Rae Yu 余蕊琪

We study the statistical properties of nonparametric distance-based (isotropic) local polynomial regression estimators of the conditional average treatment effect at the boundary, a key causal functional parameter capturing heterogeneous treatment effects in boundary discontinuity designs. We present necessary and/or sufficient conditions for identification, estimation and inference in large samples, both pointwise and uniformly along the assignment boundary. Our theoretical results highlight the crucial role played by the “regularity" of the boundary (a one-dimensional manifold) over which identification, estimation and inference is conducted. Our methods are illustrated with simulated and real-world data. Companion general-purpose software is provided.

This paper presents new uniform Gaussian strong approximations for empirical processes indexed by classes of functions based on \(d\)-variate random vectors (\(d \geq1 \)). First, a uniform Gaussian strong approximation is established for general empirical processes indexed by possibly Lipschitz functions, improving on previous results in the literature. In the setting considered by [Rio (1994)], and if the function class is Lipschitzian, our result improves the approximation rate \(n^{-1/(2d)}\) to \(n^{-1/\max\{d,2\}}\), up to a \(polylog(n)\) term, where \(n\) denotes the sample size. Remarkably, we establish a valid uniform Gaussian strong approximation at the rate \(n^{-1/2}\log n\) for \(d=2\), which was previously known to be valid only for univariate (\(d=1\)) empirical processes via the celebrated Hungarian construction [Komlós, Major, Tusnády (1975)]. Second, a uniform Gaussian strong approximation is established for multiplicative separable empirical processes indexed by possibly Lipschitz functions, which addresses some outstanding problems in the literature [Chernozhukov, Chetverikov, Kato (2014)]. Finally, two other uniform Gaussian strong approximation results are presented when the function class is a sequence of Haar basis based on quasi-uniform partitions. Applications to nonparametric density and regression estimation are discussed.

Boundary discontinuity designs are used to learn about causal treatment effects along a continuous assignment boundary that splits units into control and treatment groups according to a bivariate location score. We analyze the statistical properties of local polynomial treatment effect estimators employing location information for each unit. We develop pointwise and uniform estimation and inference methods for both the conditional treatment effect function at the assignment boundary as well as for transformations thereof, which aggregate information along the boundary. We illustrate our methods with an empirical application. Companion general-purpose software is provided.

We study the accuracy of causal trees built with common CART-type splitting rules, with and without sample splitting. Even with randomized treatment assignment and constant treatment effects, greedy partitioning can produce highly unbalanced splits and leaves with too few observations. As a result, the largest estimation error across covariates may shrink arbitrarily slowly, although average squared error can remain small. Sample splitting does not eliminate this problem. The results explain why guarantees based on average accuracy can conceal poor local estimates and clarify the importance of balanced partitions in theory for causal forests.

Uncertainty quantification in causal inference settings with random network interference is a challenging open problem. We study the large sample distributional properties of the classical difference-in-means Hajek treatment effect estimator, and propose a robust inference procedure for the (conditional) direct average treatment effect, allowing for cross-unit interference in both the outcome and treatment equations. Leveraging ideas from statistical physics, we introduce a novel Ising model capturing interference in the treatment assignment, and then obtain three main results. First, we establish a Berry-Esseen distributional approximation pointwise in the degree of interference generated by the Ising model. Our distributional approximation recovers known results in the literature under no-interference in treatment assignment, and also highlights a fundamental fragility of inference procedures developed using such a pointwise approximation. Second, we establish a uniform distributional approximation for the Hajek estimator, and develop robust inference procedures that remain valid regardless of the unknown degree of interference in the Ising model. Third, we propose a novel resampling method for implementation of robust inference procedure. A key technical innovation underlying our work is a new \textitDe-Finetti Machine that facilitates conditional i.i.d. Gaussianization, a technique that may be of independent interest in other settings.

Boundary discontinuity designs are used to learn about causal treatment effects along a continuous assignment boundary that splits units into control and treatment groups according to a bivariate location score. We study the statistical properties of a pooling based local polynomial regression estimator for boundary average treatment effect, where all observations with score within a small region covering the assignment boundary are used.

This chapter provides a review of boundary discontinuity designs, a powerful non-experimental research methodology that identifies causal effects by exploiting a thresholding treatment assignment rule based on a bivariate score and a a boundary curve. This methodology generalizes standard regression discontinuity designs based on a univariate score and scalar cutoff, having specific challenges and features related to its multi-dimensional nature. We synthesize the empirical literature by systematically reviewing over 80 empirical papers, tracing the method’s application from its formative uses to its widespread and sophisticated implementation in modern research. In addition to this empirical survey, the chapter overviews the latest methodological developments, offering a guide to the state-of-the-art in identification, estimation and inference results for boundary discontinuity designs.

The rd2d package implements estimation and inference for boundary discontinuity designs in R, Python, and Stata. It supports methods based on either bivariate location scores or signed distances to the treatment boundary, including sharp and fuzzy designs. Features include automatic bandwidth selection, bias-corrected pointwise inference, uniform confidence bands, clustered data, covariate adjustment, and checks for discrete scores. An application studies Opportunity Zone eligibility, designation, and early workplace employment growth.