Regression-Adjusted Estimation of Quantile Treatment Effects under Covariate-Adaptive Randomizations

Covariate-adaptive randomizations (CARs) have recently seen growing use in a wide variety of randomized experiments in economic research. Examples include Chong et al. (2016), Greaney et al. (2016), Jakiela and Ozier (2016), Burchardi et al. (2019), Anderson and McKenzie (2021), among many others. In CAR modeling, units are first stratified using some baseline covariates, and then, within each stratum, the treatment status is assigned (independent of covariates) to achieve the balance between the numbers of treated and control units.

In many empirical studies, apart from the average treatment effect (ATE), researchers are often interested in using randomized experiments to estimate quantile treatment effects (QTEs). The QTE has a useful role as a robustness check for the ATE and characterizes any heterogeneity that may be present in the sign and magnitude of the treatment effects according to their position within the distribution of outcomes. See, for example, Bitler et al. (2006), Muralidharan and Sundararaman (2011), Duflo et al. (2013), Banerjee et al. (2015), Crépon et al. (2015), and Campos et al. (2017).

Two practical issues arise in estimation and inference concerning QTEs under CARs. First, other covariates in addition to the strata indicators are collected during the experiment. It is possible to incorporate these covariates in the estimation of treatment effects to reduce variance and improve efficiency. In the estimation of ATE, the usual practice is to run a simple ordinary least squares (OLS) regression of the outcome on treatment status, strata indicators, and additional covariates as in the analysis of covariance (ANCOVA). Freedman (2008a, b) pointed out that such an OLS regression adjustment can degrade the precision of the ATE estimator. Lin (2013) reexamined Freedman’s critique and showed that, in order to improve efficiency, the linear regression adjustment should include a full set of interactions between the treatment status and covariates. However, because the quantile function is a nonlinear operator, even when the assignment of treatment status is completely random, a similar linear quantile regression with a full set of interaction terms is unable to provide a consistent estimate of the unconditional QTE, not to mention the improvement of estimation efficiency. Second, in order to achieve balance in the respective number of treated and control units within each stratum, treatment statuses under CARs usually exhibit a (negative) cross-sectional dependence. Standard inference procedures that rely on cross-sectional independence are therefore conservative and lack power. These two issues raise questions of how to use the additional covariates to consistently and more efficiently estimate QTE in CAR settings and how to conduct valid statistical procedures that mitigate conservatism in inference.¹¹1For example, Bugni et al. (2018) and Zhang and Zheng (2020) have shown that the usual two-sample $t$-test for inference concerning ATE and multiplier bootstrap inference concerning QTE are in general conservative under CARs.

The present paper addresses these issues by proposing a regression-adjusted estimator of the QTE, deriving its limit theory, and establishing the validity of multiplier bootstrap inference under CARs. Even under potential misspecification of the auxiliary regressions, the proposed QTE estimator is shown to maintain its consistency, and the multiplier bootstrap procedure is shown to have an asymptotic size equal to the nominal level under the null. When the auxiliary regression is correctly specified, the QTE estimator achieves minimum asymptotic variance.

We further investigate efficiency gains that materialize from the regression adjustments in three scenarios: (1) parametric regressions, (2) nonparametric regressions, and (3) regressions with regularization in high-dimensional settings. Specifically, for parametric regressions with a potentially misspecified linear probability model, we propose to compute the optimal linear coefficient by minimizing the variance of the QTE estimator. Such an adjustment is optimal within the class of linear adjustments but does not necessarily achieve the global minimum asymptotic variance. However, as no adjustment is a special case of the linear regression with all the coefficients being zero, our optimal linear adjustment is guaranteed to be weakly more efficient than the QTE estimator with no adjustments, which addresses Freedman’s critique. We also consider a potentially misspecified logistic regression with fixed-dimensional regressors and strata- and quantile-specific regression coefficients, which is then estimated by the quasi maximum likelihood estimation (QMLE). Although the QMLE does not necessarily minimize the asymptotic variance of the QTE, such a flexible logistic model can closely approximate the true specification. Therefore, in practice, the corresponding regression-adjusted QTE estimator usually has a smaller variance than that with no adjustments. Last, we propose to treat the logistic QMLE adjustments as new linear regressors and re-construct the corresponding optimal linear adjustments. We then show the QTE estimator with the new adjustments are weakly more efficient than that with both the original logistic QMLE adjustments and no adjustments.

In nonparametric regressions, we further justify the QMLE by letting the regressors in the logistic regression be a set of sieve basis functions with increasing dimension and show how such a nonparametric regression-adjusted QTE estimator can achieve the global minimum asymptotic variance. For high-dimensional regressions with regularization, we consider logistic regression under $\ell_{1}$ penalization, an approach that also achieves the global minimum asymptotic variance. All the limit theories hold uniformly over a compact set of quantile indices, implying that our multiplier bootstrap procedure can be used to conduct inference on QTEs involving single, multiple, or a continuum of quantile indices.

These results, including the limit distribution of the regression-adjusted QTE estimator and the validity of the multiplier bootstrap, provide novel contributions to the literature in three respects. First, the data generated under CARs are different from observational data as the observed outcomes and treatment statuses are cross-sectionally dependent due to the randomization schemes. Recently Bugni et al. (2018) established a rigorous asymptotic framework to study the ATE estimator under CARs and pointed out the conservatism of the two-sample t-test except for some special cases. (See Bugni et al. (2018, Remark 4.2) for more detail.) Our analysis follows this new framework, which departs from the literature of causal inference under an i.i.d. treatment structure.

Second, we contribute to the literature on causal inference under CARs by developing a new methodology that includes additional covariates in the estimation of the unconditional QTE and by establishing a general theory for regression adjustments that allow for parametric, nonparametric, and regularized estimations of the auxiliary regressions. As mentioned earlier, unlike ATE estimation, the naive linear quantile regression with additional covariates cannot even produce a consistent estimator of the QTE. Instead, we propose a new way to incorporate additional covariates based on the Neyman orthogonal moment and investigate the asymptotic properties and the efficiency gains of the proposed regression-adjusted estimator under CARs. This new machinery allows us to study the QTE regression, which is nonparametrically specified, with both linear (linear probability model) and nonlinear (logit and probit models) regression adjustments. To clarify this contribution to the literature we note that Hu and Hu (2012); Ma et al. (2015, 2020); Olivares (2021); Shao and Yu (2013); Zhang and Zheng (2020); Ye (2018); Ye and Shao (2020) considered inference of various causal parameters under CARs but without taking into account additional covariates. Bugni et al. (2018), Bugni et al. (2019), and Bugni and Gao (2021) considered saturated regressions for ATE and local ATE, which can be viewed as regression-adjustments where strata indicators are interacted with the treatment or instrument. Shao et al. (2010) showed that if a test statistic is constructed based on the correctly specified model between outcome and additional covariates and the covariates used for CAR are functions of additional covariates, then the test statistic is valid conditional on additional covariates. Bloniarz et al. (2016); Fogarty (2018); Lin (2013); Lu (2016); Lei and Ding (2021); Li and Ding (2020); Liu et al. (2020); Liu and Yang (2020); Negi and Wooldridge (2020); Ye et al. (2022); Zhao and Ding (2021) studied various estimation methods based on regression adjustments, but these studies all focused on ATE estimation. Specifically, Liu et al. (2020) considered linear adjustments for ATE under CARs in which the covariates can be high-dimensional and the adjustments can be estimated by Lasso. Ansel et al. (2018) considered regression adjustment using additional covariates for ATE and Local ATE. We differ from them by considering QTE with nonlinear adjustments such as logistic Lasso.

Third, we establish the validity of the multiplier bootstrap inference for the regression-adjusted QTE estimator under CARs. To the best of our knowledge, Shao et al. (2010) and Zhang and Zheng (2020) are the only works in the literature that studied bootstrap inference under CARs. Shao et al. (2010) considered the covariate-adaptive bootstrap for the linear regression model. Zhang and Zheng (2020) proposed to bootstrap inverse propensity score weighted (IPW) QTE estimator with the estimated target fraction of treatment even when the truth is known. They showed that the asymptotic variance of the IPW estimator is the same under various CARs. Thus, even though the bootstrap sample ignores the cross-sectional dependence and behaves as if the randomization scheme is simple, the asymptotic variance of the bootstrap analogue is still the same. We complement this research by studying the validity of multiplier bootstrap inference for our regression-adjusted QTE estimator. We establish analytically that the multiplier bootstrap with the estimated fraction of treatment is not conservative in the sense that it can achieve an asymptotic size equal to the nominal level under the null even when the auxiliary regressions are misspecified.

The present paper also comes under the umbrella of a growing literature that has addressed estimation and inference in randomized experiments. In this connection, we mention the studies of Hahn et al. (2011); Athey and Imbens (2017); Abadie et al. (2018); Tabord-Meehan (2021); Bai et al. (2021); Bai (2020); Jiang et al. (2021) among many others. Bai (2020) showed an ‘optimal’ matched-pair design can minimize the mean-squared error of the difference-in-means estimator for ATE, conditional on covariates. Tabord-Meehan (2021) designed an adaptive randomization procedure which can minimize the variance of the weighted estimator for ATE. Both works rely on a pilot experiment to design the optimal randomization. In contrast, we take the randomization scheme (i.e., CARs) as given and search for new estimators (other than difference-in-quantile and weighted estimators) for QTE that have smaller variance. In addition, our approach does not require a pilot experiment. Therefore, our and their methods are applied to different scenarios depending on the definition of ‘optimality’ and the data available, and thus, complement to each other.

From a practical perspective, our estimation and inferential methods have four advantages. First, they allow for common choices of auxiliary regressions such as linear probability, logit, and probit regressions, even though these regressions may be misspecified. Second, the methods can be implemented without tuning parameters. Third, our (bootstrap) estimator can be directly computed via the subgradient condition, and the auxiliary regressions need not be re-estimated in the bootstrap procedure, both of which save considerable computation time. Last, our estimation and inference methods can be implemented without the knowledge of the exact treatment assignment rule used in the experiment. This advantage is especially useful in subsample analysis, where sub-groups are defined using variables other than those to form the strata and the treatment assignment rule for each sub-group becomes unknown. See, for example, the anemic subsample analysis in Chong et al. (2016) and Zhang and Zheng (2020). These last three points carry over from Zhang and Zheng (2020) and are logically independent of the regression adjustments. One of our contributions is to show these results still hold for our regression-adjusted estimator.

The remainder of the paper is organized as follows. Section 2 describes the model setup and notation. Section 3 develops the asymptotic properties of our regression-adjusted QTE estimator. Section 4 studies the validity of the multiplier bootstrap inference. Section 5 considers parametric, nonparametric, and regularized estimation of the auxiliary regressions. Section 6 reports simulation results, and an empirical application of our methods to the impact of expanding access to basic bank accounts on savings is provided in Section 7. Section 8 concludes. Proofs of all results and some additional simulation results are given in the Online Supplement.

Potential outcomes for treated and control groups are denoted by $Y(1)$ and $Y(0)$, respectively. Treatment status is denoted by $A$, with $A=1$ indicating treated and $A=0$ untreated. The stratum indicator is denoted by $S$, based on which the researcher implements the covariate-adaptive randomization. The support of $S$ is denoted by $\mathcal{S}$, a finite set. After randomization, the researcher can observe the data $\{Y_{i},S_{i},A_{i},X_{i}\}_{i\in[n]}$ where $[n]=\{1,2,...n\}$, $Y_{i}=Y_{i}(1)A_{i}+Y_{i}(0)(1-A_{i})$ is the observed outcome, and $X_{i}$ contains extra covariates besides $S_{i}$ in the dataset. The support of $X$ is denoted $\text{Supp}(X)$. In this paper, we allow $X_{i}$ and $S_{i}$ to be dependent. For $i\in[n]$, let $p(s)=\mathbb{P}(S_{i}=s)$, $n(s)=\sum_{i\in[n]}1\{S_{i}=s\}$, $n_{1}(s)=\sum_{i\in[n]}A_{i}1\{S_{i}=s\}$, and $n_{0}(s)=n(s)-n_{1}(s)$. We make the following assumptions on the data generating process (DGP) and the treatment assignment rule.

Several remarks are in order. First, Assumption 1(i) allows for cross-sectional dependence among treatment statuses ($\{A_{i}\}_{i\in[n]}$), thereby accommodating many covariate-adaptive randomization schemes as discussed below. Second, although treatment statuses are cross-sectionally dependent, they are independent of the potential outcomes and additional covariates conditional on the stratum indicator $S$. Therefore, data are still experimental rather than observational. Third, Assumption 1(iii) requires the size of each stratum to be proportional to the sample size. Fourth, we can view $\pi(s)$ as the target fraction of treated units in stratum $s$. Similar to Bugni et al. (2019), we allow the target fractions to differ across strata. Just as for the overlapping support condition in an observational study, the target fractions are assumed to be bounded away from zero and one. In randomized experiments, this condition usually holds because investigators can determine $\pi(s)$ in the design stage. In fact, in most CARs, $\pi(s)=0.5$ for $s\in\mathcal{S}$. Fifth, $D_{n}(s)$ represents the degree of imbalance between the real and target factions of treated units in the $s$th stratum. Bugni et al. (2018) show that Assumption 1(iv) holds under several covariate-adaptive treatment assignment rules such as simple random sampling (SRS), biased-coin design (BCD), adaptive biased-coin design (WEI), and stratified block randomization (SBR). For completeness, we briefly repeat their descriptions below. Note we only require $D_{n}(s)/n(s)=o_{p}(1)$, which is weaker than the assumption imposed by Bugni et al. (2018) but the same as that imposed by Bugni et al. (2019) and Zhang and Zheng (2020).

Denote the $\tau$th quantile of $Y(a)$ by $q_{a}(\tau)$ for $a=0,1$. We are interested in estimating and inferring the $\tau$th quantile treatment effect defined as $q(\tau)=q_{1}(\tau)-q_{0}(\tau)$. The testing problems of interest involve single, multiple, or even a continuum of quantile indices, as in the following null hypotheses

	$\displaystyle\mathcal{H}_{0}:q(\tau)=\underline{q}\quad\text{versus}\quad q(\tau)\neq\underline{q},$
	$\displaystyle\mathcal{H}_{0}:q(\tau_{1})-q(\tau_{2})=\underline{q}\quad\text{versus}\quad q(\tau_{1})-q(\tau_{2})\neq\underline{q},\;\textrm{and}$
	$\displaystyle\mathcal{H}_{0}:q(\tau)=\underline{q}(\tau)\leavevmode\nobreak\ \forall\tau\in\Upsilon\quad\text{versus}\quad q(\tau)\neq\underline{q}(\tau)\leavevmode\nobreak\ \text{for some}\leavevmode\nobreak\ \tau\in\Upsilon,$

for some pre-specified value $\underline{q}$ or function $\underline{q}(\tau)$, where $\Upsilon$ is some compact subset of $(0,1)$. We can also test constant QTE by letting $\underline{q}(\tau)$ in the last hypothesis be a constant $\underline{q}$.

Define $m_{a}(\tau,s,x)=\tau-\mathbb{P}(Y_{i}(a)\leq q_{a}(\tau)|S_{i}=s,X_{i}=x)$ for $a=0,1$ which are the true specifications but unknown to researchers. Instead, researchers specify working models $\{\overline{m}_{a}(\tau,s,x)\}_{a=0,1}$²²2We view $\overline{m}_{a}(\cdot)$ as some function with inputs $\tau,s,x$. For example, researchers can specify a linear probability model with $\overline{m}_{a}(\tau,s,x)=\tau-x^{\top}\beta_{a,s}$, where $\beta_{a,s}$ is the linear coefficient that varies across treatment status $a$ and stratum $s$. for the true specification, which can be misspecified. Last, the researchers estimate the (potentially misspecified) working models via some forms of regression, and the estimators are denoted as $\{\widehat{m}_{a}(\tau,s,x)\}_{a=0,1}$. We also refer to $\overline{m}_{a}(\cdot)$ as the auxiliary regression.

Our regression-adjusted estimator of $q_{1}(\tau)$, denoted as $\hat{q}_{1}^{adj}(\tau)$, can be defined as

$\displaystyle\hat{q}_{1}^{adj}(\tau)=$

$\displaystyle\operatorname*{arg\,min}_{q}\sum_{i\in[n]}\left[\frac{A_{i}}{\hat{\pi}(S_{i})}\rho_{\tau}(Y_{i}-q)+\frac{(A_{i}-\hat{\pi}(S_{i}))}{\hat{\pi}(S_{i})}\widehat{m}_{1}(\tau,S_{i},X_{i})q\right],$

(3.1)

where $\rho_{\tau}(u)=u(\tau-1\{u\leq 0\})$ is the usual check function and $\hat{\pi}(s)=n_{1}(s)/n(s)$. We emphasize that $\widehat{m}_{1}(\cdot)$ may not consistently estimate the true specification $m_{1}(\cdot)$. Similarly, we can define

$\displaystyle\hat{q}_{0}^{adj}(\tau)=$

$\displaystyle\operatorname*{arg\,min}_{q}\sum_{i\in[n]}\left[\frac{1-A_{i}}{1-\hat{\pi}(S_{i})}\rho_{\tau}(Y_{i}-q)-\frac{(A_{i}-\hat{\pi}(S_{i}))}{1-\hat{\pi}(S_{i})}\widehat{m}_{0}(\tau,S_{i},X_{i})q\right].$

(3.2)

Then, our regression adjusted QTE estimator is

$\displaystyle\hat{q}^{adj}(\tau)=\hat{q}_{1}^{adj}(\tau)-\hat{q}_{0}^{adj}(\tau).$

(3.3)

Several remarks are in order. First, in observational studies with i.i.d. data and $A_{i}\perp\!\!\!\perp X_{i}|S_{i}$, Firpo (2007), Belloni et al. (2017), and Kallus et al. (2020) showed that the doubly robust moment for $q_{1}(\tau)$ is

$\displaystyle\mathbb{E}\left[\frac{A_{i}(\tau-1\{Y_{i}(1)\leq q\})}{\overline{\pi}(S_{i})}-\frac{A_{i}-\overline{\pi}(S_{i})}{\overline{\pi}(S_{i})}\overline{m}_{1}(\tau,S_{i},X_{i})\right]=0,$

(3.4)

where $\overline{\pi}(s)$ and $\overline{m}_{1}(\tau,s,x)$ are the working models for the target fraction ($\pi(s)$) and conditional probability ($m_{1}(\tau,s,x)$), respectively. Our estimator is motivated by this doubly robust moment, but our analysis differs from that for the observational data as CARs introduces cross-sectional dependence among observations. Second, as our target fraction estimator $\hat{\pi}(s)=n_{1}(s)/n(s)$ is consistent, it means $\overline{\pi}(s)$ is correctly specified as $\pi(s)$. Then, due to the double robustness, our regression adjusted estimator is consistent even when $\overline{m}_{a}(\cdot)$ is misspecified and $\widehat{m}_{a}(\cdot)$ is an inconsistent estimator of $m_{a}(\cdot)$. Third, we use the estimated target fraction $\hat{\pi}(s)$ even when $\pi(s)$ is known because this guarantees that the bootstrap inference is not conservative. Further discussion is provided after Theorem 5.

Several remarks are in order. First, Assumption 2 is standard in the quantile regression literature. We do not need $f_{a}(y|x,s)$ to be bounded away from zero because we are interested in the unconditional quantile $q_{a}(\tau)$, which is uniquely defined as long as the unconditional density $f_{a}(q_{a}(\tau))$ is positive. Second, Assumption 3(i) is high-level. If we consider a linear probability model such that $\overline{m}_{a}(\tau,s,X_{i})=\tau-X_{i}^{\top}\theta_{a,s}(\tau)$ and $\widehat{m}_{a}(\tau,s,X_{i})=\tau-X_{i}^{\top}\hat{\theta}_{a,s}(\tau)$, then Assumption 3(i) is equivalent to

$\displaystyle\sup_{\tau\in\Upsilon,a=0,1,s\in\mathcal{S}}\left|\left(\frac{\sum_{i\in I_{1}(s)}X_{i}}{n_{1}(s)}-\frac{\sum_{i\in I_{0}(s)}X_{i}}{n_{0}(s)}\right)^{\top}\left(\hat{\theta}_{a,s}(\tau)-\theta_{a,s}(\tau)\right)\right|=o_{p}(n^{-1/2}),$

which is similar to Liu et al. (2020, Assumption 3) and holds intuitively if $\hat{\theta}_{a,s}(\tau)$ is a consistent estimator of the pseudo true value $\theta_{a,s}(\tau)$. Third, Assumptions 3(ii) and 3(iii) impose mild regularity conditions on $\overline{m}_{a}(\cdot)$. Assumption 3(ii) holds automatically if $\Upsilon$ is a finite set. In general, both Assumption 3(ii) and 3(iii) hold if

$\displaystyle\sup_{a=0,1,s\in\mathcal{S},x\in\text{Supp}(X)}|\overline{m}_{a}(\tau_{2},s,x)-\overline{m}_{a}(\tau_{1},s,x)|\leq L|\tau_{2}-\tau_{1}|$

for some constant $L>0$. Such Lipschitz continuity holds for the true specification ($\overline{m}_{a}(\cdot)=m_{a}(\cdot)$) under Assumption 2. Fourth, we provide primitive sufficient conditions for Assumption 3 in Section 5.

Three remarks are in order. First, the expression for the asymptotic variance of $\hat{q}^{adj}(\tau)$ can be found in the proof of Theorem 3. It is the same whether the randomization scheme achieves strong balance⁴⁴4We refer readers to Bugni et al. (2018) for the definition of strong balance. or not. This robustness is due to the use of the estimated target fraction ($\hat{\pi}(s)$). The same phenomenon was discovered in the simplified setting by Zhang and Zheng (2020). Second, although our estimator is still consistent and asymptotically normal when the auxiliary regression is misspecified, it is meaningful to pursue the correct specification as it achieves the minimum variance. As the estimator with no adjustments can be viewed as a special case of our estimator with $\overline{m}_{a}(\cdot)=0$, Theorem 3 implies that the adjusted estimator with the correctly specified auxiliary regression is more efficient than that with no adjustments. If the auxiliary regression is misspecified, the adjusted estimator can sometimes be less efficient than the unadjusted one, which is known as the Freedman’s critique. In Section 5, we discuss how to make adjustments that do not harm the precision of the QTE estimator. Third, the asymptotic variance of $\hat{q}^{adj}(\tau)$ depends on $(f_{a}(q_{a}(\tau)),m_{a}(\tau,s,x))_{a=0,1}$, which are infinite-dimensional nuisance parameters. To conduct analytic inference, it is necessary to nonparametrically estimate these nuisance parameters, which requires tuning parameters. Nonparametric estimation can be sensitive to the choice of tuning parameters and rule-of-thumb tuning parameter selection may not be appropriate for every DGP or every quantile. The use of cross-validation in selecting the tuning parameters is possible in principle but, in practice, time-consuming. These practical difficulties of analytic methods of inference provide strong motivation to investigate bootstrap inference procedures that are much less reliant on tuning parameters.

We approximate the asymptotic distributions of $\hat{q}^{adj}(\tau)$ via the multiplier bootstrap. Let $\{\xi_{i}\}_{i\in[n]}$ be a sequence of bootstrap weights which will be specified later. Define $n_{1}^{w}(s)=\sum_{i\in[n]}\xi_{i}A_{i}1\{S_{i}=s\}$, $n_{0}^{w}(s)=\sum_{i\in[n]}\xi_{i}(1-A_{i})1\{S_{i}=s\}$, $n^{w}(s)=\sum_{i\in[n]}\xi_{i}1\{S_{i}=s\}=n_{1}^{w}(s)+n_{0}^{w}(s)$, and $\hat{\pi}^{w}(s)=n_{1}^{w}(s)/n^{w}(s)$. The multiplier bootstrap counterpart of $\hat{q}^{adj}(\tau)$ is denoted by $\hat{q}^{w}(\tau)$ and defined as

$\displaystyle\hat{q}^{w}(\tau)=\hat{q}_{1}^{w}(\tau)-\hat{q}_{0}^{w}(\tau),$

where

$\displaystyle\hat{q}_{1}^{w}(\tau)=$

$\displaystyle\operatorname*{arg\,min}_{q}\sum_{i\in[n]}\xi_{i}\left[\frac{A_{i}}{\hat{\pi}^{w}(S_{i})}\rho_{\tau}(Y_{i}-q)+\frac{(A_{i}-\hat{\pi}^{w}(S_{i}))}{\hat{\pi}^{w}(S_{i})}\widehat{m}_{1}(\tau,S_{i},X_{i})q\right],$

(4.1)

and

$\displaystyle\hat{q}_{0}^{w}(\tau)=$

$\displaystyle\operatorname*{arg\,min}_{q}\sum_{i\in[n]}\xi_{i}\left[\frac{1-A_{i}}{1-\hat{\pi}^{w}(S_{i})}\rho_{\tau}(Y_{i}-q)-\frac{(A_{i}-\hat{\pi}^{w}(S_{i}))}{1-\hat{\pi}^{w}(S_{i})}\widehat{m}_{0}(\tau,S_{i},X_{i})q\right].$

(4.2)

Two comments on implementation are noted here: (i) we do not re-estimate $\widehat{m}_{a}(\cdot)$ in the bootstrap sample, which is similar to the multiplier bootstrap procedure proposed by Belloni et al. (2017); and (ii) in Section B of the Online Supplement we propose a way to directly compute $(\hat{q}_{a}^{w}(\tau))_{a=0,1}$ from the subgradient conditions of (4.1) and (4.2), thereby avoiding the optimization. Both features considerably reduce computation time of our bootstrap procedure.

Next, we specify the bootstrap weights.

We require the bootstrap weights to be nonnegative so that the objective functions in (4.1) and (4.2) are convex. In practice, we generate $\xi_{i}$ independently from the standard exponential distribution. Assumption 5 is the bootstrap counterpart of Assumption 3. Continuing with the linear model example considered after Assumption 3, Assumption 5 requires

$\displaystyle\sup_{\tau\in\Upsilon,a=0,1,s\in\mathcal{S}}\left|\left(\frac{\sum_{i\in I_{1}(s)}\xi_{i}X_{i}}{n_{1}^{w}(s)}-\frac{\sum_{i\in I_{0}(s)}\xi_{i}X_{i}}{n_{0}^{w}(s)}\right)^{\top}\left(\hat{\theta}_{a,s}(\tau)-\theta_{a,s}(\tau)\right)\right|=o_{p}(n^{-1/2}),$

which holds if $\hat{\theta}_{a,s}(\tau)$ is a uniformly consistent estimator of $\theta_{a,s}(\tau)$.

Two remarks are in order. First, Theorem 5 shows the limit distribution of the bootstrap estimator conditional on data can approximate that of the original estimator uniformly over $\tau\in\Upsilon$. This is the theoretical foundation for the bootstrap confidence intervals and bands described in Section B in the Online Supplement. Specifically, denote $\{\hat{q}^{w,b}(\tau)\}_{b\in[B]}$ as the bootstrap estimates where $B$ is the number of bootstrap replications. Let $\widehat{\mathcal{C}}(\nu)$ and $\mathcal{C}(\nu)$ be the $\nu$th empirical quantile of the sequence $\{\hat{q}^{w,b}(\tau)\}_{b\in[B]}$ and the $\nu$th standard normal critical value, respectively. Then, we suggest using the bootstrap estimator to construct the standard error of $\hat{q}^{adj}(\tau)$ as $\hat{\sigma}=\frac{\widehat{\mathcal{C}}(0.975)-\widehat{\mathcal{C}}(0.025)}{\mathcal{C}(0.975)-\mathcal{C}(0.025)}$. Note that, unlike Hahn and Liao (2021), our bootstrap standard error is not conservative. In our context, the bootstrap estimator of $\sigma^{2}$ considered by Hahn and Liao (2021) is $\mathbb{E}^{*}(\sqrt{n}(\hat{q}^{w}(\tau)-\hat{q}^{adj}(\tau))^{2})$, where $\mathbb{E}^{*}$ is the conditional expectation given data. It is well-known that weak convergence does not imply convergence in $L_{2}$-norm, which explains why they can show their estimator is in general conservative. Instead, we use a different estimator of the standard error and can show it is consistent given weak convergence. Second, such a bootstrap approximation is consistent under CAR. Zhang and Zheng (2020) showed that for the QTE estimation without regression adjustment, bootstrapping the IPW QTE estimator with the estimated target fraction results in non-conservative inference, while bootstrapping the IPW estimator with the true fraction is conservative under CARs. As the estimator considered by Zhang and Zheng (2020) is a special case of our regression-adjusted estimator with $\widehat{m}_{a}(\cdot)=0$, we conjecture that the same conclusion holds. A proof of conservative bootstrap inference with the true target fraction is not included in the paper due to the space limit.⁶⁶6Full statements and proofs are lengthy because we need to derive the limit distributions of not only the bootstrap but also the original estimator with the true target fraction. Although the negative result is theoretically interesting, we are not aware of any empirical papers using the true target fraction while making regression adjustments. Moreover, our method is shown to have better performance than the one with the true target fraction in simulations. So the practical value of proving the negative result is limited. Our simulations confirm both the correct size coverage of our inference method using the bootstrap with the estimated target fraction and the conservatism of the bootstrap with the true target fraction. The standard error of the QTE estimator is found to be 34.9% larger on average by using the true rather than the estimated target fraction in the simulations (see Tables 1 below and 15 in the Online Supplement).

In this section, we consider two approaches to estimation for the auxiliary regressions: (1) a parametric method and (2) a nonparametric method. In Section A of the Online Supplement, we further consider a regularization method. For the parametric method, we do not require the model to be correctly specified. We propose ways to estimate the pseudo true value of the auxiliary regression. For the other two methods, we (nonparametrically) estimate the true model so that the asymptotic variance of $\hat{q}^{adj}(\tau)$ achieves its minimum based on Theorem 3. For all three methods, we verify Assumptions 3 and 5.