A Nonparametric Test of Heterogeneous Treatment Effects under Interference

A Nonparametric Test of Heterogeneous Treatment Effects under Interference

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\ifnum0=1 \addtocounter{alphasect}{1} \fi \oldsection{Introduction} The literature on causal inference focuses primarily on identifying and estimating aggregate treatment effects. However, it is increasingly recognized that these aggregate effect metrics, while useful for measuring social welfare, often overlook the variations in treatment effects. These variations are crucial for designing welfare-maximizing treatment assignment rules. For instance, a job search assistance program may show positive aggregate effects on income and welfare. However, it could also lead to significant income and welfare disparities in society, as the impact of the program may vary across individuals. This observation has sparked a significant and growing research interest in estimating and inferring heterogeneous treatment effects (HTEs). This shift in focus from aggregate treatment effects to HTEs is a significant development in the field of causal inference. The classical approach to infer HTEs involves estimating and comparing conditional average treatment effects (CATEs) of predefined subgroups in a population. The formal comparison of CATEs requires some hypothesis testing procedure. crump2006nonparametric, ding2016randomization, wager2018estimation, and sant2021nonparametric are among the few existing HTEs testing papers.

This paper introduces two nonparametric tests for inferring HTEs across subgroups while accounting for interference among economic units. According to cox1958planning, interference arises when the treatment response of one unit depends on the treatment of other units in the population. In other words, a unit's treatment response is a function of the treatment assignment --- the vector of treatments of all units. Interference may result from physical, virtual, or social connections between members of a population. While the mechanisms behind interference may vary, its presence complicates the inference of HTEs across subgroups. In the presence of interference, treatment effects may depend on pre-treatment variables, post-treatment exposure variables (which capture exposure to other units' treatment statuses), or both. For example, suppose we find that the effectiveness of a Covid-19 vaccine varies across cities. The vaccine's efficacy for an individual may be influenced by the vaccination rate among their physical contacts. It is, therefore, essential to investigate whether the observed variation in vaccine effectiveness is driven by inherent population differences across cities (e.g., genetic factors) or by differences in vaccination rates, a post-treatment exposure variable. In this example, variations in vaccination rates may mask the heterogeneity due to natural differences across cities in a standard test that does not control for the vaccination rates. This highlights that traditional tests for HTEs --- which assume no interference between units --- are likely to produce inaccurate results when interference is present.

To address the challenges of testing HTEs in the presence of interactions between units, I consider a clustered interference setting, where interference is limited to within clusters. In this setting, I introduce kernel-based test statistics for the null hypotheses of (i) constant treatment effects (CTEs) by treatment assignment for all values of the pre-treatment variables and (ii) CTEs by the pre-treatment variables for all treatment assignment vectors. Then, I recommend the holm1979simple multiple testing procedure to jointly test the null hypotheses and disentangle the source of heterogeneity in the treatment effects. The proposed test statistics are sums of the weighted $L_1$ norm of the differences in the consistent kernel estimators of CATE that characterize the null hypotheses.

The proposed testing procedure in this paper provides several practical advantages to policymakers. First, testing the null hypothesis of CTEs by treatment assignment for all covariates in isolation informs policymakers on whether to scale a program. Suppose we fail to reject the null hypothesis of CTEs across treatment assignments for all values of the pre-treatment variables. In that case, it implies that treatment spillover effects are absent, and a program can be scaled without any negative or positive externalities. Second, the null hypothesis of CTEs by the pre-treatment variables for all treatment assignment vectors in isolation helps to detect HTEs across subgroups defined by pre-treatment variables. Knowledge of a program's treatment effect variation across subgroups can guide its extension to other populations (external validation). Third, jointly testing both null hypotheses helps to disentangle the source of variation in treatment effects. Finding the drivers of the variations in treatment effects in an interconnected human society is crucial in designing welfare-maximizing treatment assignment rules.

The main theoretical results show that the proposed test statistics have an asymptotically standard normal null distribution. I prove these properties via a poissonization technique formulated by rosenblatt1975quadratic and developed by gine2003bm. The technique introduces additional randomness by assuming that the sample size is a Poisson random variable. It enables techniques that exploit Poisson processes' independent increments and infinite divisibility properties. Also, I show that the test statistics have asymptotically valid sizes and power against fixed alternatives and sequences of local alternatives. Moreover, I propose bootstrap procedures to benchmark the proposed asymptotic-based methods in finite samples. All the asymptotic results are based on a regime where the number of sampled clusters goes to infinity. As I discuss further in Section (ref), an essential feature of this regime is that it permits the safe disregard of within-cluster dependencies for local estimators in large samples.

The main contribution of this paper is twofold. First, I highlight the issue of HTEs testing in the presence of interference. As a solution, I propose testing procedures for HTEs that (i) can accommodate several forms of clustered interference and (ii) can disentangle the sources of heterogeneity. Second, I extend the CATE identification and estimation in the literature to allow for clustered interference. From the technical perspective, I prove the asymptotic properties of the proposed testing procedures in this paper using the modern Poissonization technique of gine2003bm.

The rest of the paper is organized as follows. I review the existing literature in the following subsection. Section (ref) describes the setup, the testing problem, and the test statistics. In Section (ref), I present the main asymptotic properties of the test statistics. Monte Carlo simulation design and results are in Section (ref). To illustrate the usage of the proposed tests, in Section (ref), I revisit the Chinese weather insurance policy data set in cai2015social and test for HTEs by a post-treatment variable (treatment ratio) and a pre-treatment variable (the fraction of household income from rice production). My concluding remarks are in Section (ref). All proofs, useful theorems, lemmas, and other simulation experiments are in the Appendix.

Related Literature

The nascent literature on the estimation and inference of HTEs continues to grow and spans multiple fields. The present paper falls under the arm of the literature that studies the inference of HTEs using tests based on average treatment effects (ATEs) of subgroups defined by pre-treatment variables. bitler2006mean provide an in-depth critique of this approach to testing for HTEs. They argue that heterogeneity of CATEs across subgroups often does not imply individual treatment effect variation unless one assumes constant subgroup treatment effect (CSTE). Nonetheless, regardless of the CSTE assumption, variations in ATEs of subgroups are crucial in the design of treatment assignment rules where pre-treatment variables are used to set eligibility conditions. In econometrics, tests to detect variation in ATEs across subgroups have been studied by crump2006nonparametric and lee2014multiple. Both studies abstract from interference and propose nonparametric tests to infer HTEs across predefined subgroups. In contrast, I allow clustered interference and use kernel-based estimators to construct the test statistics. To the best of my knowledge, this is the first paper to test for HTEs under any form of interference.

The problem I consider, testing for HTEs in the presence of clustered interference, is distinct from and complementary to other CATE estimation problems in the literature. abrevaya2015estimating and fan2022estimation study the identification and kernel-based estimation of CATE for traditional and high dimensional data sets, respectively. By contrast, this paper focuses on testing for HTEs or varying CATEs. Also, the identification strategy and estimation in the current paper account for within-cluster interferences, whereas the above references impose a no-interference assumption. Similarly, bargagli2020heterogeneous proposes a Network Causal Tree algorithm that seeks to find and estimate the treatment effect of subpopulations where treatment and spillover effects differ across individual, neighborhood, and network characteristics in clustered network populations. This is again distinct from my goal, which takes the subpopulations of interest as given and studies testing procedures to infer HTEs across these covariates-defined subpopulations. On the theoretical side, lee2013testing and chang2015nonparametric establish asymptotic null distributions for the $L_p$-type functions of kernel-based CATE estimators using the modern poissonization technique of gine2003bm. Allowing for clustered interference in the potential outcomes model requires a modification of the estimator of CATE to fit the current framework. In addition, the proposed test statistics in this paper are different, and the theoretical results are extensions of those in these chang2015nonparametric.

Finally, the nonparametric bootstrap algorithms proposed in this paper are similar to existing algorithms in the literature. For instance, li2009nonparametric uses a bootstrap algorithm akin to that described in Section (ref) to test for the equality of two density or conditional density functions. Also, racine1997consistent employs a residual bootstrap algorithm similar to the wild bootstrap-t procedure proposed in Section (ref) to test for the significance of pre-treatment variables in regression models. Despite the similarities, the bootstrap procedures in this paper must account for the within cluster dependencies and, as such, are novelties that fit the current setting.

\ifnum0=1 \addtocounter{alphasect}{1} \fi \oldsection{Framework }

Setup

Consider a setting where units are partitioned into clusters, such as classrooms, villages, or states. Let $N_c $ denote the number of finite units in cluster $c \in \mathcal{I}.$ Suppose that $C$ clusters are randomly drawn from the population of clusters, and let $N$ denote the number of units that make up the observed sample, i.e., $\sum_{i=1}^{C}N_c= N.$ Furthermore, assume that units in the same cluster interact arbitrarily, leading to clustered or partial interference sobel2006randomized. Let $T_{ci} \in \{0,1\}$ be the binary treatment condition of unit $i\in \{1,\dots, N\}$ in cluster $c \in \{1, \dots, C\}.$ Treatment is selected or assigned at the unit level. Thus, different clusters may have different treatment assignment vectors. The treatment vector of the units in cluster $c$ is denoted as $\mathbf{T}_c.$ Moreover, we observe the outcome variables $Y_{ci} \in \mathcal{Y}\subset \mathbbm{R},$ and a vector $X_{ci} \in \mathcal{X}\subset \mathbbm{R}^{^{d}}$ of pre-treatment variables.

Adopting the potential outcomes model of neyman1923applications and rubin1974estimating, let $Y_{ci}(\mathbf{t}_c)$ represent the potential outcome of unit $i$ in cluster $c$ when the cluster treatment assignment vector $\mathbf{T}_c$ equals $\mathbf{t}_c.$ Therefore, the potential outcomes of a unit are indexed by her cluster treatment assignment vector. Effectively, the number of potential outcomes of a unit in cluster $c$ is $2^{^{N_c}}$ (i.e., the number of all possible cluster treatment assignment vectors). Note that in the classical case of no interference, only two potential outcomes exist for each unit. Hence, allowing for (clustered) interference exacerbates the missing data problem of causal inference. As such, a salient element of causal inference in the presence of (clustered) interference is a mapping $\pi(\cdot)$, which summarizes how the (cluster) treatment vectors affect the treatment response outcome. This is commonly referred to as the exposure mapping (see manski2013identification and aronow2017estimating). Formally, I define exposure mapping as

that maps the cluster treatment vector $\mathbf{T}_{c}$ into an exposure variable $\Pi_{ci}:=\pi_{i}(\mathbf{T}_{c}) \in \mathbf{\Pi} \subset \mathbbm{R}$. The definition of $\pi_i$ is application-specific, as it must account for the within-cluster interference structures of the given problem.

Next, for a given exposure mapping, I assume that $Y_{ci}(\mathbf{t}_c)=Y_{ci}(t,\pi)$ is the potential outcome for unit $i$ if the cluster treatment vector $\mathbf{t}_c$ is such that $T_{ci} =t$ and $\pi_{i}(\mathbf{t}_c)=\pi.$ Thus, this assumption asserts that heterogeneity in treatment effects across treatment assignment vectors is analogous to heterogeneity in treatment effects across exposure variable's values (henceforth exposure values).

The following example provides a plausible specification of exposure mapping defined in (ref).

I formalize the assumptions that describe the setting as follows.

In other words, Assumption (ref) suggests that the network is a fixed population characteristic, and units do not strategically select into clusters after treatment assignment.

Assumption (ref) is embedded with a couple of information and merits a lengthy discussion. First, it restricts the nature of interference, i.e., it imposes intra-cluster interference but no inter-cluster interference. Second, this assumption limits the range of the exposure variable to a discrete finite set of $K<C$ elements; thus, the number of potential outcomes reduces significantly to $2\cdot K$. This part of the assumption also implies that units in different clusters can have the same exposure value, as a result guaranteeing that there are "enough" units for each treatment and exposure value for precise estimation of the CATEs. Finally, Assumption (ref) asserts that a unit's exposure to the treatment of other units in her cluster is in a "reduced form," and channels through which interference occurs in clusters are not distinguishable. Exploring general forms of interference with network information is a possible avenue for future research.

For each $\pi \in \mathbf{\Pi}$ and $x \in \mathcal{X},$ I define the CATE parameter as\,\, $\tau(x;\pi )\coloneq\mathbbm{E} [Y(1, \pi )|X=x] -\mathbbm{E} [Y(0, \pi )|X=x].$ The following assumptions are crucial for the identification of the CATEs.

Assumptions (ref)(ii) and (ref) are the extensions of the usual ignorability assumptions imposed on the treatment assignment mechanism under no interference, see imbens2015causal. Assumption (ref)(i) is the standard consistency condition that rules out multiple versions of the same treatment selected or assigned, and (ref)(ii) asserts that conditional on pre-treatment variables, self-selection of effective treatment, $(T, \Pi)$ is ruled out. This Assumption holds when treatment is randomly assigned at the unit level. Assumption (ref) is a modified version of the usual overlap or probabilistic assignment condition. It ensures a balance between treated and control units in each subgroup. It is particularly crucial because of the H\'ajek-type estimators of CATE I employ in this paper.

The Testing Problem

In this section, I formally describe the testing problem. As mentioned above, in the presence of interference, treatment effects may vary by either pre-treatment variables, post-treatment exposure variables, or both. Figure (ref) shows some plausible cases in a simple setting where $\mathbf{\Pi}=\{\pi_1,\pi_2\}$ and CATE is linear in a continuous $X.$ Panels (a) shows CTEs by both classes of variables. In contrast, Panels (b) and (c) show scenarios where one class of variable has HTEs, and the other class has CTEs. Panel (d) depicts the case where both classes of variables have HTEs. These facts highlight that testing for HTEs in the presence of interference requires testing for heterogeneity across the two classes of variables and the need for methods to disentangle the source of the effect heterogeneity.

To test for heterogeneity across one class of variables requires controlling for the other class of variables. Formally, I consider the following null hypotheses. The first one is the null hypothesis of constant treatment effects (CTEs) by an exposure variable while controlling for pre-treatment variables:

against the alternative hypothesis of HTEs by the exposure variable:

The null hypothesis (ref) is vital in answering several essential questions in program evaluation. For example, testing (ref) when exposure is treatment ratios helps to determine whether a program should be scaled or not; rejecting the null hypothesis implies treatment spillover effects exist and program effects may vary by program scale. Although (ref) resembles Hypothesis 2 in athey2018exact, they are not the same. The null hypothesis (ref) is a restriction on the treatment effect that there are no indirect or spillover treatment effects. In contrast, Hypothesis 2 in athey2018exact restricts the outcome of no spillovers or interference. Failure to reject Hypothesis 2 in athey2018exact implies that we fail to reject (ref); however, the converse is not valid.

The second hypothesis of interest concerns the null hypothesis of CTEs by pre-treatment variables while controlling for the post-treatment exposure variable:

against the alternative hypothesis of HTEs by pre-treatment variables:

Tests for the null hypothesis similar to (ref) exist in the literature; see crump2006nonparametric. However, this paper is the first to allow some form of interference. Testing (ref) is critical in extending existing programs to new populations; rejecting this hypothesis implies that treatment effects vary across subgroups defined by the pre-treatment variables. Therefore, a policymaker should expect different aggregate effects if the program is extended to a new population.

It is worth discussing the importance of distinguishing between the exposure variable (i.e., $\Pi$) and the pre-treatment variables in this framework. First, note that $\Pi$ is a causal variable and, as such, is used to index the potential outcomes. In contrast, the pre-treatment variables are invariant to treatment assignment, i.e., they are a priori variables. Therefore, this distinction is necessary in the potential outcome framework for identification. Secondly, distinguishing between the pre-treatment and treatment exposure variables is imperative to determine the drivers of the heterogeneity in treatment effects. It has vital implications for policymakers. For instance, if the heterogeneity is solely due to the treatment exposure variable, policymakers may have complete control over the distribution of treatment effects in the population via the treatment assignment mechanism. See han2022statistical for a statistical treatment assignment rule for homogeneous and heterogeneous populations in the presence of social interaction.

Test Statistics

The proposed test statistic for the null hypothesis $H_0^\Pi$ in (ref) is

where $\hat{\tau}(x;\pi)$ is a uniform consistent estimator of $\tau(x; \pi)$ and $\hat{w}(x, \pi, \pi')$ is the uniform consistent estimator of the inverse standard error of $\sqrt{Ch^{d}}(\hat{\tau}(x;\pi)-\hat{\tau}(x;\pi'))$ which is defined as $w(x, \pi, \pi'):= 1/\sqrt{\rho_2(x, \pi)+ \rho_2(x, \pi')-2Ch^{d}Cov(\hat{\tau}(x;\pi),\hat{\tau}(x;\pi')},$ \footnote{Since the network is clustered, $w(x, \pi, \pi'):= 1/\sqrt{\rho_2(x, \pi)+ \rho_2(x, \pi')}$ when the exposure variable is at the cluster level.} with, $\rho_2(x, \pi)$ representing the variance of $\sqrt{Ch^{d}}\hat{\tau}(x;\pi).$ Also, note that $d$ is the dimension of $X,$ $h$ is the bandwidth, and recall that $C$ is the number of sampled clusters.

I propose a nonparametric kernel-based estimator of $\tau(x; \pi)$ defined as

where $$\hat{\phi}(T_{ci},x,\pi):=\frac{T_{ci}}{\hat{P}_1(x;\pi)}-\frac{(1-T_{ci})}{\hat{P}_0(x;\pi)},$$ with $$\hat{P}_t(x;\pi):= \frac{1}{Ch^d}\sum_{c=1}^{C} \frac{1}{N_c}\sum_{i=1}^{N_c} \mathbbm{1}(\Pi_{ci}=\pi)\mathbbm{1}(T_{ci}=t)K\left(\frac{x-X_{ci}}{h}\right),\,\,\,t=0, 1, $$ and $K(\cdot)$ being a $d$-dimensional kernel function.

The test statistic for the null hypothesis $H_0^{X}$ in (ref) is defined as

where, $\hat{w}(x, x', \pi)$ is a consistent kernel estimator of ${w}(x, x', \pi),$ the inverse standard error of $\sqrt{Ch^{d}}\cdot(\hat{\tau}(x;\pi)-\hat{\tau}(x';\pi)).$ Mathematically, $w(x, x', \pi):= 1/\sqrt{\rho_2(x, x', \pi)}$ with $\rho_2(x, x', \pi):=\rho_2(x, \pi)+ \rho_2(x', \pi)-2Ch^{d}Cov(\hat{\tau}(x;\pi),\hat{\tau}(x';\pi)).$ Throughout the paper, I use the first form of $\hat{T}_2$ in (ref); however, to simplify notation, I let $\int_{\{x: x \in \mathcal{X}\}}\int_{\{x' :x' \in \mathcal{X}: x\neq x'\}}$ be equal to $\int_{ \mathcal{X}}\int_{ \mathcal{X}}.$ This should not be interpreted as integrating over the product of the covariate space as the set $\{x, x':x\neq x', x \in \mathcal{X}, x'\in \mathcal{X}\}$ may have non-zero measure.

If we test the null hypotheses $H_0^\Pi$ and $H_0^X$ simultaneously using a multiple testing procedure (MTP), we can disentangle the source of heterogeneity of treatment effects. If we fail to reject both null hypotheses, it implies CTEs by both variable classes. If we reject the null hypothesis $H_0^\Pi$ but fail to reject $H_0^X$, it suggests HTEs by exposure variable and CTEs by pre-treatment variables. In contrast, if we fail to reject the null hypothesis $H_0^\Pi$ but reject $H_0^X$, this implies CTEs by exposure variable and HTEs by pre-treatment variables. Finally, if we reject both null hypotheses, then it suggests HTEs for both variable classes. Thus, as mentioned in Remark (ref), implementing both tests in a multiple-testing framework is imperative to disentangle the source of treatment effects heterogeneity.

To control the probability of rejecting at least one null hypothesis, given that they are both true --- referred to as the family-wise error rate (FWER) --- I recommend the step-wise multiple testing procedure of holm1979simple. Let the $p_{d_1} \leq p_{d_2}$ be the ordered $p$-values, with corresponding null hypotheses $H_{0,{d_1}}$ and $H_{0,{d_2}}.$ Then the Holm step-down algorithm is as follows:

Several similar adjustments for multiple testing exist, but I opt for Holm's method due to its computational simplicity and robustness to dependencies between the two test statistics. See lehmann2022multiple for a comparison between Holm's method and other MTPs.

\ifnum0=1 \addtocounter{alphasect}{1} \fi \oldsection{ Asymptotic Results } In this section, I discuss the asymptotic properties of the proposed test statistics $\hat{T}_1$ and $\hat{T}_2$ when the null hypotheses are true and false. As mentioned in the introduction, I adopt an asymptotic regime where the number of sampled clusters $C$ goes to infinity while the units with each cluster are fixed.

I show that appropriate studentized versions of the test statistics $\hat{T}_1$ and $\hat{T}_2$ converge to the standard normal distribution under the null hypotheses. Two forms of dependencies complicate the derivation of these asymptotic properties. The first is the within-cluster dependency among observations due to interference. Secondly, the terms in the combinatorial sum or integral may also be correlated for each test statistic.

lin2000nonparametric show that when cluster sizes are finite, the most asymptotically efficient kernel estimator of the CATEs is obtained by completely ignoring the within-cluster correlation. This is commonly called the working independence approach. The intuition is that, for clusters of fixed sizes, the probability that two or more observations from the same cluster having significant kernel weights approaches zero as the bandwidth shrinks to zero (See wang2003marginal for more details).

Thus, the working independence approach provides a useful asymptotic method for handling the within-cluster correlations when the number of bounded clusters is large. In essence, it allows one to view observations --- used to estimate the CATEs --- as mutually independent in the asymptotics. Based on the foregoing argument and assuming, without loss of generality, that the clusters are of equal size $N_0,$ I rewrite the CATE kernel estimator as

where $$\hat{\phi}(T_i,x,\pi):=\frac{T_i}{\hat{P}_1(x;\pi)}-\frac{(1-T_i)}{\hat{P}_0(x;\pi)}, \,\,\,\,\,\,\, N=CN_0,$$ with $$\hat{P}_t(x;\pi):= \frac{1}{Nh^d}\sum_{i=1}^{N} \mathbbm{1}(\Pi_i=\pi)\mathbbm{1}(T_i=t)K\left(\frac{x-X_i}{h}\right),\,\,\,t=0, 1. $$ Furthermore, the kernel estimator of $w(x, \pi, \pi')$ is defined as $$\hat{w}(x, \pi, \pi'):= \frac{1}{\sqrt{\hat{\rho}_2(x, \pi)+ \hat{\rho}_2(x, \pi')}},$$ where $\hat{\rho}_2(x, \pi)$ is the kernel estimator of $\rho_2(x, \pi)$ (ignoring all covariance terms) defined as $$\hat{\rho}_2(x, \pi):=(\hat{\mu}_1(x, \pi)-\hat{\mu}_2(x,\pi))\cdot\int K(\xi)^2 d\xi,$$ and $$\hat{\mu}_1(x, \pi):=\frac{1}{Nh^d}\sum_{t \in \{0,1\}}\sum_{i=1}^N\frac{Y_i^2\mathbbm{1}(T_i=t)\mathbbm{1}(\Pi_i=\pi)K(\frac{x-X_i}{h})}{\hat{P}_t^2(x;\pi)},$$ and $$\hat{\mu}_2(x, \pi):=\frac{1}{N^2h^{2d}}\sum_{t \in \{0,1\}}\sum_{j=1}^N\sum_{i=1}^N\frac{Y_jY_i\mathbbm{1}(T_j=t)\mathbbm{1}(T_i=t)\mathbbm{1}(\Pi_i=\pi)\mathbbm{1}(\Pi_j=\pi)K(\frac{x-X_j}{h})K(\frac{x-X_i}{h})}{\hat{P}_t^3(x;\pi)}.$$ The asymptotic results for non-equal cluster sizes are a straightforward extension at the cost of additional notation. Before I state the asymptotic results, the following regularity conditions are required.

These conditions are standard in the kernel estimation literature (see lee2013testing, and chang2015nonparametric). Assumptions (ref)(a) and (b) are unnecessary for the asymptotic results. They are convenient assumptions imposing continuity on $X$ and $Y$ that help to present my main results. Assumption (ref)(c) ensures that the inverse standard error weight function is continuous and well-defined within a compact subset of $\mathcal{X}.$ Assumption (ref)(d) imposes conditions on the kernel function. Assumption (ref)(e) and (f) imposes restrictions on the underlying true data-generating process to ensure smooth and finite moments. Assumption (ref)(g) imposes standard restrictions on the choice of bandwidth. Finally, Assumption (ref)(h) ensures that the estimated weight functions are uniformly consistent.

It is worth emphasizing that for Assumption (ref)(g), the bandwidth depends on the number of clusters $C.$ For instance, if $d=1$ and a second order kernel is used (i.e., $s=2$) then the bandwidth has to be of the form $h=\kappa_0C^{-\lambda},$ where $\kappa_0$ is a positive constant and $1/4<\lambda<1/3.$ However, since $N=CN_0$ and $N_0$ is fixed in asymptotics, the restrictions in Assumption (ref)(g) can be written in terms of $N.$

Asymptotic Null Distribution and Properties of the Test Statistics

Let's focus on the test statistic $\hat{T}_1.$ Rewriting it in the form $\hat{T}_1 = \sum_{k =1}^K \sum_{j =1}^K 2^{-1}\int_{\mathcal{X}} \{\sqrt{N}|\hat{\tau}(x;\pi_k)-\hat{\tau}(x;\pi_j)|\} \cdot \allowbreak \hat{w}(x, \pi_k, \pi_j)dx,$ it resembles the test statistic $\hat{D}:=\int_{\mathcal{X}}\sqrt{N}|\hat{\tau}(x)|\hat{w}(x)dx$\footnote{ $\hat{\tau}(x)$ and $\hat{w}(x)$ are the CATE and inverse scaled standard error kernel estimators defined similar to $\hat{\tau}(x;\pi_k)$ and $\hat{w}(x, \pi_k, \pi_j)$ respectively.} proposed by chang2015nonparametric under no-interference. They show that a studentized version of $\hat{D}$ converges to the standard normal distribution. Under the working independence approach, $\hat{T}_1$ is the sum of dependent random variables $\Big\{2^{-1}\int_{\mathcal{X}} \left(\sqrt{N}|\hat{\tau}(x;\pi_k)-\hat{\tau}(x;\pi_j)|\right) \hat{w}(x, \pi_k, \pi_j): \pi_j,\pi_k \in \mathbf{\Pi}\Big\}.$ Using the poissonization technique in gine2003bm, I show that each of these random variables is asymptotically normal. See the details in Appendix (ref). Thus, as $C$ tends to infinity, $\hat{T}_1$ is the sum of $K(K-1)/2$ dependent normal random variables. It suffices to find the pairwise asymptotic covariances between terms in the sum to obtain the asymptotic normality result of $\hat{T}_1.$

To derive the formal asymptotic results, I studentized the test statistics $\hat{T}_1$ with its mean (bias) under the null and standard error. The asymptotic bias\footnote{See Appendix (ref) for the derivation.} of $\hat{T}_1$ is

where $\mathbbm{Z}_1$ is a standard normal random variable. This diverging asymptotic bias, solely determined by the data-generating process (DGP) through the bounds of $X,$ can be exactly computed.

Define $\hat{\Gamma}(x;\pi,\pi'):= \hat{\tau}(x;\pi)-\hat{\tau}(x;\pi'),$ the asymptotic variance\footnote{See Appendix (ref) for the derivation.} of $\hat{T}_1$ is

where $\int_1= \int_{\mathbbm{R}^{d}}\int_{T_0}\sum_{i=1}^K \sum_{j=1}^K \sum_{k=1}^K \sum_{l=1}^K,$ $T_0=[-1,1]^d$ and $\rho(x,t, \pi_i, \pi_j, \pi_k, \pi_l)$ is the unknown correlation between $\hat{\Gamma}(x;\pi_i,\pi_j)/\sqrt{\hat{\rho}_2(x, \pi_i, \pi_j)}$ and $\hat{\Gamma}(x;\pi_k,\pi_l)/\sqrt{\hat{\rho}_2(x, \pi_k, \pi_l)}$, (i.e., $\rho(x,t, \pi_i, \pi_j, \pi_k, \pi_l) = \mathrm{Corr}(\hat{\Gamma}(x;\pi_i,\pi_j)/\sqrt{\hat{\rho}_2(x, \pi_i, \pi_j)},\hat{\Gamma}(x;\pi_k,\pi_l)/\sqrt{\hat{\rho}_2(x, \pi_k, \pi_l)})$). $\mathbbm{Z}_1$ and $\mathbbm{Z}_2$ are mutually independent standard normal random variables. A consistent kernel estimator of $\rho(x,t, \pi_i, \pi_j, \pi_k, \pi_l)$ is defined as

where $\hat{\rho}_2(x, \pi, \pi')\coloneqq \hat{\rho}_2(x, \pi) +\hat{\rho}_2(x, \pi').$ Plug $\hat{\rho}(x, t,\pi_i,\pi_j,\pi_k,\pi_l)$ into the right hand side of ((ref)), and obtain the asymptotic variance estimator

The studentized version of the test statistic $\hat{T}_1$ is defined as $$\hat{S}_{1}\coloneqq\frac{\hat{T}_{1}-a_{_1}}{\hat{\sigma}_{1}}.$$

Similarly, the studentized version of the test statistic $\hat{T}_2$ is defined as $$\hat{S}_{2}\coloneqq\frac{\hat{T}_{2}-a_{_2}}{\hat{\sigma}_{2}},$$ where

and

where $\rho(x, t, \pi_k):= (\int K(\xi)K(\xi+t)d\xi)/(\int K(\xi)^2 d\xi),$ $\hat{\rho}_2(x,x',\pi_k)$ is the plug-in estimator of ${\rho}_2(x,x',\pi_k)$ defined as $\hat{\rho}_2(x,x',\pi_k):= \hat\rho_2(x, \pi)+ \hat\rho_2(x', \pi) -2 \hat \rho_2(x, \pi)\mathbbm{1}(t \in [-1,1]^d)\cdot\mathrm{Cov}((1-\rho(x, t, \pi_k)^2)^{1/2}\mathbbm{Z}_1 +\rho(x, t, \pi_k)\mathbbm{Z}_2,\mathbbm{Z}_2),$ and $\int_{2'}\coloneqq\int_{[-1,1]^{d}}\int_{\mathbbm{R}^{d}} \int_{\mathbbm{R}^{d}}\int_{\mathbbm{R}^{d}}\sum_{k=1}^K.$ \footnote{In the case where errors are homoskedastic, $\hat{\rho}_2(x,\pi_k) \cdot \left((\hat{\rho}_2(x,\pi_k)+\hat{\rho}_2(x',\pi_k)) (\hat{\rho}_2(x,\pi_k)+\hat{\rho}_2(x'',\pi_k))\right)^{-1/2}$ in (ref) simplifies as $\hat f(x,\pi_k)^{-1}\cdot \{(\hat f(x,\pi_k)^{-1}+\hat f(x',\pi_k)^{-1})(\hat f(x,\pi_k)^{-1}+\hat f(x'',\pi_k)^{-1})\}^{1/2}$ with $\hat f(x,\pi_k)$ the estimated joint density.} It is worth to note that the asymptotic inverse scaled standard error to compute $\hat S_2$ also becomes $$w(x, x', \pi)=\frac{1}{\sqrt{\rho_2(x, \pi)+ \rho_2(x', \pi) -2 \rho_2(x, \pi)\mathbbm{1}(t \in [-1,1]^d)\cdot\mathrm{Cov}((1-\rho(x, t, \pi_k)^2)^{1/2}\mathbbm{Z}_1 +\rho(x, t, \pi_k)\mathbbm{Z}_2,\mathbbm{Z}_2) }},$$ where $t=(x-x')/h$.\footnote{$w(x, x', \pi)$ is well defined since $Ch^{d}Cov(\hat{\tau}(x;\pi),\hat{\tau}(x';\pi))\to \rho_2(x, \pi)\mathbbm{1}(t \in [-1,1]^d)\cdot\mathrm{Cov}((1-\rho(x, t, \pi_k)^2)^{1/2}\mathbbm{Z}_1 +\rho(x, t, \pi_k)\mathbbm{Z}_2,\mathbbm{Z}_2)$ after applying a change of variable. Thus, the argument of the square root in the denominator is positive.} As such, I obtain the sample analog of $w(x, x', \pi)$ by plugging in the estimators of $\rho_2(x, \pi) $ for all $x \in \mathcal{X}.$ See Appendix (ref) for the derivation of $a_2$ and $\hat{\sigma}^2_{2}.$

Proposition (ref) suggests that critical values of the tests can be obtained from the standard normal distribution. This proposition forms an integral part of the asymptotic properties that follow.

Theorem (ref) shows that the test statistics $\hat{S}_1$ and $\hat{S}_2$ have correct sizes asymptotically under $H^{\Pi}_{0}$ and $H^{X}_{0}$ respectively. Hence, the following one-sided decision rule suffices: For $j=1,2$ reject the null hypothesis if $\hat{S}_j>z_{1-\alpha},$ where $z_{1-\alpha}, \alpha \in [0,1]$ is the $(1-\alpha)^{th}$ quantile (critical value) obtained from the standard normal distribution.

Power properties of the Test statistics

In this subsection, I investigate the power of the test statistics against a fixed and a sequence of local alternatives drifting to the null. First, I establish that $\hat{S}_1$ and $\hat{S}_2$ are consistent against the following fixed alternatives

respectively. These alternative hypotheses (ref) and (ref) are analogous to those in (ref) and (ref) respectively.

Theorem (ref) shows that the proposed test statistics have power against fixed alternatives. Next, I show that the proposed tests $\hat{S}_1$ and $\hat{S}_2$ can detect a sequence of local alternatives converging to the null hypotheses. Specifically, consider the following sequences of local alternatives

converging to the null hypotheses $H^{\Pi}_{a}$ and $H^{X}_{a}$ respectively where for $j=1,2$,\, $\delta_j(\cdot,\cdot,\cdot)$ is a real bounded function satisfying:\\ $\int_{\mathcal{X}}\sum_{\pi \in \Pi}\sum_{\pi' \in \Pi}|\delta_1(x, \pi, \pi')|w(x, \pi, \pi')dx>0,$ and $\int_{\mathcal{X}}\int_{\mathcal{X}}\sum_{\pi \in \Pi}|\delta_2(x, x', \pi)|w(x,x', \pi)dxdx' \allowbreak >0.$

Theorem (ref) demonstrates that the test statistics have statistical power greater than zero against local alternatives approaching the null at a rate slower than $C^{-1/2}h^{-d/4}.$ This rate agrees with that found in nonparametric kernel-based tests of parametric restrictions.

Bootstrap-Based Inference

I introduce bootstrap resampling procedures in this subsection to obtain the null distributions of $\hat{S}_1$ and $\hat{S}_2$. These bootstrap procedures provide a benchmark for the aforementioned asymptotic-based methods in finite samples. Specifically, I numerically compare the power and size of the asymptotic-based and bootstrap-based procedures for finite samples in Appendix (ref).

Following the recommendations of several authors, particularly davidson1999size, I propose the following bootstrap resampling procedures that impose the null hypotheses.

The Bootstrap resampling algorithm for $\hat{S}_1$

Let $\mathbf{W}_c=(\mathbf X_c, \mathbf T_c,\mathbf Y_c)$ denote the vector of variables for the $c^{th}$ cluster. Therefore, the pooled sample across all clusters can be written as $\{\mathbf W_{c}\}_{c=1}^{C}.$ I propose the following pairs cluster bootstrap-t procedure with the null $H_0^{\Pi}$ imposed to generate the null distribution of $\hat{S}_1$:

The Bootstrap resampling algorithm for $\hat{S}_2$

I propose the following wild cluster bootstrap-t procedure with the null $H_0^{X}$ imposed to generate the null distribution of $\hat{S}_2$:

\ifnum0=1 \addtocounter{alphasect}{1} \fi \oldsection{Monte Carlo Simulation} In this section, I investigate the finite sample performance of the proposed test statistics via Monte Carlo experiments. I first check the size and power of the test statistics. Then, I compare the performance with their parametric counterparts. I also report additional simulation results in Appendix (ref).

Empirical size and statistical power

All the rejection probabilities in the simulations are based on 1000 replications. In the experiments in this section, I focus on one pre-treatment variable $X_{ci}$\footnote{I extend the experiment to the case with two pre-treatment variables in Appendix (ref).} drawn from the uniform $[0, 1]$ distribution allowing for within-cluster correlations. Each cluster is assigned one of four treatment vectors $\textbf{T}(k), k=1\dots,4.$ The average of $\textbf{T}(k)$ is $\pi_k$ where $\{\pi_1,\pi_2,\pi_3,\pi_4 \}= \{0.3, 0.4, 0.5, 0.6\}=\mathbf{\Pi}.$ The realized outcome $Y_{ci}$ is constructed as follows: $$Y_{ci} = (\tau(X_{ci}, \Pi_{ci}) + U_{1ci}) \times T_{ci} + U_{0ci} \times (1 - T_{ci}),$$ where $U_{1ci}$ and $U_{0ci}$ are independent normals with a mean of zero and variance of 0.1, and $T_{ci} \in \textbf{T}(k).$ The general specification of the CATE is $$\tau(x;\pi) = \beta_0 x + \beta_1 \pi.$$ In the experiments, I compute CATEs on a uniform grid in the interval between the $10^{th}$ and $90^{th}$ percentiles of $X$. This way, I avoid the boundary bias issue associated with the kernel estimators. Integrals are computed with the composite trapezoid technique.

I adopt the following kernel function that satisfies Assumption (ref)(d):

and a bandwidth

where $\hat{s}_X$ is the sample standard deviation of $X$ and $\kappa_h$ is a constant. A similar kernel and bandwidth are employed in chang2015nonparametric. Moreover, I set $C=150$ with 10 units $(N_c=10)$ in each cluster (i.e., N=1500 units).

To compute the empirical size of $\hat{S}_1$, I fix $\beta_0=1$ and $\beta_1=0.$ Thus, $\tau(x;\pi) = x,$ and the null hypothesis $H_0^\Pi$ of CTEs by $\Pi$ is true. Note that as $\beta_1$ deviates further from $0$ in both directions, the null hypothesis deviates further away from the truth. Again, I report the empirical rejection probabilities for values of $\beta_1$ in the range $[-0.5, 0.5]$, with 0.05 increments.

In contrast, to obtain the empirical size of $\hat{S}_2$, I set $\beta_0=0,$ and $\beta_1=1.$ Thus, $\tau(x;\pi) = \pi,$ and the null hypothesis $H_0^X$ of CTEs by $X$ is true. I report the empirical rejection probabilities for each $\beta_0$ in the range $[-0.5,0.5 ]$, with 0.05 increments.

Focusing on the asymptotic-based procedure, a summary of the empirical rejection probabilities is in Figure (ref) (and in Tables (ref)--(ref) in Appendix (ref)). In each panel, the plots represent the rejection probabilities at the 1%, 5%, and 10% nominal levels. The left panel reports the empirical rejection probabilities of $\hat{S}_1,$ and the right panel reports those of $\hat{S}_2.$ When $\beta_0=\beta_1=0,$ the empirical rejection probabilities are close to the nominal probabilities, corroborating the theoretical results in Theorem (ref). On the contrary, as $\beta_0$ and $\beta_1$ deviate towards $\pm 0.5,$ the rejection probabilities approach 1, which aligns with Theorem (ref).

Similarly, using the bootstrap procedures, I summarize the empirical rejection probabilities in Figure (ref) (and in Tables (ref)--(ref) in Appendix (ref)). The results are based on 399 bootstrap resamples. Compared to the asymptotic-based empirical rejection probabilities, the differences in the probabilities are negligible. This suggests that the test statistics under the null hypotheses "converge in bootstrap distribution" to the standard normal distribution for large sample sizes.

Parametric Testing and Misspecification

The Monte Carlo experiment in this subsection seeks to show that parametric tests of $H_0^\Pi$ and $H_0^X$ may be misleading because parametric models are always misspecified to a certain degree. I only report the results of the asymptotic-based procedure to save space. I generate sample data $\{Y_i,X_i,T_i, \Pi_i\}$ of size N=600 ($C=60$ and $N_c=10$), and $\Pi_i \in \{0.3,0.4,0.5,0.6\}$. The outcome is of the form: $$Y_{ci} = (\tau(X_{ci}, \Pi_{ci}) + U_{1ci}) \times T_{ci} + U_{0ci} \times (1 - T_{ci}),$$ where $$\tau(x;\pi) =30\cdot \cos(2\cdot3.142\cdot x)\cdot (\pi^2-\pi).$$ I keep the remaining design as in Section (ref) above. This new CATE specification is highly non-linear in $X$ and $\Pi.$

I estimate the following linear regression model using ordinary least squares:

The parameters $\beta_4$ and $\beta_5$ measures HTEs across $X$ and $\Pi$ respectively. The estimation results in Table (ref) show that $\beta_4$ and $\beta_5$ are insignificant when we use clustered standard errors. It leads to the wrong conclusion that treatment effects do not vary by the exposure variable $\Pi,$ and the pre-treatment variable $X$.

Next, I test the null hypotheses using this paper's proposed nonparametric test statistics. I use the kernel function in (ref) and the bandwidth formula in (ref). Table (ref) summarizes the results of the two tests at different bandwidth choices (different $\kappa_h$ in the bandwidth formula in (ref)). The results unequivocally reject the null hypotheses of CTEs by the exposure variable $\Pi,$ and the pre-treatment variable $X$. This serves as a stark reminder of the potential for misspecification of the functional form of the conditional mean in parametric models to lead to an erroneous inference of HTEs, underscoring the need for the nonparametric testing procedures in this paper.

\ifnum0=1 \addtocounter{alphasect}{1} \fi \oldsection{Empirical Application} In this section, I use the experimental data from cai2015social to demonstrate the usage of the proposed test statistics. This experiment was implemented to help determine whether farmers' understanding of a weather insurance policy affects purchasing decisions. The authors examine the impact of two types of information sessions on insurance adoption among 5335 households in 185 small rice-producing villages (47 administrative villages) in 3 regions in the Jiangxi province in China. They show that the type of information session directly affects participants' adoption and significantly affects the adoption decision of participants' friends.

The data includes each household's network information (each household has at most five friends) and additional pre-treatment information such as age, gender, rice production area, risk aversion score, the fraction of household income from rice production, and others. The outcome of interest is binary: whether or not a household buys the insurance policy. For each village in the experiment, there were two rounds of information sessions offered to introduce the insurance product. Households are randomly assigned to rounds.\footnote{Each household can only participate in one of the two rounds.} In each round, two sessions were held simultaneously: a simple session (with less information) and an intensive session. Households are randomly assigned to sessions. Participants have to make a purchase decision on the spot in each session. Since the second round of information sessions were held three days after the first, cai2015social argue that information may spill to households participating in the second round of information sessions from their friends who attended the first round. That is, the insurance adoption of those in the first round depends only on their treatment (participation) status, but the insurance adoption of participants in the second round depends on their treatment and the treatment statuses of their friends in the first round. Table (ref) shows the result of a nonlinear probability (logit) model of insurance adoption on "Second-round" --- a binary variable which is one if households participated in the second round session and zero otherwise --- among households in the simple sessions. The coefficients of "Second-round" indicate that the probability of adoption among the second-round participants in the simple sessions is higher than their first-round counterparts. This is evidence of information spillover.

Furthermore, the analyses in cai2015social rule out any form of between-administrative village diffusion of information, which makes the data fit into the clustered network setting of this paper. Figure (ref) from bargagli2020heterogeneous shows a network plot of the data for the 47 administrative villages.

In addition, cai2015social show that while farmers are influenced by their friends who attended the first round of intensive sessions, they are unaffected by friends who attended the first round of simple sessions. They also show that people are less influenced by their friends when they have the same education about the insurance products (e.g., information from first-round households in the intensive session does not affect the purchasing decision of second-round households in the intensive session). These observations are crucial for defining the post-treatment exposure variable in this application.

To illustrate the use of the proposed test statistics, I focus on the second-round participants in each administrative village. I define the post-treatment exposure variable as the fraction of households who attended the first-round intensive session in an administrative village. I also focus on one pre-treatment variable: the fraction of household income from rice production.\footnote{The fraction of household income from rice production should significantly influence a household's decision to purchase insurance. Households with a small fraction of income from rice production are unlikely to buy insurance, those with a moderate fraction are more likely to purchase it, and households with a large fraction are also unlikely to buy --- because they may already have an insurance policy --- ceteris paribus.} A household is considered "treated" if it attended an intensive session, while those that participated in the simple session are labeled "untreated." The outcome of interest is the second-round participants' adoption of the insurance policy.

There are 2653 households in my restricted sample (second-round participants with no missing values for the four variables). To ensure balance in the post-treatment exposure variable (treatment ratio), I binarize the treatment ratio: households from villages with the fraction of first-round intensive session participants less than 0.22 constitute the first group, and those from villages with the fraction of first-round intensive session participants greater than 0.22 make up the second group. There are 1488 and 1239 households in the two groups, respectively.\footnote{Note that the threshold for the binarization is purely based on sample size considerations. I chose the threshold so the two groups will have sufficient units.} Thus, technically, the exposure variable is a threshold function of the treatment ratio.

Using this sample, I test the two null hypotheses. Figure (ref) shows the estimated average treatment effects against the fraction of income from rice production for the two exposure groups. Visually, the estimated ATEs seem to vary with the exposure variable and the fraction of household income from rice production. However, are these variations statistically significant? Using the kernel function defined in (ref) and the bandwidth formula in (ref), the results in Table (ref) show that the proposed asymptotic-based tests fail to reject the null hypotheses of CTEs by the treatment ratio variable and the fraction of household income from rice production at different bandwidths and conventional significance levels. Moreover, based on Holm's MTP, I jointly fail to reject the two null hypotheses at the different bandwidths and conventional significance levels.

Based on Figure (ref), there is a wider gap between the ATEs for the two categories of treatment ratio when the fraction of rice production lies between 0.5 and 0.8. Thus, to check the robustness of the proposed testing procedures, I restrict the sample to these households and re-implement the tests. Figure (ref) shows a plot of the estimated ATEs against the fraction of income from rice production using the new sample. The test results in Table (ref) show that ATE varies with the treatment ratio (at the conventional significance levels) but is constant across the fraction of income from rice production as expected.

\ifnum0=1 \addtocounter{alphasect}{1} \fi \oldsection{Conclusion} The nonparametric tests I develop in this paper not only allow for valid inference for heterogeneous treatment effects in the presence of clustered interference but also play a crucial role in disentangling the source of variation in the treatment effects. This unique feature sets the proposed tests apart from existing procedures. The test statistics are sums of weighted $L_1$-norm pairwise differences in consistent nonparametric kernel estimators of conditional average treatment effects. Applying the poissonization technique in gine2003bm, I show that the test statistics are asymptotically normally distributed and have correct sizes in large samples. Moreover, they are consistent under fixed alternatives and have nonzero power against local alternatives, drifting to the null. I provide Monte Carlo results that corroborate the theoretical findings. On the applied side, I illustrate the usage of the proposed tests with the data from cai2015social. I find no evidence of heterogeneous treatment effects across the values of both a post-treatment exposure variable and a pre-treatment variable: the fraction of household income from rice production.

There is room for several extensions. It will be interesting to provide theoretical guarantees for the proposed bootstrap methods in the paper. For instance, Table (ref) in Appendix (ref) provides simulation evidence that suggests that the bootstrap algorithm of $\hat{S}_1$ achieves asymptotic refinement. It will be insightful to theoretically investigate if the proposed bootstrap algorithms achieve asymptotic refinement over asymptotic-based tests. Since it is never trivial to show bootstrap refinements even in more standard tests, I defer such a study to future research. Secondly, it is valuable to provide HTEs testing procedures in networks that are not clustered (single un-clustered networks). owusu2023randomization studies randomization tests for HTEs in single un-clustered networks. The procedures in this companion paper are, however, only applicable to experimental data.

\ifnum0=1 \addtocounter{alphasect}{1} \fi \oldsection{Acknowledgments} This is a revised version of the first Chapter of my Ph.D. Dissertation at McMaster University, owusu2023treatment. It has benefited from feedback of Youngki Shin, Jeffrey S. Racine, Micheal Veall, Saraswata Chaudhuri, Sukjin Han, Emmanuel S. Tsyawo, Monika A. M\'arquez, Antoine A. Djogbenou, and participants at the 32nd Annual Meeting of Midwest Econometrics Group and the 56th Annual Conference of the Canadian Economics Association. This work was made possible by the advanced research computing platform provided by the Digital Research Alliance of Canada (formerly Compute Canada). All errors are my own.

\ifnum0=1 \addtocounter{alphasect}{1} \fi \oldsection{Appendix}