# Robust inference for the treatment effect variance in experiments using
machine learning ††thanks: Email<EMAIL_ADDRESS>I would
like to thank Abhishek Ananth, Xu Cheng, Juan Estrada, Wayne Gao, Guido
Imbens, Toru Kitagawa, Francesca Molinari, Mikkel Plagborg-Møller, Ulrich
Müller, Megan Reed, Jörg Stoye, Mark Watson, and Kaspar Wuthrich, for helpful
comments and suggestions. I would also like to thank seminar participants at
the New York Econometrics Camp 2022, NYU, the Canadian Economic Association
2022, the International Association for Applied Econometrics Conference 2022,
Syracuse, LACEA/LAMES 2022, Facebook/Meta Core Data Science Group, the CFE-CM
Statistics Conference 2022, the European Winter Meeting of the Econometric
Society 2022, and the Duke Microeconometrics Conference for the Class of 2020
and 2021.

Alejandro Sánchez-Becerra
Emory QTM

(First Version: February 12, 2022
This version: )

###### Abstract

Experimenters often collect baseline data to study heterogeneity. I propose
the first valid confidence intervals for the VCATE, the treatment effect
variance explained by observables. Conventional approaches yield incorrect
coverage when the VCATE is zero. As a result, practitioners could be prone to
detect heterogeneity even when none exists. The reason why coverage worsens at
the boundary is that all efficient estimators have a locally-degenerate
influence function and may not be asymptotically normal. I solve the problem
for a broad class of multistep estimators with a predictive first stage. My
confidence intervals account for higher-order terms in the limiting
distribution and are fast to compute. I also find new connections between the
VCATE and the problem of deciding whom to treat. The gains of targeting
treatment are (sharply) bounded by half the square root of the VCATE. Finally,
I document excellent performance in simulation and reanalyze an experiment
from Malawi.

Keywords: Debiased machine learning, treatment effect heterogeneity,
experiments, non-standard inference, variance decomposition
JEL Codes C14, C21, C55, C90

##  1 Introduction

In recent years, there has been a rapid expansion of experiments to evaluate
public policy programs and corporate initiatives. There is also more evidence
that the effectiveness of a program can vary across individuals. For instance,
Dizon-Ross (2019) studies a population of low-income parents in Malawi with
large misperceptions about their children’s school performance. She finds that
a simple intervention can bridge these gaps and that the Conditional Average
Treatment Effect (CATE) varies by children’s initial scores. In practice, even
though researchers collect baseline surveys with many characteristics, the
CATE is typically estimated via regressions with one or two interactions, thus
underutilizing the full set of variables. The promise of leveraging the vast
and readily available baseline data has sparked more applications of
supervised machine learning (Crépon et al., 2021; Davis and Heller, 2020;
Deryugina et al., 2019). Methods such as LASSO, neural networks, random
forests, or boosting are data-driven and allow for more variables and
flexibility.

In this paper, I focus on the unconditional variance of the CATE, the VCATE,
which measures the dispersion of treatment effects predicted by a set of
baseline characteristics. The VCATE has a clear interpretation, even if the
CATE is nonlinear or depends on many characteristics. Chernozhukov et al.
(2022a) and Ding et al. (2019) separately propose estimators with a
misspecification robust interpretation, whereas Levy et al. (2021) propose an
efficient estimator. Despite these recent advances, there are currently no
valid confidence intervals for the VCATE. In fact, Levy et al. (2021) study
the performance of confidence intervals based on the efficient influence
function, i.e., the conventional way. They find that coverage degrades near
the boundary, reaching a low of 32% in simulations. They speculate that poor
coverage is due to the degeneracy of the efficient influence function when the
VCATE is zero. As a result, conventional guarantees for $\sqrt{n}-$asymptotic
normality do not apply to this part of the parameter space. Moreover, a VCATE
close to zero is economically meaningful because it could reflect null
effects, low effect heterogeneity, or irrelevant covariates. For such
situations, which are common in practice, conventional approaches to inference
could be misleading.

This paper provides fresh insights regarding the VCATE and proposes a solution
for inference in experiments. I propose (a) novel ways to interpret the VCATE
for decision-making by deriving sharp bounds on the population gains of
personalized treatment assignment, (b) novel estimators of the VCATE that are
both misspecification-robust and efficient, combining the best features of
previous approaches, and (c) novel confidence intervals that are shape-
adaptive and fast to compute.

I break down why conventional confidence intervals have incorrect size. I show
that the boundary inference problem can manifest even when the CATE is linear
and univariate. I solve the problem for the linear case by proposing adaptive
confidence intervals that meet the high-level conditions outlined in Andrews
et al. (2020). I then show that these can be readily extended to the class of
nonlinear models in Chernozhukov et al. (2022a) that combine regression
adjustments and a machine learning first stage. In addition to providing new
confidence intervals, my paper has novel implications for estimation by
showing that only a subset of these multi-step estimators is efficient.
Adaptive inference and regression adjustments work well for various predictive
models under weak assumptions.111While I specialize my results to the VCATE,
my inference approach relies on more general principles: I use knowledge of
the limiting distribution function, conditional on the cross-fitted estimates.
Correct coverage follows from verifying high-level assumptions that can be
satisfied by a wide array of machine learning method used in the first step.
In principle, my approach could extend to other non-standard inference
problems. In the fully nonparametric case, I use a conservative procedure with
valid coverage over multiple sample splits.222This adjustment is in the spirit
of (Chernozhukov et al., 2022a), who propose robust t-tests assuming a
conditionally normal distribution. Their results are not directly applicable
here due to the boundary inference problem. However, I apply the principles
behind their “median-parameter” confidence intervals to my adaptive intervals.
I derive the local power curve for the associated tests of homogeneity and
their relationship to the tests in Crump et al. (2008) and Ding et al. (2019).
I also propose confidence intervals for settings with cluster dependence.

I document excellent root mean square error (RMSE) performance and coverage in
simulations using LASSO, even in high dimensions. I benchmark my multi-step
approach against a two-step debiased machine learning estimator. As predicted
by theory, all approaches are asymptotically normal, efficient, and have good
coverage in highly heterogeneous designs. However, when the VCATE is zero or
close to zero, coverage of two-step alternatives can be as low as 45%. By
contrast, my adaptive intervals produce coverage at the intended 95% level and
better RMSE at all regions of the parameter space. I study the robustness of
the multi-step approach in both the theory and simulations. I consider
situations where the predictive component is misspecified or slow to converge.
I also discuss issues related to uniform vs. pointwise coverage.

I apply my approach to data from Dizon-Ross (2019), an information experiment
with low-income parents in Malawi who had at least two school-age children.
The intervention redesigned the way in which parents received information
about their children’s school performance. The endline survey measured
parental beliefs about student grades and asked parents to allocate tickets to
a scholarship between their children. Dizon-Ross (2019) presented graphs with
a non-parametric CATE by baseline test scores, which had an approximately
linear shape, and separately tested for significance using a regression with
an interaction. I use LASSO to compute the VCATE for the two outcomes
(parental beliefs and lottery allocations) by different characteristics of
students, parents, and households. To make the results interpretable, I focus
on the standard deviation of the CATE, i.e., $\sqrt{VCATE}$, and normalize it
by the standard deviation of the outcome in the control group.

My approach allows us to quantify the magnitude of effect heterogeneity. I
find that the treatment effect heterogeneity explained by test scores is
equivalent to 40% of the standard deviation (SD) of the beliefs of the control
group, and 16% of the SD of the control group lottery allocation. I also find
that the effect heterogeneity collectively explained by other student
variables (grade, age, gender, attendance, and educational expenditures) is
comparable to 11% of the SD of beliefs in the control group. The VCATE of
beliefs by student variables is significant at the 5% level, but the VCATE of
lottery outcomes by student variables is not. The combined VCATE associated
with student scores and 12 other key characteristics has a similar value to
the VCATE with only scores. Despite being conservative, the intervals for the
VCATE are short in length in this empirical example. Using my new welfare
bounds $(-|ATE|+\sqrt{VCATE+ATE^{2}})/2$, I predict that targeted
interventions using the baseline covariates have a maximum added benefit of
7.9% SD and 7.4% SD (standard deviations of the outcome for the control group)
on beliefs and lottery allocations, respectively.

Researchers should focus on the VCATE because it is a model-free quantity with
good properties: it is well-defined even if the CATE is continuous or
discrete, and it weakly increases when researchers add more covariates to
their analysis. Researchers can test for homogeneity by evaluating whether
confidence intervals for the VCATE include zero. In addition to testing, by
quantifying the VCATE, researchers can compare the magnitude of heterogeneity
relative to a benchmark, such as the variance at baseline, the VCATE for
different covariates, or experiments in other sites.

###  1.1 Contribution

My first main contribution is to show that the VCATE provides a bound for the
welfare gains of policy targeting. A policymaker might decide to use the
information from the CATE to design personalized treatment recommendations
(Manski, 2004; Athey and Wager, 2021; Kitagawa and Tetenov, 2018; Mbakop and
Tabord-Meehan, 2021). One can measure utilitarian welfare by computing the
expected outcome under different policies. Such policies can be further
constrained to a class that respects budget limits, incentive compatibility,
or fairness considerations (Viviano and Bradic, 2023; Sun, 2021). I show that
the difference in mean outcomes between a targeted policy and a non-targeted
policy using only the average treatment effect (ATE), is bounded by
$\sqrt{VCATE}/2$. For instance, under homogeneity ($VCATE=0$), there are no
gains from targeting. I show that this bound holds in the population
regardless of the choice of policy class and the underlying distribution.
Furthermore, the bound is sharp in the sense that it holds exactly for at
least one policy and distribution.

The proposed bound on utilitarian welfare communicates information to
practitioners about whether a targeting exercise is even worth pursuing,
without needing to solve the targeting problem itself. The VCATE can be a
supplemental quantity reported in regression analyses, or a benchmark for
analysts choosing the optimal policy. If the VCATE is very low, practitioners
may consider expanding the set of covariates in the analysis. To derive the
bound, I use a constructive approach to solve the most adversarial
distribution. I also prove a more general bound
$(-|ATE|+\sqrt{VCATE+ATE^{2}})/2$, and show that the distribution that leads
to a maximum welfare gain is one where the CATE has binary support and mean
zero. The gains from targeting easily diminish if the value of the ATE is
relatively higher than the VCATE.

My second contribution is related to efficient estimation and robust
inference. New theory is required here because of a unique feature of the
VCATE: the efficient influence function is degenerate when the CATE is
homogeneous (Levy et al., 2021). Classical results by Newey (1990) show that
any regular, efficient estimator can be decomposed as
$\frac{1}{n}\sum_{i=1}^{n}\varphi_{i}+R_{n}$, where $n$ is the sample size,
$\\{\varphi_{i}\\}_{i=1}^{n}$ are a set of i.i.d. mean-zero influence
functions, and $R_{n}$ is a residual with higher-order terms that are
$o_{p}(n^{-1/2})$.333Many standard estimators can achieve this property, e.g.,
the “debiased machine learning” estimator (Chernozhukov et al., 2018) or the
targeted maximum likelihood estimator in Levy et al. (2021). Conventional
approaches assume that $\mathbb{V}(\varphi)>0$ and in this case, the
estimation error converges at $\sqrt{n}$ to
$\mathcal{N}(0,\mathbb{V}(\varphi_{i}))$ by the CLT. However, when the VCATE
is zero, $\mathbb{V}(\varphi_{i})=0$ as well. Hence, the limiting distribution
is dominated by the higher terms in $R_{n}$, which may not be asymptotically
normal. Therefore, while $\sqrt{n}-$estimation is still possible, t-tests that
plug in an estimate of $\mathbb{V}(\varphi)$ may have incorrect coverage. By
contrast, other common quantities such as the average treatment effect (ATE)
or the local average treatment effect (LATE) do not have this problem because
they satisfy $\mathbb{V}(\varphi_{i})>0$ uniformly (Chernozhukov et al.,
2018).

I start by analyzing a simple two step estimator, assuming that the CATE is
linear in covariates and can be estimated from a regression. I show that the
limiting distribution of the VCATE estimator can be written as a linear
combination of a Chi-square that converges at $n$-rate and a normal
distribution that converges at $\sqrt{n}-$rate. The weights are determined by
the value of the VCATE, which means that the shape of the distribution changes
depending on the region of the parameter space. At the boundary, it behaves
like a rescaled chi-square, is $O_{p}(n^{-1})$, and confidence intervals with
normal critical values will have incorrect coverage. For values of the VCATE
bounded away from zero, the distribution is asymptotically normal as in the
classical results.

In the linear case, I construct adaptive confidence intervals that account for
the higher terms of the distribution. I apply the framework of Andrews et al.
(2020) to show that this produces uniform, exact coverage when the linear
model is correctly specified.444This type of strategy has proven effective to
deal with other non-standard problems where the shape of the limiting
distribution depends on an unknown parameter, such as the AR coefficient in a
time series, the effect parameter under weak instruments, or the quasi-
likelihood ratio test for nonlinear regression (Andrews et al., 2020). The
intervals are fast to compute because the expressions are all analytic. When
there is a single covariate, I also show that a homogeneity test that
evaluates whether zero is contained in the confidence intervals is
algebraically identical to (i) a test of whether the interaction in the
regression model is equal to zero, and (ii) the single-covariate homogeneity
test of Crump et al. (2008). The test is also asymptotically equivalent to
Ding et al. (2019). However, these other tests only apply to series estimators
and are not nested with mine in the multivariate and non-parametric cases.

I extend my results to the class of nonlinear models proposed by Chernozhukov
et al. (2022a). Chernozhukov et al. (2022a) showed that in experiments with
known assignment probabilities, their models produce a meaningful pseudo-VCATE
even if the functional form is misspecified. The pseudo-VCATE is non-negative,
weakly lower than the VCATE, and converges to the true value under mild
conditions on the estimated CATE.555This monotonicity property means that in
experiments the pseudo-VCATE will not falsely detect heterogeneity, even if
the machine learning stage is misspecified. Chernozhukov et al. (2022a) argue
that the pseudo-VCATE might be of independent interest as a measure of model
fit.666 Ding et al. (2019) also define a similar pseudo-VCATE based on
randomization inference. They describe a three-step estimator with a machine
learning/prediction first-stage, a regression second-stage, and a sample-
variance third stage. Related multi-step estimators have also been considered
in other work (Guo et al., 2021).

To the best of my knowledge, there are no existing asymptotic results for the
multi-step VCATE estimator proposed in Chernozhukov et al. (2022a). I fill in
that gap by proving two sets of results. First, I show that all the estimators
in their class converge to the true VCATE at least at $\sqrt{n}$-rate, are
$o_{p}(n^{-1/2})$ at the boundary (as in the simple linear model), and have
the convenient property that they are always non-negative. This builds on the
asymptotic expansion for the linear case I introduced above. Second, I prove
that only a subset of the Chernozhukov et al. (2022a) estimators are
efficient, i.e. converge at $\sqrt{n}-$ to an average of i.i.d. efficient
influence functions. The key ingredient is to prove a novel finite-sample
equivalence result. I find that the first order conditions of the regression
step and the bias-correction component of the VCATE influence function are in
fact identical, given a particular decomposition of the nuisance functions.
The asymptotic results follow from fairly standard assumptions on convergence
rates (Chernozhukov et al., 2018; Belloni et al., 2017). To get the limiting
distribution, the only meaningful extra assumption is that the estimated CATE
has bounded kurtosis (thin tails).

I show that extending the adaptive confidence intervals (CIs) to multi-step
estimators is straightforward. The procedure randomly splits the data into
subsets or folds and estimates the nuisance functions and the VCATE on
different folds. To compute the confidence intervals for a particular fold,
the researcher can treat the second-stage regression as if the variables were
given, and then construct the CIs as in the simple case. I construct median
confidence intervals (CIs) to aggregate information across multiple folds. I
show that the single fold procedure produces uniform, exact coverage for the
pseudo-VCATE and point-wise, exact coverage for the VCATE for all points in
the parameter space, at a nominal level $1-\alpha$. The multifold CIs have
pointwise conservative coverage.

Furthermore, the probability that the true VCATE is below the confidence
interval bounds is uniformly bounded by $\alpha$ in large samples. This result
applies to the single and multifold CIs and does not require that the first-
stage estimates to converge. Instead it relies on the fact that in experiments
the pseudo-VCATE is weakly lower than the true VCATE. Tests for homogeneity
(whether zero is contained in the CI) belong to this broader class of tests.
Having uniform size control for this class of one-sided tests means that my
tests of homogeneity are robust.

This paper is also related to a growing literature on debiased-machine
learning (Chernozhukov et al., 2022b, c; Belloni et al., 2014, 2017;
Chernozhukov et al., 2018), semiparametric efficiency (Newey, 1990), uniform
inference for non-standard problems (Andrews et al., 2020), and tests of
treatment effect homogeneity (Ding et al., 2019; Crump et al., 2008; Heckman
et al., 1997; Bitler et al., 2017). My approach combines results from these
literatures by addressing a boundary inference problem with a machine learning
stage, and applying techniques of uniform inference. A related literature also
focuses on confidence intervals around point-predictions of the CATE (Athey et
al., 2019; Semenova and Chernozhukov, 2021), rather than overall measures of
dispersion.

Section 2 provides key definitions, introduces the welfare bound, and presents
a version of the adaptive confidence intervals for the univariate regression
case. Section 3 frames the inference problem in a more general setting, and
extends the adaptive confidence intervals for VCATE estimation with a machine
learning first stage. Section 4 presents the large sample theory. Section 6
introduces the simulations. Section 7 applies my approach to an empirical
example from Malawi. Section 8 concludes.

##  2 Overview of framework

Consider a program evaluation setting in which an individual is assigned to
either a treatment $(D=1)$ or a control group $(D=0)$. The outcome of interest
$Y$ depends on the treatment status. I denote the potential outcome under
treatment and control status as $Y_{1}$ and $Y_{0}$, respectively, and the
treatment effect as $Y_{1}-Y_{0}$. The conditional average treatment effect
(CATE) given covariates $X$ is defined as

$\tau(x):=\mathbb{E}[Y_{1}-Y_{0}\mid X=x],$

and the average treatment effect (ATE) is defined as
$\tau_{av}:=\mathbb{E}[Y_{1}-Y_{0}]$. This paper proposes an estimator of the
variance of the CATE (VCATE) defined as

$V_{\tau}:=\mathbb{V}(\tau(X)).$

The variance $V_{\tau}$ measures the dispersion of treatment effects that can
be attributed to observable characteristics $X$. The value of $V_{\tau}$
depends on the choice of covariates. To understand how different covariates
might impact the VCATE, let
$V_{\tau}^{\prime}=\mathbb{V}(\mathbb{E}[Y_{1}-Y_{0}\mid X^{\prime}])$ be the
VCATE for a different set of covariates $X^{\prime}$.

######  Lemma 1.

If $X$ is $X^{\prime}$-measurable, then $V_{\tau}\leq
V_{\tau}^{\prime}\leq\mathbb{V}(Y_{1}-Y_{0})$.

Lemma 1 shows that the VCATE has the following monotonicity property: if the
researcher adds more covariates to the analysis, or breaks down an existing
covariate into more categories, then the VCATE will be weakly larger.

The propensity score, $p(x)$, is defined as follows

$p(x):=\mathbb{P}(D=1\mid X=x).$ (1)

I restrict attention to experimental settings where $p(x)$ is known. The CATE
can be identified under further assumptions.

######  Assumption 1.

(i) Stable unit treatment value assumption (SUTVA), $Y=Y_{1}D+(1-D)Y_{0}$ (ii)
Strong overlap, there is a constant $\delta\in(0,1/2)$ such that
$\mathbb{P}(\delta<p(X)<1-\delta)=1$, (iii) Selection on observables,
$Y_{1},Y_{0}\ \rotatebox[origin={c}]{90.0}{$\models$}\ D\mid X$.

Assumption 1.(i) formalizes the idea that the researcher can only observe
either $Y_{1}$ or $Y_{0}$, but not both, for any particular individual.
Assumption (ii) holds in randomized controlled trials with treatment
probabilities bounded away from $\\{0,1\\}$. Assumption (iii) states that an
individual’s treatment probability depends on $X$ but not their potential
outcomes. Let $\mu_{d}(x)$ be the conditional mean of $Y$ given $X$ and a
fixed value of $d\in\\{0,1\\}$,

$\mu_{d}(x):=\mathbb{E}[Y\mid D=d,X=x].$ (2)

Under Assumption 1, $\mathbb{E}[Y_{d}\mid X=x]=\mu_{d}(x)$, and hence
$\tau(x)=\mu_{1}(x)-\mu_{0}(x)$. This means that the VCATE is identified, with
$V_{\tau}=\mathbb{V}(\mu_{1}(X)-\mu_{0}(X))$.

###  2.1 The VCATE and policy targeting

Practitioners can use estimates of $\tau(X)$ to decide whom to treat in future
interventions (Athey and Wager, 2021; Kitagawa and Tetenov, 2018; Manski,
2004). Program managers can target the treatment recipients based on their
initial covariates. However, whether targeting can substantially improve
average outcomes depends on the dispersion of $\tau(x)$. I show that a simple
function of the VCATE bounds the marginal gains of targeting.

Let $\gamma$ denote the joint distribution of $(X,Y_{1},Y_{0})$, $\mathcal{X}$
a set containing the support of $X$,
$\tau_{\gamma}(x):=\mathbb{E}_{\gamma}[Y_{1}-Y_{0}\mid X=x]$ the CATE given
$\gamma$. A function which maps $x$ to a probability of treatment $\pi(x)$ is
known as a statistical allocation rule (Manski, 2004). Furthermore, I denote
the set of all possible allocation rules by $\Pi$, which contains all
functions $\\{\pi:\mathcal{X}\to[0,1]\\}$. The set $\Pi$ includes many well-
known assignment rules. For instance, it includes the “non-targeted” policy
which assigns everyone to treatment if
$\mathbb{E}_{\gamma}[Y_{1}]>\mathbb{E}_{\gamma}[Y_{0}]$ and to the control
group otherwise. Moreover, the average outcome under rule $\pi$ is
$\mathbb{E}_{\gamma}[\pi(X)Y_{1}+(1-\pi(X))Y_{0}]$, and the marginal benefit
compared to the non-targeted policy is defined as
$\mathcal{U}_{\gamma}(\pi):=\mathbb{E}_{\gamma}[\pi(X)Y_{1}+(1-\pi(X))Y_{0}]-\max\\{\mathbb{E}_{\gamma}[Y_{1}],\mathbb{E}_{\gamma}[Y_{0}]\\}$.

######  Theorem 1.

Let $\Gamma$ denote the set of distributions such that
$\mathbb{V}_{\gamma}[\tau_{\gamma}(X)]=V_{\tau}$. For all $\gamma\in\Gamma$
and $\pi\in\Pi$,

$\mathcal{U}_{\gamma}(\pi)\leq\underbrace{\left(\begin{array}[]{c}\text{Welfare}\\\
\text{Optimal}\\\
\text{Targeting}\end{array}\right)}_{\sup_{\pi\in\Pi}\mathbb{E}_{\gamma}[\pi(X)Y_{1}+(1-\pi(X))Y_{0}]}-\underbrace{\left(\begin{array}[]{c}\text{Welfare}\\\
\text{No}\\\
\text{Targeting}\end{array}\right)}_{\max\\{\mathbb{E}_{\gamma}[Y_{1}],\mathbb{E}_{\gamma}[Y_{0}]\\}}\leq\frac{1}{2}\sqrt{V_{\tau}}.$

The bound is sharp in the sense that
$\mathcal{U}_{\gamma}(\pi)=\frac{1}{2}\sqrt{V_{\tau}}$ for at least one
$\gamma\in\Gamma$ and $\pi\in\Pi$.

######  Theorem 2.

Consider distributions where $\mathbb{E}_{\gamma}[\tau_{\gamma}(X)]=\tau_{av}$
and $\mathbb{V}_{\gamma}[\tau_{\gamma}(X)]=V_{\tau}$, then
$U_{\gamma}(\pi)\leq\frac{1}{2}\left(-|\tau_{av}|+\sqrt{V_{\tau}+\tau_{av}^{2}}\right)$.
This bound is sharp over this subset of distributions.

Theorem 1 shows the VCATE provides a welfare bound over the superset of policy
classes. Furthermore, any type of restrictions on $\Pi$ such as budget
constraints or incentive compatibility will achieve utilitarian welfare gains
that are weakly lower than $\frac{1}{2}\sqrt{V_{\tau}}$. The bound in Theorem
1 and the generalization in Theorem 2 provide simple bounds on the prospective
gains of targeting, without needing to solve for $\pi(x)$. The bounds are most
informative when $V_{\tau}$ is low. For instance, when $V_{\tau}=0$ there is
no heterogeneity explained by the observables $X$ and therefore there are no
gains from targeting. However, the fact that the bound is sharp does not imply
that it is always achievable for every $\gamma\in\Gamma$, and when $V_{\tau}$
is high it is still be necessary to optimize $\pi(x)$ to determine whether
personalized offers are worthwhile.

Finding the bound in Theorem 1 relies on two important insights. On one hand,
the optimal policy in $\Pi$ treats an individual if and only if $\tau(x)\geq
0$ (Kitagawa and Tetenov, 2018). Substituting the optimal policy,
$\sup_{\pi\in\Pi}\mathcal{U}_{\gamma}(\pi)$ is equal to
$\mathbb{E}_{\gamma}[\max\\{\tau_{\gamma}(X),0\\}]-\max\\{\mathbb{E}_{\gamma}[\tau_{\gamma}(X)],0\\}$.
On the other hand, to avoid optimizing over all $\gamma\in\Gamma$, I break the
problem down into equivalence classes based on the moments of the negative,
zero, and positive components of the CATE. I use a constructive approach to
derive the most “adversarial” distribution. The upper bound is achieved when
the CATE has a binary support, which is partly why the bounds in Theorems 1
and 2 have simple closed forms.

######  Corollary 1.

Let $\kappa_{1},\kappa_{2}\in\mathbb{R}$ and define a new outcome
$\tilde{Y}=\kappa_{1}+\kappa_{2}Y$. The maximum welfare gain for the
transformed outcome is
$\frac{|\kappa_{2}|}{2}(-|\tau_{av}|+\sqrt{V_{\tau}+(\tau_{av})^{2}})$.

Corollary 1 shows that the welfare bound is invariant to location shifts in
the outcome, and grows linearly with scale shifts. This result implies that
transformations that change the sign, e.g. $\kappa_{2}=-1$, do not change the
value of the welfare bound. Consequently, the bound applies regardless of
whether the welfare objective is to increase a desirable outcome or to
decrease an undesirable outcome.

###  2.2 Inference using regressions

Consider a simple situation where $X$ is real valued, the treatment $D$ is
experimentally assigned with constant probability, and $U$ is a mean zero
error term. The researcher runs the following linear regression,

$Y=c_{1}+c_{2}X+\beta_{1}D+\beta_{2}DX+U,\quad\mathbb{E}[(1,X,D,DX)^{\prime}U]=0.$
(3)

Define the auxiliary quantities $\tau^{*}(x):=\beta_{1}+\beta_{2}x$ and
$V_{x}:=\mathbb{V}(X)$. The pseudo-VCATE is defined as

$V_{\tau}^{*}:=\mathbb{V}(\tau^{*}(X))=\beta_{2}^{2}V_{x}.$ (4)

The pseudo-VCATE has a close connection to the VCATE. If the linear model
describes the conditional mean $\mu_{d}(x)$, then $\tau^{*}(x)=\tau(x)$ and
$V_{\tau}=V_{\tau}^{*}$. For instance, in models with binary $X$, the
functional form is correctly specified and $V_{\tau}=V_{\tau}^{*}$. For now,
assume that the pseudo-VCATE and the VCATE coincide. In later sections, I
analyze models that allow for misspecification.

Consider a sequence of distributions
$\\{\gamma_{n}\\}_{n=1}^{\infty}\in\Gamma^{\infty}$. I index the regression
coefficients and model variances by the sample size $n$ as $\beta_{2n}$ and
$(V_{xn},V_{\tau n})$, respectively. Define an estimator of the VCATE as
$\widehat{V}_{\tau n}=\widehat{\beta}_{2n}^{2}\widehat{V}_{xn}$, where
$\widehat{\beta}_{2n}$ is the least squares estimator of (3) and
$\widehat{V}_{xn}:=\frac{1}{n}\sum_{i}X_{i}^{2}-\left[\frac{1}{n}\sum_{i}X_{i}\right]^{2}$.
With some algebraic manipulations the estimation error can be decomposed as

$\widehat{V}_{\tau n}-V_{\tau
n}^{*}=V_{xn}(\widehat{\beta}_{2n}-\beta_{2n})^{2}\left(\frac{\widehat{V}_{xn}}{V_{xn}}\right)+2\beta_{2n}V_{xn}(\widehat{\beta}_{2n}-\beta_{2n})\left(\frac{\widehat{V}_{xn}}{V_{xn}}\right)+\beta_{2n}^{2}V_{xn}\left(\frac{\widehat{V}_{xn}}{V_{xn}}-1\right).$
(5)

To derive the asymptotic distribution we can apply the central limit theorem
to individual components. For generality, I state joint convergence to a
normal distribution as an assumption. This holds as a special case if the
observations are i.i.d. and key moments of the distribution are bounded, but
may also hold under other forms of dependence. I defer stating primitive
conditions until Section 3.2.

######  Assumption 2.

There is a sequence of distributions
$\\{\gamma_{n}\\}_{n=1}^{\infty}\in\Gamma^{\infty}$ with associated quantities
$\\{V_{xn},V_{\tau n}^{*},\beta_{2n},\Omega_{n}\\}_{n=1}^{\infty}$, which are
related by the identity $V_{\tau n}^{*}=\beta_{2n}^{2}V_{xn}$, and satisfy the
following properties: (i) $V_{xn}>0$, (ii) $V_{\tau n}^{*}$ is contained in a
bounded subset of $[0,\infty)$, and (iii) $\Omega_{n}$ is a positive definite
matrix with eigenvalues bounded way from zero and a finite upper bound. There
is a sequence of estimators $\\{\widehat{V}_{xn},\widehat{V}_{\tau
n},\widehat{\beta}_{2n},\widehat{\Omega}_{n}\\}_{n=1}^{\infty}$ which satisfy
$\widehat{V}_{\tau n}=\widehat{\beta}_{2n}^{2}\widehat{V}_{xn}$. As
$n\to\infty$, $\widehat{\Omega}_{n}\to^{p}\Omega_{n}$, and

$\Omega_{n}^{-1/2}\sqrt{n}\begin{pmatrix}\sqrt{V_{xn}}(\widehat{\beta}_{2n}-\beta_{2n})\\\
\frac{\widehat{V}_{xn}}{V_{xn}}-1\end{pmatrix}\to^{d}Z_{n}\sim\mathcal{N}(0,I_{2\times
2}).$ (6)

The normalization by $V_{xn}$ is intended to align with the decomposition in
(5). The $2\times 2$ matrix $\Omega_{n}$ is an estimator of the covariance
matrix. I present Assumption 2 as a triangular array because it makes it
easier to formalize discussions of uniform coverage over the parameter space.
Assumption 2 allows for cases where $V_{\tau n}^{*}$ is arbitrarily close to
or includes zero. Let $\Omega^{1/2}$ denote the Cholesky decomposition of a
matrix $\Omega$. The estimator of the VCATE converges to the empirical process
$G$, defined as

$G(n,V_{\tau}^{*},\Omega,z,\zeta):=\frac{(e_{1}^{\prime}\Omega^{1/2}z)^{2}}{n}+2\zeta\sqrt{\frac{V_{\tau}}{n}^{*}}(e_{1}^{\prime}\Omega^{1/2}z)+\frac{V_{\tau}^{*}}{\sqrt{n}}(e_{2}^{\prime}\Omega^{1/2}z),$
(7)

where $z\in\mathbb{R}^{2}$, $\zeta\in\\{-1,1\\}$, $e_{1}=[1,0]^{\prime}$ and
$e_{2}=[0,1]^{\prime}$.

######  Lemma 2.

Suppose that Assumption 2 holds, then $\widehat{V}_{\tau n}-V_{\tau
n}^{*}=O_{p}\left(\max\left\\{\frac{1}{n},\sqrt{\frac{V_{\tau
n}^{*}}{n}}\right\\}\right)$, and there exists a sequence of
$\zeta_{n}\in\\{-1,1\\}$, such that

$\widehat{V}_{\tau n}-V_{\tau n}^{*}=G(n,V_{\tau
n}^{*},\Omega_{n},Z_{n},\zeta_{n})+o_{p}\left(\frac{1}{n}\right)+o_{p}\left(\sqrt{\frac{V_{\tau
n}^{*}}{n}}\right)+o_{p}\left(\frac{V_{\tau n}^{*}}{\sqrt{n}}\right).$ (8)

Lemma 2 shows that the limiting distribution of $\widehat{V}_{\tau n}$ is a
linear combination of a Chi-square and a normal, whose weights depend on the
value of $V_{\tau n}^{*}$. The relative magnitude of $V_{\tau n}^{*}$
determines the fit of the normal approximation. In the heterogeneous case,
$V_{\tau n}^{*}\geq\delta>0$, $\sqrt{n}G$ converges to a normal as
$n\to\infty$ because the first term in (7) is asymptotically negligible.
However, when $V_{\tau n}^{*}=0$, only the first term remains and $nG$
converges to a non-central Chi-Square distribution, which is asymmetric. Using
normal critical values here (even if everything else was known) would produce
distorted coverage. Furthermore, when $V_{\tau n}^{*}=0$ the rate of
convergence is $n$, which is faster than $\sqrt{n}$, and hence the estimator
is “super consistent” near the boundary. The error is dominated by the first
stage sampling uncertainty in estimating the nuisance parameter
$\beta_{2n}^{2}$, which converges at $n$ rate.

In practice, all three components in (7) contribute to the limiting
distribution, and this information can be used for inference. I propose an
analytic approach based on the quantiles of the empirical process that can
deliver exact coverage. Let $F_{n,V_{\tau}^{*},\Omega,\zeta}(v)$ be the
conditional CDF of the empirical process, defined as

$F_{n,V_{\tau}^{*},\Omega,\zeta}(v)=\mathbb{P}(G(n,V_{\tau}^{*},\Omega,Z,\zeta)\leq
v),\qquad Z\sim\mathcal{N}(0,I_{2\times 2}),\quad v\in\mathbb{R}.$ (9)

Based on this CDF we can construct a test statistic,

$F_{n,V_{\tau}^{*},\widehat{\Omega}_{n},\zeta}(\widehat{V}_{\tau
n}-V_{\tau}^{*}),$

indexed by unknown values of $(V_{\tau}^{*},\zeta)$ and substituting the
estimated covariance matrix $\widehat{\Omega}_{n}$. By construction, the test
statistic is contained in $[0,1]$. Similarly, I construct critical values as
functions of the parameters for a nominal level $\alpha$, as follows

$\displaystyle q_{\alpha/2}(n,V_{\tau}^{*},\Omega,\zeta)$
$\displaystyle:=\min\left\\{\alpha/2,F_{n,V_{\tau}^{*},\Omega,\zeta}(0)\right\\}$
(10) $\displaystyle q_{1-\alpha/2}(n,V_{\tau}^{*},\Omega,\zeta)$
$\displaystyle:=1-\alpha+\min\left\\{\alpha/2,F_{n,V_{\tau}^{*},\Omega,\zeta}(0)\right\\}$

The difference in the critical values is $(1-\alpha)$ to achieve the desired
coverage. The lower critical value is the minimum of the $\alpha/2$ percentile
and $0$. This adjustment is meant to increase the power of tests of
homogeneity (see Remark 2). I propose an adaptive confidence interval by
substituting the $\widehat{\Omega}_{n}$, $n$, and $\widehat{V}_{\tau n}$ into
the following formula

$\displaystyle\begin{split}\widehat{CI}_{\alpha
n}=\bigg{\\{}&V_{\tau}^{*}\in\mathbb{R}_{+},\zeta\in\\{-1,1\\}:\\\
&F_{n,V_{\tau}^{*},\widehat{\Omega}_{n},\zeta}(\widehat{V}_{\tau
n}-V_{\tau}^{*})\in\left[q_{\alpha/2}(n,V_{\tau}^{*},\widehat{\Omega}_{n},\zeta),q_{1-\alpha/2}(n,V_{\tau}^{*},\widehat{\Omega}_{n},\zeta)\right]\bigg{\\}}.\end{split}$
(11)

The set $\widehat{CI}_{\alpha n}$ can be constructed via a grid search between
$0$ and an arbitrarily high value, to test whether a particular $V_{\tau}^{*}$
satisfies the inequality constraints. The procedure achieves correct
asymptotic size because the test statistic converges to a uniform random
variable in $[0,1]$ for each value of $V_{\tau}^{*}$. In general, the
distribution in (8) depends on the value of $\zeta$ and I obtain a
conservative interval in (11) by considering the union of intervals with
different values of $\zeta$. Moreover, if the off-diagonal element of
$\Omega_{n}$ is zero, then the distribution of the empirical process in (7)
does not depend on the value of $\zeta$. This property is plausible and I
introduce primitive conditions that satisfy it in Section 4. Under those
conditions the confidence interval has exact asymptotic coverage .

The procedure is fast because at each point in the grid the researcher
evaluates the condition in (11), using the same estimate of
$(\widehat{V}_{\tau n},\widehat{\Omega}_{n})$. The critical values can be
computed numerically from the quantiles of a generalized Chi-square with
distribution $F$, which are available in most statistical software packages.

######  Remark 1 (Equivalence of homogeneity test, $\beta_{2}=0$).

Researchers can test for homogeneity by evaluating whether
$0\in\widehat{CI}_{\alpha n}$. By definition,
$e_{1}^{\prime}\widehat{\Omega}_{n}^{1/2}Z=\sqrt{\Omega_{n,11}}Z_{1}$, where
$\Omega_{n,11}$ is the upper-left entry. Under the null, the test statistic is
$F_{n,0,\widehat{\Omega}_{n},\zeta}(v)=\mathbb{P}(\widehat{\Omega}_{n,11}Z_{1}^{2}/n\leq
v)$, which is the CDF of a rescaled Chi-square distribution with one degree of
freedom. Furthermore, the critical values are $\\{0,1-\alpha\\}$, given that
$F_{n,0,\widehat{\Omega}_{n},\zeta}(0)=0$. Neither quantity depends on the
choice of $\zeta$. Because of the normalization in (6), we can choose
$\widehat{\Omega}_{n,11}=\widehat{V}_{xn}\widehat{\mathbb{V}}(\widehat{\beta}_{2n})$,
where $\widehat{\mathbb{V}}(\widehat{\beta}_{2n})$ is an estimate of the
asymptotic variance of $\widehat{\beta}_{2n}$ such as the robust sandwich
estimator. Therefore, evaluating
$F_{n,0,\widehat{\Omega}_{n},\zeta}(\widehat{V}_{\tau n})\in[0,1-\alpha]$ is
algebraically equivalent to a test of whether
$n(\widehat{\beta}_{2n}^{2}\widehat{V}_{xn}/(\widehat{V}_{xn}\widehat{\mathbb{V}}(\widehat{\beta}_{2n}))=n\widehat{\beta}_{2n}^{2}/\widehat{\mathbb{V}}(\widehat{\beta}_{2n})$
exceeds the $1-\alpha$ quantile of a Chi-square with one degree of freedom.
This is identical to a test of $\beta_{2}=0$ in the regression in (3).

######  Remark 2 (Adjusting critical values).

The critical value $q_{\alpha/2}(V_{\tau}^{*},\Omega,n,\zeta)$ in (10) is
constructed to guarantee that $\widehat{V}_{\tau n}\in\widehat{CI}_{\alpha
n}$. In this case, $\widehat{V}_{\tau n}$ belongs to the CI if and only if
$F_{n,\widehat{V}_{\tau n},\widehat{\Omega}_{n},\zeta}(0)$ is contained in the
critical region for some $\zeta\in\\{-1,1\\}$. The unadjusted CI with critical
values $\left\\{\alpha/2,1-\alpha/2\right\\}$ is not guaranteed to contain the
test statistic.777For example, suppose that $\widehat{V}_{\tau n}=0$. Then the
empirical process has a Chi-square distribution for $V_{\tau}=0$. Since the
unadjusted critical value is bounded away from zero, $\widehat{V}_{\tau n}$
would not be contained in the unadjusted CI. Another rationale for doing the
adjustment in (10), is to increase the power of the test of homogeneity,
$0\in\widehat{CI}_{\alpha n}$, relative to a test based on the unadjusted CI.
The unadjusted test has correct size but the rejection region is
discontinuous: it rejects when the test statistics is very close to zero or
when it exceeds a threshold. Instead, the adjusted test shifts the critical
region left and has the form of a Chi-squared test. It only rejects the null
if the test statistic is larger than $1-\alpha$, which is a threshold that is
smaller than $1-\alpha/2$ for the unadjusted CI.

######  Remark 3 (Comparison to other tests of homogeneity).

Crump et al. (2008) suggest estimating $\widehat{\mu}_{d}(x)$ by a series
estimator with $K$ terms, for subsamples $D=d\in\\{0,1\\}$. They propose a
bias-corrected Wald statistic, which takes the form
$\widehat{T}_{n}^{series}:=\\{[\widehat{\xi}_{1}-\widehat{\xi}_{0}]^{\prime}[\widehat{\mathbb{V}}(\widehat{\xi}_{1}-\widehat{\xi}_{0})]^{-1}[\widehat{\xi}_{1}-\widehat{\xi}_{0}]-(K-1)\\}/\sqrt{2(K-1)}$,
where $(\widehat{\xi}_{1},\widehat{\xi}_{0})$ are non-intercept coefficients
associated with $\widehat{\mu}_{1}(x)$ and $\widehat{\mu}_{0}(x)$,
respectively and $K$ is the number of covariates. For regressions with
univariate $X$ as in (3), $K=2$ and
$\widehat{T}_{n}^{series}=(n\widehat{\beta}_{2n}^{2}/\widehat{\mathbb{V}}(\widehat{\beta}_{2n})-1)/\sqrt{2}$.
Essentially this is just a transformation of the test statistic proposed
above, which will produce the same acceptance/rejection result for
significance level $\alpha$ (using the critical values in their equation
3.11). Ding et al. (2019) study a framework with a fixed population where the
only source of randomness is the experimental assignment of offers. They
propose a similar Wald estimator, but replace estimates of
$(\widehat{\xi}_{1},\widehat{\xi}_{0})$ and the asymptotic variance with
randomization inference counterparts. In samples with large $n$, this leads to
very similar test statistics, but may produce slightly different results in
small samples.

The approach that I introduce in the following section differs substantially
in the way that I handle multivariate cases. For $K>2$, the approaches are
non-nested because I use sample splitting and consider a wider range of
methods to estimate $\widehat{\mu}_{d}(x)$ than series estimators.

##  3 Inference for nonparametric CATE

In this section I provide an overview of the inference problems associated
with efficient estimators of the VCATE and how to solve them for the
nonlinear/high-dimensional case. Let $\\{Y_{i},D_{i},X_{i}\\}$ be i.i.d.. As
shown in Newey (1990), efficient estimators can be decomposed as

$\sqrt{n}(\widehat{V}_{\tau n}-V_{\tau
n})=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\varphi_{i}+\underbrace{\
\text{Residual}_{n}\ }_{o_{p}(1)},$ (12)

where $\varphi_{i}$ is an i.i.d. realization from the efficient influence
function with mean zero, and the residual becomes asymptotically negligible as
$n\to\infty$. The semiparametric lower bound is $\mathbb{V}(\varphi_{i})$. Let
$\eta(\cdot)$ be a set of nuisance functions defined as

$\eta(x):=(\tau(x),\mu_{0}(x),p(x),\tau_{av}).$ (13)

Levy et al. (2021) showed that the efficient influence function for the VCATE
is equal to $\varphi_{i}=\varphi(Y_{i},D_{i},X_{i},\eta)-V_{\tau n}$, where
$\varphi$ is defined as

$\varphi(y,d,x,\eta):=(\tau(x)-\tau_{av})^{2}+2(\tau(x)-\tau_{av})\left[\frac{d(y-\mu_{0}(x)-\tau(x))}{p(x)}-\frac{(1-d)(y-\mu_{0}(x))}{1-p(x)}\right].$
(14)

By (12), all efficient estimators –regardless of their form– are
$\sqrt{n}$–asymptotically equivalent to
$\frac{1}{n}\sum_{i=1}^{n}\varphi_{i}$. Let
$\varphi_{i}=\varphi(Y_{i},D_{i},X_{i},\eta(X_{i}))$ be a realization of the
efficient influence function. If $\mathbb{V}(\varphi_{i})>0$, then

$\mathbb{V}(\varphi_{i})^{-1}\times\sqrt{n}(\widehat{V}_{\tau n}-V_{\tau
n})=\mathbb{V}(\varphi_{i})^{-1}\left[\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\varphi_{i}\right]+o_{p}(1)\to^{p}\mathcal{N}(0,1).$

In this case, any confidence interval based on normal critical values and a
consistent estimator of $\mathbb{V}(\varphi_{i})$, produces valid coverage.
For common functionals such as the ATE, $\mathbb{V}(\varphi_{i})>0$. However,
this cannot be guaranteed for the VCATE.

######  Lemma 3.

Let $\sigma_{d}^{2}(x):=\mathbb{V}(Y\mid D=d,X=x)$.

$\mathbb{V}(\varphi_{i})=\mathbb{V}((\tau(X)-\tau_{av})^{2})+4\mathbb{E}\left[(\tau(X)-\tau_{av})^{2}\left(\frac{\sigma_{1}^{2}(X)}{p(X)}+\frac{\sigma_{0}^{2}(X)}{p(X)}\right)\right].$
(15)

Both of the inner terms in (15) are multiplied by $(\tau(x)-\tau)$. When the
VCATE is zero, $\tau(x)=\tau$ almost surely, and the influence function is
degenerate. The condition that $\mathbb{V}(\varphi_{i})>0$ does not hold
uniformly over all $V_{\tau}$ in the parameter space. In this case, the
distribution of $\sqrt{n}(\widehat{V}_{\tau n}-V_{\tau n})$ is dominated by
the higher order terms of the residual (12), and the CLT cannot be applied to
guarantee normality near the boundary. The linear estimator discussed in the
previous section is just one example. Moreover, if the tails of $\tau(x)$ are
thin, then the value of $\mathbb{V}(\varphi_{i})$ is also small near the
boundary.

######  Corollary 2.

If $\mathbb{E}[(\tau(X)-\tau)^{4}]\leq\kappa^{2}V_{\tau}^{2}$ for
$\kappa\in\mathbb{R}_{+}$, then

$\mathbb{V}(\varphi_{i})\leq\kappa^{2}V_{\tau}^{2}+4\kappa
V_{\tau}\sqrt{\mathbb{E}\left[\left(\frac{\sigma_{1}^{2}(X)}{p(X)}+\frac{\sigma_{0}^{2}(X)}{1-p(X)}\right)\right]}.$

Corollary 2 shows that the variance of the efficient influence function is
bounded by a quantity that scales up or down proportional to the value of
$V_{\tau}$. Consequently, when $\mathbb{V}(\varphi_{i})$ is relatively small,
the higher order terms in the residual may still dominate.

###  3.1 Pseudo-VCATE, regressions, and efficiency

A robust way to introduce nonlinearity is to consider a regression with real-
valued basis functions $M(x)$ and $S(x)$. For now, I will leave these
unspecified but in the next section I will show how they can be estimated non-
parametrically in a first stage.

$Y=\underbrace{c_{0}+M(X)c_{1}+[D-p(X)]\beta_{1}+[D-p(X)]S(X)\beta_{2}}_{W(X,D)^{\prime}\theta}+U$
(16)

with weights $\lambda(X):=[p(X)(1-p(X))]^{-1}$, regressors $W(X,D)$, and
parameters $\theta=[c_{0},c_{1},\beta_{1},\beta_{2}]$. This specification
accommodates experiments with heterogeneous assignment probabilities.888When
$p(x)=1/2$ this produces exactly the same coefficients as a regression of $Y$
on $(1,M(X),D,D\times S(X))$, but differs when the probabilities are
heterogeneous. If $M(X)=S(X)=X$ as well, this reduces to (3). Consider the
following minimizers:

$\Theta^{*}:=\arg\min_{\theta\in\mathbb{R}^{4}}\
\mathbb{E}[\lambda(X)(Y-W(X,D)^{\prime}\theta)^{2}].$ (17)

Chernozhukov et al. (2022a) showed that if $\mathbb{V}(S(X))>0$,
$\mathbb{E}[S(X)]=0$, and the vector $(\beta_{1},\beta_{2})$ are part of the
solution to (17), then $(\beta_{1},\beta_{2})$ are also the intercept and
slope of the best linear projection of $\tau(X)$ on $S(X)$.999When
$\mathbb{V}(S(X))=0$, $\beta_{2}$ does not have a unique solution in (17), but
$V_{\tau}^{*}=\beta_{2}^{2}\mathbb{V}(S(X))=0\leq V_{\tau}$ is still the best
linear projection, regardless of the value of $\beta_{2}$. Hence the pseudo-
VCATE has an upper bound, $V_{\tau}^{*}=\beta_{2}^{2}\mathbb{V}(S(X))\leq
V_{\tau}$. Because of this bound, if $V_{\tau}=0$, then the pseudo-VCATE
$(V_{\tau}^{*})$ will not falsely detect heterogeneity even if $S(X)$ is
misspecified. If anything, poor choices of $S(X)$ will possibly understate the
amount of heterogeneity. When $\tau(X)$ is spanned by $S(X)$,
$V_{\tau}^{*}=V_{\tau}$ and the two notions coincide.

To obtain a feasible estimator we define
$\widehat{S}(x):=S(x)-\frac{1}{n}\sum_{i=1}^{n}S(X_{i})$, and compute
$\widehat{W}(x,d)$ by substituting $\widehat{S}(x)$ in (16). Now consider a
value of $\widehat{\theta}_{n}$ that minimizes
$\frac{1}{n}\sum_{i=1}^{n}\lambda(X_{i})(Y_{i}-\widehat{W}(X_{i},D_{i})^{\prime}\theta)^{2}$,
by solving the first order condition

$\mathcal{Q}(\widehat{\theta}_{n}):=\frac{1}{n}\sum_{i=1}^{n}\lambda(X_{i})(Y-\widehat{W}(X_{i},D_{i})^{\prime}\widehat{\theta}_{n})\widehat{W}(X_{i},D_{i})^{\prime}.$
(18)

The regression parameters can be used to construct the CATE and other nuisance
functions. For a given $\theta\in\mathbb{R}^{4}$,

$\tilde{\eta}_{\theta}(x):=\begin{pmatrix}\tilde{\tau}_{\theta}(x)\\\
\tilde{\mu}_{0,\theta}(x)\\\ \tilde{p}_{\theta}(x)\\\
\tilde{\tau}_{av,\theta}\end{pmatrix}=\begin{pmatrix}(\widehat{W}(x,1)-\widehat{W}(x,0))^{\prime}\theta\\\
\widehat{W}(x,0)^{\prime}\theta\\\ p(x)\\\
\frac{1}{n}\sum_{i=1}^{n}\tilde{\tau}_{\theta}(X_{i})\end{pmatrix}.$ (19)

Lemma 4 shows that the sample variance of the estimated CATE can be
interpreted as an estimator that plugs-in (19) to the efficient influence
function in (14).

######  Lemma 4.

Define $\widehat{V}_{xn}=\frac{1}{n}\sum_{i=1}^{n}\widehat{S}(X_{i})^{2}$. Let
$\varphi$, $\tilde{\eta}_{\theta}$ and $\mathcal{Q}(\theta)$ be defined as in
(14), (18), and (19), respectively. If
$\widehat{\theta}_{n}=(\widehat{c}_{1n},\widehat{c}_{2n},\widehat{\beta}_{1n},\widehat{\beta}_{2n})$
solves $\mathcal{Q}(\widehat{\theta}_{n})=0$, then $\widehat{V}_{\tau
n}:=\widehat{\beta}_{2n}^{2}\widehat{V}_{xn}=\frac{1}{n}\sum_{i=1}^{n}\varphi(Y_{i},D_{i},X_{i},\tilde{\eta}_{\widehat{\theta}_{n}})$.

Mechanically, the influence function can be decomposed into primary and bias-
correction components. As an intermediate step for Lemma 4, I show that the
bias-correction terms and the fourth component of (18) are proportional to
each other. The optimal $\widehat{\theta}_{n}$ implicitly sets the average
bias correction to zero. Intuitively, the linear model minimizes the covariate
imbalances between the treatment and control group in-sample. Lemma 4 suggests
that $\widehat{V}_{\tau n}$ could be asymptotically efficient if
$\tilde{\eta}_{\widehat{\theta}_{n}}$ is sufficiently close to $\eta$. In
Section 4.1, I show that my proposed semiparametric estimator can indeed
achieve this.

As a preliminary step, it is necessary to determine which $S(x)$ and $M(x)$
ensure that $\tilde{\eta}_{\theta}=\eta$. Not all choices achieve this
property.101010This point highlights that while all regressions of the form in
(16) proposed by Chernozhukov et al. (2022a) estimate an interpretable
$V_{\tau}^{*}$ –regardless of the choice of $M(x)$–, not every regression in
this class is efficient. The functions $S(x)$, and $M(x)$ in particular, both
affect efficiency. However, if they are chosen in such a way that
$W(x,d)^{\prime}\theta=\mu_{d}(x)$ for some $\theta\in\mathbb{R}^{4}$, then
that’s sufficient to guarantee that $\eta_{\theta}=\eta$. Lemma 5 shows that
any $\theta$ with this property is also a solution to the regression problem,
and provides guidance on the choice of $S(x)$ and $M(x)$.

######  Lemma 5.

Let $\Theta^{*}$ be the optimizer set defined in (17). If (i)
$\mathbb{E}[S(X)]=0$ and (ii) $W(x,d)^{\prime}\theta=\mu_{d}(x)$ for some
$\theta\in\mathbb{R}^{4}$, then $\theta\in\Theta^{*}$. Conditions (i) can be
satisfied by setting $S(x)=\tau(x)-\mathbb{E}[\tau]$. Condition (ii) can be
satisfied by setting $M(x)=\mu_{0}(x)+p(x)\tau(x)$. In this special case,
$\theta=(0,1,\mathbb{E}[\tau(X)],1)^{\prime}\in\Theta^{*}$.

Lemma 5 provides efficient choices of $S(x)$ and $M(x)$ that can be expressed
in terms of conditional moments, and that for this choice, the optimal
$\theta$ has a known, simple form. In practice, $S(x)$ and $M(x)$ can be
estimated non-parametrically.

###  3.2 Multi-step approach

My proposed procedure randomly partitions the observations
$\mathcal{I}_{n}:=\\{1,\ldots,n\\}$ into $K$ folds of equal size $n_{k}:=n/K$.
Denote the observations in each fold by $\mathcal{I}_{nk}$, so that
$\bigcup_{k=1}^{K}\mathcal{I}_{nk}=\mathcal{I}_{n}$, and let
$\mathcal{I}_{-nk}:=\mathcal{I}_{n}\backslash\mathcal{I}_{nk}$ be the set of
observations that are not in fold $k$. In a slight abuse of notation, I use
$\mathcal{I}_{-nk}$ when defining conditional expectations, to denote the full
set of random variables associated with observations not included in fold $k$.
For simplicity, I also label the fold of observation $i\in\mathcal{I}_{nk}$,
by $k_{i}$.

Let
$\widehat{\eta}_{-k}(x):=(\widehat{\tau}_{-k}(x),\widehat{\mu}_{0,-k}(x),p(x),\widehat{\tau}_{-k,av})$
denote a prediction of the nuisance function $\eta(x)$ over the set
$\mathcal{I}_{-nk}$, using the researcher’s preferred prediction algorithm.
This could include traditional methods such as linear regression, or more
modern “machine learning” approaches such as LASSO, neural networks, or random
forests. The only function that is known in advance is the propensity score,
since I restrict attention to randomized experiments. Guided by Lemma 5,
define

$\displaystyle\begin{split}S_{-k}(x)&:=\widehat{\tau}_{-k}(x)-\mathbb{E}[\widehat{\tau}_{-k}(X_{i})\mid\mathcal{I}_{-nk}],\\\
M_{-k}(x)&:=\widehat{\mu}_{0,-k}(x)+p(x)\widehat{\tau}_{-k}(x),\\\
\lambda(x)&:=[p(x)(1-p(x))]^{-1},\\\
W_{i}^{\prime}&:=\begin{bmatrix}1&M_{-k_{i}}(X_{i})&(D_{i}-p(X_{i}))&(D_{i}-p(X_{i}))S_{-k_{i}}(X_{i})\end{bmatrix}\end{split}$
(20)

Consider a regression with weights $\lambda(X_{i})$, parameters
$\theta:=(c_{1},c_{2},\beta_{1},\beta_{2})$, and

$Y_{i}=W_{i}^{\prime}\theta_{nk}+U_{i},\qquad\mathbb{E}[W_{i}U_{i}\mid\mathcal{I}_{-nk}]=0$
(21)

In practice, $\mathbb{E}[\widehat{\tau}_{-k}(X_{i})\mid\mathcal{I}_{-nk}]$
needs to be estimated, and I use a sample analog:

$\displaystyle\begin{split}\widehat{S}_{-k}(x)&:=\widehat{\tau}_{-k}(x)-\widehat{\tau}_{nk,av},\qquad\widehat{\tau}_{nk,av}:=\frac{1}{n_{k}}\sum_{i\in\mathcal{I}_{nk}}\widehat{\tau}_{-k}(X_{i}),\\\
\widehat{W}_{i}^{\prime}&:=\begin{bmatrix}1&M_{-k_{i}}(X_{i})&(D_{i}-p(X_{i}))&(D_{i}-p(X_{i}))\widehat{S}_{-k_{i}}(X_{i})\end{bmatrix},\\\
\widehat{\theta}_{nk}&:=\left[\frac{1}{n}\sum_{i\in\mathcal{I}_{nk}}\widehat{W}_{i}\widehat{W}_{i}^{\prime}\right]^{-1}\left[\frac{1}{n}\sum_{i\in\mathcal{I}_{nk}}\widehat{W}_{i}Y_{i}\right].\end{split}$
(22)

Let
$\widehat{\theta}_{nk}=(\widehat{c}_{1nk},\widehat{c}_{2nk},\widehat{\beta}_{1nk},\widehat{\beta}_{2nk})$
be the estimator over the subsample $\mathcal{I}_{nk}$. The fold-specific
variance of $\widehat{S}_{-k_{i}}(x)$ is defined as

$\widehat{V}_{xnk}:=\frac{1}{n_{k}}\sum_{i\in\mathcal{I}_{nk}}\widehat{S}_{-k_{i}}(X_{i})^{2}.$
(23)

The estimator of the VCATE for fold $k$ is

$\widehat{V}_{\tau nk}:=\widehat{\beta}_{2nk}^{2}\widehat{V}_{xnk}.$ (24)

In this case $\widehat{V}_{xnk}$ can be viewed as a preliminary estimate of
the VCATE using the data in $\mathcal{I}_{-nk}$, whereas $\widehat{V}_{\tau
nk}$ is a regression-adjusted estimator that fits the sample
$\mathcal{I}_{nk}$. This adjustment will produce better results, with a
pseudo-VCATE interpretation even if the first step $\widehat{S}_{-k_{i}}(x)$
function is noisy, misspecified, or slow to converge to $\tau(x)$. The
estimator in (23) belongs to the class of multi-step estimators defined in
Chernozhukov et al. (2022a). I add a restriction on the choice of $M_{-k}(x)$,
guided by Lemma 5, to ensure asymptotic efficiency.

To quantify the uncertainty in $(\widehat{\theta}_{nk},\widehat{V}_{xnk})$ I
compute a robust (sandwich) estimator. I start by defining two auxiliary
residuals,
$\widehat{T}_{i}:=\widehat{V}_{xnk}^{-1}\widehat{S}_{-k_{i}}(X_{i})^{2}-1$ and
$\widehat{U}_{i}:=Y_{i}-\widehat{W}_{i}^{\prime}\widehat{\theta}_{nk}$. Let
$\widehat{\Pi}_{nk}$ be a $4\times 4$ diagonal matrix with diagonal entries
$(1,1,1,\widehat{V}_{xnk}^{-1/2})$. Researchers can compute estimators of the
individual components of the sandwich form $\widehat{J}_{nk}$,
$\widehat{H}_{nk}$, and a selection matrix $\Upsilon$ defined as follows

$\widehat{J}_{nk}:=\begin{bmatrix}\frac{1}{n_{k}}\sum_{i\in\mathcal{I}_{k}}\lambda(X_{i})\widehat{\Pi}_{nk}\widehat{W}_{i}\widehat{W}_{i}^{\prime}\widehat{\Pi}_{nk}^{\prime}&0\\\
0&1\end{bmatrix},\qquad\Upsilon:=\begin{bmatrix}0&0&0&1&0\\\
0&0&0&0&1\end{bmatrix}$ (25)
$\widehat{H}_{nk}:=\frac{1}{n_{k}}\sum_{i\in\mathcal{I}_{k}}\begin{bmatrix}\lambda(X_{i})^{2}\left[\widehat{U}_{i}^{2}\widehat{\Pi}_{nk}\widehat{W}_{i}\widehat{W}_{i}^{\prime}\widehat{\Pi}_{nk}^{\prime}\right]&\lambda(X_{i})\left[\widehat{U}_{i}\widehat{\Pi}_{nk}\widehat{W}_{i}\widehat{T}_{i}\right]\\\
\lambda(X_{i})\left[\widehat{U}_{i}\widehat{W}_{i}^{\prime}\widehat{\Pi}_{nk}^{\prime}\widehat{T}_{i}\right]&\widehat{T}_{i}^{2}\end{bmatrix}.$
(26)

The sandwich covariance estimator is

$\widehat{\Omega}_{nk}=\Upsilon\widehat{J}_{nk}^{-1}\widehat{H}_{nk}\widehat{J}_{nk}^{-1}\Upsilon^{\prime}.$
(27)

The population covariance matrix is

$\Omega_{nk}=\mathbb{V}\left(\begin{bmatrix}\lambda(X_{i})(D_{i}-p(x_{i}))V_{xnk}^{-1/2}S_{-k}(X_{i})U_{i}\\\
V_{xnk}^{-1}S_{-k}(X_{i})^{2}\end{bmatrix}\mid\mathcal{I}_{-nk}\right).$ (28)

When the VCATE is zero, then $\widehat{V}_{xnk}$ (as a consistent estimator of
$V_{\tau n}$) should converge to zero along the asymptotic sequence. To
prevent asymptotic degeneracy, we need to rescale the estimands along the
lines of Assumption 2. The random variable $V_{xnk}^{-1/2}S_{-k}(X_{i})$ is
normalized to (conditionally) have variance one by design, even if
$S_{-k}(X_{i})$ converges to zero. This requires two much weaker conditions:
(i) that $V_{xnk}>0$, i.e. there is some noise in estimating the CATE;111111I
also propose an extension that allows for $V_{xnk}=0$ in Remark 6. (ii)
$\Omega_{nk}$ has eigenvalues bounded away from zero. To ensure this, the
tails of $S_{-k}(X_{i})$ need to be thin.121212One sufficient additional
restriction is that $\mathbb{E}[U_{i}\mid X_{i},D_{i},\mathcal{I}_{-nk}]=0$
(the model is correctly specified), $\mathbb{V}(U_{i}\mid
X_{i},D_{i}\mid\mathcal{I}_{-nk})$ is bounded away from zero, and
$S_{-k}(X_{i})$ has bounded kurtosis. In that case the off-diagonal elements
of $\Omega_{nk}$ are zero and the diagonals are uniformly bounded. Positive-
definiteness may also hold in a neighborhood where the nuisance functions are
close to the true value and $\mathbb{E}[U_{i}\mid
X_{i},D_{i},\mathcal{I}_{-nk}]\approx 0$.

######  Assumption 3 (Moment Bounds).

Suppose that there exists a constant $\delta\in(0,1)$ such that for each fold
$k$, almost surely, (i)
$\mathbb{E}[M_{-k}(X_{i})^{2}\lambda(X_{i})\mid\mathcal{I}_{-nk}]-\mathbb{E}[M_{-k}(X_{i})\lambda(X_{i})\mid\mathcal{I}_{-nk}]\mathbb{E}[\lambda(X_{i})\mid\mathcal{I}_{-nk}]\geq\delta$,
(ii) $\mathbb{E}[M_{-k}(X_{i})^{4}\mid\mathcal{I}_{-nk}]\leq 1/\delta$, (iii)
$\mathbb{E}[U_{i}^{4}\mid\mathcal{I}_{-nk}]\leq(1/\delta)$, and (iv)
$\mathbb{E}[S_{-k}(X_{i})^{4}\mid\mathcal{I}_{-nk}]\leq(1/\delta)\mathbb{E}[S_{-k}^{2}\mid\mathcal{I}_{-nk}]^{2}$,
(v) $\mathbb{E}[\widehat{\tau}_{-k}(X)^{2}\mid\mathcal{I}_{-nk}]<1/\delta$.

Assumption 3.(i) is a rank condition that ensures that the auxiliary regressor
$M_{-k}(X_{i})$ is not degenerate. Assumption 3.(ii) ensures that the second-
moment of the candidate regressor $M_{-k}(X_{i})$ is bounded. Assumption
3.(iii) is a standard condition indicating that the fourth moment of the
residuals are bounded. Assumption 3.(iv) is a bounded kurtosis condition
indicating that the out-of-sample, machine learning predictions of $\tau(x)$
have thin tails. Finally, Assumption 3.(v) is a bound on the variance of the
first-stage VCATE.

######  Assumption 4 (Non-degeneracy).

The following properties hold almost surely over sequences of random data
realizations $\\{\mathcal{I}_{n1},\ldots,\mathcal{I}_{nK}\\}_{n=1}^{\infty}$.
Conditional on $\mathcal{I}_{-nk}$: (i) $V_{xnk}>0$, (ii) $V_{xnk}$ has a
finite upper bound, (iii) $V_{\tau nk}^{*}:=\beta_{2nk}^{2}V_{xnk}$ is
contained in a bounded subset of $[0,\infty)$, and (iv) $\Omega_{nk}$ defined
in (28) is a positive definite matrix with bounded eigenvalues.

######  Assumption 5 (Random Sampling).

The observations $\\{Y_{0i},Y_{1i},D_{i},X_{i}\\}_{i}^{n}$ are i.i.d. across
$i$ for fixed $n$, and drawn from a sequence of data generating processes
$\\{\gamma_{n}\\}_{n=1}^{\infty}$.

Theorem 3 shows how these primitive conditions imply an analog of Assumption 2
for the cross-fitted case.

######  Theorem 3.

Consider a sequence of random data realizations
$\\{\mathcal{I}_{n1},\ldots,\mathcal{I}_{nK}\\}_{n=1}^{\infty}$ with
associated quantities $\\{V_{xnk},V_{\tau
nk}^{*},\beta_{2nk},\Omega_{nk}\\}_{n=1}^{\infty}$ for each $k$, as well as a
sequence of estimators
$\\{\widehat{\beta}_{2nk},\widehat{V}_{xnk},\widehat{V}_{\tau
nk}^{*},\widehat{\Omega}_{nk}\\}_{n=1}^{\infty}$ computed from (22), (23),
(24), and (27), respectively. Suppose that these quantities satisfy
Assumptions 1.(ii), 3, 4, and 5. Then as $n_{k}\to\infty$, for all
$k\in\\{1,\ldots,K\\}$, Conditional on a sequence of $\mathcal{I}_{-nk}$,

  1. (i)

$\Omega_{nk}^{-1/2}\sqrt{n_{k}}\begin{pmatrix}\sqrt{V_{xnk}}(\widehat{\beta}_{2nk}-\beta_{2nk})\\\
\frac{\widehat{V}_{xnk}}{V_{xnk}}-1\end{pmatrix}\mid\mathcal{I}_{-nk}\to
Z_{nk}\sim\mathcal{N}(0,I_{2\times 2}).$

  2. (ii)

$\widehat{\Omega}_{nk}\to^{p}\Omega_{nk}$.

Theorem 3 presents a central limit theorem for the components of
$\widehat{V}_{\tau nk}^{*}=\widehat{\beta}_{2nk}^{2}\widehat{V}_{xnk}$,
properly rescaled and conditional on $\mathcal{I}_{-nk}$. This result holds
regardless of whether the nuisance parameters are properly specified and
primarily relies on the independence of the folds. By Lemma 2, conditional on
$\mathcal{I}_{-nk}$,

$\widehat{V}_{\tau nk}-V_{\tau nk}^{*}=G(n_{k},V_{\tau
nk}^{*},\Omega_{nk},Z_{nk})+o_{p}\left(\frac{1}{n_{k}}\right)+o_{p}\left(\sqrt{\frac{V_{\tau
nk}^{*}}{n_{k}}}\right)+o_{p}\left(\frac{V_{\tau
nk}^{*}}{\sqrt{n_{k}}}\right).$ (29)

Then it is possible to construct adaptive confidence intervals, substituting
the sample size $n_{k}$ and estimated statistics $(\widehat{V}_{\tau
nk},\widehat{\Omega}_{nk})$.

$\displaystyle\begin{split}\widehat{CI}_{\alpha
nk}=\bigg{\\{}&V_{\tau}^{*}\in\mathbb{R}_{+},\zeta\in\\{-1,1\\}:\\\
&F_{n_{k},V_{\tau}^{*},\widehat{\Omega}_{nk},\zeta}(\widehat{V}_{\tau
nk}-V_{\tau}^{*})\in\left[q_{\alpha/2}(n_{k},V_{\tau}^{*},\widehat{\Omega}_{nk},\zeta),q_{1-\alpha/2}(n_{k},V_{\tau}^{*},\widehat{\Omega}_{nk},\zeta)\right]\bigg{\\}}.\end{split}$
(30)

The confidence intervals take the same form as in the regression case in (11),
except that now the inputs are obtained from the cross-fitted regression step.
The confidence interval is fast to compute because $(\widehat{V}_{\tau
nk},\widehat{\Omega}_{nk})$ only needs to be computed once. It is worth noting
that because the confidence interval only uses information in fold
$\mathcal{I}_{nk}$, the effective sample size is $n_{k}$. While this does not
affect the nominal asymptotic size of the confidence interval, it may affect
the power of tests against specific alternatives.

####  3.2.1 Ensemble estimator

We can construct an “ensemble” to aggregate across folds, defined as follows

$\widehat{V}_{\tau
n}:=\frac{1}{K}\sum_{k=1}^{K}\widehat{\beta}_{2nk}^{2}\widehat{V}_{xnk}.$ (31)

In Section 4.1, I show that this ensemble estimator is efficient.

####  3.2.2 Splitting uncertainty and median intervals

So far in this section we have used the data from a single split or fold of
the data. However, the choice of fold $k$ or the particular split may lead to
different values of $\widehat{V}_{\tau nk}$ and hence distinct confidence
intervals. Chernozhukov et al. (2022a) propose an aggregation procedure based
on “median parameter” confidence intervals, inspired by false-discovery rate
adjustments. Their proposed conditional t-tests are not directly applicable
here because $\widehat{V}_{\tau nk}$ conditionally converges to a generalized
Chi-square. However, I show that the basic idea can still be adapted.

Let $K$ be the total number of folds, obtained across one or more splits of
the data. For instance, a 2-fold sample with 10 splits would have $K=20$, Let
$\inf\widehat{CI}_{\alpha nk}$ and $\sup\widehat{CI}_{\alpha nk}$ denote the
lower and upper bounds of $\widehat{CI}_{\alpha nk}$, respectively, and
$\text{Med}_{K}\\{\cdots\\}$ denote the median over a set indexed by
$k=\\{1,\ldots,K\\}$. If $K$ is even then two quantities might be tied for the
median, and in that case I compute their midpoint. The multifold confidence
interval is defined as

$\widehat{CI}_{\alpha
n}^{\text{multifold}}=\left[\text{Med}_{K}\left\\{\inf\widehat{CI}_{\frac{\alpha}{2}nk}\right\\},\text{Med}_{K}\left\\{\sup\widehat{CI}_{\frac{\alpha}{2}nk}\right\\}\right].$
(32)

Intuitively, the $K$ fold-specific intervals “vote” to include a particular
value, and $V_{\tau}^{*}\in\widehat{CI}_{\alpha n}^{\text{multifold}}$ only if
there is a majority vote. The “median” interval $\widehat{CI}_{\alpha
n}^{\text{multifold}}$ contains values within the median lower bound and the
median upper bound across folds. To control the overall false discovery rate,
I adjust the nominal size to $\alpha/2$. This adjustment produces a
conservative interval because it assumes a worst-case dependence structure
between the folds and the splits, regardless of the size of $K$. In some
instances, the asymptotic coverage probability may be strictly higher than
$(1-\alpha)$, particularly when there is a lot of heterogeneity.131313For
instance, given a single split, Theorem 6 implies that the
$\\{\sqrt{n_{k}}(\widehat{V}_{\tau nk}-V_{\tau})\\}_{k=1}^{K}$ are
asymptotically uncorrelated. However, near the boundary, the estimators
converge at a rate faster than $\sqrt{n}$ and their relative dependence
structure at that rate is unclear. At the boundary, with low effect
heterogeneity or none at all, it is much harder to asses the dependence
structure between the fold-specific estimators. One of the benefits of using a
worst-case approach is that it provides coverage guarantees under weak
assumptions. Moreover, the empirical example illustrates that even though
these intervals are conservative, they may have a short length in practice.

##  4 Large Sample Theory

###  4.1 $\sqrt{n}$-Consistency, Efficiency, and Boundary Rates

Let $\gamma\in\Gamma$ denote a probability distribution over i.i.d
observations $(Y_{1i},Y_{0i},D_{i},X_{i})$. I use the notation
$\mathbb{E}_{\gamma}[\cdot]$ and $\mathbb{P}_{\gamma}(\cdot)$ to denote the
expectation and probability under $\gamma$, respectively. Let $S_{-k}(x)$ be
the function defined in (20). The true value of the CATE and VCATE is given by
$\tau_{\gamma}$ and $V_{\tau}(\gamma)$, respectively. The pseudo-VCATE is
given by

$V_{\tau}^{*}(\gamma,\mathcal{I}_{-nk}):=V_{\tau}(\gamma)-\inf_{(\beta_{1},\beta_{2})\in\mathbb{R}^{2}}\mathbb{E}_{\gamma}[(\tau_{\gamma}(X)-\beta_{1}-\beta_{2}S_{-k}(X))^{2}\mid\mathcal{I}_{-nk}].$
(33)

Define the estimation error of the CATE in the $L_{2}$ norm as

$\omega\left(\gamma\right):=\sqrt{\mathbb{E}_{\gamma}[\|\widehat{\tau}_{-k}(X)-\tau_{\gamma}(X)\|^{2}]}.$
(34)

We can bound the difference between the pseudo-VCATE and its true value:

######  Theorem 4 (Bias of the pseudo-VCATE).

Under the distribution $\gamma\in\Gamma$,

$\mathbb{E}_{\gamma}[|V_{\tau}(\gamma)-V_{\tau}^{*}(\gamma,\mathcal{I}_{-nk})|]\leq\min\left\\{16\times\omega(\gamma)^{2},V_{\tau}(\gamma)\right\\}.$
(35)

Theorem 4 derives a non-asymptotic bound for the VCATE as the minimum of two
key quantities: (i) the conditional $L_{2}$ error between the candidate
function and the true CATE, and (ii) the true value of the VCATE. This proof
only relies on the definition in (33). For instance, when
$V_{\tau}(\gamma)=0$, then
$V_{\tau}(\gamma)-V_{\tau}^{*}(\gamma,\mathcal{I}_{-nk})=0$, regardless of
whether $\widehat{\tau}_{-k}(\cdot)$ is properly specified. The difference
between the two quantities is also small if $\omega(\gamma)$ is sufficiently
close to zero. In the multi-step approach, $\omega(\gamma)$ captures the
first-stage uncertainty from estimating the CATE, which decreases with sample
size. I consider the following convergence condition.

######  Assumption 6 (Convergence CATE).

$\sqrt{n_{k}}\omega(\gamma_{n})^{2}=o(1)$ as $n\to\infty$.

Assumption 6 imposes an $L_{2}$ consistency condition on the CATE. A large
class of machine learning models can meet this requirement. For example,
Bickel et al. (2009) and Belloni et al. (2014) evaluate rates of convergence
under sparse models, Chen and White (1999) for neural networks, and Wager and
Walther (2015) for regression trees and random forest.

######  Theorem 5 (Faster than $\sqrt{n}$ convergence near boundary).

Consider a sequence of data generating processes
$\\{\gamma_{n}\\}_{n=1}^{\infty}$ where $V_{\tau}(\gamma_{n})\to 0$ as
$n\to\infty$ and Assumptions 1, 3, 4, 5 and 6 hold. Define
$\Delta_{nk}:=\left(\widehat{V}_{\tau nk}-V_{\tau}(\gamma_{n})\right)$ for the
estimator defined in (24). Then (i) $\sqrt{n_{k}}\Delta_{nk}=o_{p}(1)$, and
(ii) if in addition $n_{k}^{1/2+\rho}V_{\tau}(\gamma_{n})=o(1)$ for
$\rho\in[0,1/2)$, then $n_{k}^{1/2+\rho}\Delta_{nk}=o_{p}(1)$.

Theorem 5 shows that multi-step estimators of the VCATE converge to zero
faster than $\sqrt{n_{k}}$ near the boundary. I formalize “near” by
considering sequences of distributions where the VCATE approaches zero.
Theorem 5 relies on the non-asymptotic bound in Theorem 4, the normal
approximation in Theorem 3, and the empirical process in Lemma 2. There is no
requirement on the rate of convergence of $\widehat{\mu}_{-0k}(\cdot)$ (and
consequently on the generated regressor $M_{-k_{i}}(\cdot)$), only an
assumption that $p(x)$ is known and that the CATE is estimated at a
sufficiently fast rate. Furthermore, if the true CATE is nearly flat in the
sense that for $\rho\in[0,1/2)$, then
$n_{k}^{1/2+\rho}V_{\tau}(\gamma_{n})=o(1)$ (or even exactly equal to zero),
then the estimator has a faster rate guarantee.

To prove efficiency we have the stronger requirement that all the nuisance
functions converge to their true value in the $L_{4}$ norm and at
$n_{k}^{1/4}$ rate in the $L_{2}$ norm.

######  Assumption 7 (Regularity conditions).

Define the residuals $U_{i}=Y_{i}-\mathbb{E}_{\gamma_{n}}[Y_{i}\mid
D_{i},X_{i}]$. (i) $\mathbb{E}_{\gamma_{n}}[\|Y_{i}\|^{4}]$,
$\mathbb{E}_{\gamma_{n}}[\|U_{i}\|^{4}]$,
$\mathbb{E}_{\gamma_{n}}[\|\eta(X_{i})\|^{4}]$, (ii)
$\mathbb{E}_{\gamma_{n}}[\|\widehat{\eta}_{-k}(X_{i})\|^{4}]$ are uniformly
bounded, (iii)
$\mathbb{E}_{\gamma_{n}}\left[\|\widehat{\eta}_{-k}(X_{)}-\eta(X_{i})\|^{4}\right]\to
0$, (iv)
$\sqrt{n_{k}}\mathbb{E}_{\gamma_{n}}[\|\widehat{\eta}_{-k}(X_{i})-\eta(X_{i})\|^{2}]=o(1)$
for all $k\in\\{1,\ldots,K\\}$.

The next step is to show that the estimation error of the fold-specific VCATE
converges at $\sqrt{n_{k}}$ to an average of efficient influence functions.

######  Theorem 6 ($\sqrt{n}$ Consistency and Efficiency).

Consider a sequence of data generating processes
$\\{\gamma_{n}\\}_{n=1}^{\infty}$ where $V_{\tau}(\gamma_{n})\to 0$ as
$n\to\infty$ and Assumptions 1, 3, 4, 5, 6, and 7 hold. Then

$\sqrt{n_{k}}(\widehat{V}_{\tau
nk}-V_{\tau}(\gamma_{n}))=\frac{1}{\sqrt{n_{k}}}\sum_{i\in\mathcal{I}_{nk}}\varphi_{i}+o_{p}(1).$

Theorem 6 shows that the fold-specific estimator converges at
$\sqrt{n_{k}}$-rate to an average of i.i.d influence function. This requires
standard regularity conditions. The proof of Theorem 6 is non-standard due to
the multi-step nature of the procedure. I start by applying Lemma 4, which
shows hows to write $\widehat{V}_{\tau nk}$ as an average of estimated
influence functions. I break down the proof into sequences where
$V_{\tau}(\gamma_{n})$ converges to zero and those where it’s bounded away
from zero. For the first part, I leverage (a) the boundary convergence result
in Theorem 5, (b) the bound for $\mathbb{V}(\varphi_{i})$ in Lemma 2. For the
second part, I provide a novel decomposition of regression adjusted nuisance
functions. The key is to prove that the regression parameters
$\widehat{\theta}_{nk}$ converge at $n_{k}^{1/4}$ rate to the values in Lemma
5 for sequences where $V_{\tau}(\gamma_{n})\to V_{\tau}>0$. Once in this form,
the rest of the proof relies on a traditional Taylor expansion argument.

The ensemble estimator $\widehat{V}_{\tau n}$ combines information from the
whole sample. By definition $n=n_{k}\times K$ and $K$ is finite, which means
that algebraically $\sqrt{n}(\widehat{V}_{\tau
n}-V_{\tau}(\gamma_{n}))=\frac{\sqrt{n_{k}}}{\sqrt{K}}\sum_{k=1}^{K}(\widehat{V}_{\tau
nk}-V_{\tau}(\gamma_{n}))$, and by Theorem 6,

$\sqrt{n}(\widehat{V}_{\tau
n}-V_{\tau}(\gamma_{n}))=\left[\frac{1}{\sqrt{n_{k}K}}\sum_{k=1}^{K}\sum_{i\in\mathcal{I}_{nk}}\varphi_{i}+\frac{1}{\sqrt{K}}\sum_{k=1}^{K}o_{p}(1)\right]=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\varphi_{i}+o_{p}(1).$

This means that aggregating the estimators restores full efficiency,
satisfying the property described in (12).

###  4.2 Asymptotic Coverage

I start by showing that the single fold confidence interval has uniform
coverage for the pseudo-VCATE, and exact coverage under an additional
assumption.

######  Assumption 8 (Exact coverage condition).

Let $\Omega_{nk,12}$ be the off-diagonal element of $\Omega_{nk}$. For each
$t>0$,
$\underset{n\to\infty}{\lim\sup}\sup_{\gamma\in\Gamma}\mathbb{P}_{\gamma}(\sqrt{V_{\tau}^{*}(\gamma,\mathcal{I}_{-nk})}|\Omega_{nk,12}|>t)=0$.

Assumption 8 states that the product of the pseudo-VCATE and the off-diagonal
element of the limiting covariance matrix in (28) needs to converge to zero
uniformly.

######  Theorem 7 (Uniform Coverage of Pseudo-VCATE).

Let $\Gamma$ denote a set of distributions, constrained in such a way that
Assumptions 1, 3, 4, and 5 hold. Let $\widehat{CI}_{\alpha nk}$ and
$V_{\tau}^{*}(\gamma,\mathcal{I}_{-nk})$ be defined as in (30) and (33),
respectively. Then

$1-\alpha\leq\underset{n\to\infty}{\lim\inf}\inf_{\gamma\in\Gamma}\mathbb{P}_{\gamma}\left(V_{\tau}^{*}(\gamma,\mathcal{I}_{-nk})\in\widehat{CI}_{\alpha
nk}\right)$ (36)

If Assumption 8 also holds, then

$\underset{n\to\infty}{\lim\sup}\sup_{\gamma\in\Gamma}\mathbb{P}_{\gamma}\left(V_{\tau}^{*}(\gamma,\mathcal{I}_{-nk})\in\widehat{CI}_{\alpha
nk}\right)\leq 1-\alpha.$ (37)

Theorem 7 shows that the confidence intervals always have uniform coverage of
the pseudo-VCATE of at least $(1-\alpha)$.141414The theorem only uses
Assumptions 1, 3, 4, and 5 to verify normality in Assumption 2. A broad class
of confidence intervals of the form in (11) constructed from regression
adjusted estimators will satisfy these uniformity properties. The key is to
prove that the confidence intervals yield coverage under arbitrary sequences
of distributions, which includes cases where $V_{\tau n}^{*}$ is either equal
to zero or approaches zero as $n\to\infty$. The proof builds on the
approximation of Lemma 2 and shows that for every sequence, the test statistic
for a particular $\zeta_{n_{k}}\in\\{-1,1\\}$ converges to a uniform
distribution. This sequential characterization suffices to apply generic
results in Andrews et al. (2020), which guarantee uniform coverage even in non
standard cases like this one. Coverage over the pseudo-VCATE holds regardless
of whether the nuisance functions are slow to converge or even misspecified.

The intervals are in general conservative because we’re not plugging in the
unknown $\zeta$, and instead define a robust confidence interval as the union
of CIs with given $\zeta\in\\{-1,1\\}$. However, the key insight is that
$\zeta$ only affects the coverage when the pseudo-VCATE is bounded away from
zero. If Assumption 8 holds, the value of $\zeta$ doesn’t enter the asymptotic
distribution of the estimator. I show that this condition holds automatically
if the nuisance functions converge to their true value at a sufficiently fast
rate.

######  Lemma 6 (Verify Exact Coverage).

Let $\Gamma$ denote a set of distributions that satisfy Assumptions 1, 3, 4,
5, 6, and 7. Then Assumption 8 also holds.

As a special case, when the model is correctly specified, i.e.
$W_{i}^{\prime}\theta=\mu_{d}(x)$ for some $\theta\in\mathbb{R}^{4}$, then
$\Omega_{nk,12}=0$ by construction. Lemma 6 states that we only need a model
that is correctly specified asymptotically, given the rates in Assumptions 6
and 7. Then for non-boundary cases, $\Omega_{nk}$ converges to the population
analog under correct specification. These conditions also imply point-wise
coverage of the true VCATE.

######  Theorem 8 (Pointwise, Exact Coverage of VCATE).

Let $\Gamma$ denote a set of distributions that satisfy Assumptions 1, 3, 4,
5, 6, and 7. Then

$\inf_{\gamma\in\Gamma}\underset{n\to\infty}{\lim\inf}\
\mathbb{P}_{\gamma}\left(V_{\tau}(\gamma)\in\widehat{CI}_{\alpha
nk}\right)=\sup_{\gamma\in\Gamma}\underset{n\to\infty}{\lim\sup}\
\mathbb{P}_{\gamma}\left(V_{\tau}(\gamma)\in\widehat{CI}_{\alpha
nk}\right)=1-\alpha.$

Theorem 8 shows that if the nuisance functions converge at a sufficiently fast
rate, then the proposed intervals achieve point-wise exact coverage. The
confidence intervals provide correct size coverage for all regions of the
parameter space, including $V_{\tau}(\gamma)=0$.

Proving uniform coverage of the VCATE (rather than the pseudo-VCATE) is more
challenging in the non-parametric case without much stronger conditions on the
convergence rates of the nuisance functions. The lack of uniformity stems from
a difficulty in controlling the ratio
$\sqrt{n_{k}}\omega(\gamma_{n})/\sqrt{V_{\tau}(\gamma_{n})}$, which measures
the relative error in estimating the CATE vs. the overall level of the VCATE.
By the bound in (4), this ratio is easy to control when
$n_{k}V_{\tau}(\gamma_{n})=o(1)$ (near homogeneity) or
$V_{\tau}(\gamma_{n})\to V_{\tau}>0$ (strong heterogeneity). However, it is
possible to construct sequences, e.g., $n_{k}V_{\tau}(\gamma_{n})\to v>0$,
where $(\widehat{V}_{\tau nk}-V_{\tau}^{*}(\gamma_{n},\mathcal{I}_{-nk}))$
converges to zero at a faster or comparable rate to the error of the pseudo-
VCATE. There may be distortions in coverage in smaller samples. I illustrate
this issue in the simulations.

######  Remark 4 (Uniform inference for one-sided tests).

Uniform inference is only challenging for two-sided tests. If instead, the
researcher is only interested in left-sided tests, then uniform inference is
still possible. To do so, we can make explicit use of the inequality
$V_{\tau}^{*}(\gamma,\mathcal{I}_{-nk})\leq V_{\tau}(\gamma)$. If
$V_{\tau}(\gamma)<\inf_{V_{\tau}^{*}}\widehat{CI}_{\alpha nk}$ (the lower
bound of the CI), then
$V_{\tau}^{*}(\gamma,\mathcal{I}_{-nk})\notin\widehat{CI}_{\alpha nk}$.
Therefore, for all $\gamma\in\Gamma$,

$\mathbb{P}_{\gamma}\left(V_{\tau}(\gamma)\geq\inf\widehat{CI}_{\alpha
nk}\right)\leq\mathbb{P}_{\gamma}\left(V_{\tau}^{*}(\gamma,\mathcal{I}_{-nk})\notin\widehat{CI}_{\alpha
nk}\right).$ (38)

I prove a weaker uniformity result for one-sided tests building on Theorem 7.

######  Corollary 3.

If Assumptions 1, 3, 4, and 5 hold, then

$\underset{n\to\infty}{\lim\sup}\sup_{\gamma\in\Gamma}\mathbb{P}_{\gamma}\left(V_{\tau}(\gamma)<\inf\widehat{CI}_{\alpha
nk}\right)\leq\underset{n\to\infty}{\lim\sup}\sup_{\gamma\in\Gamma}\mathbb{P}_{\gamma}\left(V_{\tau}^{*}(\gamma,\mathcal{I}_{-nk})\notin\widehat{CI}_{\alpha
nk}\right)\leq\alpha.$

Corollary 3 is empirically relevant for interpreting confidence intervals that
do not include zero. It states that the asymptotic probability of having
$V_{\tau}(\gamma)\in\left[0,\inf\widehat{CI}_{\alpha nk}\right)$ is uniformly
less than $\alpha$. Tests of homogeneity belong to this class and therefore
have the correct size when $V_{\tau}(\gamma)=0$. Moreover, the result in
Corollary 3 is much stronger because it guarantees that a broader class of
one-sided tests also has the correct size. It is important to emphasize that I
do not impose any assumptions on rates of convergence of
$(\widehat{\eta}_{-k}(x)-\eta(x))$, but only the inequality on the pseudo-
VCATE. Consequently, while estimating $\mu(x)$ and $\tau(x)$ may be important
for increasing the power of tests of homogeneity, it is not necessary for
controlling their size.

###  4.3 Multifold Coverage

The multi-fold confidence interval covers the VCATE asymptotically.

######  Theorem 9.

Let $\Gamma$ be a set of distributions that satisfy Assumptions 1, 3, 4, 5.
Then

$\underset{n\to\infty}{\lim\sup}\sup_{\gamma\in\Gamma}\mathbb{P}_{\gamma}\left(V_{\tau}(\gamma)<\inf\widehat{CI}_{\alpha
nk}^{multifold}\right)\leq\alpha.$

If Assumptions 6, and 7 also hold, then

$\sup_{\gamma\in\Gamma}\underset{n\to\infty}{\lim\sup}\
\mathbb{P}_{\gamma}\left(V_{\tau}(\gamma)\notin\widehat{CI}_{\alpha
n}^{\text{multifold}}\right)\leq\alpha.$

The first part of Theorem 9 shows that the multifold CI uniformly controls the
size of one-sided tests. The second part shows that if the nuisance functions
converge to their true value asymptotically, then the multifold confidence
interval provides point-wise size-control for two-sided tests. Coverage of the
true parameter will be weakly larger that $(1-\alpha)$ asymptotically.

###  4.4 Power

The test of homogeneity has power against local alternatives.

######  Lemma 7.

Consider a sequence of distributions $\\{\gamma_{n}\\}_{n=1}^{\infty}$ and
$\\{\mathcal{I}_{-nk}\\}_{n=1}^{\infty}$, where
$\Omega_{nk}\to\Omega_{\infty}$ and
$n_{k}V_{\tau}^{*}(\gamma_{n},\mathcal{I}_{-nk})=v+o(1)$, for
$v\in[0,\infty)$. Assume that 1, 3, 4, and 5 hold. Let $\Omega_{\infty,11}$ be
the upper-left entry of $\Omega_{\infty}$, $\Phi(\cdot)$ be the standard
normal CDF, and $z_{1-\alpha}$ be the $(1-\alpha)-$quantile. Then

$\lim_{n\to\infty}\mathbb{P}_{\gamma_{n}}(0\notin\widehat{CI}_{\alpha
nk}\mid\mathcal{I}_{-nk})=1-\Phi\left(z_{1-\alpha}-\frac{\sqrt{v}}{\sqrt{\Omega_{\infty,11}}}\right)+\Phi\left(-\frac{\sqrt{v}}{\sqrt{\Omega_{\infty,11}}}\right).$

Lemma 7 computes the power curve for a sequence of local alternatives. When
$v=0$ the power is equal to $\alpha$, whereas when $v\to\infty$ the power
tends to one. This shows that tests of homogeneity have local power the null.
When the pseudo-VCATE is bounded away from zero, the test rejects with
probability approaching one.

##  5 Extensions

######  Remark 5 (Clustered Standard Errors).

In some cases, assuming that units $i$ are independent may be strong. For
example, in Dizon-Ross (2019) units are randomized at the household level, and
it is reasonable to expects that units within a household have correlated
outcomes and covariates. To deal with this dependence structure, suppose that
the sample can be partitioned into $C$ clusters, $c\in\\{1,\ldots,C\\}$, which
are independent and identically distributed. The researcher can compute
$\widehat{\beta}_{2nk}$, $\widehat{V}_{\tau nk}$ and $\widehat{CI}_{\alpha
nk}$ via cross-fitting by randomly partitioning entire clusters rather than
the individual observations.

######  Lemma 8.

Let $\\{r_{nk}\\}_{n=1}^{\infty}$ be a sequence of positive scalars. Suppose
that $V_{xnk}>0$, $\Omega_{nk}$ is positive definite with positive
eigenvalues, and that conditional on a sequence
$\\{\mathcal{I}_{-nk}\\}_{n=1}^{\infty}$,
$r_{nk}^{-1}\widehat{\Omega}_{nk}\to^{p}\Omega_{nk}$, $n_{k}/r_{nk}\to\infty$,
and

$\Omega_{nk}^{-1/2}\sqrt{\frac{n_{k}}{r_{nk}}}\begin{pmatrix}\sqrt{V_{xn}}(\widehat{\beta}_{2nk}-\beta_{2nk})\\\
\frac{\widehat{V}_{xnk}}{V_{xnk}}-1\end{pmatrix}\mid\mathcal{I}_{-nk}\to^{d}Z_{n}\sim\mathcal{N}(0,I_{2\times
2}).$ (39)

Then $\widehat{CI}_{\alpha nk}$, substituting the arguments
$(n_{k},\widehat{V}_{\tau nk},\widehat{\Omega}_{nk})$, satisfies Theorem 7.

Lemma 8 proposes high-level conditions that ensure that confidence intervals
have correct coverage. The quantity $\sqrt{n_{k}/r_{nk}}$ is the effective
rate of convergence, which features prominently in problems with cluster
dependence (MacKinnon et al., 2022). For example, if the observations are
fully correlated within clusters and the clusters have equal size, then
$r_{nk}$ is the cluster size, $n_{k}/r_{nk}=C$, and the estimators in (8)
converge at $\sqrt{C}$ rate (the total number of clusters). The analyst does
not need to specify the quantity $r_{nk}$ to apply the procedure, but merely
specify an estimator of the covariance matrix that meets the rate requirement.
Under minor modifications to the existing proofs, we can also prove analogs of
Theorems 8 and 9.

We can construct estimators that satisfy Lemma 8. Let $\mathcal{I}_{nkc}$ be
the set of units in fold $k$ and cluster $c$, and let $\mathcal{C}_{nk}$ be
the indexes of the clusters selected for fold $k$.

$\widehat{H}_{nk}^{cluster}:=\frac{1}{n_{k}}\sum_{c\in\mathcal{C}_{nk}}\left(\sum_{i\in\mathcal{I}_{nkc}}\begin{bmatrix}\lambda(X_{i})\widehat{U}_{i}\widehat{W}_{i}\\\
\widehat{T}_{i}\end{bmatrix}\right)\left(\sum_{i\in\mathcal{I}_{nkc}}\begin{bmatrix}\lambda(X_{i})\widehat{U}_{i}\widehat{W}_{i}\\\
\widehat{T}_{i}\end{bmatrix}\right)^{\prime}.$

The clustered standard errors are
$\widehat{\Omega}_{nk}^{cluster}=\widehat{\Upsilon}_{nk}\widehat{J}_{nk}^{-1}\widehat{H}_{nk}^{cluster}\widehat{J}_{nk}^{-1}\widehat{\Upsilon}_{nk}^{\prime}$,
where $\widehat{J}_{nk},\widehat{\Upsilon}_{nk}$ are computed as outlined in
(25).

######  Remark 6 (Confidence intervals when $V_{xnk}=0$).

When the conditional mean is constant, i.e. $\mu_{d}(x)=\mathbb{E}[Y_{d}]$,
prediction models with corner solutions like LASSO may estimate a constant
conditional mean, i.e. $\widehat{\mu}_{d,-k}(x)=\widehat{\mu}_{d,av}$,
$\widehat{\tau}_{-k}(x)=\widehat{\mu}_{1,-k}(x)-\widehat{\mu}_{0,-k}(x)$ is
constant, and consequently $V_{xnk}=0$.151515It is still possible to have
$V_{xnk}>0$ almost surely even if $V_{\tau}=0$, as long as $\mu_{0}(x)$ is not
constant. This violates Assumption 4.(i), and it is challenging to construct a
confidence interval with exact coverage. One alternative is to construct an
ensemble of sparse and non-sparse estimators of the CATE in the first-stage.
Another alternative is to use degenerate confidence intervals:

$\widehat{CI}_{\alpha nk}^{0}=\begin{cases}\widehat{CI}_{\alpha nk}&\text{if
}V_{xnk}\neq 0,\\\ [0,0]&\text{if }V_{xnk}=0.\end{cases}$ (40)

The confidence intervals collapse to zero when the $\widehat{\tau}_{-k}(x)$
prediction is degenerate. For example, in LASSO researchers can check whether
the coefficients are zero, in tree-based methods when there are no splits, or
whether $\widehat{V}_{xnk}=0$. We can also define an analogous multifold
confidence interval.

$\widehat{CI}_{\alpha
n}^{\text{0,multifold}}=\left[\text{Med}_{K}\left\\{\inf\widehat{CI}_{\frac{\alpha}{2}nk}\right\\},\text{Med}_{K}\left\\{\sup\widehat{CI}_{\frac{\alpha}{2}nk}\right\\}\right].$
(41)

I study the asymptotic properties of these confidence intervals.

######  Lemma 9.

Let $\Gamma$ denote a set of distributions that satisfy Assumptions 1, 3, and
5. Suppose that Assumption 4 holds, except for the requirement that
$V_{xnk}=0$. Then (i)

$\underset{n\to\infty}{\lim\inf}\inf_{\gamma\in\Gamma}\
\mathbb{P}_{\gamma}\left(V_{\tau}^{*}(\gamma,\mathcal{I}_{-nk})\in\widehat{CI}_{\alpha
nk}^{0}\right)\geq 1-\alpha$ (42)

(ii) If Assumptions 6, and 7 also hold, then

$\inf_{\gamma\in\Gamma}\underset{n\to\infty}{\lim\inf}\
\mathbb{P}_{\gamma}\left(V_{\tau}(\gamma)\in\widehat{CI}_{\alpha
nk}^{0}\right)\geq 1-\alpha$ (43)
$\inf_{\gamma\in\Gamma}\underset{n\to\infty}{\lim\inf}\
\mathbb{P}_{\gamma}\left(V_{\tau}(\gamma)\in\widehat{CI}_{\alpha
nk}^{0,multifold}\right)\geq 1-\alpha.$ (44)

To prove this result I focus on the coverage for subsequences where
$V_{xnk}=0$ and $V_{xnk}>0$, and apply the results for conservative coverage
results in Andrews et al. (2020). In subsequences where $V_{xnk}=0$, then
$V_{\tau}^{*}(\gamma,\mathcal{I}_{-nk})=0$ which means that coverage of the
pseudo-VCATE is equal to one. In subsequences where $V_{xnk}>0$ and assuming
that $\Omega_{nk}$ has eigenvalues bounded away from zero, then we can apply
similar arguments as before to prove $1-\alpha$ coverage. To prove point-wise
coverage, I separate the cases where $V_{\tau}(\gamma)=0$ and
$V_{\tau}(\gamma)>0$. In the latter case, I show that $V_{xnk}$ is point-wise
bounded away from zero, though not uniformly. The proof of Lemma 9 does not
rely on the i.i.d. assumption, and can also accommodate cluster dependence. In
the empirical example, I compute confidence intervals with clustered standard
errors and degenerate CATE predictions.

When there’s more heterogeneity and the nuisance functions are estimated
accurately, then $V_{xnk}>0$ with high probability. However, when $V_{\tau
n}\approx 0$ and $\mathbb{V}(\mu_{0}(X))\approx 0$, then procedures like LASSO
may imply $V_{xnk}=0$ (Fu and Knight, 2000), which means that marginally
heterogeneous CATEs could be estimated as homogeneous. This could be impact
the power of tests of homogeneity. The size for two-sided tests is not
uniformly bounded. Furthermore, the multifold confidence interval allows for
some quantification of uncertainty across folds/splits: the CI is degenerate
only if more than half the fold/split-specific CIs are degenerate.

Furthermore, the degenerate CI has correct size control for one-sided tests.

######  Corollary 4.

Under the assumptions of Lemma 9.(i),

$\underset{n\to\infty}{\lim\sup}\sup_{\gamma\in\Gamma}\mathbb{P}_{\gamma}\left(V_{\tau}(\gamma)<\inf\widehat{CI}_{\alpha
nk}^{0}\right)\leq\alpha.$ (45)
$\underset{n\to\infty}{\lim\sup}\sup_{\gamma\in\Gamma}\mathbb{P}_{\gamma}\left(V_{\tau}(\gamma)<\inf\widehat{CI}_{\alpha
nk}^{0,multifold}\right)\leq\alpha.$ (46)

The tests of homogeneity have the correct size when $V_{\tau}(\gamma)=0$.
Corollary 4 guarantees that the probability of falsely rejecting a class of
one-sided test is uniformly bounded in large samples.

######  Remark 7 (Monotonic Transformations).

It may be useful to report the standard deviation of the CATE, which is
$\sqrt{VCATE}$. I propose the following confidence interval:

$\widehat{CI}_{\alpha
n}^{0,multifold,sqrt}=\left\\{\sqrt{V_{\tau}^{*}}:V_{\tau}^{*}\in\widehat{CI}_{\alpha
n}^{0,multifold}\right\\}.$ (47)

Since the square root is a strictly increasing transformation and the VCATE is
non-negative, then $\sqrt{V_{\tau}(\gamma)}\in\widehat{CI}_{\alpha
n}^{0,multifold,sqrt}$ if and only $V_{\tau}(\gamma)\in\widehat{CI}_{\alpha
n}^{0,multifold}$. Since the events are equivalent, the transformed confidence
interval preserves the coverage probabilities and will have valid coverage by
Lemma 9.

##  6 Simulations

I use a simulation design to study the properties of the VCATE estimators. The
baseline covariates are distributed as
$[X_{0},X_{1}]\in\mathcal{N}(0,\Sigma_{x})$, where $\rho=0.5$ and

$\Sigma_{x}=\begin{bmatrix}I_{J\times J}&\rho I_{J\times J}\\\ \rho I_{J\times
J}&I_{J\times J}\end{bmatrix}.$

The random variables $X_{0}$ and $X_{1}$ are standard normal vectors of
dimension $J$. The covariance between pairs of components $X_{0j}$ and
$X_{1j^{\prime}}$ is equal to $\rho=0.5$ for when $j=j^{\prime}$, but zero
otherwise. The outcome is generated from a model where $Y=DY_{0}+(1-D)$, $D$
is generated by a Bernoulli draw with probability $0.5$, and

$\displaystyle\begin{split}Y_{0}&=c+\beta_{0}^{\prime}X_{0}+U_{0}\sqrt{\tilde{\sigma}_{0}^{2}+\kappa_{0}^{\prime}X_{0}X_{0}^{\prime}\kappa_{0}}\\\
Y_{1}&=(c+\tau)+\beta_{0}^{\prime}X_{0}+\beta_{\tau}^{\prime}X_{1}+U_{1}\sqrt{\tilde{\sigma}_{1}^{2}+\kappa_{1}^{\prime}X_{1}X_{1}^{\prime}\kappa_{1}},\end{split}$
(48)

where $c,\tau\in\mathbb{R}$,
$\beta_{0},\beta_{\tau},\kappa_{0},\kappa_{1}\in\mathbb{R}^{p}$. The errors
$(U_{0},U_{1})$ are independent of the covariates
$(U_{0},U_{1})\rotatebox[origin={c}]{90.0}{$\models$}(X_{0},X_{1})$, and
distributed as standard normals
$[U_{0},U_{1}]^{\prime}\in\mathcal{N}(0_{2\times 1},I_{2})$. The key model
quantities have closed-form expressions. The conditional means at baseline and
the CATE are given by $\mu_{1}(x)=\alpha+\beta_{0}^{\prime}x_{0}$ and
$\tau(x)=\tau+\beta_{\tau}^{\prime}Z_{1}$, respectively. The conditional
variances are
$\sigma_{d}^{2}(x)=\kappa_{d}^{\prime}x_{d}x_{d}^{\prime}\kappa_{d}$ for
$d\in\\{0,1\\}$. This formulation incorporates heteroskedasticity. Covariates
that influence the outcomes at baseline may also affect the treatment effects.

The regressors are constructed in such a way that
$\mathbb{E}[X_{d}X_{d}^{\prime}]=I_{p}$ for $d\in\\{0,1\\}$. This implies
simple expressions for the variances of the model,
$\mathbb{V}(U_{d})=\tilde{\sigma}_{d}^{2}+\kappa_{d}^{\prime}\kappa_{d}$,

$V_{\tau}=\beta_{\tau}^{\prime}\beta_{\tau},\qquad\mathbb{V}(Y_{0})=\beta_{0}^{\prime}\beta_{0}+\tilde{\sigma}_{d}^{2}+\kappa_{0}^{\prime}\kappa_{0},$
$\mathbb{V}(Y_{1})=\beta_{0}^{\prime}\beta_{0}+\beta_{\tau}^{\prime}\beta_{\tau}+2(1-\rho)\beta_{0}^{\prime}\beta_{\tau}+\tilde{\sigma}_{1}^{2}+\kappa_{1}^{\prime}\kappa_{1},$

I choose an approximately sparse specification for (48) where the coefficients
decay exponentially at a rate of decay of $\lambda=0.7$. Let
$\ell_{j}=\sqrt{\left(\frac{1-\lambda}{1-\lambda^{J}}\right)\lambda^{1-j}}$ be
a geometric sequence, which satisfies
$\sum_{j=1}^{J}\left(\frac{1-\lambda}{1-\lambda^{J}}\right)\lambda^{1-j}=1$.
Given user-specified parameters
$(V_{\mu},V_{\tau},\sigma_{0}^{2},\sigma_{1}^{2})$, the coefficients for the
entries $j\in\\{1,\ldots,J\\}$ are determined by
$\beta_{0,j}=\ell_{j}\sqrt{V_{\mu}}$,
$\beta_{\tau,j}=\ell_{j}\sqrt{V_{\tau}}$,
$\kappa_{d,j}=\ell_{j}\sqrt{\sigma_{d}^{2}-\tilde{\sigma}_{d}^{2}}$, for
$d\in\\{0,1\\}$. Since
$\sum_{j=1}^{J}\left(\frac{1-\lambda}{1-\lambda^{J}}\right)\lambda^{1-j}=1$,
then $\beta_{0}^{\prime}\beta_{0}=V_{\mu}$,
$\beta_{\tau}\beta_{\tau}=V_{\tau}$, and
$\beta_{0}^{\prime}\beta_{\tau}=\sqrt{V_{\mu}V_{\tau}}$. We can obtain
analogous expressions for the variances of the unobserved components, so that
$\tilde{\sigma}_{d}^{2}+\kappa_{0}^{\prime}\kappa_{0}=\sigma_{d}^{2}$ for
$d\in\\{0,1\\}$.

I choose an average effect size of $\tau=0.15$, that is coherent with the
recent meta-analyses of economic experiments in Vivalt (2015). To make sure
that the magnitudes are interpretable, I normalize the coefficients so that
the variance for the control group is $\mathbb{V}(Y_{0})=1$, by setting set
$c=1$, $\sigma_{d}=0.7$, $\tilde{\sigma}_{d}=0.21$, and $V_{\mu}=0.3$. The
design is easy to scale for different values of $V_{\tau}$ and $J$. My design
is similar to that in Belloni et al. (2014) but I choose $\Sigma_{x}$ and the
sparsity structure in such a way that $V_{\tau}$ has a closed form expression.
I use LASSO to estimate $\mu_{1}(x)$ and $\mu_{0}(x)$, tuned via cross-
validation. The coefficients of this model are consistent given this sparse
linear structure, even in high dimensions. I randomly simulate 2000 datasets
to compute each of the estimators, and split them into $K=2$ folds.

Figure 1: (Density of estimators) The figure shows the distributions of
$\widehat{V}_{\tau n}$ (multi-step) and $\widehat{V}_{\tau n}^{\text{two-
step}}$, defined in (31) and (49), respectively, for $n=2500$, $K=2$ folds,
and a single split. The horizontal panels show designs with homogeneity
($V_{\tau}=0$, left), moderate heterogeneity ($V_{\tau}=0.5$, middle) and
high-heterogeneity ($V_{\tau}=1$, right). The vertical panels have a low
dimension case with $2J=10$ (top), and a high-dimension case with $2J=40$
(bottom).

Figure 1 considers a simulation with $n=2500$. The figure displays a density
plot for the multi-step estimator, $\widehat{V}_{\tau n}$ defined in (31), and
a two-step debiased machine learning estimator computed as:

$\widehat{V}_{\tau n}^{\text{two-
step}}:=\frac{1}{n}\sum_{i=1}^{n}\varphi(Y_{i},D_{i},X_{i},\widehat{\eta}_{-k_{i}}(X_{i})),$
(49)

where
$\widehat{\eta}_{-k}(x)=(\widehat{\tau}_{-k}(x),\widehat{\mu}_{0,-k}(x),p(x),\widehat{\tau}_{n,av})$
and
$\widehat{\tau}_{n,av}=\frac{1}{n}\sum_{i=1}^{n}\widehat{\tau}_{-k_{i}}(X_{i})$.
When $V_{\tau}>0$ the efficient influence function is non-degenerate. In high-
heterogeneity regimes both converge to the same limiting
distribution.161616This is shown in Theorem 6 for the multistep approach and
can be shown for the two-step using standard arguments, e.g. Chernozhukov et
al. (2018). However, when $V_{\tau}=0$, the influence function is degenerate
and they may converge at different rates. We see that the multi-step approach
is much more precise. This can be explained by the fast boundary convergence
rates derived in Theorem 5. The two step approach can also produce negative
estimates of $V_{\tau}$, which is an undesirable feature, whereas the multi-
step estimator is always non-negative. Both estimators have higher bias when
the dimension increases because there is more first-stage noise.

Figure 2: (Root Mean Square Error) The figure shows the root mean-square error
of $\widehat{V}_{\tau n}$ (multi-step) and $\widehat{V}_{\tau n}^{\text{two-
step}}$, defined in (31) and (49), respectively, for different sample sizes,
$K=2$ folds, and a single split. The ‘oracle” estimator is constructed by
substituting $\widehat{\eta}_{-k}(X_{i})=\eta(X_{i})$ in (49). The horizontal
panels show designs with homogeneity ($V_{\tau}=0$, left), moderate
heterogeneity ($V_{\tau}=0.5$, middle) and high-heterogeneity ($V_{\tau}=1$,
right). The vertical panels have a low dimension case with $2J=10$ (top), and
a high-dimension case with $2J=40$ (bottom).

Figure 2 plots the root mean-square error (RMSE) of $\widehat{V}_{\tau n}$ and
$\widehat{V}_{\tau n}^{two-step}$ for different sample sizes. I compute the
semiparametric efficiency bound by computing the RMSE of an “oracle” estimator
that substitutes $\widehat{\eta}_{-k}(X_{i})=\eta(X_{i})$ in (49). The results
show that as the sample size increases, both estimators achieve a higher level
of accuracy and their variance approaches the semi-parametric lower bound (the
RMSE of the oracle). As expected by Corollary 2, the semiparametric lower
bound is zero at the boundary. The differences in RMSE shorten with higher
$V_{\tau}$ and in lower dimensional settings $(2J=10)$ (which have lower
first-stage noise).

Figure 3: (Coverage of $V_{\tau}$) The figure shows the 95% coverage
probabilities at different sample sizes, for the multi-fold CIs and single
fold confidence intervals defined in (40) and (41), respectively, for $K=2$
folds, single split. The two-step CIs are constructed as
$(1/n)\sum_{i=1}^{n}\widehat{\varphi}_{i}\pm
1.96\sqrt{\smash[b]{\widehat{V}_{\varphi}/n}}$, where $\widehat{\varphi}_{i}$
is the summand in (49) and $\widehat{V}_{\varphi}$ is an estimate of its
sample variance. The horizontal panels show a regime with homogeneity
($V_{\tau}=0$), moderate heterogeneity $(V_{\tau}=0.5)$ and high-heterogeneity
$(V_{\tau}=1)$. The vertical panels show low dimension case with $2J=10$
(top), a high-dimension case with $2J=40$ (bottom). The dotted vertical lines
denote the true value of the VCATE.

Figure 3 shows the coverage of $V_{\tau}$ for the different proposed
confidence intervals (CIs). For the multi-step approach, I consider the single
splits CIs in (40) and the conservative multi-fold CIs from and (41). The two-
step CIs are constructed as $\frac{1}{n}\sum_{i=1}^{n}\widehat{\varphi}_{i}\pm
1.96\sqrt{\widehat{V}_{\varphi}/n}$, where $\widehat{\varphi}_{i}$ is the
summand in (49) and $\widehat{V}_{\varphi}$ is an estimate of its sample
variance. The coverage of the two-step approach is very low under homogeneity,
and there is no improvement as sample size increases when $V_{\tau}=0$. The
coverage of the two-step estimator only improves with higher $n$, in high
heterogeneity designs. By contrast, both multi-step approaches cover the
parameter at the intended level, and coverage improves with higher sample
size. For fixed $n$, coverage degrades for both cases when the number of
covariates is higher.

Figure 4 explores the differences in covering the VCATE vs the pseudo-VCATE
when $n=2500$ and $2J=10$ for a fine-grained set of values of $V_{\tau}$.
Panel (a) reflects a dip in coverage close to the boundary. My theory predicts
that the multistep CIs have exact coverage when $V_{\tau}=0$, but may not
cover uniformly close to the boundary (see discussion after Theorem 8). The
mulit-fold CIs have conservative coverage. Conversely, Figure 4, Panel (b)
shows the multi-step CIs always uniformly cover the pseudo-VCATE, as predicted
by theory. This provides a robustness guarantee for how to interpret the CIs.
The two-step approach has much lower coverage and no guarantees when
$V_{\tau}=0$ in either panel.

Figure 4: (Coverage of $V_{\tau}$ vs $V_{\tau}^{*}$) The figure shows the
coverage of the VCATE (left) and the fold-specific pseudo-VCATE (right) for a
set of fine-grained values of the true $V_{\tau}$, given $n=2500$, $K=2$,
single split, and $2J=10$. The multi-fold CIs and single fold CIS, defined in
(40) and (41), respectively. The two-step CIs are constructed as
$(1/n)\sum_{i=1}^{n}\widehat{\varphi}_{i}\pm
1.96\sqrt{\smash[b]{\widehat{V}_{\varphi}/n}}$, where $\widehat{\varphi}_{i}$
is the summand in (49) and $\widehat{V}_{\varphi}$ is an estimate of its
sample variance. For the multi-fold and two-step approaches in the right
panel, I report coverage of the median $V_{\tau nk}^{*}$ across folds. Figure
5: (Power and size) Panel(a) shows the power of tests of homogeneity (whether
zero is contained in the confidence interval) for a set of fine-grained values
of the true $V_{\tau}$, given $n=2500$, $K=2$, single split, and $2J=10$.
Panel (b) shows the size of one-sided tests, i.e., the probability that the
VCATE is strictly below the CI bounds. The multi-fold CIs and single fold CIs
are defined in (40) and (41), respectively. The two-step CIs are constructed
as $(1/n)\sum_{i=1}^{n}\widehat{\varphi}_{i}\pm
1.96\sqrt{\smash[b]{\widehat{V}_{\varphi}/n}}$, where $\widehat{\varphi}_{i}$
is the summand in (49) and $\widehat{V}_{\varphi}$ estimates its sample
variance.

Figure 5.(left) shows the power of tests of homogeneity in a simulation with
$n=2500$. The multi-step, single fold approach has correct size control and
has local power, in line with the result of Lemma 7. The power of the test
using the multifold approach is similar to using a single fold. The right
panel shows the probability that the VCATE is strictly below the CI bounds. As
predicted by theory, this probability is uniformly bounded by $\alpha=0.05$
for the single and multifold approaches (see Corollaries 3 and 4, and Theorem
9). The two-step approach has a non-monotonic power curve with incorrect size.
The size of one-sided tests in the right panel is uniformly bounded by
$\alpha=0.05$, though this may be partly the fact that $\widehat{V}_{\tau
n}^{two-step}$ can take negative values and has a negative bias (see Figure
1).

##  7 Empirical Example

In this section, I illustrate my approach using data from a large-scale
information experiment conducted by Dizon-Ross (2019). The study, which
covered 39 school districts, involved an intervention to provide low-income
parents of at least two children with information about their children’s
school performance. Half the households where assigned to the information
intervention and the rest were assigned to the control group. Dizon-Ross
(2019) showed that, at baseline, parents faced large information gaps
regarding their children’s grades and class ranking. Even though schools
produced a report card, 60% of parents were unaware of their child’s
performance. Many parents reported that they did not receive the report card
(children either lost them or did not take them home), or had trouble
interpreting the report card structure, primarily due to low literacy levels.

The intervention was designed to present details of their children’s school
performance in an easily accessible way. Dizon-Ross (2019) showed that the
information gaps (the difference between believed and true test scores) went
down as a result of the intervention, and the amount of updating varied
depending on students’ initial test scores. Dizon-Ross (2019) also introduced
a real-stakes scenario where parents received a series of lottery tickets for
a scholarship paying for four years of high school. Parents had to decide how
to allocate tickets between two siblings. If there were more than two siblings
residing in the household, the survey team selected two at random. The results
showed that parents allocated tickets towards their better performing child.

To test for heterogeneity, Dizon-Ross (2019) ran a linear regression of
parental beliefs on initial scores, treatment, and an interaction as in (3),
and reported estimates $\bar{X}=46.8$ (on a scale of 100) and
$(\widehat{\beta}_{1},\widehat{\beta}_{2})=(-25.9,0.40)$ in their Tables 1 and
2, respectively. The coefficient $\widehat{\beta}_{2}$ captures how much the
treatment effects vary (on a $0$ to $100$ scale) for a each additional point
in students initial scores. The VCATE combines information about the
coefficient and the initial variability in scores. From the data, I also
estimate the variance of the control $\widehat{V}_{x}=305.53$, and estimate
the VCATE as $\widehat{\beta}_{2}^{2}\widehat{V}_{x}=48.89$. Taking the square
root and normalizing by the standard deviation of the outcome for the control
group ($\widehat{V}_{(Y\mid D=0)}=311.72$), produces $0.40$. This means that
the magnitude of treatment effect heterogeneity explained by scores is
comparable to 40% of the standard deviation of beliefs in the control group.

The VCATE can also help us understand the magnitude of treatment effect
heterogeneity in the experiment using multiple covariates. I use LASSO for the
first-stage predictions, two folds per split with 20 splits, and estimate
$\\{\widehat{\Omega}_{nk}\\}_{k=1}^{K}$ using clustered standard errors at the
household level and the formulas for the confidence intervals defined in (41).
The results are not very sensitive to the number of splits. For ease of
exposition, I report point-estimates and CIs for
$\sqrt{\frac{V_{\tau}}{\mathbb{V}(Y_{0})}}$, which is the standard deviation
of the CATE divided by the standard deviation of the outcome for the control
group. I compute confidence intervals for the square root via the
transformation proposed in (47).

N Clusters Estimate 95 Welfare Bounds (HH) $\sqrt{V_{\tau}/\mathbb{V}(Y_{0})}$
% C.I. Thm. 1 Thm. 2 Panel (a): Endline Beliefs.
$\frac{ATE}{\sqrt{\mathbb{V}(Y_{0})}}$: -0.42 (0.03) Scores 5244 2626 0.40
[0.32, 0.48] 0.201 0.081 Parent Years of Education 5208 2608 0.05 [0.00, 0.13]
0.023 0.001 Scores + Parents’ Education 5208 2608 0.40 [0.32, 0.48] 0.199
0.079 Above Median Educ. Expenses 5244 2626 0.06 [0.00, 0.15] 0.028 0.002
Respondent Variables 4722 2365 0.03 [0.00, 0.12] 0.015 0.001 Household
Variables 5244 2626 0.02 [0.00, 0.10] 0.009 0.000 Student Variables 4959 2532
0.11 [0.04, 0.20] 0.057 0.007 All variables 4464 2278 0.40 [0.31, 0.48] 0.199
0.079 Panel (b): Secondary School Lottery.
$\frac{ATE}{\sqrt{\mathbb{V}(Y_{0})}}$: 0.00 (0.00) Scores 5258 2629 0.16
[0.07, 0.24] 0.078 0.078 Parent Years of Education 5222 2611 0.00 [0.00, 0.00]
0.000 0.000 Scores + Parents’ Education 5222 2611 0.16 [0.07, 0.24] 0.078
0.078 Above Median Educ. Expenses 5258 2629 0.03 [0.00, 0.07] 0.015 0.015
Respondent Variables 4736 2368 0.00 [0.00, 0.00] 0.000 0.000 Household
Variables 5258 2629 0.00 [0.00, 0.00] 0.000 0.000 Student Variables 4971 2535
0.06 [0.00, 0.15] 0.029 0.027 All variables 4476 2281 0.15 [0.05, 0.24] 0.076
0.074

Table 1: (Empirical Estimates) Each panel computes the ATE and VCATE
normalized by the standard deviation of each outcome for the control group.
The $N$ varies depending on the missing values for the covariates and the
outcome. Each line within panels (a) and (b) considers 8 different sets of
covariates measured at baseline which include test scores, years of parental
education, an indicator for whether annual educational expenditures the
previous year (uniforms, fees, school supplies) are above the median, other
respondent variables (gender, age, is literate, is farmer), household
variables (number of kids, single-parent), and student variables (grade, age,
gender, attendance). I estimated clustered covariance matrices at the
household level. The function $\mu_{d}(x)$ is computed using LASSO with
10-fold cross-validation, and the estimates are computed using 2-fold cross-
fitting with 20 splits. The point estimates are the median values of
$\widetilde{V}_{\tau n}$ across splits. I compute the bound in Column 7, using
the ATE from the corresponding subsample.

Table 1 computes the ATE and the $\sqrt{VCATE}$ for two outcomes (parental
beliefs and lottery allocations) and 8 different sets of covariates. Panel (a)
shows that, on average, parents downgrade their beliefs about test scores by
42% of the standard deviation (SD) of the beliefs of the control group. The
treatment effect heterogeneity explained by test scores is equivalent to 40%
of the standard deviation (SD) of the beliefs of the control group. This is
statistically significant at the 5% level and has a comparable magnitude to
the ATE. The confidence intervals are relatively short in length. However,
applying the bounds from Theorem 2 and Corollary 1 shows that differentiating
treatment offers based on scores could further lower beliefs by at most 8.1%
SDs of the beliefs in the control group. In this case the ATE is already
fairly high compared to $\sqrt{V_{\tau}}$, so in spite of the large
heterogeneity, the marginals gains from targeting would be modest. Panel (b)
presents the results for the secondary school lottery. The ATE is estimated
precisely at zero, because the lottery tickets had to be divided as a zero sum
between the siblings. The VCATE measures how much the dispersion in the
allocation depends on the covariates. The standard deviation of the VCATE
explained by initial scores is 16% of the SD of the control group lottery
allocation, and the maximum welfare gains are around 7.8% SD.

The student variables (grade, age, gender, attendance, and educational
expenditures) collectively explain 11% of the SD of parental beliefs. This is
statistically significant at the 5% level. The magnitude is around a fourth of
the variation for test scores, and the maximum welfare gain from targeting is
0.7% of the SD of parental beliefs in the control group. I find that other
subsets of covariates do not produce statistically significant estimates of
the VCATE at the 5% level. The added welfare of personalizing treatment
assignment using these covariates is also very low.

The estimates that use all the covariates are computed over a smaller
subsample with non-missing values across all variables. Despite the large
number of variables and the smaller sample, the estimates of the VCATE remain
relatively stable across specifications. The VCATE computed from a rich set of
respondent, household, and student covariates has a comparable magnitude to
the VCATE that only includes student scores. The confidence intervals are also
similar. The estimates of the maximum welfare gains from targeting using all
covariates are 7.9% SD for beliefs and 7.4% SD for lottery outcomes,
respectively, which are similar to the welfare gains computed using only
scores.

##  8 Conclusion

I propose an efficient estimator of the variance of treatment effects that can
be attributed to baseline characteristics and propose novel adaptive
confidence intervals that produce valid coverage. I analyze issues of non-
standard inference that arise in this context, and how to address them. I also
explore the economic significance of the VCATE for policymakers and
researchers, by showing that the $\sqrt{VCATE}/2$ bounds the marginal gains of
targeted policies. Overall, this paper proposes a broadly applicable approach
to measure treatment effect heterogeneity in experiments.

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##  Appendix A Critical Values

Define the correlation $\rho=\Omega_{12}/\sqrt{\Omega_{11}\Omega_{22}}$. By
definition of the Cholesky decomposition,
$(e_{1}^{\prime}\Omega^{1/2}Z)=\sqrt{\Omega_{11}}Z_{1},$ and
$(e_{2}^{\prime}\Omega^{1/2}Z)=\sqrt{\Omega_{22}(1-\rho^{2})}Z_{2}+\rho\sqrt{\Omega_{22}}Z_{1}$.
Substituting these terms into the expression for $G$

$\displaystyle G(n,V_{\tau}^{*},\Omega,Z,\zeta)$
$\displaystyle=\underbrace{\frac{\Omega_{11}}{n}}_{\nu_{1}}Z_{1}^{2}+\underbrace{2\left[\zeta\sqrt{\frac{V_{\tau}^{*}\Omega_{11}}{n}}+\frac{V_{\tau}^{*}}{2\sqrt{n}}\rho\sqrt{\Omega_{22}}\right]}_{\kappa_{1}}Z_{1}+\underbrace{\frac{V_{\tau}^{*}}{\sqrt{n}}\left[\sqrt{\Omega_{22}(1-\rho^{2})}\right]}_{\kappa_{2}}Z_{2}.$

This has a quadratic form,
$G(n,V_{\tau}^{*},\Omega,z,\zeta)=\nu_{1}\left(Z_{1}+\frac{\kappa_{1}}{2\nu_{1}}\right)^{2}+\kappa_{2}Z_{2}-\frac{\kappa_{1}^{2}}{4\nu_{1}}$,
which fits the form of a generalized chi-square (Das and Geisler, 2021). To
compute critical values we compute feasible analogs
$(\widehat{\nu}_{1},\widehat{\kappa}_{1},\widehat{\kappa}_{2})$ from an
estimate of $\Omega$.

##  Appendix B Proofs Main Document

  * Proof of Lemma 1.

Define $\tilde{\tau}(X^{\prime}):=\mathbb{E}[Y_{1}-Y_{0}\mid X^{\prime}]$.
Since $X$ is $X^{\prime}$-measurable, then by the law of iterated
expectations, $\mathbb{E}[\tilde{\tau}(X^{\prime})\mid X]=\tau(X)$. By the law
of total variance
$V_{\tau}^{\prime}=\mathbb{V}(\tau(X))+\mathbb{E}[\mathbb{V}(\tilde{\tau}(X^{\prime})\mid
X)]\geq V_{\tau}$. We can prove the upper bound by setting
$X^{\prime}=Y_{1}-Y_{0}$. ∎

  * Proof of Theorem 1.

The result is a special case of Theorem 2. The most adversarial distribution
is one where $\tau_{av}=0$. ∎

  * Proof of Theorem 2.

Our goal is to find $\mathcal{R}:=\sup_{\gamma\in\Gamma}\sup_{\pi\in\Pi}\
\mathcal{U}_{\gamma}(\pi)$, and to prove that
$\mathcal{U}_{\gamma}(\pi)=\mathcal{R}$ for at least one $\gamma\in\Gamma$ and
$\pi\in\Pi$.

Define two random variables $T:=\mathbb{E}_{\gamma}[Y_{1}-Y_{0}\mid X=x]$ and
$M:=\mathbbm{1}\\{T>0\\}$. By adding/subtracting $\mathbb{E}_{\gamma}[Y_{0}]$
and applying the law of iterated expectations,
$\mathcal{U}_{\gamma}(\pi)=\mathbb{E}_{\gamma}[\pi(X)T]-\max\\{0,\mathbb{E}_{\gamma}[T]\\}$.
The optimal policy $\pi^{*}(X)=\mathbbm{1}\\{T\geq 0\\}$, which belongs to
$\Pi$, i.e. the “first-best” (Kitagawa and Tetenov, 2018). This means that
$\mathcal{R}=\sup_{\gamma\in\Gamma}\
\mathbb{E}_{\gamma}[\max\\{0,T\\}]-\max\\{0,\mathbb{E}_{\gamma}[T]\\}$.

Step 1: (Problem Equivalence)  The set of distributions $\Gamma$ is very
large. Instead I focus on the problem over a set of equivalence classes. For
$m\in\\{0,1\\}$, define the moments $p_{m}=\mathbb{P}_{\gamma}(M=m)$,
$\tau_{m}=\mathbb{E}_{\gamma}[T\mid M=m]$, and
$\omega_{m}=\mathbb{V}_{\gamma}(T\mid M=m)$. By definition,
$\mathcal{U}_{\gamma}(\pi^{*})=p_{1}\tau_{1}-\max\\{0,\tau_{av}\\}$, where
$p_{1}\tau_{1}$ is the proportion of people that benefit from treatment times
their conditional mean. Let $\Delta:=\tau_{1}-\tau_{0}$ be the mean difference
between those that benefit from the program and those that do not. By
definition $\Delta>0$ because $\tau_{1}>0$ and $\tau_{0}\leq 0$. The
conditional treatment effect is $\mathbb{E}_{\gamma}[T\mid
M]=\tau_{0}+M\Delta$, and $\tau_{av}=\tau_{0}+p_{1}\Delta$. Rearranging these
expressions, $\tau_{1}=\tau_{av}+(1-p_{1})\Delta$ and
$\mathcal{U}_{\gamma}(\pi^{*})=p_{1}(\tau_{av}+(1-p_{1})\Delta)-\max\\{0,\tau_{av}\\}$.
By definition $\mathbb{V}_{\gamma}(\mathbb{E}_{\gamma}[T\mid
M])=\Delta^{2}p_{1}(1-p_{1})$. By applying the law of total variance
$V_{\tau}=p_{1}\omega_{1}^{2}+(1-p_{1})\omega_{0}^{2}+\Delta^{2}(p_{1})(1-p_{1})$.
Then

$\displaystyle\begin{split}\mathcal{R}&=\sup_{\\{p_{1},\tau_{0},\Delta,\omega_{0}^{2},\omega_{1}^{2}\\}}p_{1}\tau_{av}+p_{1}(1-p_{1})\Delta-\max\\{0,\tau_{av}\\},\\\
s.t.\qquad&p_{1}\in[0,1],\tau_{0}\leq
0,\tau_{0}+\Delta>0,\Delta>0,\omega_{1}^{2},\omega_{0}^{2}\geq
0,\tau_{0}+p_{1}\Delta=\tau_{av},\\\
&p_{1}\omega_{1}^{2}+(1-p_{1})\omega_{0}^{2}+p_{1}(1-p_{1})\Delta^{2}=V_{\tau}.\end{split}$
(50)

The values of $\tau_{av}$ and $V_{\tau}$ impose the following constraints on
the feasible set:

| $V_{\tau}=0$ | $V_{\tau}>0$
---|---|---
$\tau_{av}\leq 0$ |  $p_{1}=0$, $\tau_{0}=\tau_{av}$, $\omega_{0}=0$ | $p_{1}\in[0,1)$
$\tau_{av}>0$ |  $p_{1}=1$, $\tau_{1}=\tau_{av}$, $\omega_{1}=0$ | $p_{1}\in(0,1]$

By the form of the objective, if $p_{1}\in\\{0,1\\}$ then $\mathcal{R}=0$.
Therefore, if $V_{\tau}=0$, then $\mathcal{R}=0$ and the result of the theorem
holds. Any value of the remaining parameters that satisfies the sign
constraints will be feasible. Without loss, we focus on $V_{\tau}>0$.

Step 2: (Optimum has binary support) Consider a situation where
$(p_{1},\omega_{1}^{2},\omega_{0}^{2})$ are fixed and $p_{1}\in(0,1)$. From
the variance equation,
$\Delta^{*}=\sqrt{\frac{V_{\tau}-p_{1}\omega_{1}^{2}-(1-p_{1})\omega_{0}^{2}}{p_{1}(1-p_{1})}}$,
and from the mean $\tau_{0}^{*}=\tau_{av}-p_{1}\Delta^{*}$ and
$\tau_{1}^{*}=\tau_{av}+(1-p_{1})\Delta^{*}$.

A solution is feasible as long as $\omega_{1}^{2},\omega_{0}^{2}\geq 0$,
$p_{1}\omega_{1}^{2}+(1-p_{1})\omega_{0}^{2}\leq V_{\tau}$, $\tau_{0}^{*}\leq
0$, $\tau_{1}^{*}>0$, and $\Delta^{*}=\tau_{1}^{*}-\tau_{0}^{*}>0$. The
optimal $\tau_{0}^{*}$ is strictly increasing in
$(\omega_{1}^{2},\omega_{0}^{2})$ and $\tau_{1}^{*}$ is strictly decreasing.
If a given value of $(\omega_{1}^{2},\omega_{0}^{2})$ is feasible, then
another candidate which shrinks it to zero will still satisfy the constraints.
Moreover, the objective function is strictly increasing in $\Delta^{*}$, which
in turn is strictly decreasing in $(\omega_{1}^{2},\omega_{0}^{2})$. Therefore
the optimum is $(\omega_{1}^{2},\omega_{0}^{2})=(0,0)$, i.e. binary support
for the CATE, $\Delta^{*}=\sqrt{V_{\tau}/p_{1}(1-p_{1})}$,
$\tau_{0}^{*}=\tau_{av}-\sqrt{p_{1}/(1-p_{1})}\sqrt{V_{\tau}}$, and
$\tau_{1}^{*}=\tau_{av}+\sqrt{(1-p_{1})/p_{1}}\sqrt{V_{\tau}}$.

Only some values of $p_{1}$ are feasible, i.e. satisfy the constraints
$\Delta^{*}>0$, $\tau_{0}^{*}\leq 0$, and $\tau_{1}^{*}>0$. The function
$p_{1}/(1-p_{1})$ is strictly increasing in $p$ with a range in $(0,\infty)$.
The feasible values depend on the sign of $\tau_{av}$. If (i) $\tau_{av}\leq
0$, then $p_{1}\in\left(0,\frac{V_{\tau}}{\tau_{av}^{2}+V_{\tau}}\right]$. If
(ii) $\tau_{av}>0$,
$p_{1}\in\left[\frac{\tau_{av}^{2}}{\tau_{av}^{2}+V_{\tau}},1\right)$.

Step 3: (Solve points of support) For case (i) the objective function is
$\mathcal{R}=\max_{p_{1}\in\left[0,\frac{V_{\tau}}{\tau^{2}+V_{\tau}}\right]}\quad
p_{1}\tau_{av}+\sqrt{p_{1}(1-p_{1})}\sqrt{V_{\tau}}$. The unique solution to
the FOC is
$p_{1}^{*}=(1/2)-(1/2)\sqrt{\tau_{av}^{2}/(\tau_{av}^{2}+V_{\tau})}$, and is
interior because
$\sqrt{\tau_{av}^{2}/(\tau_{av}^{2}+V_{\tau})}\geq\tau_{av}^{2}/(\tau_{av}^{2}+V_{\tau})$.
By strict concavity, it is a global optimum, and
$\mathcal{R}=\tau_{av}/2-(\tau_{av}|\tau_{av}|)/(2\sqrt{\tau_{av}^{2}+V_{\tau}})+V_{\tau}/(2\sqrt{\tau_{av}^{2}+V_{\tau}})$.
Since $\tau_{av}\leq 0$, this can be simplified to
$\frac{1}{2}(-|\tau_{av}|+\sqrt{\tau_{av}^{2}+V_{\tau}})$. For case (ii),
$\tau_{av}>0$, the objective function is
$(p_{1}-1)\tau_{av}+\sqrt{p_{1}(1-p_{1})}\sqrt{V}_{\tau}$, subject to $1\geq
p_{1}\geq\frac{\tau_{av}^{2}}{\tau_{av}^{2}+V_{\tau}}$. The unique solution to
the FOC is
$p_{1}^{*}=(1/2)+(1/2)\sqrt{\tau_{av}^{2}/(\tau_{av}^{2}+V_{\tau})}$, it is
interior because
$p_{1}^{*}\geq\frac{1}{2}+\frac{1}{2}\frac{\tau_{av}^{2}}{\tau_{av}^{2}+V_{\tau}}\geq\frac{\tau_{av}^{2}}{\tau_{av}^{2}+V_{\tau}}$,
and produces the desired $\mathcal{R}$.

∎

  * Proof of Corollary 1.

The potential outcomes under a linear transformation are
$\kappa_{1}+\kappa_{2}Y_{0}$ and $\kappa_{1}+\kappa_{2}Y_{1}$, respectively.
The treatment effect is $\kappa_{2}(Y_{1}-Y_{0})$, which does not depend on
$\kappa_{1}$, and the transformed CATE is $\kappa_{2}\tau(x)$. The ATE is
$\kappa_{2}\tau_{av}$ and the VCATE is $\kappa^{2}V_{\tau}$. The result
follows from substituting the transformed values into Theorem 2 and
factorizing $|\kappa_{2}|$.

∎

  * Proof of Lemma 2 .

We decompose $\widehat{V}_{\tau n}=\widehat{\beta}_{2n}^{2}\widehat{V}_{xn}$
into components that map into those of Assumption 2, by centering the key
terms.

$\displaystyle\widehat{V}_{\tau n}-V_{\tau
n}^{*}=\widehat{\beta}_{2n}^{2}\widehat{V}_{xn}-\beta_{2n}^{2}V_{xn}=(\widehat{\beta}_{2n}-\beta_{2n}+\beta_{2n})^{2}\frac{V_{xn}}{V_{xn}}(\widehat{V}_{xn}-V_{xn}+V_{xn})-\beta_{2n}^{2}V_{xn}$
(51)
$\displaystyle=[\sqrt{V}_{xn}(\widehat{\beta}_{2n}-\beta_{2n})]^{2}+2[\sqrt{V_{xn}}\beta_{2n}][\sqrt{V}_{xn}(\widehat{\beta}_{2n}-\beta_{2n})]+[V_{xn}\beta_{2}^{2}]\left(\frac{\widehat{V}_{xn}}{V_{xn}}-1\right)$
$\displaystyle\quad+[\sqrt{V_{xn}}(\widehat{\beta}_{2n}-\beta_{2n})]^{2}\left(\frac{\widehat{V}_{xn}}{V_{xn}}-1\right)+2[\sqrt{V_{xn}}\beta_{2n}][\sqrt{V_{xn}}(\widehat{\beta}_{2n}-\beta_{2nk})]\left(\frac{\widehat{V}_{xn}}{V_{xn}}-1\right).$

Let
$\widehat{Z}_{n}=n^{1/2}\Omega_{n}^{-1/2}[\sqrt{V}_{xn}(\widehat{\beta}_{2n}-\beta_{2n}),\widehat{V}_{xn}/V_{xn}-1]^{\prime}$
be a normalized statistic, and let $e_{1}=[1,0]^{\prime}$,
$e_{2}=[0,1]^{\prime}$ be vectors that select the first and second
coordinates, respectively. We can always find a $\zeta_{n}\in\\{-1,1\\}$ that
solves $\zeta_{n}\sqrt{V_{\tau n}}=\beta_{2n}\sqrt{V_{xn}}$. If $V_{\tau n}>0$
this is the sign of $\beta_{2n}$ but otherwise any $\zeta_{n}\in\\{-1,1\\}$
will solve the equation. By substituting the definition of $\widehat{Z}_{n}$,
$V_{\tau n}=\beta_{2n}^{2}V_{xn}$, and

$\displaystyle\begin{split}\widehat{V}_{\tau n}-V_{\tau
n}^{*}&=\frac{(e_{1}^{\prime}\Omega_{n}\widehat{Z}_{n})^{2}}{n}+2\zeta_{n}\sqrt{V_{\tau
n}^{*}}\left(\frac{e_{1}^{\prime}\Omega_{n}^{1/2}\widehat{Z}_{n}}{\sqrt{n}}\right)+V_{\tau
n}^{*}\frac{e_{2}^{\prime}\Omega_{n}^{1/2}\widehat{Z}_{n}}{\sqrt{n}}\\\
&\quad+\frac{(e_{1}^{\prime}\Omega_{n}^{1/2}\widehat{Z}_{n})^{2}(e_{2}^{\prime}\Omega_{n}^{1/2}\widehat{Z}_{n})}{n^{3/2}}+2\zeta_{n}\sqrt{V_{\tau
n}^{*}}\frac{(e_{1}^{\prime}\Omega_{n}^{1/2}\widehat{Z}_{n})(e_{2}^{\prime}\Omega_{n}^{1/2}\widehat{Z}_{n})}{n}.\end{split}$
(52)

By Assumption 2, $\widehat{Z}_{n}\to Z_{n}+o_{p}(1)$. Moreover, since
$\Omega_{n}$ has bounded eigenvalues,
$\Omega_{n}^{1/2}\widehat{Z}_{n}=\Omega_{n}^{1/2}Z_{n}+o_{p}(1)$. This means
that

$\displaystyle\widehat{V}_{\tau n}-V_{\tau n}^{*}$
$\displaystyle=\frac{(e_{1}^{\prime}\Omega_{n}^{1/2}Z_{n})^{2}}{n}+2\zeta_{n}\sqrt{\frac{V_{\tau
n}^{*}}{n}}e_{1}^{\prime}\Omega_{n}^{1/2}Z_{n}+\frac{V_{\tau
n}^{*}}{\sqrt{n}}e_{2}^{\prime}\Omega_{n}^{1/2}Z_{n}\ +\ \text{Residual}_{n},$
(53)