…
For non-randomized studies, the regression discontinuity design (RDD) can be used to identify and estimate causal effects from a "locally-randomized" subgroup of subjects, under relatively mild conditions. However, current models focus causal inferences on the impact of the treatment (versus non-treatment) variable on the mean of the dependent variable, via linear regression. For RDDs, we propose a flexible Bayesian nonparametric regression model that can provide accurate estimates of causal effects, in terms of the predictive mean, variance, quantile, probability density, distribution functio…
How should one assess the credibility of assumptions weaker than statistical independence, like quantile independence? In the context of identifying causal effects of a treatment variable, we argue that such deviations should be chosen based on the form of selection on unobservables they allow. For quantile independence, we characterize this form of treatment selection. Specifically, we show that quantile independence is equivalent to a constraint on the average value of either a latent propensity score (for a binary treatment) or the cdf of treatment given the unobservables (for a continuous …
One of the most popular methodologies for estimating the average treatment effect at the threshold in a regression discontinuity design is local linear regression (LLR), which places larger weight on units closer to the threshold. We propose a Gaussian process regression methodology that acts as a Bayesian analog to LLR for regression discontinuity designs. Our methodology provides a flexible fit for treatment and control responses by placing a general prior on the mean response functions. Furthermore, unlike LLR, our methodology can incorporate uncertainty in how units are weighted when estim…
Explanations of the internal validity of regression discontinuity designs (RDD) generally appeal to the idea that RDDs are ``as good as" random near the treatment cut point. Cattaneo, Frandsen, and Titiunik (2015) are the first to take this justification to its full conclusion and propose estimating the RDD local average treatment effect (LATE) the same as one would a randomized experiment. This paper explores the implications of analyzing an RDD as a local random experiment when the running variable is a test score. I derive a formula for the bias in the LATE estimate estimated using the loca…
In medical settings, treatment assignment may be determined by a clinically important covariate that predicts patients' risk of event. There is a class of methods from the social science literature known as regression discontinuity (RD) designs that can be used to estimate the treatment effect in this situation. Under certain assumptions, such an estimand enjoys a causal interpretation. However, few authors have discussed the use of RD for censored data. In this paper, we show how to estimate causal effects under the regression discontinuity design for censored data. The proposed estimation pr…
Estimation of a treatment effect by a regression discontinuity design faces a severe challenge when the running variable contains measurement errors since the errors smoothen the discontinuity on which the identification depends. The existing studies show that the variance of the measurement errors plays a vital role in both bias correction and identification under such situations. However, the methodologies to estimate the variance from data are relatively undeveloped. This paper proposes two estimators for the variance of measurement errors of running variables of sharp regression continuity…
The regression discontinuity (RD) design is a popular approach to causal inference in non-randomized studies. This is because it can be used to identify and estimate causal effects under mild conditions. Specifically, for each subject, the RD design assigns a treatment or non-treatment, depending on whether or not an observed value of an assignment variable exceeds a fixed and known cutoff value. In this paper, we propose a Bayesian nonparametric regression modeling approach to RD designs, which exploits a local randomization feature. In this approach, the assignment variable is treated as a c…
Analyses of environmental phenomena often are concerned with understanding unlikely events such as floods, heatwaves, droughts or high concentrations of pollutants. Yet the majority of the causal inference literature has focused on modelling means, rather than (possibly high) quantiles. We define a general estimator of the population quantile treatment (or exposure) effects (QTE) -- the weighted QTE (WQTE) -- of which the population QTE is a special case, along with a general class of balancing weights incorporating the propensity score. Asymptotic properties of the proposed WQTE estimators ar…
Consider sensitivity analysis for estimating average treatment effects under unmeasured confounding, assumed to satisfy a marginal sensitivity model. At the population level, we provide new representations for the sharp population bounds and doubly robust estimating functions, recently derived by Dorn, Guo, and Kallus. We also derive new, relaxed population bounds, depending on weighted linear outcome quantile regression. At the sample level, we develop new methods and theory for obtaining not only doubly robust point estimators for the relaxed population bounds with respect to misspecificatio…
We study identification and estimation in the Regression Discontinuity Design (RDD) with a multivalued treatment variable. We also allow for the inclusion of covariates. We show that without additional information, treatment effects are not identified. We give necessary and sufficient conditions that lead to identification of LATEs as well as of weighted averages of the conditional LATEs. We show that if the first stage discontinuities of the multiple treatments conditional on covariates are linearly independent, then it is possible to identify multivariate weighted averages of the treatment e…
Causal Inference for Heterogeneous Extreme Quantiles with Heavy-Tailed Outcomes
Nanjing University of Information Science and Technology（南京信息工程大学）；Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）；Rice University（莱斯大学）；Beijing Normal-Hong Kong Baptist University（北京师范大学-香港浸会大学联合国际学院）；School of Statistics, East China Normal University（华东师范大学统计学院）
The regression discontinuity design (RDD) is a quasi-experimental design that can be used to identify and estimate the causal effect of a treatment using observational data. In an RDD, a pre-specified rule is used for treatment assignment, whereby a subject is assigned to the treatment (control) group whenever their observed value of a specific continuous variable is greater than or equal to (is less than) a fixed threshold. Sharp RDDs occur when guidelines are strictly adhered to and fuzzy RDDs occur when the guidelines or not strictly adhered to. In this paper, we take a rigorous decision th…
We propose a novel targeted maximum likelihood estimator (TMLE) for quantiles in semiparametric missing data models. Our proposed estimator is locally efficient, $\sqrt{n}$-consistent, asymptotically normal, and doubly robust, under regularity conditions. We use Monte Carlo simulation to compare our proposed method to existing estimators. The TMLE is superior to all competitors, with relative efficiency up to three times smaller than the inverse probability weighted estimator (IPW), and up to two times smaller than the augmented IPW. This research is motivated by a causal inference research qu…
We propose a new general approach for estimating the effect of a binary treatment on a continuous and potentially highly skewed response variable, the generalized quantile treatment effect (GQTE). The GQTE is defined as the difference between a function of the quantiles under the two treatment conditions. As such, it represents a generalization over the standard approaches typically used for estimating a treatment effect (i.e., the average treatment effect and the quantile treatment effect) because it allows the comparison of any arbitrary characteristic of the outcome's distribution under the…
This paper shows that the Conditional Quantile Treatment Effect on the Treated can be identified using a combination of (i) a conditional Distributional Difference in Differences assumption and (ii) an assumption on the conditional dependence between the change in untreated potential outcomes and the initial level of untreated potential outcomes for the treated group. The second assumption recovers the unknown dependence from the observed dependence for the untreated group. We also consider estimation and inference in the case where all of the covariates are discrete. We propose a uniform infe…
In non-experimental settings, the Regression Discontinuity (RD) design is one of the most credible identification strategies for program evaluation and causal inference. However, RD treatment effect estimands are necessarily local, making statistical methods for the extrapolation of these effects a key area for development. We introduce a new method for extrapolation of RD effects that relies on the presence of multiple cutoffs, and is therefore design-based. Our approach employs an easy-to-interpret identifying assumption that mimics the idea of "common trends" in difference-in-differences de…
Regression discontinuity (RD) is a widely used quasi-experimental design for causal inference. In the standard RD, the assignment to treatment is determined by a continuous pretreatment variable (i.e., running variable) falling above or below a pre-fixed threshold. In the case of the corporate sector purchase programme (CSPP) of the European Central Bank, which involves large-scale purchases of securities issued by corporations in the euro area, such a threshold can be defined in terms of an ordinal running variable. This feature poses challenges to RD estimation due to the lack of a meaningfu…
Datasets from field experiments with covariate-adaptive randomizations (CARs) usually contain extra covariates in addition to the strata indicators. We propose to incorporate these additional covariates via auxiliary regressions in the estimation and inference of unconditional quantile treatment effects (QTEs) under CARs. We establish the consistency and limit distribution of the regression-adjusted QTE estimator and prove that the use of multiplier bootstrap inference is non-conservative under CARs. The auxiliary regression may be estimated parametrically, nonparametrically, or via regulariza…
Evaluating Dynamic Conditional Quantile Treatment Effects with Applications in Ridesharing
DiDi Chuxing（滴滴出行）；University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）；Shanghai University of Finance and Economics（上海财经大学）；London School of Economics and Political Science（伦敦政治经济学院）
The Regression Discontinuity (RD) design is a quasi-experimental design which emulates a randomised study by exploiting situations where treatment is assigned according to a continuous variable as is common in many drug treatment guidelines. The RD design literature focuses principally on continuous outcomes. In this paper we exploit the link between the RD design and instrumental variables to obtain a causal effect estimator, the risk ratio for the treated (RRT), for the RD design when the outcome is binary. Occasionally the RRT estimator can give negative lower confindence bounds. In the Bay…
This paper studies the validity of nonparametric tests used in the regression discontinuity design. The null hypothesis of interest is that the average treatment effect at the threshold in the so-called sharp design equals a pre-specified value. We first show that, under assumptions used in the majority of the literature, for \emph{any} test the power against any alternative is bounded above by its size. This result implies that, under these assumptions, any test with nontrivial power will exhibit size distortions. We next provide a sufficient strengthening of the standard assumptions under wh…
