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Mediation analysis is a powerful tool for studying causal pathways between exposure, mediator, and outcome variables of interest. While classical mediation analysis using observational data often requires strong and sometimes unrealistic assumptions, such as unconfoundedness, Mendelian Randomization (MR) avoids unmeasured confounding bias by employing genetic variations as instrumental variables. We develop a novel MR framework for mediation analysis with genome-wide associate study (GWAS) summary data, and provide solid statistical guarantees. Our framework employs carefully crafted estimating equations, allowing for different sets of genetic variations to instrument the exposure and the mediator, to efficiently integrate information stored in three independent GWAS. As part of this endeavor, we demonstrate that in mediation analysis, the challenge raised by instrument selection goes beyond the well-known winner's curse issue, and therefore, addressing it requires special treatment. We then develop bias correction techniques to address the instrument selection issue and commonly encountered measurement error bias issue. Collectively, through our theoretical investigations, we show that our framework provides valid statistical inference for both direct and mediation effects with enhanced statistical efficiency compared to existing methods. We further illustrate the finite-sample performance of our approach through simulation experiments and a case study.

Understanding treatment effect heterogeneity has become an increasingly popular task in various fields, as it helps design personalized advertisements in e-commerce or targeted treatment in biomedical studies. However, most of the existing work in this research area focused on either analyzing observational data based on strong causal assumptions or conducting post hoc analyses of randomized controlled trial data, and there has been limited effort dedicated to the design of randomized experiments specifically for uncovering treatment effect heterogeneity. In the manuscript, we develop a framework for designing and analyzing response adaptive experiments toward better learning treatment effect heterogeneity. Concretely, we provide response adaptive experimental design frameworks that sequentially revise the data collection mechanism according to the accrued evidence during the experiment. Such design strategies allow for the identification of subgroups with the largest treatment effects with enhanced statistical efficiency. The proposed frameworks not only unify adaptive enrichment designs and response-adaptive randomization designs but also complement A/B test designs in e-commerce and randomized trial designs in clinical settings. We demonstrate the merit of our design with theoretical justifications and in simulation studies with synthetic e-commerce and clinical trial data.

Randomized experiments have been the gold standard for assessing the effectiveness of a treatment or policy. The classical complete randomization approach assigns treatments based on a prespecified probability and may lead to inefficient use of data. Adaptive experiments improve upon complete randomization by sequentially learning and updating treatment assignment probabilities. However, their application can also raise fairness and equity concerns, as assignment probabilities may vary drastically across groups of participants. Furthermore, when treatment is expected to be extremely beneficial to certain groups of participants, it is more appropriate to expose many of these participants to favorable treatment. In response to these challenges, we propose a fair adaptive experiment strategy that simultaneously enhances data use efficiency, achieves an envy-free treatment assignment guarantee, and improves the overall welfare of participants. An important feature of our proposed strategy is that we do not impose parametric modeling assumptions on the outcome variables, making it more versatile and applicable to a wider array of applications. Through our theoretical investigation, we characterize the convergence rate of the estimated treatment effects and the associated standard deviations at the group level and further prove that our adaptive treatment assignment algorithm, despite not having a closed-form expression, approaches the optimal allocation rule asymptotically. Our proof strategy takes into account the fact that the allocation decisions in our design depend on sequentially accumulated data, which poses a significant challenge in characterizing the properties and conducting statistical inference of our method. We further provide simulation evidence to showcase the performance of our fair adaptive experiment strategy.

Barseghyan and Molinari (2023) give sufficient conditions for semi-nonparametric point identification of parameters of interest in a mixture model of decision-making under risk, allowing for unobserved heterogeneity in utility functions and limited consideration. A key assumption in the model is that the heterogeneity of risk preferences is unobservable but context-independent. In this comment, we build on their insights and present identification results in a setting where the risk preferences are allowed to be context-dependent.

Developments in genome-wide association studies and the increasing availability of summary genetic association data have made the application of two-sample Mendelian Randomization (MR) with summary data increasingly popular. Conventional two-sample MR methods often employ the same sample for selecting relevant genetic variants and for constructing final causal estimates. Such a practice often leads to biased causal effect estimates due to the well known "winner's curse" phenomenon. To address this fundamental challenge, we first examine its consequence on causal effect estimation both theoretically and empirically. We then propose a novel framework that systematically breaks the winner's curse, leading to unbiased association effect estimates for the selected genetic variants. Building upon the proposed framework, we introduce a novel rerandomized inverse variance weighted estimator that is consistent when selection and parameter estimation are conducted on the same sample. Under appropriate conditions, we show that the proposed RIVW estimator for the causal effect converges to a normal distribution asymptotically and its variance can be well estimated. We illustrate the finite-sample performance of our approach through Monte Carlo experiments and two empirical examples.

This paper discusses the R package lpcde, which stands for local polynomial conditional density estimation. It implements the kernel-based local polynomial smoothing methods introduced in Cattaneo, Chandak, Jansson, Ma (2024( for statistical estimation and inference of conditional distributions, densities, and derivatives thereof. The package offers mean square error optimal bandwidth selection and associated point estimators, as well as uncertainty quantification based on robust bias correction both pointwise (e.g., confidence intervals) and uniformly (e.g., confidence bands) over evaluation points. The methods implemented are boundary adaptive whenever the data is compactly supported. The package also implements regularized conditional density estimation methods, ensuring the resulting density estimate is non-negative and integrates to one. We contrast the functionalities of lpcde with existing R packages for conditional density estimation, and showcase its main features using simulated data.

We begin by introducing a class of conditional density estimators based on local polynomial techniques. The estimators are boundary adaptive and easy to implement. We then study the (pointwise and) uniform statistical properties of the estimators, offering characterizations of both probability concentration and distributional approximation. In particular, we establish uniform convergence rates in probability and valid Gaussian distributional approximations for the Studentized t-statistic process. We also discuss implementation issues such as consistent estimation of the covariance function for the Gaussian approximation, optimal integrated mean squared error bandwidth selection, and valid robust bias-corrected inference. We illustrate the applicability of our results by constructing valid confidence bands and hypothesis tests for both parametric specification and shape constraints, explicitly characterizing their approximation errors. A companion R software package implementing our main results is provided.

We introduce an Attention Overload Model that captures the idea that alternatives compete for the decision maker's attention, and hence the attention that each alternative receives decreases as the choice problem becomes larger. Using this nonparametric restriction on the random attention formation, we show that a fruitful revealed preference theory can be developed and provide testable implications on the observed choice behavior that can be used to (point or partially) identify the decision maker's preference and attention frequency. We then enhance our attention overload model to accommodate heterogeneous preferences. Due to the nonparametric nature of our identifying assumption, we must discipline the amount of heterogeneity in the choice model: we propose the idea of List-based Attention Overload, where alternatives are presented to the decision makers as a list that correlates with both heterogeneous preferences and random attention. We show that preference and attention frequencies are (point or partially) identifiable under nonparametric assumptions on the list and attention formation mechanisms, even when the true underlying list is unknown to the researcher. Building on our identification results, for both preference and attention frequencies, we develop econometric methods for estimation and inference that are valid in settings with a large number of alternatives and choice problems, a distinctive feature of the economic environment we consider. We provide a software package in R implementing our empirical methods, and illustrate them in a simulation study.

This paper investigates the large sample properties of local regression distribution estimators, which include a class of boundary adaptive density estimators as a prime example. First, we establish a pointwise Gaussian large sample distributional approximation in a unified way, allowing for both boundary and interior evaluation points simultaneously. Using this result, we study the asymptotic efficiency of the estimators, and show that a carefully crafted minimum distance implementation based on "redundant" regressors can lead to efficiency gains. Second, we establish uniform linearizations and strong approximations for the estimators, and employ these results to construct valid confidence bands. Third, we develop extensions to weighted distributions with estimated weights and to local $L^{2}$ least squares estimation. Finally, we illustrate our methods with two applications in program evaluation: counterfactual density testing, and IV specification and heterogeneity density analysis. Companion software packages in Stata and R are available.

Density estimation and inference methods are widely used in empirical work. When the underlying distribution has compact support, conventional kernel-based density estimators are no longer consistent near or at the boundary because of their well-known boundary bias. Alternative smoothing methods are available to handle boundary points in density estimation, but they all require additional tuning parameter choices or other typically ad hoc modifications depending on the evaluation point and/or approach considered. This article discusses the R and Stata package lpdensity implementing a novel local polynomial density estimator proposed and studied in Cattaneo, Jansson, and Ma (2020, 2021), which is boundary adaptive and involves only one tuning parameter. The methods implemented also cover local polynomial estimation of the cumulative distribution function and density derivatives. In addition to point estimation and graphical procedures, the package offers consistent variance estimators, mean squared error optimal bandwidth selection, robust bias-corrected inference, and confidence bands construction, among other features. A comparison with other density estimation packages available in R using a Monte Carlo experiment is provided.

This paper introduces an intuitive and easy-to-implement nonparametric density estimator based on local polynomial techniques. The estimator is fully boundary adaptive and automatic, but does not require pre-binning or any other transformation of the data. We study the main asymptotic properties of the estimator, and use these results to provide principled estimation, inference, and bandwidth selection methods. As a substantive application of our results, we develop a novel discontinuity in density testing procedure, an important problem in regression discontinuity designs and other program evaluation settings. An illustrative empirical application is given. Two companion Stata and R software packages are provided.

Inverse Probability Weighting (IPW) is widely used in empirical work in economics and other disciplines. As Gaussian approximations perform poorly in the presence of "small denominators," trimming is routinely employed as a regularization strategy. However, ad hoc trimming of the observations renders usual inference procedures invalid for the target estimand, even in large samples. In this paper, we first show that the IPW estimator can have different (Gaussian or non-Gaussian) asymptotic distributions, depending on how "close to zero" the probability weights are and on how large the trimming threshold is. As a remedy, we propose an inference procedure that is robust not only to small probability weights entering the IPW estimator but also to a wide range of trimming threshold choices, by adapting to these different asymptotic distributions. This robustness is achieved by employing resampling techniques and by correcting a non-negligible trimming bias. We also propose an easy-to-implement method for choosing the trimming threshold by minimizing an empirical analogue of the asymptotic mean squared error. In addition, we show that our inference procedure remains valid with the use of a data-driven trimming threshold. We illustrate our method by revisiting a dataset from the National Supported Work program.

We study the implications of including many covariates in a first-step estimate entering a two-step estimation procedure. We find that a first order bias emerges when the number of \textit{included} covariates is "large" relative to the square-root of sample size, rendering standard inference procedures invalid. We show that the jackknife is able to estimate this "many covariates" bias consistently, thereby delivering a new automatic bias-corrected two-step point estimator. The jackknife also consistently estimates the standard error of the original two-step point estimator. For inference, we develop a valid post-bias-correction bootstrap approximation that accounts for the additional variability introduced by the jackknife bias-correction. We find that the jackknife bias-corrected point estimator and the bootstrap post-bias-correction inference perform excellent in simulations, offering important improvements over conventional two-step point estimators and inference procedures, which are not robust to including many covariates. We apply our results to an array of distinct treatment effect, policy evaluation, and other applied microeconomics settings. In particular, we discuss production function and marginal treatment effect estimation in detail.

This paper illustrates how one can deduce preference from observed choices when attention is not only limited but also random. In contrast to earlier approaches, we introduce a Random Attention Model (RAM) where we abstain from any particular attention formation, and instead consider a large class of nonparametric random attention rules. Our model imposes one intuitive condition, termed Monotonic Attention, which captures the idea that each consideration set competes for the decision-maker's attention. We then develop revealed preference theory within RAM and obtain precise testable implications for observable choice probabilities. Based on these theoretical findings, we propose econometric methods for identification, estimation, and inference of the decision maker's preferences. To illustrate the applicability of our results and their concrete empirical content in specific settings, we also develop revealed preference theory and accompanying econometric methods under additional nonparametric assumptions on the consideration set for binary choice problems. Finally, we provide general purpose software implementation of our estimation and inference results, and showcase their performance using simulations.