Inference on the Distribution of Individual Treatment Effects in Nonseparable Triangular Models

Inference on the Distribution of Individual Treatment Effects in Nonseparable Triangular ModelsDate:

Introduction

Heterogeneity of individual treatment effects (ITEs), including scenarios with endogenous treatment, has received substantial attention in the literature. When ITEs are heterogeneous, the econometrician is often interested in the properties of their distribution, e.g., the CDF and quantile function, as they contain important policy-relevant information beyond average treatment effects. Recently, using a triangular model with binary endogenous treatment, vuong2017counterfactual and \citet*[FVX, hereafter]{feng2019estimation} established nonparametric identification of heterogeneous ITEs and proposed their nonparametric estimation. The estimated ITEs (also referred to as pseudo ITEs) can be used further to estimate the distribution of the ITEs.

In this paper, we develop the asymptotic theory of the empirical CDF and quantile function of the nonparametrically estimated (pseudo) ITEs, which has been lacking in the literature so far. Such results are nontrivial because of the multi-step nonparametric estimation procedure required for their construction. We further use the results to develop easy-to-implement nonparametric bootstrap methods for inference on the CDF and quantile function of the ITE distribution. Our methods can be used, e.g., for inference on the proportion of the population with positive or negative ITEs and the dispersion of ITEs as measured by the interquartile range (IQR). Moreover, our procedure can be used to compare the ITE distributions between different sub-populations. E.g., one can use our results to test whether the distribution of the ITEs in one sub-population stochastically dominates that for another sub-population.

Suppose that the econometrician observes data on an outcome variable, a binary endogenous treatment, a binary instrument, and exogenous covariates. We assume that the outcome variable and the endogenous treatment are generated from the nonseparable nonparametric triangular model of \citetalias{vuong2017counterfactual} that satisfies the rank invariance assumption. We further assume that the econometrician uses the nonparametric method of \citetalias{feng2019estimation} to construct pseudo ITEs as the estimates of the true ITEs for each individual. In the next step, the econometrician uses the pseudo ITEs to construct the empirical CDF or quantile function as the estimates of the true ITE CDF or quantile function, respectively. The second step can be performed for the entire sample or in sub-groups determined by chosen values of discretely distributed exogenous covariates. E.g., the econometrician can perform the second step by gender, education levels, income quartiles, etc., as well as intersections of such groups.

The first contribution of the paper is to show that the properly scaled difference between the empirical CDF of the pseudo ITEs and the CDF of the true ITEs weakly converges to a tight Gaussian process, with a similar result holding for the empirical quantile function of the pseudo ITEs. Importantly, we show that due to the two-step estimation, the asymptotic variances of the empirical CDF and quantile function of pseudo ITEs are “inflated” relative to their infeasible counterparts based on true unobserved ITEs.

For our second contribution, we use the weak convergence results to develop bootstrap inference methods for the CDF and quantile function of the distribution of the ITE. Both pointwise confidence intervals and uniform confidence bands (UCBs) are considered, as the pointwise confidence interval is useful, e.g., for inference on the percentage of the population with positive ITEs and the IQR, while the UCB is useful for inference on the entire CDF or quantile function and comparing the distributions of the ITEs between different sub-populations.\footnote{A UCB is a collection of random intervals that cover the unknown curve of interest simultaneously over a range of values with a pre-specified confidence level.} Our method for constructing confidence intervals for the percentages has the desirable range-preserving property: the bootstrap percentile confidence intervals are always sub-intervals of $\left[0,1\right]$.\footnote{See, e.g., Efron1994.}

Our proposed inference methods exhibit excellent finite-sample performance in Monte Carlo simulations. We further demonstrate their practical value by revisiting a well-known empirical application: the effect of participation in 401(k) retirement programs on personal savings, see, e.g., \citet*{Chernozhukov2006} and \citetalias{feng2019estimation}, where our methods can be used to conduct valid inference on important distributional features such as the proportion of individuals with positive ITEs and stochastic dominance relationships between the distributions of ITEs in different subpopulations. In the case of 401(k) programs, our method reveals rich features of the ITE distributions. For instance, the 95% confidence interval for the proportion of households with a positive ITE is $\left[0.851,0.919\right]$, suggesting that program participation increased savings for the majority of households, though a nontrivial minority experienced negative effects. Moreover, for young individuals (with age in the first quartile), the 95% confidence interval is $\left[0.706,0.884\right]$, suggesting that up to 29.4% of young individuals may experience negative ITEs. The median ITE has a 95% confidence interval of $\left[6.96,9.74\right]$ thousand dollars, indicating a significantly positive central tendency of the treatment effect distribution. The 95% confidence interval for the IQR, $\left[16.68,23.38\right]$, underscores substantial heterogeneity in the ITEs. A subgroup analysis reveals that as income or age increases, the ITE distribution shifts to the right, with both the median and the quartiles moving upward, and the spread of the distribution widening. The UCBs of the quantile functions further indicate that, across all quantiles between the $0.2$ and $0.9$ levels, the ITE is consistently larger for higher-income groups than for lower-income groups.

Our paper contributes to the growing literature on causal inference methods that emphasize heterogeneous treatment effects (see, e.g., angrist2004treatment,Heckman1997,Heckman2006 among others). The \citetalias{vuong2017counterfactual} model we employ belongs to a broad class of triangular models widely used for causal inference.\footnote{See, e.g., abrevaya2021estimation,Chesher2003,Chesher2005,d2015identification,Imbens2009,Jun2011,Newey1999,Torgovitsky2015,Vytlacil2007, among others.} \citetalias{vuong2017counterfactual} showed the identification of the “counterfactual mappings”, which can be used to obtain the counterfactual outcome for each individual. \citetalias{feng2019estimation} proposed convenient extremum estimators for the counterfactual mappings and established their asymptotic properties. Using estimated/pseudo ITEs, \citetalias{feng2019estimation} also proposed a kernel estimator for the probability density function (PDF) of the ITE distribution. The asymptotic theory of the density estimator was further developed in \citet*[MMY, hereafter]{Ma2023}. \citetalias{Ma2023} showed that this estimator converges at the optimal rate (Stone1982), established its asymptotic normality, and proposed a bootstrap-based UCB for inference on the density function of the ITE distribution. Our paper continues this line of research by developing corresponding inference methods for the CDF and quantile function of the ITE distribution.\footnote{Like MMY, our paper also contributes to the literature of multi-step nonparametric estimation using nonparametrically generated variables. See, e.g., Ma2019 and Mammen2012 among others.} Combined with the results in \citetalias{Ma2023}, the econometrician can use our results to characterize the commonly used distributional features for the ITE. The methods for inference on the proportion of positive/negative ITEs, the median, the IQR and also the stochastic order relation between ITE distributions cannot be derived from the results on PDF estimation and inference in \citetalias{Ma2023}. E.g, when comparing two distributions, first-order stochastic dominance is evident when one quantile function lies entirely above the other, even though their PDFs may still intersect.

While our results are complementary to \citetalias{feng2019estimation} and \citetalias{Ma2023}, their derivation employs different techniques from those used in \citetalias{Ma2023}. The main difference is that the density estimator in \citetalias{feng2019estimation} and \citetalias{Ma2023} is a differentiable function of the pseudo ITEs. \citetalias{Ma2023} utilizes this fact and U-process theory to establish its properties. On the other hand, the empirical CDF estimator we focus on here is non-differentiable, and we use the approach of van2007empirical instead. One should also note that the CDF-based approach developed here is tuning-parameter-free, unlike the PDF-based approach in \citetalias{feng2019estimation} and \citetalias{Ma2023}.\footnote{See liu2022sample and liu2024tuning among others for recent examples of tuning-free methods in the causal inference literature.}

A related strand of literature is concerned with quantile treatment effects (QTEs). When the treatment is endogenous, QTEs are often estimated using the local quantile treatment effect (LQTE) model (Abadie2002,Froelich2013) or the instrumental variable quantile regression (IVQR) model (\citealp*{Chernozhukov2005,chernozhukov2006instrumental}). Unlike the LQTE model, the approach of \citetalias{vuong2017counterfactual} and \citetalias{feng2019estimation} allows for the identification and estimation of ITEs for the entire population rather than just for compliers. This is possible due to somewhat stronger assumptions of \citetalias{vuong2017counterfactual}, such as the rank invariance condition enabling the identification of ITEs. Nevertheless, we believe that the ability to estimate effects for a broader population can be important in practice.\footnote{Neither LQTE nor IVQR can identify the ITE distribution without the rank invariance condition. An alternative strand of the literature avoids the rank invariance assumption and employs a copula-based approach to derive sharp bounds on the ITE distribution, typically in the context of randomized experiments or under selection-on-observables assumptions (see, e.g., fan2009partial,fan2010sharp,fan2012confidence,firpo2019partial among others).} Moreover, the approach of \citetalias{feng2019estimation} is computationally attractive as it only involves a one-dimensional optimization problem.

The rest of the paper is organized as follows. Section (ref) reviews the model and the identification and estimation of ITEs as proposed in \citetalias{vuong2017counterfactual} and \citetalias{feng2019estimation}. Section (ref) shows the asymptotic normality and weak convergence results for the empirical CDF and quantiles of the pseudo ITEs. Section (ref) describes the construction of bootstrap percentile confidence intervals and bootstrap UCBs for the ITE CDF and quantiles. Section (ref) presents extensions of the methods proposed in the preceding section, including inference on the ITE distributions of broader subgroups and the differences of ITE quantiles of subgroups. Section (ref) provides numerical evidence that shows the validity of the asymptotic theory of Section (ref) and evaluates the finite sample performances of the inference methods proposed in Section (ref). Section (ref) revisits the empirical application in \citetalias{feng2019estimation}, which assesses the effect of participation in the 401(k) retirement program on savings. Proofs of all main results are presented in an online appendix.\cprotect\footnote{The appendix is available at \url{https://ruc-econ.github.io/ITE_CDF_app_V3.pdf}.}

Notation. We use “$a\coloneqq b$” to denote “$a$ is defined by $b$”, and “$a\eqqcolon b$” is understood as “$b$ is defined by $a$”. The closed interval $\left[a-b,a+b\right]$ is denoted as $a\pm b$ . Let $\mathrm{sgn}\left(u\right)\coloneqq2\times\mathbbm{1}\left(u>0\right)-1$ denote the left continuous sign function, where $\mathbbm{1}\left(\cdot\right)$ denotes the indicator function. For $a\in\mathbb{R}$, let $\left\lceil a\right\rceil \coloneqq\min\left\{ z\in\mathbb{Z}:z\geq a\right\} $ be the smallest integer greater than or equal to $a$. Let $a^{\top}$ denote the transpose of $a$. For a positive integer $T$, $\left[T\right]\coloneqq\left\{ 1,...,T\right\} $. Let $\mathscr{S}_{V}$ denote the support of the distribution of a random vector $V$, and let $\mathscr{S}_{V\mid W=w}$ denote the support of the conditional distribution of $V$ given $W=w$. The conditional CDF and PDF of the distribution of $V$ given $W=w$ are denoted as $F_{V\mid W}\left(\cdot\mid w\right)$ and $f_{V\mid W}\left(\cdot\mid w\right)$, respectively. Convergence in distribution in the general sense (van2000asymptotic) is denoted as “$\rightsquigarrow$”. Let $\ell^{\infty}\left[a,b\right]$ denote the set of bounded real-valued functions on the closed interval $\left[a,b\right]$. For any $f\in\ell^{\infty}\left[a,b\right]$, let $\left\Vert f\right\Vert _{\left[a,b\right]}\coloneqq\sup_{t\in\left[a,b\right]}\left|f\left(t\right)\right|$ denote the sup-norm of $f$ on $\left[a,b\right]$. Let $C\left[a,b\right]$ denote the set of continuous functions on $\left[a,b\right]$. Let $D\left[a,b\right]$ denote the set of c�dl�g functions on $\left[a,b\right]$ (i.e., for all $f\in D\left[a,b\right]$, $f$ is right continuous at each point in $\left[a,b\right)$ and has a left limit at each point in $\left(a,b\right]$). All the three spaces are endowed with the sup-norm metric. Let $\mathit{BL}_{1}\left(\mathbb{D}\right)$ be the collection of real valued functions defined on a Banach space $\mathbb{D}$ (endowed with a norm $\left\Vert \cdot\right\Vert $) that satisfy the following condition: $h\in\mathit{BL}_{1}\left(\mathbb{D}\right)$ if and only if $\left|h\left(x\right)-h\left(y\right)\right|\leq\left\Vert x-y\right\Vert $ for all $x,y\in\mathbb{D}$ and $\sup_{x\in\mathbb{D}}\left|h\left(x\right)\right|\leq1$.

Model and estimation of ITEs

For completeness, in Section (ref), we review the model setup and assumptions of \citetalias{vuong2017counterfactual} and \citetalias{feng2019estimation}. Similarly, in Section (ref), we review the definition of ITEs, the additional assumption imposed by \citetalias{Ma2023}, and the estimation method of \citetalias{feng2019estimation}. The main objects of interest, the ITE CDF and quantile function as well as their estimators are defined in Section (ref).

Triangular model

Let $Y$ be a continuously distributed outcome variable and let $D$ be an endogenous binary treatment variable. The model assumes that $Y$ and $D$ are determined by the following outcome and selection equations:

In the outcome equation ((ref)), $X$ is a vector of observed explanatory variables (covariates), $\epsilon$ is the unobserved scalar-valued disturbance, and $g$ is an unknown function. The right hand side of ((ref)) is of a completely nonseparable form.\footnote{The outcome model ((ref)) does not assume additive or weak separability (see, e.g., Vytlacil2007). See Section 2.2 of \citetalias{vuong2017counterfactual} and abrevaya2021estimation for examples of nonseparable specifications.} The selection equation ((ref)) has the form of a latent index model, where $Z$ is a binary instrument (or instrumental variable, IV) excluded from the outcome equation, $\eta$ is the unobserved scalar-valued cost of the treatment to the individual, $s$ is an unknown function, and $s\left(Z,X\right)$ is understood as the benefit from the treatment. The treatment is taken up if the net utility from taking up the treatment is positive.

Let $Y\left(d,x\right)\coloneqq g\left(d,x,\epsilon\right)$ and $D\left(z,x\right)\coloneqq\mathbbm{1}\left(\eta\leq s\left(z,x\right)\right)$ denote the potential outcome and treatment, and $\mathsf{co}_{x}$ denote the “complier” event “$X=x$ and $D\left(0,x\right)<D\left(1,x\right)$”. Lastly, let $\mathscr{S}_{Y\left(d,x\right)\mid\mathsf{co}_{x}}$ and $f_{Y\left(d,x\right)\mid\mathsf{co}_{x}}$ denote the support and Lebesgue density of the conditional distribution of $Y\left(d,x\right)$ given $\mathsf{co}_{x}$. The assumptions on the data generating process (DGP) from \citetalias{vuong2017counterfactual} and \citetalias{feng2019estimation} are summarized as follows.

The monotonicity of $g\left(d,x,\cdot\right)$ in Part (a) imposes rank invariance on the potential outcomes. Part (b) is the IV exogeneity assumption and Part (c) is the IV relevance assumption. Given the assumption in Part (b) and equations ((ref))--((ref)), $Z$ is independent of $\left(Y\left(1,x\right),Y\left(0,x\right),D\left(1,x\right),D\left(0,x\right)\right)$ conditionally on $X=x$. Part (c) and equation ((ref)), imply the monotonicity assumption of potential treatments: $D\left(0,x\right)\leq D\left(1,x\right)$.\footnote{See, e.g., Vytlacil2002. Note also that the independence and monotonicity assumptions jointly have testable implications Kitagawa2015.} Parts (d,e) are mild regularity conditions. The support condition in Part (f) is crucial for the identification result of Lemma 1 of \citetalias{vuong2017counterfactual} and is related to the effectiveness of the IV.\footnote{See Section 2.1 of \citetalias{vuong2017counterfactual}. In particular, Part (f) is satisfied if the conditional distribution of $\left(\epsilon,\eta\right)$ given $X=x$ has a rectangular support for all $x\in\mathscr{S}_{X}$.} Parts (a,c,d) together with equations ((ref))--((ref)) ensure that the conditional distribution of $Y\left(d,x\right)$ given $\mathsf{co}_{x}$ is absolutely continuous with respect to the Lebesgue measure, and thus the existence of a continuous and bounded Lebesgue density $f_{Y\left(d,x\right)\mid\mathsf{co}_{x}}$ is guaranteed. Given the conditions of Parts (a,d), $\mathscr{S}_{Y\left(d,x\right)\mid X=x}$ is a compact interval. Moreover, Lemma 1 of \citetalias{vuong2017counterfactual} shows that $\mathscr{S}_{Y\left(d,x\right)\mid X=x}=\mathscr{S}_{Y\mid D=d,X=x}$ and, therefore, the end points $\underline{y}_{dx}$ and $\overline{y}_{dx}$ of $\mathscr{S}_{Y\left(d,x\right)\mid X=x}$ are identifiable and estimable. Part (h) assumes that $\underline{y}_{dx}$ and $\overline{y}_{dx}$ are known, however, in practice, $\underline{y}_{dx}$ and $\overline{y}_{dx}$ can be estimated by the minimum and the maximum of the observed outcomes, respectively.\footnote{As discussed in \citetalias{feng2019estimation}, Parts (g,h,i) can be relaxed at the cost of technical complications. See Section 3 therein.}

ITEs and their estimation

The ITE is defined as

Note that $\varDelta$ is random conditionally on $X$ due to the unobserved $\epsilon$, i.e., the treatment effects vary among individuals with the same observed characteristics. Since the disturbances $\epsilon$ and $\eta$ are allowed to be correlated conditionally on $X$, whether or not individuals select into treatment can be correlated with the gain from treatment.\footnote{The property is referred to as “essential heterogeneity” in the causal inference literature. See, e.g., Heckman2006.}

Let $\varDelta_{x}\left(e\right)\coloneqq g\left(1,x,e\right)-g\left(0,x,e\right)$. As discussed in \citetalias{Ma2023}, the assumptions imposed in the preceding section alone are insufficient to ensure that the conditional distribution of $\varDelta$ given $X=x$ is absolutely continuous with respect to the Lebesgue measure. Therefore, as in \citetalias{Ma2023}, we introduce the following seemingly minimal assumption which guarantees that the conditional distribution of $\varDelta$ given $X=x$ has a continuous PDF denoted as $f_{\varDelta\mid X}\left(\cdot\mid x\right)$. Let $\left(\underline{\epsilon}_{x},\overline{\epsilon}_{x}\right)$ be the end points of $\mathscr{S}_{\epsilon\mid X=x}$; that is, $\underline{\epsilon}_{x}<\overline{\epsilon}_{x}$ and $\mathscr{S}_{\epsilon\mid X=x}=\left[\underline{\epsilon}_{x},\overline{\epsilon}_{x}\right]$.

Note that the knowledge of the partition introduced in Assumption (ref) is not required for the implementation of our methods.

Denote $d'\coloneqq1-d$, and let $g^{-1}\left(d',x,\cdot\right)$ be the inverse function of $g\left(d',x,\cdot\right)$. For $y\in\mathscr{S}_{Y\left(d',x\right)\mid X=x}$, define the corresponding counterfactual mapping $\phi_{dx}\left(y\right)\coloneqq g\left(d,x,g^{-1}\left(d',x,y\right)\right)$, i.e., $\phi_{dx}\left(y\right)$ is the counterfactual outcome if the observed treatment status $d'$ were $d$. Using the counterfactual mappings, we can write the ITE as

Lemma 1 of \citetalias{vuong2017counterfactual} provides a constructive nonparametric identification result for the counterfactual mappings. This result and ((ref)) establish the identification of the distribution of $\varDelta$.

Next, we review the estimation procedure of \citetalias{feng2019estimation}. Lemma 1 of \citetalias{feng2019estimation} shows that $\phi_{dx}\left(y\right)$ is the unique minimizer of the strictly convex function $\varUpsilon_{dx}\left(\cdot,y\right)$, where

The fact that $\phi_{dx}\left(y\right)$ uniquely minimizes $\varUpsilon_{dx}\left(\cdot,y\right)$ motivates using an extremum estimator for its estimation.

Since estimation is performed for each given value of $x\in\mathscr{S}_{X}$, we make the following assumption, which allows us to treat the sample size $n_{x}$ of a sub-sample with the covariate values being $x$ as non-random. It is a simplification that does not affect the properties of the estimation and inference procedures.

Let $\widehat{\varUpsilon}_{dx}^{\left(-i\right)}\left(t,y\right)$ denote the leave-$i$-out sample analogue of $\varUpsilon_{dx}\left(t,y\right)$:

The leave-$i$-out nonparametric estimator of $\phi_{dx}\left(y\right),d\in\left\{ 0,1\right\} $, can be constructed as

One can now estimate the ITEs by replacing $\phi_{dx}(y)$ in ((ref)) with its leave-$i$-out nonparametric estimator $\widehat{\phi}_{dx}^{\left(-i\right)}\left(y\right)$:

Using these estimated/pseudo ITEs, one can estimate various features of the distribution of $\varDelta$.

Empirical CDF and quantile function of pseudo ITEs

We estimate the conditional CDF $F_{\varDelta\mid X}\left(\cdot\mid x\right)$ given $X=x$ of ITEs using the empirical CDF of the pseudo ITEs $\left\{ \widehat{\varDelta}_{i}\right\} _{i=1}^{n_{x}}$:

Related quantities of practical interest are, e.g., the proportion $F_{\varDelta\mid X}\left(0\mid x\right)$ of population with positive ITEs or the proportion $1-F_{\varDelta\mid X}\left(0\mid x\right)$ of population with negative ITEs.

For $\tau\in\left(0,1\right)$, the $\tau$-th quantile of the ITE distribution conditional on $X=x$ is defined as $Q_{\varDelta\mid X}\left(\tau\mid x\right)\coloneqq\inf\left\{ y\in\mathbb{R}:F_{\varDelta\mid X}\left(y\mid x\right)\geq\tau\right\} $. We estimate $Q_{\varDelta\mid X}\left(\tau\mid x\right)$ using the corresponding empirical quantile of the pseudo ITEs $\left\{ \widehat{\varDelta}_{i}\right\} _{i=1}^{n_{x}}$:

The econometrician may be interested in the conditional median $Q_{\varDelta\mid X}\left(0.5\mid x\right)$ as a measure of centrality of the ITE distribution or the conditional population IQR

as a measure of dispersion.

Asymptotic properties

Section (ref) presents the asymptotic theory for the ITE CDF estimator ((ref)) and discusses the key steps in the proof. Section (ref) presents the asymptotic theory for the quantile estimator ((ref)).

Asymptotic Gaussianity of the empirical CDF

Let $\text{\ensuremath{\left[\underline{v}_{x},\overline{v}_{x}\right]}}$ be any inner closed sub-interval of $\mathscr{S}_{\varDelta\mid X=x}$. Denote

Our first result is that the process $S_{F}\left(\cdot\mid x\right)$, as a map from the underlying probability space into $\ell^{\infty}\left[\underline{v}_{x},\overline{v}_{x}\right]$, converges in distribution to a tight Gaussian process. The asymptotic normality of $S_{F}\left(v\mid x\right)$ for any fixed $v\in\left[\underline{v}_{x},\overline{v}_{x}\right]$ immediately follows from this result.

Before we discuss the key steps in the proof of the convergence in distribution result for $S_{F}\left(\cdot\mid x\right)$, we introduce the following notations. Let

By Kosorok2007 and Kosorok2007, we have

where $\left\{ \mathbb{B}_{0}\left(t\right):t\in\left[0,1\right]\right\} $ is a standard Brownian bridge, whose sample path is continuous almost surely. Therefore, $\mathbb{H}_{x}$ concentrates on $C\left[\underline{\epsilon}_{x},\overline{\epsilon}_{x}\right]\subseteq\ell^{\infty}\left[\underline{\epsilon}_{x},\overline{\epsilon}_{x}\right]$ (i.e., $\Pr\left[\mathbb{H}_{x}\in C\left[\underline{\epsilon}_{x},\overline{\epsilon}_{x}\right]\right]=1$) and $\mathbb{H}_{x}$ is a tight random element in $\ell^{\infty}\left[\underline{\epsilon}_{x},\overline{\epsilon}_{x}\right]$ (i.e., for every $\varepsilon>0$, there exists a compact set $K\subseteq\ell^{\infty}\left[\underline{\epsilon}_{x},\overline{\epsilon}_{x}\right]$ such that $\mathrm{Pr}\left[\mathbb{H}_{x}\notin K\right]\leq\varepsilon$).

The following notations are used to define an intermediate surrogate for $\widehat{F}_{\varDelta\mid X}\left(v\mid x\right)$. Let

Then, let \[ \widecheck{F}{}_{\varDelta\mid X}\left(v\mid x\right)\coloneqq\frac{1}{n_{x}}\sum_{i=1}^{n_{x}}\sum_{d\in\left\{ 0,1\right\} }\mathbbm{1}\left(\varDelta_{i}+\frac{H_{x}\left(\epsilon_{i}\right)}{\varsigma_{dx}\left(\epsilon_{i}\right)}\leq v\right)\mathbbm{1}\left(D_{i}=d'\right) \] be the intermediate surrogate of $\widehat{F}_{\varDelta\mid X}\left(v\mid x\right)$. In the appendix, using the Bahadur-type representation result given by Lemma 2 in \citetalias{Ma2023}, we show that

uniformly in $v\in\text{\ensuremath{\left[\underline{v}_{x},\overline{v}_{x}\right]}}$.

Let \[ \widetilde{F}_{\varDelta\mid X}\left(v\mid x\right)\coloneqq\frac{1}{n_{x}}\sum_{i=1}^{n_{x}}\mathbbm{1}\left(\varDelta_{i}\leq v\right),\,v\in\mathbb{R}, \] be the infeasible estimator using the true ITEs. Define the operator $\varPsi_{dx}:\ell^{\infty}\left[\underline{\epsilon}_{x},\overline{\epsilon}_{x}\right]\rightarrow\ell^{\infty}\left[\underline{v}_{x},\overline{v}_{x}\right]$ by

Then, in the appendix, we show that

uniformly in $v\in\left[\underline{v}_{x},\overline{v}_{x}\right]$. Note that ((ref)) and the continuous mapping theorem (CMT, see, e.g., Kosorok2007) imply $\left\Vert H_{x}\right\Vert _{\left[\underline{\epsilon}_{x},\overline{\epsilon}_{x}\right]}\rightarrow_{p}0$. Also, it is clear that all sample paths of $H_{x}$ reside in the space $D\left[\underline{\epsilon}_{x},\overline{\epsilon}_{x}\right]$. To establish the result in ((ref)), since the function class \[ \left\{ e\mapsto\mathbbm{1}\left(\varDelta_{x}\left(e\right)+\frac{h\left(e\right)}{\varsigma_{dx}\left(e\right)}\leq v\right):\left(v,h\right)\in\left[\underline{v}_{x},\overline{v}_{x}\right]\times D\left[\underline{\epsilon}_{x},\overline{\epsilon}_{x}\right]\right\} \] does not satisfy the bounded complexity (Donsker) condition, we follow the arguments of van2007empirical, which make use of ((ref)) and also the fact that the limit $\mathbb{H}_{x}$ concentrates on the much smaller separable Banach space $C\left[\underline{\epsilon}_{x},\overline{\epsilon}_{x}\right]$ . Now by using ((ref)) and ((ref)), we obtain the following approximation for $S_{F}\left(v\mid x\right)$:

uniformly in $v\in\left[\underline{v}_{x},\overline{v}_{x}\right]$.

Let $\left\{ \mathbb{B}_{1}\left(t\right):t\in\left[0,1\right]\right\} $ be a standard Brownian bridge and define the Gaussian process \[ \mathbb{F}_{1}\left(v\mid x\right)\coloneqq\mathbb{B}_{1}\left(F_{\varDelta\mid X}\left(v\mid x\right)\right),\,v\in\left[\underline{v}_{x},\overline{v}_{x}\right]. \] Since $\mathbb{B}_{1}$ has continuous sample paths almost surely, under the model assumptions, $\mathbb{F}_{1}\left(\cdot\mid x\right)$ concentrates on $C\left[\underline{v}_{x},\overline{v}_{x}\right]$. By the functional central limit theorem (see, e.g., van2000asymptotic),

In the appendix, we show that $\varPsi_{dx}$ is Hadamard differentiable (see, e.g., van2000asymptotic for the definition) at $\varDelta_{x}$ with derivative denoted by $\psi_{dx}$. By the functional delta method (see, e.g., van2000asymptotic), we have

uniformly in $v\in\left[\underline{v}_{x},\overline{v}_{x}\right]$. We can show that the leading term on the right hand side of ((ref)) is uncorrelated with the first term on the right hand side of ((ref)). Before characterizing its limiting distribution, we introduce the following notations. Let \[ f_{\left(\epsilon,D\right)\mid X}\left(e,d\mid x\right)\coloneqq f_{\epsilon\mid\left(D,X\right)}\left(e\mid d,x\right)\mathrm{Pr}\left[D=d\mid X=x\right] \] denote the conditional density of $\left(\epsilon,D\right)$ given $X=x$, and also let

Let $\left\{ \mathbb{B}_{2}\left(t\right):t\in\left[0,1\right]\right\} $ be a standard Brownian bridge that is independent of $\left\{ \mathbb{B}_{1}\left(t\right):t\in\left[0,1\right]\right\} $. Define the Gaussian process \[ \mathbb{F}_{2}\left(v\mid x\right)\coloneqq\sqrt{p_{1\mid x}^{-1}+p_{0\mid x}^{-1}}\left\{ \sum_{j=1}^{m}\omega_{x,j}\left(v\right)\mathbb{B}_{2}\left(F_{\epsilon\mid X}\left(\varDelta_{x,j}^{-1}\left(v\right)\mid x\right)\right)\right\} ,\,v\in\left[\underline{v}_{x},\overline{v}_{x}\right]. \] It is clear that under the model assumptions, $\mathbb{F}_{2}\left(\cdot\mid x\right)$ also concentrates on $C\left[\underline{v}_{x},\overline{v}_{x}\right]$. Then we can show that the leading term on the right hand side of ((ref)) also converges in distribution:

Now it follows from ((ref)), ((ref)), ((ref)), and ((ref)) that $S_{F}\left(\cdot\mid x\right)$ converges in distribution to a tight Gaussian process in $\ell^{\infty}\text{\ensuremath{\left[\underline{v}_{x},\overline{v}_{x}\right]}}$. We present it as the first main result of this paper in the following theorem.

Asymptotic Gaussianity of the empirical quantiles

The estimator $\widehat{Q}_{\varDelta\mid X}\left(\cdot\mid x\right)$ of the ITE quantile function defined in ((ref)) is a left continuous step function on $\left(0,1\right)$: for $\tau\in\left(0,1\right)$,

where $\widehat{\varDelta}_{\left\langle 1\right\rangle }\leq\cdots\leq\widehat{\varDelta}_{\left\langle n_{x}\right\rangle }$ are the order statistics corresponding to the pseudo ITEs. Then, we can show that the quantile estimator also has an asymptotically normal distribution. This result is presented in the following corollary to Theorem (ref).

Bootstrap inference

It has been discussed in Remarks (ref), (ref) and (ref) that bootstrapping seems to be a feasible approach to estimate the asymptotic distributions. In Section (ref), we discuss the construction and the algorithms of the bootstrap-based confidence intervals and UCBs. Section (ref) is devoted to the presentation of the results showing the asymptotic validity of the inference methods proposed in Section (ref).

Constructing bootstrap confidence intervals and UCBs

A nonparametric bootstrap sample $\left\{ W_{i}^{\dagger}\coloneqq\left(Y_{i}^{\dagger},D_{i}^{\dagger},Z_{i}^{\dagger}\right)^{\top}\right\} _{i=1}^{n_{x}}$ consists of $n_{x}$ independent draws from the original sample $\left\{ W_{i}\right\} _{i=1}^{n_{x}}$ with replacement. Let $\widehat{\varUpsilon}_{dx}^{\left(-i\right)\dagger}\left(t,y\right)$ denote the bootstrap analogue of $\widehat{\varUpsilon}_{dx}^{\left(-i\right)}\left(t,y\right)$, i.e., $\widehat{\varUpsilon}_{dx}^{\left(-i\right)\dagger}\left(t,y\right)$ is given by the right hand side of ((ref)) with $\left\{ W_{j}\right\} _{j\in\left[n_{x}\right]\setminus\left\{ i\right\} }$ replaced by $\left\{ W_{j}^{\dagger}\right\} _{j\in\left[n_{x}\right]\setminus\left\{ i\right\} }$. Let $\widehat{\phi}_{dx}^{\left(-i\right)\dagger}\left(y\right)$ be the bootstrap analogue of $\widehat{\phi}_{dx}^{\left(-i\right)}\left(y\right)$ defined by \[ \widehat{\phi}_{dx}^{\left(-i\right)\dagger}\left(y\right)\coloneqq\underset{t\in\left[\underline{y}_{dx},\overline{y}_{dx}\right]}{\arg\min}\widehat{\varUpsilon}_{dx}^{\left(-i\right)\dagger}\left(t,y\right). \] Similarly, we construct the bootstrap analogues \[ \widehat{\varDelta}_{i}^{\dagger}\coloneqq D_{i}^{\dagger}\left(Y_{i}^{\dagger}-\widehat{\phi}_{0x}^{\left(-i\right)\dagger}\left(Y_{i}^{\dagger}\right)\right)+\left(1-D_{i}^{\dagger}\right)\left(\widehat{\phi}_{1x}^{\left(-i\right)\dagger}\left(Y_{i}^{\dagger}\right)-Y_{i}^{\dagger}\right). \] and

Let $v$ be an interior point of $\mathscr{S}_{\varDelta\mid X=x}$. Let $\mathrm{Pr}_{\dagger}\left[\cdot\right]$ denote the conditional probability given the original sample. Now we construct the (asymptotically valid) bootstrap confidence interval for $F_{\varDelta\mid X}\left(v\mid x\right)$. For $p\in\left(0,1\right)$, let

be the $p$-th quantile of the resampling distribution of $\widehat{F}_{\varDelta\mid X}^{\dagger}\left(v\mid x\right)$ (i.e., the conditional distribution of $\widehat{F}_{\varDelta\mid X}^{\dagger}\left(v\mid x\right)$ given the original data). Note that the resampling distribution of $\widehat{F}_{\varDelta\mid X}^{\dagger}\left(v\mid x\right)$ can be easily simulated. The bootstrap percentile confidence interval with nominal coverage probability $1-\alpha$ for $F_{\varDelta\mid X}\left(v\mid x\right)$ is given by $\left[s_{F,\alpha/2}\left(v\mid x\right),s_{F,1-\alpha/2}\left(v\mid x\right)\right]$. The following algorithm summarizes the procedure that uses simulations to calculate the confidence interval $\left[s_{F,\alpha/2}\left(v\mid x\right),s_{F,1-\alpha/2}\left(v\mid x\right)\right]$. Let $B$ denote the number of bootstrap replications.

For any $\tau\in\left(0,1\right)$, it is also straightforward to construct a bootstrap confidence interval for the $\tau$-th quantile $Q_{\varDelta\mid X}\left(\tau\mid x\right)$ by adapting the preceding algorithm. For $\tau\in\left(0,1\right)$, denote

where $\widehat{\varDelta}_{\left\langle 1\right\rangle }^{\dagger}\leq\cdots\leq\widehat{\varDelta}_{\left\langle n_{x}\right\rangle }^{\dagger}$ are the order statistics corresponding to the pseudo ITEs from the bootstrap sample. Let $\widehat{\mathit{IR}}_{\varDelta\mid X=x}^{\dagger}\coloneqq\widehat{Q}_{\varDelta\mid X}^{\dagger}\left(0.75\mid x\right)-\widehat{Q}_{\varDelta\mid X}^{\dagger}\left(0.25\mid x\right)$ be the bootstrap analogue of the estimated IQR. For $p\in\left(0,1\right)$, let

be the $p$-th quantiles of the resampling distributions of $\widehat{Q}_{\varDelta\mid X}^{\dagger}\left(\tau\mid x\right)$ and $\widehat{\mathit{IR}}_{\varDelta\mid X=x}^{\dagger}$. Similarly, these resampling distributions can be simulated. The bootstrap percentile confidence intervals for the quantile and the IQR are given by $\left[s_{Q,\alpha/2}\left(\tau\mid x\right),s_{Q,1-\alpha/2}\left(\tau\mid x\right)\right]$ and $\left[s_{\mathit{IR},\alpha/2},s_{\mathit{IR},1-\alpha/2}\right]$. The following algorithm summarizes the simulation procedure for calculating these confidence intervals.

Next, we consider constructing bootstrap UCBs for the CDF over any inner closed sub-interval $\left[\underline{v}_{x},\overline{v}_{x}\right]$ of $\mathscr{S}_{\varDelta\mid X=x}$. Denote

For $p\in\left(0,1\right)$, let

be the $p$-th quantile of the resampling distribution of $\left\Vert S_{F}^{\dagger}\left(\cdot\mid x\right)\right\Vert _{\left[\underline{v}_{x},\overline{v}_{x}\right]}$. Then, we construct the UCB with the nominal coverage probability $1-\alpha$ from the following continuum

of random intervals using the critical value $s_{F,1-\alpha}^{\mathsf{unif}}$. The following discretization algorithm summarizes the simulation procedure for computing the bootstrap UCB $\left\{ \mathit{CB}_{F}\left(v\mid x\right):v\in\left[\underline{v}_{x},\overline{v}_{x}\right]\right\} $ for the ITE CDF. Let $T$ be a large positive integer and let $\mathcal{V}_{x}\coloneqq\left\{ v_{x}^{\left(1\right)},...,v_{x}^{\left(T\right)}\right\} $ be equally spaced grid points in $\left[\underline{v}_{x},\overline{v}_{x}\right]$.

Similarly, we can also construct bootstrap UCBs for the ITE quantile function over the range $\left[\underline{\tau},\overline{\tau}\right]$ for any $0<\underline{\tau}<\overline{\tau}<1$. Let

The bootstrap UCB with the nominal coverage probability $1-\alpha$ is given by the continuum of intervals

where $s_{Q,1-\alpha}^{\mathsf{unif}}$ is the $\left(1-\alpha\right)$-th quantile of the resampling distribution of $\left\Vert S_{Q}^{\dagger}\left(\cdot\mid x\right)\right\Vert _{\left[\underline{\tau},\overline{\tau}\right]}$. We summarize the procedure for computing $\left\{ \mathit{CB}_{Q}\left(\tau\mid x\right):\tau\in\left[\underline{\tau},\overline{\tau}\right]\right\} $ in the following algorithm. Let $T$ be a large positive integer and let $\mathcal{T}\coloneqq\left\{ \tau^{\left(1\right)},...,\tau^{\left(T\right)}\right\} $ be equally spaced grid points in $\left[\underline{\tau},\overline{\tau}\right]$.

Next, we consider variable-width UCBs that are based on studentized statistics. One of the advantages of variable-width UCBs is that they adjust to local variability and are narrower where the function is estimated more precisely, i.e., the estimator has a smaller pointwise variance. We follow the approach of Chernozhukov2018 to construct a variable-width UCB. Recall that $s_{Q,p}\left(\tau\mid x\right)$ is defined to be the $p$-th quantile of the resampling distribution of $\widehat{Q}_{\varDelta\mid X}^{\dagger}\left(\tau\mid x\right)$. Then it is clear that $\sqrt{n_{x}}\left(s_{Q,p}\left(\tau\mid x\right)-\widehat{Q}_{\varDelta\mid X}\left(\tau\mid x\right)\right)$ is the $p$-th quantile of the resampling distribution of $S_{Q}^{\dagger}\left(\tau\mid x\right)$. In the proof of Corollary (ref), we show that $\sqrt{n_{x}}\left(s_{Q,p}\left(\tau\mid x\right)-\widehat{Q}_{\varDelta\mid X}\left(\tau\mid x\right)\right)$ consistently estimates the $p$-th quantile of $\mathbb{Q}\left(\tau\mid x\right)\sim\mathrm{N}\left(0,V_{Q}\left(\tau\mid x\right)\right)$. Therefore, a consistent estimator of $V_{Q}\left(\tau\mid x\right)$ is given by \[ n_{x}\left(\frac{s_{Q,0.75}\left(\tau\mid x\right)-s_{Q,0.25}\left(\tau\mid x\right)}{z_{0.75}-z_{0.25}}\right)^{2}, \] where $z_{p}$ denotes the $p$-th quantile of $\mathrm{N}\left(0,1\right)$ and $s_{Q,0.75}\left(\tau\mid x\right)-s_{Q,0.25}\left(\tau\mid x\right)$ is the IQR of the resampling distribution of $\widehat{Q}_{\varDelta\mid X}^{\dagger}\left(\tau\mid x\right)$. Let \[ \tilde{s}_{Q,p}^{\mathsf{unif}}\coloneqq\inf\left\{ u\in\mathbb{R}:\mathrm{Pr}_{\dagger}\left[\underset{\tau\in\left[\underline{\tau},\overline{\tau}\right]}{\sup}\frac{\left|\widehat{Q}_{\varDelta\mid X}^{\dagger}\left(\tau\mid x\right)-\widehat{Q}_{\varDelta\mid X}\left(\tau\mid x\right)\right|}{\left(s_{Q,0.75}\left(\tau\mid x\right)-s_{Q,0.25}\left(\tau\mid x\right)\right)/\left(z_{0.75}-z_{0.25}\right)}\leq u\right]\geq p\right\} \] be the quantile of the resampling distribution of the supremum of the studentized version of $\left|S_{Q}^{\dagger}\left(\cdot\mid x\right)\right|$. A variable-width UCB is given by the continuum $\left\{ \widetilde{\mathit{CB}}_{Q}\left(\tau\mid x\right):\tau\in\left[\underline{\tau},\overline{\tau}\right]\right\} $ of intervals, where

A procedure to calculate the variable-width UCB consists of steps that are adaptations of those in Algorithms (ref) and (ref). We summarize the procedure in the following algorithm.

A variable-width UCB for the CDF can be defined analogously. The procedure for computation is similar to Algorithm (ref). We omit the details for simplicity.

Now it remains to show the asymptotic validity of these inference methods. We will show that the validity results essentially follow from bootstrap analogues of Theorem (ref) and Corollary (ref).

Asymptotic validity

Let $\mathrm{E}_{\dagger}\left[\cdot\right]$ denote the conditional expectation given the original sample. Suppose that $\mathbb{W}_{n_{x}}$ is a map (from the underlying probability space) into some Banach space $\mathbb{D}$. $\mathbb{W}_{n_{x}}$ depends on the bootstrap sample, and let $\mathbb{W}$ be a tight random element in $\mathbb{D}$, we use “$\mathbb{W}_{n_{x}}\rightsquigarrow_{\dagger}\mathbb{W}$ in $\mathbb{D}$” to denote convergence in distribution conditional on the original data: “$\mathbb{W}_{n_{x}}\rightsquigarrow_{\dagger}\mathbb{W}$ in $\mathbb{D}$” is understood as \[ \underset{h\in\mathit{BL}_{1}\left(\mathbb{D}\right)}{\sup}\left|\mathrm{E}_{\dagger}\left[h\left(\mathbb{W}_{n_{x}}\right)\right]-\mathrm{E}\left[h\left(\mathbb{W}\right)\right]\right|\rightarrow_{p}0, \] as $n_{x}\uparrow\infty$ (see van2000asymptotic). The following result shows that for any inner closed sub-interval $\text{\ensuremath{\left[\underline{v}_{x},\overline{v}_{x}\right]}}$ of $\mathscr{S}_{\varDelta\mid X=x}$, the bootstrap analogue $S_{F}^{\dagger}\left(\cdot\mid x\right)$ of $S_{F}\left(\cdot\mid x\right)$, defined by ((ref)), as a map from the underlying probability space into $\ell^{\infty}\left[\underline{v}_{x},\overline{v}_{x}\right]$ converges in distribution to the same limiting random element $\mathbb{F}\left(\cdot\mid x\right)$. It can be viewed as a bootstrap analogue of Theorem (ref)(i).

The asymptotic validity of the confidence interval $\left[s_{F,\alpha/2}\left(v\mid x\right),s_{F,1-\alpha/2}\left(v\mid x\right)\right]$ for $F_{\varDelta\mid X}\left(v\mid x\right)$ and the UCB $\left\{ \mathit{CB}_{F}\left(v\mid x\right):v\in\left[\underline{v}_{x},\overline{v}_{x}\right]\right\} $ for $F_{\varDelta\mid X}\left(v\mid x\right)$ over $v\in\left[\underline{v}_{x},\overline{v}_{x}\right]$ essentially follows from the stochastic convergence results ((ref)) and ((ref)) stated in the preceding remark and also the fact that the Kolmogorov-Smirnov distance between the distribution of $S_{F}\left(v\mid x\right)$ (or $\left\Vert S_{F}\left(\cdot\mid x\right)\right\Vert _{\left[\underline{v}_{x},\overline{v}_{x}\right]}$) and the distribution of $\mathbb{F}\left(v\mid x\right)$ (or $\left\Vert \mathbb{F}\left(\cdot\mid x\right)\right\Vert _{\left[\underline{v}_{x},\overline{v}_{x}\right]}$) converges to zero, which follows from van2000asymptotic and the continuity of the CDF of $\left\Vert \mathbb{F}\left(\cdot\mid x\right)\right\Vert _{\left[\underline{v}_{x},\overline{v}_{x}\right]}$. We present the asymptotic validity results in the following corollary. For simplicity, we give the result for the constant-width UCB only. The validity of the variable-width UCB follows from similar arguments.

Similarly, we can show a bootstrap analogue of Corollary (ref)(i). By using this result and similar arguments as those used in the proof of Corollary (ref), we can show the asymptotic validity of the bootstrap percentile confidence intervals $\left[s_{Q,\alpha/2}\left(\tau\mid x\right),s_{Q,1-\alpha/2}\left(\tau\mid x\right)\right]$ and $\left[s_{\mathit{IR},\alpha/2},s_{\mathit{IR},1-\alpha/2}\right]$ for the quantile $Q_{\varDelta\mid X}\left(\tau\mid x\right)$ and the IQR defined by ((ref)), and also the UCB $\left\{ \mathit{CB}_{Q}\left(\tau\mid x\right):\tau\in\left[\underline{\tau},\overline{\tau}\right]\right\} $ for $Q_{\varDelta\mid X}\left(\tau\mid x\right)$ over $\tau\in\left[\underline{\tau},\overline{\tau}\right]$. These results are summarized in the following corollary.

Extensions

This section is devoted to the presentation of several useful extensions to the results and algorithms given in the preceding section. Section (ref) considers inference on the ITE distribution conditional on a sub-vector of the covariate vector $X$.

In many empirical applications, the econometrician is interested in analyzing and comparing heterogeneous treatment effects in subgroups corresponding to different covariate values. Let $x_{1}$ and $x_{2}$ be two different values in $\mathscr{S}_{X}$. It would be of interest to compare the two ITE distributions “$\varDelta$ given $X=x_{1}$” versus “$\varDelta$ given $X=x_{2}$”. To this end, being interested in comparing central tendencies (or dispersions), one can employ the estimation and inference methods proposed in the preceding section and compare the confidence intervals for $Q_{\varDelta\mid X}\left(0.5\mid x_{1}\right)$ and $Q_{\varDelta\mid X}\left(0.5\mid x_{2}\right)$ (or those for $\mathit{IR}_{\varDelta\mid X=x_{1}}$ and $\mathit{IR}_{\varDelta\mid X=x_{2}}$). Another more transparent approach is to construct confidence intervals for the differences $Q_{\varDelta\mid X}\left(0.5\mid x_{1}\right)-Q_{\varDelta\mid X}\left(0.5\mid x_{2}\right)$ or $\mathit{IR}_{\varDelta\mid X=x_{1}}-\mathit{IR}_{\varDelta\mid X=x_{2}}$. One may be also interested in making judgement about equality of the entire ITE distributions, rather than comparing certain summary measures. This can be facilitated by computing and comparing the UCBs of $Q_{\varDelta\mid X}\left(\cdot\mid x_{1}\right)$ and $Q_{\varDelta\mid X}\left(\cdot\mid x_{2}\right)$. Similarly, one can also refer to an estimate and a UCB of the quantile difference function $Q_{\varDelta\mid X}\left(\cdot\mid x_{1}\right)-Q_{\varDelta\mid X}\left(\cdot\mid x_{2}\right)$. E.g., a constant quantile difference function suggests that the two ITE distributions are the same up to a location shift and a monotonic quantile difference function suggests that one ITE distribution is more dispersed than the other. In Section (ref), we present results and algorithms related to the problem of inference on quantile differences.

Conditioning on sub-vectors of the covariates

Suppose that $\tilde{X}$ is a sub-vector of $X$ and let $\tilde{X}_{i}$ denote the corresponding sub-vector of $X_{i}$. Let $A$ be a subset of $\mathscr{S}_{\tilde{X}}$. Let $F_{\varDelta\mid\tilde{X}}\left(v\mid A\right)\coloneqq\mathrm{Pr}\left[\varDelta\leq v\mid\tilde{X}\in A\right]$ be the conditional CDF of $\varDelta$ given $\tilde{X}\in A$. For $\tau\in\left(0,1\right)$, let $Q_{\varDelta\mid\tilde{X}}\left(\tau\mid A\right)\coloneqq\inf\left\{ y\in\mathbb{R}:F_{\varDelta\mid\tilde{X}}\left(y\mid A\right)\geq\tau\right\} $ denote the $\tau$-th quantile. Note that $A$ can be taken to be $\mathscr{S}_{\tilde{X}}$ such that $F_{\varDelta\mid\tilde{X}}\left(\cdot\mid A\right)$ equals the unconditional CDF $F_{\varDelta}$. Similarly, let \[ \mathit{IR}_{\varDelta\mid\tilde{X}\in A}\coloneqq Q_{\varDelta\mid\tilde{X}}\left(0.75\mid A\right)-Q_{\varDelta\mid\tilde{X}}\left(0.25\mid A\right) \] be the IQR of the conditional distribution of $\varDelta$ given $\tilde{X}\in A$. We consider the problem of estimation and inference for $F_{\varDelta\mid\tilde{X}}\left(v\mid A\right)$, $Q_{\varDelta\mid\tilde{X}}\left(\tau\mid A\right)$ and $\mathit{IR}_{\varDelta\mid\tilde{X}\in A}$.

Our sample consists of i.i.d. observations $\left\{ W_{i}\right\} _{i=1}^{n_{A}}$ with observed covariates $X_{i}$ satisfying $\tilde{X}_{i}\in A$, where we redefine $W_{i}$ as $W_{i}\coloneqq\left(Y_{i},D_{i},Z_{i},X_{i}^{\top}\right)^{\top}$ collecting the observed variables from the $i$-th individual for notational convenience. Under this sampling assumption, the probability masses of $X$ are given by $\left\{ \mathrm{Pr}\left[X=x\mid\tilde{X}\in A\right]:x\in\mathscr{S}_{X\mid\tilde{X}\in A}\right\} $, where $\mathscr{S}_{X\mid\tilde{X}\in A}$ denotes the conditional support of $X$ given $\tilde{X}\in A$. For each $x\in\mathscr{S}_{X\mid\tilde{X}\in A}$, we redefine $\widehat{\varUpsilon}_{dx}^{\left(-i\right)}\left(t,y\right)$ as

i.e., the leave-$i$-out sample analogue of the right hand side of ((ref)) using $\left\{ W_{i}\right\} _{i=1}^{n_{A}}$ as the sample. The leave-$i$-out nonparametric estimator $\widehat{\phi}_{dx}^{\left(-i\right)}\left(y\right)$ of $\phi_{dx}\left(y\right)$ can be defined similarly as $\widehat{\phi}_{dx}^{\left(-i\right)}\left(y\right)\coloneqq\textrm{argmin}_{t\in\left[\underline{y}_{dx},\overline{y}_{dx}\right]}\widehat{\varUpsilon}_{dx}^{\left(-i\right)}\left(t,y\right)$. We redefine $\widehat{\varDelta}_{i}$ as the pseudo ITE

for the $i$-th individual in the sample.

Let

be the nonparametric estimator of $F_{\varDelta\mid\tilde{X}}\left(v\mid A\right)$ using the pseudo ITEs defined by ((ref)). For each $\tau\in\left(0,1\right)$, let

be the estimated quantile, where $\widehat{\varDelta}_{\left\langle 1\right\rangle }\leq\cdots\leq\widehat{\varDelta}_{\left\langle n_{A}\right\rangle }$ are the order statistics corresponding to $\left\{ \widehat{\varDelta}_{i}\right\} _{i=1}^{n_{A}}$. Similarly, we let $\widehat{\mathit{IR}}_{\varDelta\mid\tilde{X}\in A}\coloneqq\widehat{Q}_{\varDelta\mid\tilde{X}}\left(0.75\mid A\right)-\widehat{Q}_{\varDelta\mid\tilde{X}}\left(0.25\mid A\right)$ be the estimator of $\mathit{IR}_{\varDelta\mid\tilde{X}\in A}$. Let $\left[\underline{v}_{A},\overline{v}_{A}\right]$ be an inner closed sub-interval of $\mathscr{S}_{\varDelta\mid\tilde{X}\in A}$. Let

and let $S_{Q}\left(\tau\mid A\right)$ be defined analogously. By using the same arguments as those in the proof of Theorem (ref)(i), we can show that $S_{F}\left(\cdot\mid A\right)$ converges in distribution to a tight Gaussian process in $\ell^{\infty}\left[\underline{v}_{A},\overline{v}_{A}\right]$. An analogous result can be established for $S_{Q}\left(\cdot\mid A\right)$ that takes values in $\ell^{\infty}\left[\underline{\tau},\overline{\tau}\right]$.

A nonparametric bootstrap sample $\left\{ W_{i}^{\dagger}\right\} _{i=1}^{n_{A}}$ is obtained by independently drawing $n_{A}$ observations from the original sample $\left\{ W_{i}\right\} _{i=1}^{n_{A}}$ and let $Y_{i}^{\dagger},D_{i}^{\dagger},Z_{i}^{\dagger}$ and $X_{i}^{\dagger}$ be the corresponding components of the vector $W_{i}^{\dagger}$. By replacing $\left\{ W_{j}\right\} _{j\in\left[n_{A}\right]\setminus\left\{ i\right\} }$ on the right hand side of ((ref)) with $\left\{ W_{j}^{\dagger}\right\} _{j\in\left[n_{A}\right]\setminus\left\{ i\right\} }$, we get the bootstrap analogue $\widehat{\varUpsilon}_{dx}^{\left(-i\right)\dagger}\left(t,y\right)$ of $\widehat{\varUpsilon}_{dx}^{\left(-i\right)}\left(t,y\right)$. Let $\widehat{\phi}_{dx}^{\left(-i\right)\dagger}\left(y\right)\coloneqq\textrm{argmin}_{t\in\left[\underline{y}_{dx},\overline{y}_{dx}\right]}\widehat{\varUpsilon}_{dx}^{\left(-i\right)\dagger}\left(t,y\right)$ be the bootstrap analogue of $\widehat{\phi}_{dx}^{\left(-i\right)}\left(y\right)$ and by using this counterfactual mapping estimator from the bootstrap sample and replacing $\left(Y_{i},D_{i},X_{i}\right)$ and $\left(\widehat{\phi}_{0X_{i}}^{\left(-i\right)},\widehat{\phi}_{1X_{i}}^{\left(-i\right)}\right)$ on the right hand side of ((ref)) with their bootstrap analogues, we construct the pseudo ITEs $\left\{ \widehat{\varDelta}_{i}^{\dagger}\right\} _{i=1}^{n_{A}}$ from the bootstrap sample. Let

be bootstrap analogues of $\widehat{F}_{\varDelta\mid\tilde{X}}\left(v\mid A\right)$, $\widehat{Q}_{\varDelta\mid\tilde{X}}\left(\tau\mid A\right)$ and $\widehat{\mathit{IR}}_{\varDelta\mid\tilde{X}\in A}$. Note that we have $\widehat{Q}_{\varDelta\mid\tilde{X}}^{\dagger}\left(\tau\mid A\right)=\widehat{\varDelta}_{\left\langle \left\lceil \tau n_{A}\right\rceil \right\rangle }^{\dagger}$, where $\widehat{\varDelta}_{\left\langle 1\right\rangle }^{\dagger}\leq\cdots\leq\widehat{\varDelta}_{\left\langle n_{A}\right\rangle }^{\dagger}$ are the order statistics corresponding to $\left\{ \widehat{\varDelta}_{i}^{\dagger}\right\} _{i=1}^{n_{A}}$. Bootstrap percentile confidence intervals for $F_{\varDelta\mid\tilde{X}}\left(v\mid A\right)$, $Q_{\varDelta\mid\tilde{X}}\left(\tau\mid A\right)$ and $\mathit{IR}_{\varDelta\mid\tilde{X}\in A}$ can be defined by using the $\left(\alpha/2\right)$-th and the $\left(1-\alpha/2\right)$-th quantiles of the resampling distributions of $\widehat{F}_{\varDelta\mid\tilde{X}}^{\dagger}\left(v\mid A\right)$, $\widehat{Q}_{\varDelta\mid\tilde{X}}^{\dagger}\left(\tau\mid A\right)$ and $\widehat{\mathit{IR}}_{\varDelta\mid\tilde{X}\in A}^{\dagger}$ as the end points.

The end points of these bootstrap confidence intervals can be easily estimated by Monte Carlo simulations. It is straightforward to adapt Algorithms (ref) and (ref) to obtain bootstrap percentile confidence intervals. In the first two steps, in the $r$-th bootstrap replication, we independently draw a bootstrap sample $\left\{ W_{i}^{\dagger\left(r\right)}\right\} _{i=1}^{n_{A}}$ and compute the pseudo ITEs $\left\{ \widehat{\varDelta}_{i}^{\dagger\left(r\right)}\right\} _{i=1}^{n_{A}}$ using the procedure described in the preceding paragraph. Then by using the formulae given by ((ref)) with $\left\{ \widehat{\varDelta}_{i}^{\dagger}\right\} _{i=1}^{n_{A}}$ replaced by $\left\{ \widehat{\varDelta}_{i}^{\dagger\left(r\right)}\right\} _{i=1}^{n_{A}}$, we can easily compute $\widehat{F}_{\varDelta\mid\tilde{X}}^{\dagger\left(r\right)}\left(v\mid A\right)$ and $\widehat{Q}_{\varDelta\mid\tilde{X}}^{\dagger\left(r\right)}\left(\tau\mid A\right)=\widehat{\varDelta}_{\left\langle \left\lceil \tau n_{A}\right\rceil \right\rangle }^{\dagger\left(r\right)}$, where $\widehat{\varDelta}_{\left\langle 1\right\rangle }^{\dagger\left(r\right)}\leq\cdots\leq\widehat{\varDelta}_{\left\langle n_{A}\right\rangle }^{\dagger\left(r\right)}$ are the order statistics corresponding to $\left\{ \widehat{\varDelta}_{i}^{\dagger\left(r\right)}\right\} _{i=1}^{n_{A}}$. The rest of the steps are identical to those in Algorithms (ref) and (ref).

The UCBs ((ref)) and ((ref)) constructed in Section (ref) can also be easily extended. A bootstrap UCB for $F_{\varDelta\mid\tilde{X}}\left(v\mid A\right)$ over $v\in\left[\underline{v}_{A},\overline{v}_{A}\right]$ with nominal coverage probability $1-\alpha$ centers around $\widehat{F}_{\varDelta\mid\tilde{X}}\left(v\mid A\right)$ and has radius given by the $\left(1-\alpha\right)$-th quantile of the resampling distribution of $\left\Vert \widehat{F}_{\varDelta\mid\tilde{X}}^{\dagger}\left(\cdot\mid A\right)-\widehat{F}_{\varDelta\mid\tilde{X}}\left(\cdot\mid A\right)\right\Vert _{\left[\underline{v}_{A},\overline{v}_{A}\right]}$. A bootstrap UCB for $Q_{\varDelta\mid\tilde{X}}\left(\tau\mid A\right)$ over $\tau\in\left[\underline{\tau},\overline{\tau}\right]$ can be constructed analogously. A straightforward adaptation leads to the construction of a variable-width bootstrap UCB for $Q_{\varDelta\mid\tilde{X}}\left(\cdot\mid A\right)$ similar to ((ref)).

We again easily adapt Algorithms (ref) and (ref). The first two or three steps are the same as those in the algorithms for computing the bootstrap percentile confidence intervals. Then, we compute $\widehat{F}_{\varDelta\mid\tilde{X}}^{\dagger\left(r\right)}\left(v\mid A\right)-\widehat{F}_{\varDelta\mid\tilde{X}}\left(v\mid A\right)$ for $\left(r,v\right)\in\left[B\right]\times\mathcal{V}_{A}$, where $\mathcal{V}_{A}\coloneqq\left\{ v_{A}^{\left(1\right)},...,v_{A}^{\left(T\right)}\right\} $ are equally spaced grid points in $\left[\underline{v}_{A},\overline{v}_{A}\right]$ and $\widehat{Q}_{\varDelta\mid\tilde{X}}^{\dagger\left(r\right)}\left(\tau\mid A\right)-\widehat{Q}_{\varDelta\mid\tilde{X}}\left(\tau\mid A\right)$ for $\left(r,\tau\right)\in\left[B\right]\times\mathcal{T}$. The simulated critical values are given by the $\left(1-\alpha\right)$-th empirical quantiles of \[ \left\{ \max_{v\in\mathcal{V}_{A}}\left|\widehat{F}_{\varDelta\mid\tilde{X}}^{\dagger\left(r\right)}\left(v\mid A\right)-\widehat{F}_{\varDelta\mid\tilde{X}}\left(v\mid A\right)\right|\right\} _{r=1}^{B}\textrm{ and }\left\{ \max_{\tau\in\mathcal{T}}\left|\widehat{Q}_{\varDelta\mid\tilde{X}}^{\dagger\left(r\right)}\left(\tau\mid A\right)-\widehat{Q}_{\varDelta\mid\tilde{X}}\left(\tau\mid A\right)\right|\right\} _{r=1}^{B}, \] respectively. As those in Algorithms (ref) and (ref), the UCBs are collections of intervals centered around $\left\{ \widehat{F}_{\varDelta\mid\tilde{X}}\left(v\mid A\right)\right\} _{v\in\mathcal{V}_{A}}$ and $\left\{ \widehat{Q}_{\varDelta\mid\tilde{X}}\left(\tau\mid A\right)\right\} _{\tau\in\mathcal{T}}$ with radii given by these critical values. The variable-width counterparts can be computed analogously.

Let $S_{F}^{\dagger}\left(v\mid A\right)$ be the bootstrap analogue of ((ref)) defined analogously to ((ref)). Similarly, let $S_{Q}^{\dagger}\left(\tau\mid A\right)$ denote the bootstrap analogue of $S_{Q}\left(\tau\mid A\right)$. To justify the validity of the inference methods just proposed, we can use the same arguments as those in the proofs of Theorem (ref) and Corollary (ref)(i) to show that $S_{F}^{\dagger}\left(\cdot\mid A\right)$ and $S_{Q}^{\dagger}\left(\cdot\mid A\right)$ converge in distribution conditionally on the original data to the same limits as those of $S_{F}\left(\cdot\mid A\right)$ and $S_{Q}\left(\cdot\mid A\right)$. The asymptotic validity follows from these results and arguments in the proofs of Corollaries (ref) and (ref).

Comparison of ITE distributions

Let $A_{0}$ and $A_{1}$ be two disjoint subsets of $\mathscr{S}_{\tilde{X}}$ respectively. We consider the problem of comparing the ITE distributions conditional on $\tilde{X}\in A_{0}$ and $\tilde{X}\in A_{1}$ respectively. Let $\delta\left(\tau\right)\coloneqq Q_{\varDelta\mid\tilde{X}}\left(\tau\mid A_{1}\right)-Q_{\varDelta\mid\tilde{X}}\left(\tau\mid A_{0}\right)$ for $\tau\in\left[\underline{\tau},\overline{\tau}\right]$ denote the difference of the $\tau$-th quantiles. In empirical applications, it may be interesting to learn about $\delta\left(\tau\right)$. E.g., we can conclude which subgroup of individuals tend to have a larger median effect by constructing a confidence interval for $\delta\left(0.5\right)$ and drawing inference on the sign of $\delta\left(0.5\right)$. Similarly, the difference of dispersions of ITE distributions can be measured by $\mathit{IR}_{\varDelta\mid\tilde{X}\in A_{1}}-\mathit{IR}_{\varDelta\mid\tilde{X}\in A_{0}}=\delta\left(0.75\right)-\delta\left(0.25\right)$ and knowledge about the sign of this quantity is useful in determining which subgroup of individuals tend to have more dispersed ITEs.

Our sample is the union of two independent samples $\left\{ W_{0,i}\right\} _{i=1}^{n_{0}}$ and $\left\{ W_{1,i}\right\} _{i=1}^{n_{1}}$. Let $n\coloneqq n_{0}+n_{1}$ be the sample size. Let $\widehat{\delta}\left(\tau\right)\coloneqq\widehat{Q}_{\varDelta\mid\tilde{X}}\left(\tau\mid A_{1}\right)-\widehat{Q}_{\varDelta\mid\tilde{X}}\left(\tau\mid A_{0}\right)$ be the estimator of $\delta\left(\tau\right)$ based on ((ref)) defined in the preceding subsection. Under the additional assumption that the limits of $n_{0}/n$ and $n_{1}/n$ as $n_{0},n_{1}\uparrow\infty$ exist, we can show that $\sqrt{n}\left(\widehat{\delta}-\delta\right)$ converges in distribution in $\ell^{\infty}\left[\underline{\tau},\overline{\tau}\right]$ to the sum of two independent tight Gaussian processes. Let $\widehat{\delta}^{\dagger}\left(\tau\right)\coloneqq\widehat{Q}_{\varDelta\mid\tilde{X}}^{\dagger}\left(\tau\mid A_{1}\right)-\widehat{Q}_{\varDelta\mid\tilde{X}}^{\dagger}\left(\tau\mid A_{0}\right)$ denote the bootstrap analogue of $\widehat{\delta}\left(\tau\right)$ constructed from bootstrap samples $\left\{ W_{0,i}^{\dagger}\right\} _{i=1}^{n_{0}}$ and $\left\{ W_{1,i}^{\dagger}\right\} _{i=1}^{n_{1}}$ of $\left\{ W_{0,i}\right\} _{i=1}^{n_{0}}$ and $\left\{ W_{1,i}\right\} _{i=1}^{n_{1}}$. We can show that $\sqrt{n}\left(\widehat{\delta}^{\dagger}-\widehat{\delta}\right)$ converges in distribution conditionally on the original data to the same limiting tight Gaussian process. The asymptotic validity of all inference methods follow from these results. Bootstrap percentile confidence intervals for $\delta\left(\tau\right)$ (or $\delta\left(0.75\right)-\delta\left(0.25\right)$) can be defined by using the $\left(\alpha/2\right)$-th and $\left(1-\alpha/2\right)$-th quantiles of the resampling distribution of $\widehat{\delta}^{\dagger}\left(\tau\right)$ (or $\widehat{\delta}^{\dagger}\left(0.75\right)-\widehat{\delta}^{\dagger}\left(0.25\right)$) as the end points. We summarize the procedure for computing these confidence intervals in the following algorithm.

In applications, one may also be interested in comparing the entire ITE distributions of two subgroups. To this end, one can use a UCB for $\delta\left(\tau\right)$ over $\tau\in\left[\underline{\tau},\overline{\tau}\right]$ with $\underline{\tau}$ and $\overline{\tau}$ chosen to be close to 0 and 1 (e.g., $\left[\underline{\tau},\overline{\tau}\right]=\left[0.1,0.9\right]$). It is straightforward to extend the method proposed in Section (ref). The desired UCB with nominal coverage probability $1-\alpha$ centers around $\widehat{\delta}\left(\tau\right)$ and has radius given by the $\left(1-\alpha\right)$-th quantile of the resampling distribution of $\left\Vert \widehat{\delta}^{\dagger}-\widehat{\delta}\right\Vert _{\left[\underline{\tau},\overline{\tau}\right]}$. We summarize the procedure for this UCB in the following algorithm.

A variable-width UCB for the quantile difference function can be constructed by following the approach of Chernozhukov2018 and using the calculations in Algorithms (ref) and (ref). The following algorithm summarizes the procedure.

We can use the UCB constructed by Algorithm (ref) or Algorithm (ref) to test the equality of the two ITE distributions. The null hypothesis in this case is “$\mathrm{H}_{0}^{\mathsf{a}}$: $\delta\left(\tau\right)=0$, for all $\tau\in\left[\underline{\tau},\overline{\tau}\right]$” and the alternative hypothesis is “$\mathrm{H}_{1}^{\mathsf{a}}$: $\delta\left(\tau\right)\neq0$ for some unknown $\tau\in\left[\underline{\tau},\overline{\tau}\right]$”. We do not reject $\mathrm{H}_{0}^{\mathsf{a}}$ if the zero function $\left[\underline{\tau},\overline{\tau}\right]\ni\tau\mapsto0$ is covered by the confidence band (i.e., $\sup_{\tau\in\mathcal{T}}\left|\widehat{\delta}\left(\tau\right)\right|\leq s_{\delta,\left\langle \left\lceil B\left(1-\alpha\right)\right\rceil \right\rangle }^{\dagger}$) and reject $\mathrm{H}_{0}^{\mathsf{a}}$ otherwise ($\sup_{\tau\in\mathcal{T}}\left|\widehat{\delta}\left(\tau\right)\right|>s_{\delta,\left\langle \left\lceil B\left(1-\alpha\right)\right\rceil \right\rangle }^{\dagger}$). Note that the asymptotic validity of the UCB immediately implies the asymptotic validity of the test.

In empirical applications, it can be interesting to learn whether the conditional ITE distribution given $\tilde{X}\in A_{0}$ is the same as the conditional distribution given $\tilde{X}\in A_{1}$ up to a location shift (i.e., $\delta:\left[\underline{\tau},\overline{\tau}\right]\rightarrow\mathbb{R}$ is some unknown constant function) or there is also difference in dispersions. This testing “equality up to a location shift” problem is a generalization of equality testing. Let $\gamma\left(\tau\right)\coloneqq\delta\left(\tau\right)-\left(\int_{\underline{\tau}}^{\overline{\tau}}\delta\left(t\right)\mathrm{d}t\right)/\left(\overline{\tau}-\underline{\tau}\right)$ for $\tau\in\left[\underline{\tau},\overline{\tau}\right]$. The problem can be formulated as testing the null hypothesis “$\mathrm{H}_{0}^{\mathsf{b}}$: $\gamma\left(\tau\right)=0$, for all $\tau\in\left[\underline{\tau},\overline{\tau}\right]$” against the alternative hypothesis “$\mathrm{H}_{1}^{\mathsf{b}}$: $\gamma\left(\tau\right)\neq0$, for some unknown $\tau\in\left[\underline{\tau},\overline{\tau}\right]$”. Let $\widehat{\gamma}\left(\tau\right)\coloneqq\widehat{\delta}\left(\tau\right)-\left(\int_{\underline{\tau}}^{\overline{\tau}}\widehat{\delta}\left(t\right)\mathrm{d}t\right)/\left(\overline{\tau}-\underline{\tau}\right)$ be the estimator of $\gamma\left(\tau\right)$. The bootstrap analogue $\widehat{\gamma}^{\dagger}\left(\tau\right)$ of $\widehat{\gamma}\left(\tau\right)$ is defined analogously.\footnote{It follows from the continuity of the map $f\mapsto f-\left(\int_{\underline{\tau}}^{\overline{\tau}}f\left(t\right)\mathrm{d}t\right)/\left(\overline{\tau}-\underline{\tau}\right)$ and CMT that $\sqrt{n}\left(\widehat{\gamma}-\gamma\right)$ (or $\sqrt{n}\left(\widehat{\gamma}^{\dagger}-\widehat{\gamma}\right)$) converges in distribution (conditionally on the original data) to a tight Gaussian process.} Similarly, an asymptotically valid test of equality up to a location shift can be based on using an asymptotically valid UCB for $\gamma\left(\tau\right)$ over $\tau\in\left[\underline{\tau},\overline{\tau}\right]$, whose construction is a straightforward extension of the UCB for $\delta\left(\tau\right)$ over $\tau\in\left[\underline{\tau},\overline{\tau}\right]$. For practical computation, we can easily adapt Algorithm (ref) or Algorithm (ref). Steps 1-3 are the same as those in Algorithm (ref). Then, we compute $\left\{ \left|\widehat{\gamma}^{\dagger\left(r\right)}\left(\tau\right)-\widehat{\gamma}\left(\tau\right)\right|\right\} _{\left(r,\tau\right)\in\left[B\right]\times\mathcal{T}}$ and order $\left\{ \max_{\tau\in\mathcal{T}}\left|\widehat{\gamma}^{\dagger\left(r\right)}\left(\tau\right)-\widehat{\gamma}\left(\tau\right)\right|\right\} _{r=1}^{B}$ to get the order statistics $s_{\gamma,\left\langle 1\right\rangle }^{\dagger}\leq\cdots\leq s_{\gamma,\left\langle B\right\rangle }^{\dagger}$ and the critical value $s_{\gamma,\left\langle \left\lceil B\left(1-\alpha\right)\right\rceil \right\rangle }^{\dagger}$. We reject $\mathrm{H}_{0}^{\mathsf{b}}$ if $\sup_{\tau\in\mathcal{T}}\left|\widehat{\gamma}\left(\tau\right)\right|>s_{\gamma,\left\langle \left\lceil B\left(1-\alpha\right)\right\rceil \right\rangle }^{\dagger}$.

We can also use a one-sided UCB to test the hypothesis that the conditional ITE distribution given $\tilde{X}\in A_{0}$ stochastically dominates the conditional distribution given $\tilde{X}\in A_{1}$, which can be formulated as testing “$\mathrm{H}_{0}^{\mathsf{c}}$: $\delta\left(\tau\right)\leq0$, for all $\tau\in\left[\underline{\tau},\overline{\tau}\right]$” against the alternative hypothesis “$\mathrm{H}_{1}^{\mathsf{c}}$: $\delta\left(\tau\right)>0$, for some unknown $\tau\in\left[\underline{\tau},\overline{\tau}\right]$”. Let $\dot{s}_{\delta,1-\alpha}^{\mathsf{unif}}$ denote the $\left(1-\alpha\right)$-th quantile of the resampling distribution of $\sup_{\tau\in\left[\underline{\tau},\overline{\tau}\right]}\left\{ \widehat{\delta}^{\dagger}\left(\tau\right)-\widehat{\delta}\left(\tau\right)\right\} $. A one-sided bootstrap UCB is given by $\left\{ \left[\widehat{\delta}\left(\tau\right)-\dot{s}_{\delta,1-\alpha}^{\mathsf{unif}},\infty\right):\tau\in\left[\underline{\tau},\overline{\tau}\right]\right\} $. We accept $\mathrm{H}_{0}^{\mathsf{c}}$ if the constant zero function is covered by the UCB (i.e., the lower bound of the UCB is smaller than zero for all $\tau$). We can show that under $\mathrm{H}_{0}^{\mathsf{c}}$, \[ \Pr\left[\underset{\tau\in\left[\underline{\tau},\overline{\tau}\right]}{\sup}\widehat{\delta}\left(\tau\right)>\dot{s}_{\delta,1-\alpha}^{\mathsf{unif}}\right]\leq\Pr\left[\underset{\tau\in\left[\underline{\tau},\overline{\tau}\right]}{\sup}\left\{ \widehat{\delta}\left(\tau\right)-\delta\left(\tau\right)\right\} >\dot{s}_{\delta,1-\alpha}^{\mathsf{unif}}\right], \] and the right hand side of the inequality converges to $\alpha$ as $n_{0},n_{1}\uparrow\infty$. This result shows that the proposed test is asymptotically valid. We can easily adapt Algorithm (ref) for practical computation of the critical value $\dot{s}_{\delta,1-\alpha}^{\mathsf{unif}}$. Steps 1-4 are the same as those in Algorithm (ref). Then, we order $\left\{ \max_{\tau\in\mathcal{T}}\left\{ \widehat{\delta}^{\dagger\left(r\right)}\left(\tau\right)-\widehat{\delta}\left(\tau\right)\right\} \right\} _{r=1}^{B}$ to get the corresponding order statistics $\dot{s}_{\delta,\left\langle 1\right\rangle }^{\dagger}\leq\cdots\leq\dot{s}_{\delta,\left\langle B\right\rangle }^{\dagger}$. The critical value is given by $\dot{s}_{\delta,\left\langle \left\lceil B\left(1-\alpha\right)\right\rceil \right\rangle }^{\dagger}$. We reject $\mathrm{H}_{0}^{\mathsf{c}}$ if $\sup_{\tau\in\mathcal{T}}\widehat{\delta}\left(\tau\right)>\dot{s}_{\delta,\left\langle \left\lceil B\left(1-\alpha\right)\right\rceil \right\rangle }^{\dagger}$.

Monte Carlo simulations

Section (ref) examines the quality of the Gaussian approximation to the finite sample distributions of the estimators proposed in Section (ref). The Gaussian approximation is justified by the asymptotic results in Sections (ref) and (ref). Section (ref) provides simulation results to assess the finite sample performances of the inference methods proposed in Section (ref).

We consider the same DGP as in the simulation section of \citetalias{feng2019estimation}. The same DGP is also used in the simulations in \citetalias{Ma2023}. The outcome and treatment status are generated by $Y=\left(\epsilon+1\right)^{2+D}$ and $D=\mathbbm{1}\left(-0.5+0.5\cdot Z+\eta\geq0\right)$, where $\left(\epsilon,\eta\right)=\left(\varPhi\left(U\right),\varPhi\left(V\right)\right)$, $\left(U,V\right)$ follow a mean-zero bivariate normal distribution with $\mathrm{Var}\left[U\right]=\mathrm{Var}\left[V\right]=1$ and $\mathrm{Cov}\left[U,V\right]=0.3$. Here, $\varPhi$ denotes the CDF of $\mathrm{N}\left(0,1\right)$. The IV is generated by $Z=\mathbbm{1}\left(N>0\right)$, where $N\sim\mathrm{N}\left(0,1\right)$ is independent of $\left(\epsilon,\eta\right)$. It is straightforward to check that the ITE is given by $\varDelta=\epsilon\left(\epsilon+1\right)^{2}$, where $\epsilon=\varPhi\left(U\right)$ follows a uniform distribution on $\left[0,1\right]$. Therefore, the support of $\varDelta$ is $\left[0,4\right]$. Throughout the simulations, the number of Monte Carlo replications is set to $1,000$, and the number of bootstrap replications is set to 500. Let $n$ denote the sample size in each of the Monte Carlo replications.

Validity of the asymptotic theory

To avoid redundancy, we focus on estimating the $\tau$-th quantile $Q_{\varDelta}\left(\tau\right)$ using the empirical quantiles of pseudo ITEs and omit the results that assess the quality of the estimator of the cumulative probabilities. In Figure (ref), each histogram displays realizations of $\widehat{Q}_{\varDelta}\left(\tau\right)$, the $\tau$-th empirical quantile of pseudo ITEs, computed over $1,000$ simulation replications. The solid curve in each panel represents the large sample density of $\widehat{Q}_{\varDelta}\left(\tau\right)$, i.e., the Gaussian density with mean $Q_{\varDelta}\left(\tau\right)$ and variance $V_{Q}\left(\tau\right)/n$, as characterized by Corollary (ref)(ii), for $\tau\in\{0.25,0.5,0.75\}$ and for $n=250$ and $500$. Figure (ref) displays analogous results for more extreme quantiles, with $\tau\in\{0.1,0.9\}$. Both figures demonstrate close agreement between the simulated distributions of $\widehat{Q}_{\varDelta}\left(\tau\right)$ and the corresponding large sample Gaussian distributions for moderate sample sizes across a range of probability levels, including relatively extreme levels such as $0.1$ and $0.9$.

Finite sample performances of the inference methods

This section evaluates the finite sample performances of the confidence intervals and UCBs proposed in Algorithms (ref) to (ref). We consider the same DGP as in the preceding subsection and examine the inference methods for four target parameters: (i) bootstrap percentile confidence intervals for the cumulative probabilities $F_{\varDelta}\left(v\right)$ for fixed values of $v$; (ii) bootstrap UCBs for the CDF (values $F_{\varDelta}\left(v\right)$ of the CDF over a range of $v$'s); (iii) bootstrap percentile confidence intervals for the ITE quantiles $Q_{\varDelta}\left(\tau\right)$ for fixed values of $\tau$; (iv) bootstrap UCBs for the quantile function (the values $Q_{\varDelta}\left(\tau\right)$ of the quantile function over a range of $\tau$'s). The sample sizes considered are $n=250$, $500$ and $1,000$.

Table (ref) reports the pointwise coverage probabilities and the expected lengths of the bootstrap percentile confidence interval (denoted as BP) proposed in Algorithm (ref) for the cumulative probabilities $F_{\varDelta}\left(v\right)$, at $v\in\{0.5,1,2,3,3.5\}$. For comparison, the table also includes a “naive” confidence interval (NAI), which is constructed using the standard error $\sqrt{\widehat{F}_{\varDelta}\left(v\right)\left(1-\widehat{F}_{\varDelta}\left(v\right)\right)/n}$ and accounts only for the component $V_{1}\left(v\right)$ of the asymptotic variance given in Theorem (ref)(ii), ignoring the ITE estimation error. The results show that the bootstrap percentile confidence interval for $F_{\varDelta}\left(v\right)$ described in Algorithm (ref) provides coverage probabilities close to the nominal level across all values of $v$ and sample sizes. In contrast, the “naive” confidence intervals severely undercover, highlighting the importance of accounting for the estimation error captured by $V_{2}\left(v\right)$, which may contribute more to the asymptotic variance than the canonical sampling variation $V_{1}\left(v\right)$.

Table (ref) reports the simultaneous coverage probabilities of the constant-width UCB from Algorithm (ref) and the variable-width UCB, constructed analogously to Algorithm (ref), for the CDF $F_{\varDelta}$ over equally spaced grid points in the intervals $[0.04,3.96]$ and $[0.10,3.90]$ respectively with the step size $0.01$. For comparison, we include Interpolated BP which constructs a band by interpolating the pointwise bootstrap percentile confidence intervals in Algorithm (ref). Table (ref) shows that the UCBs lead to good simultaneous coverage. Although the interpolated BP intervals perform well pointwise (as in Table (ref)), they perform poorly for uniform coverage. We also calculate the average expected widths of the two confidence bands and show the results in Table (ref).\footnote{The average expected width is computed by first averaging the widths in all simulation replications at each grid point and then averaging over all grid points in the given range.}

Table (ref) presents results showing the finite sample performance of the bootstrap percentile confidence intervals (Algorithm (ref)) for the $\tau$-th quantile of the ITEs, with $\tau\in\left\{ 0.1,0.25,0.5,0.75,0.9\right\} $, and the interquartile range (IQR). Table (ref) examines the UCB for the quantile function $Q_{\varDelta}$ over equally spaced grid points in the intervals $[0.05,0.95]$ and $[0.2,0.8]$, with the step size $0.01$. Similar to the results discussed in the preceding paragraph, Table (ref) confirms that the bootstrap percentile confidence intervals for ITE quantiles and the IQR achieve good pointwise coverage, while Table (ref) shows that the UCBs for the quantile function, both the constant-width UCB from Algorithm (ref) and the variable-width UCB from Algorithm (ref), provide reliable simultaneous coverage. It is worth noting that all of the bootstrap percentile confidence intervals and UCBs exhibit good coverage accuracy, even in relatively small samples ($n=250$). When the sample size $n=500$ or $1000$, the variable-width UCB appears narrower than the constant-width counterpart.

Empirical application: 401(k) program and savings

We revisit the empirical application of \citetalias{feng2019estimation} and conduct inference on the distribution of ITEs of participating in 401(k) retirement programs on personal savings. Following \citetalias{feng2019estimation}, the outcome variable is family net financial assets; the treatment indicator reflects participation in 401(k) programs; the IV is eligibility for 401(k); and the covariates include categorical variables for income and age (each grouped into four categories based on distributional quartiles), an indicator for marital status, and a dummy for family size less than 3. We show that many of the qualitative statements in the empirical application sections of \citetalias{feng2019estimation} can be confirmed by using the inference methods proposed in this paper. At the same time, our CDF-based approach allows one to directly target important distributional characteristics, such as the proportion of individuals with positive ITEs, and conduct valid inference.

Table (ref) reports the 95% confidence intervals for three features of the ITE distribution: the proportion of positive ITEs ($\Pr[\varDelta>0]$), the median, and the interquartile range (IQR). For the full sample, the confidence interval for the proportion of positive ITEs is $\left[0.851,0.919\right]$, indicating that while most households benefited, a non-negligible fraction experienced negative effects. Note that the \citetalias{feng2019estimation} estimate for the same feature is 0.917, which is near the right boundary of our 95% confidence interval. Thus, our result suggests that the proportion of individuals with negative ITEs may be larger than that reported in \citetalias{feng2019estimation}. In particular, at the 5% significance level, we cannot reject the null hypothesis that 14.9% of individuals experience a negative ITE. The median ITE has a confidence interval of $\left[6.96,9.74\right]$ (in thousands of dollars), confirming a significantly positive center of the treatment effect distribution. The IQR, with a confidence interval of $\left[16.68,23.38\right]$, reveals considerable variation in treatment effects across households.

Subsample analysis based on covariate categories reveals notable patterns. The proportion of individuals with positive ITEs tends to increase with income and age, but remains relatively stable across groups defined by marital status and family size. Regarding the median impact of the program, even in the two subgroups that benefit the least-- the lowest income group and the youngest age group--the median ITE remains significantly positive. In terms of dispersion, the IQR of the ITE distribution increases substantially with income and age. Married individuals also exhibit greater dispersion in their ITE distribution than unmarried individuals. These findings suggest that treatment effect heterogeneity is more pronounced among higher-income, older, and married subpopulations.

Our subsample analysis also suggests that a larger proportion of young individuals may have negative ITEs than that reported in \citetalias{feng2019estimation}. According to their estimates, 15.93% of young individuals (with age in the first quartile) have negative effects. However, our 95% confidence interval suggests that 29.4% of young individuals may experience negative ITEs.

Table (ref) summarizes how the ITE distribution varies with each of the four covariates by reporting confidence intervals for differences in three representative quantiles ($\tau=0.25,0.5,0.75$) and for the difference in the IQR of the ITE distribution between groups $A_{1}$ and $A_{0}$, computed using Algorithm (ref). Parallel to Figures 4-7 of \citetalias{feng2019estimation}, Figure (ref) visualizes the quantile functions $Q_{\Delta\mid\tilde{X}}\left(\cdot\mid A_{0}\right)$ and $Q_{\Delta\mid\tilde{X}}\left(\cdot\mid A_{1}\right)$ together with their $95\%$ variable-width UCBs (Algorithm (ref) with range $\left[\underline{\tau},\overline{\tau}\right]$ =$\left[0.1,0.9\right]$). Panels (a) and (b) of Figures (ref) indicate that the ITE distribution shifts to the right and becomes more dispersed as income and age increase. A similar but weaker pattern is observed in Panel (c), where marital status changes from unmarried to married. By contrast, family size shows little influence on the ITE distribution as Panel (d) shows.

Figure (ref) depicts the estimator of the quantile difference function $Q_{\Delta\mid\tilde{X}}\left(\cdot\mid A_{1}\right)-Q_{\Delta\mid\tilde{X}}\left(\cdot\mid A_{0}\right)$ on $\left[\underline{\tau},\overline{\tau}\right]$ =$\left[0.1,0.9\right]$ and its $95\%$ UCB (Algorithm (ref) with $\left[\underline{\tau},\overline{\tau}\right]=\left[0.1,0.9\right]$). Panel (a) of Figure (ref) suggests that the ITE distribution for individuals with above the median income stochastically dominates that for individuals with below the median income. Similarly, Panel (b) of Figure (ref) suggests that the ITE distribution for older individuals (age above the median) stochastically dominates that for younger individuals (age below the median). Furthermore, Panel (c) suggests that the ITE distribution for married individuals may stochastically dominate that for unmarried individuals, with particularly clear dominance in the upper tail. On the other hand, Panel (d) shows that we cannot reject the null hypothesis of equality in the ITE distributions between individuals with larger and smaller family sizes (family size above or below three).

Declaration of generative AI and AI-assisted technologies in the manuscript preparation process

During the preparation of this work, the authors used AI-assisted technologies for language refinement and readability improvements. After using these tools, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article.