# Design-Based Multi-Way Clustering

Luther Yap

###### Abstract

This paper extends the design-based framework to settings with multi-way
cluster dependence, and shows how multi-way clustering can be justified when
clustered assignment and clustered sampling occurs on different dimensions, or
when either sampling or assignment is multi-way clustered. Unlike one-way
clustering, the plug-in variance estimator in multi-way clustering is no
longer conservative, so valid inference either requires an assumption on the
correlation of treatment effects or a more conservative variance estimator.
Simulations suggest that the plug-in variance estimator is usually robust, and
the conservative variance estimator is often too conservative.

##  1 Introduction

There have been thousands of papers that adjust standard errors for clustering
in linear regression. To address the issue of when such clustering is
appropriate, Abadie et al. (2023) extended the design-based framework of
Abadie et al. (2020) to show how clustering on a particular dimension, such as
state, is justified when there is clustered sampling or assignment on that
dimension. However, their framework and theory apply only to one-way
clustering, which leaves an open question on how papers that clustered on
multiple dimensions, such those applying the plug-in variance of Cameron et
al. (2011), henceforth CGM, can be justified. There are situations where
sampling and assignment occur on different clustering dimensions (e.g., Gruber
and Madrian (1995)). There are also sampling (e.g., Hersch (1998)) or
assignment (e.g., Nunn and Wantchekon (2011)) mechanisms that are multi-way
clustered. Then, there is an open question of whether the results of Abadie et
al. (2023) generalize to multi-way clustering settings. This paper fills the
gap.

Until recently, accounting for the large sample behavior of design-based
settings with multi-way clustering has been a difficult problem. Asymptotic
theory for variables that have multi-dimensional dependence has thus far
relied on separate exchangeability (e.g., Davezies et al. (2018)). Separate
exchangeability implies that the marginal distributions of clusters are
exchangeable. (MacKinnon et al., 2021) However, by construction, separate
exchangeability is violated in a design-based framework because the residual
$u_{i}=W_{i}u_{i}(1)+(1-W_{i})u_{i}(0)$ depends on nonstochastic potential
residual $u_{i}(w)$ and treatment $W_{i}$. Hence, even if the treatments
$W_{i}$’s were identically distributed over clusters, the marginal
distribution of $u_{i}$ cannot be the same as the $W_{i}$’s are weighted
differently. Hence, limit theory that accommodates heterogeneity of clusters
and observations is required. By building on the central limit theorem in Yap
(2023), this paper obtains results on large-sample behavior of standard
estimators in this environment.

This paper shows that while most results in the one-way design-based setup
generalize to multi-way clustering, there are some nuances in multi-way
clustering. As we would expect, when sampling and assignment occur on
different clustering dimensions, it is necessary to cluster on both dimensions
to obtain valid inference. When sampling is two-way clustered, it is necessary
to cluster on both dimensions for valid inference. When assignment is one-way
clustered, Abadie et al. (2023) showed that the standard Liang and Zeger
(1986) plug-in variance estimator is conservative, even though it is still
necessary to cluster in some way for valid inference. Similarly, in several
data-generating processes with multi-way clustered assignment, the CGM plug-in
estimator is conservative. However, unlike one-way clustering, the CGM
estimator is no longer always conservative. In fact, it is possible to
construct a data-generating process that makes the CGM variance estimator
anticonservative when there is multiway sampling.

In response to the anticonservativeness of the CGM estimator in design-based
settings, there are two approaches that empirical researchers may take. The
first approach is to make an assumption on how the individual treatment
effects are correlated within the same cluster: CGM is conservative when the
correlation is positive, which is reasonable in most applications. The second
approach is to remain agnostic and to use CGM2, a more conservative version of
the CGM variance estimator proposed by Davezies et al. (2018). Since the
simulations show that CGM2 is often unnecessarily conservative, making an
assumption on the correlation is usually the more reasonable approach.

##  2 Setting

We have a sequence of populations indexed by $k$, each with $n_{k}$ units.
Each unit is indexed by $i=1,\cdots,n_{k}$, partitioned into clusters on two
dimensions $G,H$, indexed by $g$ and $h$. Let $m=(g,h)$ denote the
intersection of two cluster indices that are nonempty. Let $g_{ki},h_{ki}$
denote the cluster that unit $i$ belongs to on the respective dimensions, and
$m_{ki}$ its intersection of clusters. For treatment variable $W\in\\{0,1\\}$,
we have the potential outcome is denoted $y_{ki}(w)$ that is nonstochastic.

We are interested in the population average treatment effect (ATE):

$\displaystyle\tau_{k}$
$\displaystyle=\frac{1}{n_{k}}\sum_{i=1}^{n_{k}}(y_{ki}(1)-y_{ki}(0))$

With $\alpha_{k}:=(1/n_{k})\sum_{i=1}^{n_{k}}y_{ki}(0)$ and residuals
$U_{ki}:=Y_{ki}-\alpha_{k}-\tau_{k}W_{ki}$, the potential residuals are
denoted:

$\displaystyle u_{ki}(1)$ $\displaystyle=y_{ki}(1)-(\alpha_{k}+\tau_{k})$
$\displaystyle u_{ki}(0)$ $\displaystyle=y_{ki}(0)-\alpha_{k}$

Let $R_{ki}$ denote the sampling indicator, so observation $i$ in population
$k$ is observed when $R_{ki}=1$. Similarly, $W_{ki}$ denotes the treatment
indicator. Hence, for $R_{ki}=1$, we observe the triple
$\\{Y_{ki},W_{ki},m_{ki}\\}$, with
$Y_{ki}=W_{ki}y_{ki}(1)+(1-W_{ki})y_{ki}(0)$. The high-level assumption on
assignment and sampling is that
$(R_{ki},W_{ki})\perp\\!\\!\\!\perp(R_{kj},W_{kj})$ whenever they do not share
any cluster, and that sampling and assignments are independent, as stated in
Assumption 1.

######  Assumption 1.

$(R_{ki},W_{ki})\perp\\!\\!\\!\perp(R_{kj},W_{kj})$ if $g_{ki}\neq g_{kj}$ and
$h_{ki}\neq h_{kj}$. Further,
$\\{R_{ki}\\}_{i=1}^{n_{k}}\perp\\!\\!\\!\perp\\{W_{ki}\\}_{i=1}^{n_{k}}$. For
all $i$, there exists some $b_{k1}$ and $b_{k0}$ such that
$E\left[R_{ki}W_{ki}\right]=b_{k1}$ and
$E\left[R_{ki}\left(1-W_{ki}\right)\right]=b_{k0}$.

$b_{k1}$ is the probability that an individual is observed and treated;
$b_{k0}$ is the expected probability that an individual is observed and
untreated.

The above assumption can be generated from several design-based settings. One
possibility is to adapt the data-generating process from Abadie et al. (2023)
for one-way clustering, just that the sampling and assignment processes are
allowed to occur on different clustering dimensions. This environment strictly
generalizes their setting: their setting is a special case where both sampling
and assignment occur on the same dimension, and everyone is in their own
cluster on the other dimension. Another possibility is to have multi-way
clustering occur on either the assignment or sampling dimension, and
independence over units on the other dimension. I detail both possibilities in
the next two subsections, and provide empirical examples.

###  2.1 Sampling and Assignment on Different Dimensions

Without loss of generality, suppose there is clustered sampling on the $G$
dimension. Here, every $G$ cluster is independently sampled with probability
$q_{k}$. If a cluster is sampled, then units within the cluster are sampled
independently with probability $p_{k}$. Clustered assignment can occur on the
$H$ dimension. Every cluster on the $H$ dimension has an assignment
probability $B_{kh}$, drawn from a distribution with mean $\mu_{k}$ and
$\sigma_{k}^{2}$. Then, for a given cluster $h$, units within that cluster are
independently assigned treatment with probability $B_{kh}$.

An empirical example is Gruber and Madrian (1995), who study the effect of
health insurance on retirement. We may want to cluster by household and state-
year cell. They use the CPS data which has a sample that is negligibly small
relative to the superpopulation of households, and each household reveals data
on employment in previous years (where they could have moved across different
states). It then seems reasonable to believe that sampling occurs on the
household dimension. Assignment occurs on the state dimension because
insurance is affected by state policy. Since the same incumbent party has
differing policies over states, and the same state has differing policies when
the party differs over time, assignment to health insurance varies by state
clusters.

###  2.2 Multiway Clustering on Assignment

This subsection explains a possible mechanism for multiway clustering on the
assignment dimension. Since the setup for multiway sampling is analogous, its
exposition is omitted for brevity.

Data is generated by independently drawing $A_{kg}\in[0,1]$, $B_{kh}\in[0,1]$
and $e_{ki}\sim U[0,1]$, with
$W_{ki}=1\left\\{e_{i}<A_{g(i)}B_{h(i)}\right\\}$. The random variables
$A_{kg}$ and $B_{kh}$ have means $\mu_{Ak},\mu_{Bk}$ and variances
$\sigma_{Ak}^{2},\sigma_{Bk}^{2}$ respectively. This process nests several
cases. If assignment is one-way clustered, then we can simply set $A_{kg}=1$.
A special case is where assignment occurs at the intersection level, and we
need both dimensions $G$ and $H$ to be assigned treatment for the unit to be
treated. Then, $A_{kg},B_{kh}\in\\{0,1\\}$. This mechanism also nests the
assignment mechanism where a unit is treated whenever either of its clustering
dimensions is treated. To see this, observe that the unit is untreated only if
both its clusters are untreated, so we can simply switch the labels of
treatment and non-treatment.

As an example of multi-way assignment, Nunn and Wantchekon (2011) studied the
effect of slave trade on trust today. They clustered by ethnic groups and
district. The unit of observation is an individual. Both these dimensions
affect assignment because people of similar ethnicities have similar history
of slave trade. District is relevant since people in the same location are
more likely hit by slave trade.

An example of multi-way sampling is from Hersch (1998), who was interested in
the effect of injury risk on wages. She clustered by industry and occupation.
When sampling from a large number of industries and occupations, we would
observe an individual only when both his industry and his occupation were
sampled.

##  3 Least Squares Estimator and Variance

Let $N_{k}:=\sum_{i=1}^{n_{k}}R_{ki}$ denote the number of sampled units.
Further, $N_{k1}:=\sum_{i=1}^{n_{k}}R_{ki}W_{ki}$ and
$N_{k0}:=\sum_{i=1}^{n_{k}}R_{ki}(1-W_{ki})$. The least squares estimator is:

$\displaystyle\hat{\tau}_{k}$
$\displaystyle=\frac{1}{N_{k1}}\sum_{i=1}^{n_{k}}R_{ki}W_{ki}Y_{ki}-\frac{1}{N_{k0}}\sum_{i=1}^{n_{k}}R_{ki}(1-W_{ki})Y_{ki}$

The rest of this section first shows how the estimator is asymptotically
normal, and derives an expression for its asymptotic variance. Then, it
presents the asymptotic limits of various commonly-used variance estimators.

###  3.1 Large Sample Properties of OLS Estimator

To state the main result, I first define a few terms. Let $\mathcal{N}_{i}$
denote the neighborhood of $i$, which is the set of observations that are
plausibly correlated with $i$. In the multi-way clustering context,
$\mathcal{N}_{i}:=\\{j:g_{ki}=g_{kj}\\}\cup\\{j:h_{ki}=h_{kj}\\}$. For
$C\in\\{G,H\\}$, let $\mathcal{N}^{C}_{c}$ denote the set of observations in
cluster $c$, and $N^{C}_{c}:=|\mathcal{N}^{C}_{c}|$ denote the number of
observations in cluster $c$ on the $C$ dimension. Further, define $\xi_{ki}$
as the demeaned residual for individual $i$ that features in the variance of
$\hat{\tau}_{k}$:

$\displaystyle\xi_{ki}$
$\displaystyle:=\frac{1}{b_{k1}}\left(R_{ki}W_{ki}-b_{k1}\right)u_{ki}(1)-\frac{1}{b_{k0}}\left(R_{ki}\left(1-W_{ki}\right)-b_{k0}\right)u_{ki}(0)$

######  Assumption 2.

Let $Q_{ki}$ denote $R_{ki}W_{ki}$, $R_{ki}(1-W_{ki})$ or $\xi_{ki}$. For some
positive $K_{0}<\infty$, $C\in\\{G,H\\}$ and
$\lambda_{k}=Var(\sum_{i=1}^{n_{k}}Q_{ki})$, the following hold:

  1. 1.

$|y_{ki}(w)|\leq K_{0}$.

  2. 2.

$\frac{1}{\lambda_{k}}\operatornamewithlimits{max}_{c}(N^{C}_{c})^{2}\rightarrow
0$.

  3. 3.

$\frac{1}{\lambda_{k}}\sum_{c}(N^{C}_{c})^{2}\leq K_{0}$.

These regularity conditions are required in the multi-way clustering context
so that the central limit theorem (CLT) from Yap (2023) can be applied. These
regularity conditions require bounded potential outcomes, the contribution of
the largest cluster to the total variance be small, and a summability
condition.

######  Theorem 1.

Under Assumption 1 and Assumption 2,

$\frac{\sqrt{N_{k}}(\hat{\tau}_{k}-\tau_{k})}{\sqrt{v_{k}}}\xrightarrow{d}N(0,1)$
(1)

where

$\displaystyle v_{k}$
$\displaystyle:=\frac{N_{k}}{n_{k}^{2}}\sum_{i=1}^{n_{k}}\sum_{j\in\mathcal{N}_{i}}E[\xi_{ki}\xi_{kj}]$

and

$\displaystyle E\left[\xi_{ki}\xi_{kj}\right]$
$\displaystyle=\left(\frac{1}{b_{k1}^{2}}E\left[R_{ki}R_{kj}\right]E\left[W_{ki}W_{kj}\right]-1\right)u_{ki}(1)u_{kj}(1)$
$\displaystyle\qquad+\left(\frac{1}{b_{k0}^{2}}E\left[R_{ki}R_{kj}\right]E\left[\left(1-W_{ki}\right)\left(1-W_{kj}\right)\right]-1\right)u_{ki}(0)u_{kj}(0)$
$\displaystyle\qquad-\left(\frac{1}{b_{k1}b_{k0}}E\left[R_{ki}R_{kj}\right]E\left[W_{ki}\left(1-W_{kj}\right)\right]-1\right)u_{ki}(1)u_{kj}(0)$
$\displaystyle\qquad-\left(\frac{1}{b_{k1}b_{k0}}E\left[R_{ki}R_{kj}\right]E\left[\left(1-W_{ki}\right)W_{kj}\right]-1\right)u_{ki}(0)u_{kj}(1)$

In Theorem 1, the variance expression is a function of expectations of $R$ and
$W$ cross products across different units. These cross-products have different
expressions, depending on the environment that generated the data, as stated
in Lemmas 1 and 2 below.

######  Lemma 1.

Suppose Assumption 1 holds, there is clustered sampling on dimension $G$, and
clustered assignment on dimension $H$. Then, $g_{ki}\neq g_{kj}$ implies
$E\left[R_{ki}R_{kj}\right]=p_{k}^{2}q_{k}^{2}$ and $h_{ki}\neq h_{kj}$
implies $E\left[W_{ki}W_{kj}\right]=\mu^{2}$. If $g_{ki}=g_{kj}$,
$E\left[R_{ki}R_{kj}\right]=q_{k}p_{k}^{2}$. If $h_{ki}=h_{kj}$,
$E\left[W_{ki}W_{kj}\right]=\sigma^{2}+\mu^{2}$,
$E\left[W_{ki}\left(1-W_{kj}\right)\right]=\mu\left(1-\mu\right)-\sigma^{2}$,
and
$E\left[\left(1-W_{ki}\right)\left(1-W_{kj}\right)\right]=\left(1-\mu\right)^{2}+\sigma^{2}$.

######  Lemma 2.

Suppose Assumption 1 holds and there is multiway clustering in assignment.
Then, if $m_{ki}=m_{kj}$, then
$E\left[W_{ki}W_{kj}\right]=\left(\mu_{A}^{2}+\sigma_{A}^{2}\right)\left(\mu_{B}^{2}+\sigma_{B}^{2}\right)$.
If $g_{ki}=g_{kj},h_{ki}\neq h_{kj}$, then
$E\left[W_{ki}W_{kj}\right]=\left(\mu_{A}^{2}+\sigma_{A}^{2}\right)\mu_{B}^{2}$.
If $g_{ki}\neq g_{kj}$ and $h_{ki}\neq h_{kj}$, then
$E\left[W_{ki}W_{kj}\right]=\mu_{A}^{2}\mu_{B}^{2}$.

Multi-way sampling can be treats $R$ analogously to $W$ in Lemma 2.

###  3.2 Variance Estimators

The true variance as stated above is a function of potential residuals, so a
direct plug-in is infeasible. Variance estimators hence use a feasible analog.
Define:

$\displaystyle\eta_{ki}$
$\displaystyle:=R_{ki}\left(\frac{W_{ki}}{b_{k1}}-\frac{1-W_{ki}}{b_{k0}}\right)U_{ki}$
$\displaystyle\hat{\eta}_{ki}$
$\displaystyle:=R_{ki}\left(\frac{W_{ki}}{\hat{b}_{k1}}-\frac{1-W_{ki}}{\hat{b}_{k0}}\right)\hat{U}_{ki}$
$\displaystyle\hat{U}_{ki}$
$\displaystyle:=Y_{ki}-\hat{\alpha}_{k}-\hat{\tau}_{k}W_{ki}$

Observe that $\xi_{ki}=\eta_{ki}-E[\eta_{ki}]$, and
$E[\eta_{ki}]=u_{ki}(1)-u_{ki}(0)$. The common variance estimators include the
heteroskedasticity-robust variance estimator attributed to Eicker (1967),
Huber (1967) and White (1980) (EHW), and the one-way cluster robust variance
estimator attributed to Liang and Zeger (1986) (LZ). The CGM estimator
(Cameron et al., 2011) is the sum of LZ estimators on the two different
dimensions minus the LZ estimator of the intersection to avoid double-counting
terms. The CGM2 estimator (Davezies et al., 2018) uses the sum without
subtracting the intersection so terms are double-counted. To be precise,

$\displaystyle\hat{V}_{EHW}$
$\displaystyle:=\frac{1}{N_{k}}\sum_{i=1}^{n_{k}}\hat{\eta}_{ki}^{2}$
$\displaystyle\hat{V}_{LZC}$
$\displaystyle:=\frac{1}{N_{k}}\sum_{i=1}^{n_{k}}\sum_{j\in\mathcal{N}^{C}_{c_{ki}}}\hat{\eta}_{ki}\hat{\eta}_{kj}$
$\displaystyle\hat{V}_{CGM}$
$\displaystyle:=\frac{1}{N_{k}}\sum_{i=1}^{n_{k}}\sum_{j\in\mathcal{N}_{i}}\hat{\eta}_{ki}\hat{\eta}_{kj}=\hat{V}_{LZG}+\hat{V}_{LZH}-\hat{V}_{LZM}$
$\displaystyle\hat{V}_{CGM2}$
$\displaystyle:=\frac{1}{N_{k}}\sum_{i=1}^{n_{k}}\sum_{j\in\mathcal{N}_{i}}\hat{\eta}_{ki}\hat{\eta}_{kj}+\frac{1}{N_{k}}\sum_{i=1}^{n_{k}}\sum_{j\in\mathcal{N}^{M}_{m_{ki}}}\hat{\eta}_{ki}\hat{\eta}_{kj}=\hat{V}_{LZG}+\hat{V}_{LZH}$

where, $\hat{V}_{LZG}$ and $\hat{V}_{LZH}$ denote the LZ estimator on the G
and H dimensions respectively, while $\hat{V}_{LZM}$ is the LZ estimator
treating each intersection $m$ as a single cluster.

######  Assumption 3.

For $C,C^{\prime}\in\\{G,H\\}$, let
$\lambda^{C}_{k}:=\sum_{i=1}^{n_{k}}\sum_{j\in\mathcal{N}^{C}_{c_{ki}}}E[\eta_{ki}\eta_{kj}]$.
Then,
$\frac{1}{\lambda^{C}_{k}}\operatornamewithlimits{max}_{c}(N^{C^{\prime}}_{c})^{2}\rightarrow
0$ and $\frac{1}{\lambda^{C}_{k}}\sum_{c}(N^{C^{\prime}}_{c})^{2}\leq K_{0}$.

The conditions in Assumption 3 are required so that the asymptotic error
incurred by using $\hat{V}$ relative to the true $V$ converges to zero. Since
the strategy for showing such convergence is similar to Yap (2023), an
analogous summability condition and a condition on the largest cluster having
a negligible contribution to the variance are required.

######  Theorem 2.

Under Assumption 1, Assumption 2 and Assumption 3, for $LZC\in\\{LZG,LZH\\}$,

$\displaystyle\frac{\hat{V}_{EHW}}{V_{k}^{EHW}}\xrightarrow{p}1\qquad\frac{\hat{V}_{LZC}}{V_{k}^{LZC}}\xrightarrow{p}1\qquad\frac{\hat{V}_{CGM}}{V_{k}^{CGM}}\xrightarrow{p}1\qquad\frac{\hat{V}_{CGM2}}{V_{k}^{CGM2}}\xrightarrow{p}1$

where

$\displaystyle V_{k}^{EHW}$
$\displaystyle:=\frac{1}{N_{k}}\sum_{i=1}^{n_{k}}E[\eta_{ki}^{2}]$
$\displaystyle V_{k}^{LZC}$
$\displaystyle:=\frac{1}{N_{k}}\sum_{i=1}^{n_{k}}\sum_{j\in\mathcal{N}^{C}_{c_{ki}}}E[\eta_{ki}\eta_{kj}]$
$\displaystyle V_{k}^{CGM}$
$\displaystyle:=\frac{1}{N_{k}}\sum_{i=1}^{n_{k}}\sum_{j\in\mathcal{N}_{i}}E[\eta_{ki}\eta_{kj}]$
$\displaystyle V_{k}^{CGM2}$
$\displaystyle:=\frac{1}{N_{k}}\sum_{i=1}^{n_{k}}\sum_{j\in\mathcal{N}_{i}}E[\eta_{ki}\eta_{kj}]+\frac{1}{N_{k}}\sum_{i=1}^{n_{k}}\sum_{j\in\mathcal{N}^{M}_{m_{ki}}}E[\eta_{ki}\eta_{kj}]$
$\displaystyle E[\eta_{ki}\eta_{kj}]$
$\displaystyle=E[\xi_{ki}\xi_{kj}]+E[\eta_{ki}]E[\eta_{kj}]$

The result allows us to observe differences in variance estimators in
Corollary 1.

######  Corollary 1.

The differences between the limit of the variance estimators and the true
variance are as follows:

$\displaystyle V_{k}^{CGM}-v_{k}$
$\displaystyle=\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{i}}E\left[\eta_{ki}\right]E\left[\eta_{kj}\right]$
$\displaystyle V_{k}^{EHW}-v_{k}$
$\displaystyle=\frac{N_{k}}{n_{k}^{2}}\sum_{i}E\left[\eta_{ki}\right]^{2}-\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{i}\backslash\left\\{i\right\\}}E\left[\xi_{ki}\xi_{kj}\right]$
$\displaystyle V_{k}^{LZG}-v_{k}$
$\displaystyle=-\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{i}\backslash\mathcal{\mathcal{N}}_{g(i)}^{G}}E\left[\xi_{ki}\xi_{kj}\right]+\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{\mathcal{N}}_{g(i)}^{G}}E\left[\eta_{ki}\right]E\left[\eta_{kj}\right]$
$\displaystyle V_{k}^{CGM2}-v_{k}$
$\displaystyle=\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{\mathcal{N}}_{m(i)}^{M}}E\left[\xi_{ki}\xi_{kj}\right]+\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{\mathcal{N}}_{h(i)}^{H}}E\left[\eta_{ki}\right]E\left[\eta_{kj}\right]+\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{\mathcal{N}}_{g(i)}^{G}}E\left[\eta_{ki}\right]E\left[\eta_{kj}\right]$

In general, we cannot guarantee that $V_{k}^{CGM}-v_{k}$ is positive. The
simulation presents one example where this difference is negative.
Nonetheless, observe that, since $u_{ki}(1)-u_{ki}(0)=E[\eta_{ki}]$,

$\displaystyle u_{ki}(1)-u_{ki}(0)$
$\displaystyle=[y_{ki}(1)-(\alpha_{k}+\tau_{k})]-[y_{ki}(0)-\alpha_{k}]$
$\displaystyle=y_{ki}(1)-y_{ki}(0)-\tau_{k}$
$\displaystyle=:\tau_{ki}-\tau_{k}$

Hence, $V_{k}^{CGM}-v_{k}$ will be positive whenever
$\sum_{j\in\mathcal{N}_{i}}(\tau_{ki}-\tau_{k})(\tau_{kj}-\tau_{k})\geq 0$
i.e., when the sum of the correlation in the treatment effects for
observations that are plausibly correlated is positive. This assumption is
implied by the treatment effects of all units who are plausibly correlated
having a positive correlation (i.e.,
$(\tau_{ki}-\tau_{k})(\tau_{kj}-\tau_{k})\geq 0$ for all
$j\in\mathcal{N}_{i}$). However, the requirement for $V_{k}^{CGM}-v_{k}$ is
weaker than this uniform assumption on the correlation, because we can have
some negative correlations — we merely require their overall sum to be weakly
positive. The assumption that
$\sum_{j\in\mathcal{N}_{i}}(\tau_{ki}-\tau_{k})(\tau_{kj}-\tau_{k})\geq 0$ is
reasonable in many empirical situations, as we would usually expect treatment
effects of neighbors to be positively rather than negatively correlated.
However, this assumption cannot be written in terms of $R_{ki}$ and $W_{ki}$
that our design-based setting has restrictions on. Hence, nothing in the
design-based setting necessarily implies that CGM is conservative.

The asymptotic underestimation of the EHW estimator comes from failing to
account for correlations in $\xi$, but it may be compensated by the nonzero
mean of $\eta$. A similar interpretation is obtained in that underestimation
comes from the failure to account for correlations in H that are not part of
the g cluster. This underestimation may be offset by
$\sum_{i}\sum_{j\in\mathcal{\mathcal{N}}_{g(i)}^{G}}E\left[\eta_{ki}\right]E\left[\eta_{kj}\right]\geq
0$, which is weakly positive because the object can be written as a sum of
outer products.

Finally, it can be shown that CGM2 is conservative.

######  Corollary 2.

$V_{k}^{CGM2}-v_{k}\geq 0$.

The CGM2 difference above is guaranteed to be positive, because it can be
written as an additive function of one-way clustered objects. Its
conservativeness arises from double-counting the intersection of clusters, and
the nonzero mean of $\eta$.

Consequently, in contrast to the plug-in (LZ) estimator in one-way clustering
being conservative, the plug-in (CGM) estimator in two-way clustering is no
longer conservative. This leaves researchers with two options: they can either
assume that
$\sum_{j\in\mathcal{N}_{i}}(\tau_{ki}-\tau_{k})(\tau_{kj}-\tau_{k})\geq 0$, or
they can use CGM2.

##  4 Simulations

This section describes and presents results for simulations to illustrate how
the theoretical results perform in practice. The simulations use the following
procedure:

  1. 1.

Construct a population of size $n_{k}$. Each unit in the population is a tuple
of $(g(i),h(i),Y_{ki}(1),Y_{ki}(0))$.

  2. 2.

For every simulation $s$:

    1. (a)

Draw random variables $(R_{ki},W_{ki})$ for every unit and hence construct a
sample of observations with $R_{ki}=1$. For every unit in the sample,
$(g(i),h(i),W_{ki},Y_{ki})$ is observed.

    2. (b)

Calculate $\hat{\tau}$ from the regression.

    3. (c)

Calculate the variance estimates for various procedures and construct a
confidence interval to determine if the CI covers $\tau$.

  3. 3.

Aggregate the coverage over the simulations.

Then, every design modifies some part of the procedure. The following parts
can be modified:

  1. 1.

How clusters $(g,h)$ are assigned in the population.

    1. (a)

Balanced: Use $n_{k}=1e6$, with $1000$ $G$ clusters and $1000$ $H$ clusters,
so there is one observation for every intersection.

    2. (b)

Staircase: Let $k_{1}$ denote an odd number, and $(g,h)$ describe a cluster
intersection that is in cluster $g$ on the $G$ dimension and in cluster $h$ on
the $H$ dimension. In the population, individuals can only belong to cluster
intersections of the form $(k_{1},k_{1})$, $(k_{1},k\pm 1)$ and $(k_{1}\pm
1,k_{1})$. The mass of the population in $(k_{1},k_{1})$ is four times larger
than cluster intersections of the other four forms, which are of equal mass.

  2. 2.

How potential outcomes and hence $\tau_{ki}$ are constructed in the
population. Use $u_{i}\sim N(0,0.1)$ so that $Y_{ki}=\tau_{ki}W_{ki}+u_{i}$.

    1. (a)

Gvar: $\tau_{ki}=\tau_{g(i)}+\tau_{h(i)}$, where $\tau_{g}=\pm 2$ with equal
probability and $\tau_{h}=\pm 1/2$ with equal probability.

    2. (b)

Hvar: $\tau_{ki}=\tau_{g(i)}+\tau_{h(i)}$, where $\tau_{h}=\pm 2$ with equal
probability and $\tau_{g}=\pm 1/2$ with equal probability.

    3. (c)

same: $\tau_{ki}=\tau_{g(i)}+\tau_{h(i)}$, where $\tau_{g}=\pm 1$ with equal
probability and $\tau_{h}=\pm 1$ with equal probability.

    4. (d)

constant: $\tau_{ki}=1$.

    5. (e)

oddeven: $\tau_{ki}=1$ if $g(i)$ and $h(i)$ are both odd, and $-1$ otherwise.

  3. 3.

How $R_{ki}$’s are generated for every sample.

    1. (a)

$p_{k}$ individual sampling probability

    2. (b)

$q_{k}$ cluster sampling probability for $G$

    3. (c)

Multiway Sampling: The intersection $(g,h)$ is sampled when $A^{sam}_{kg}=1$
and $B^{sam}_{kh}=1$, each independently drawn from $Be(0.25)$. Then,
$p_{k}=0.25$.

  4. 4.

How $W_{ki}$’s are generated for every sample.

    1. (a)

Hway: Assignment probability $B_{kh}$ for every cluster is drawn from
$U[0,1]$. $W_{ki}=1$ with probability $B_{kh(i)}$.

    2. (b)

none: $W_{ki}=1$ with probability 1/2 so $W_{ki}$ is iid.

    3. (c)

AND: $A_{kg}\in\\{0,1\\}$ and $B_{kh}\in\\{0,1\\}$ are drawn independently,
and each takes the value 1 with probability $1/\sqrt{2}$. Then,
$W_{ki}=A_{kg(i)}B_{kh(i)}$.

The staircase design for constructing the clusters is artificially designed so
that $V_{CGM,k}-v_{k}\leq 0$. Details are in Appendix A. I use $nsim=5000$
simulation draws and report eight informative designs.

Table 1: Comparison of Standard Methods for Different Designs

| EHWCov | LZGCov | LZHCov | CGMCov | CGM2Cov | EHWVar | LZGVar | LZHVar | CGMVar | CGM2Var
---|---|---|---|---|---|---|---|---|---|---
D1 | 0.7736 | 0.9880 | 0.9884 | 0.9990 | 0.9994 | 4e-04 | 0.0018 | 0.0018 | 0.0032 | 0.0036
D2 | 0.7836 | 0.8518 | 0.9986 | 0.9994 | 0.9996 | 8e-04 | 0.0011 | 0.0064 | 0.0067 | 0.0076
D3 | 0.3258 | 0.8802 | 0.8790 | 0.9680 | 0.9706 | 4e-04 | 0.0054 | 0.0054 | 0.0103 | 0.0107
D4 | 0.2542 | 0.9200 | 0.9336 | 0.9874 | 0.9882 | 2e-04 | 0.0050 | 0.0054 | 0.0103 | 0.0104
D5 | 0.9562 | 0.9502 | 0.9540 | 0.9476 | 0.9946 | 0e+00 | 0.0000 | 0.0000 | 0.0000 | 0.0000
D6 | 0.3000 | 0.9608 | 0.9898 | 0.9988 | 0.9990 | 1e-04 | 0.0025 | 0.0040 | 0.0065 | 0.0066
D7 | 0.9902 | 1.0000 | 0.9966 | 1.0000 | 1.0000 | 9e-04 | 0.0048 | 0.0012 | 0.0051 | 0.0060
D8 | 0.2434 | 0.9568 | 0.9556 | 0.9396 | 0.9950 | 3e-04 | 0.0143 | 0.0144 | 0.0125 | 0.0287

| (g,h) | $\tau$ | R | W
---|---|---|---|---
D1 | balanced | same | $q_{k}=1,p_{k}=1$ | AND
D2 | balanced | Hvar | $q_{k}=1,p_{k}=1$ | AND
D3 | balanced | same | multiway | none
D4 | balanced | Hvar | $q_{k}=0.05,p_{k}=1$ | Hway
D5 | balanced | constant | $q_{k}=1,p_{k}=1$ | AND
D6 | balanced | Hvar | $q_{k}=0.1,p_{k}=1$ | none
D7 | balanced | Gvar | $q_{k}=1,p_{k}=1$ | Hway
D8 | staircase | oddeven | multiway | none

Table 1 reports the results for various designs (D). D1 shows that there is
massive over-coverage in two-way assignment. However, by virtue of clustered
assignment, if we used the EHW standard errors, we would not have correct
coverage. D2 shows how $\tau$ matters. Even though there is multiway
assignment, because there is more variation in $\tau$ on the $H$ dimension, it
suffices to cluster on $H$. D3 shows that when there is multiway sampling, we
need to cluster on both dimensions to have valid confidence intervals — doing
so on just one dimension will under-cover. D4 reflects some necessity for
considering sampling and assignment. When sampling and assignment occur on
different clustering dimensions, if we had just clustered on either one, we
would not have 0.95 coverage, so we need to cluster on both.

D5 is an example when CGM yields correct coverage, namely, when $\tau$ is
constant. D6 shows that when there is clustered sampling on $G$ but no
clustered assignment, it suffices to cluster on $G$. If we had clustered on
$H$ too, the variance would have been twice as large. D7 shows that there is
no need to cluster on $G$ when there is clustered assignment only on $H$. The
variance is four times as large when clustering on the unnecessary dimension.
D8 provides the counterexample where CGM under-covers when there is clustered
sampling. The details of this construction is given in Appendix A.

Overall, the simulations illustrate the theoretical results that sampling and
assignment matter. We can get exact coverage with multiway sampling or when
treatment effects are constant. CGM can under-cover in multi-way sampling,
while CGM2 is always conservative. In fact, the conservativeness of CGM2 can
be quite substantial: CGM2 variance is five times what is required for valid
inference in D7, for instance. While D8 shows how CGM may be anticonservative,
it is usually possible to rule out such designs by institutional details or
some economic model.

Design-based inference fundamentally relies on the institutional knowledge of
how sampling and assignment occur. This knowledge then informs what dimensions
the researcher should cluster on, if at all. When moving from one-way
clustering to multi-way clustering, this paper has shown how such demands on
institutional knowledge still apply. To use the plug-in variance estimator in
multi-way settings, however, researchers need to make an additional assumption
on the correlation of treatment effects within clusters. Without such an
assumption, valid inference may require a variance estimator that is
unnecessarily conservative.

##  Appendix A CGM Counterexample

This section theoretically constructs a counterexample such that the CGM
estimator is anticonservative. Due to Corollary 1,
$V_{k}^{CGM}-v_{k}=(N_{k}/n_{k}^{2})\sum_{i}\sum_{j\in\mathcal{N}_{i}}E[\eta_{ki}]E[\eta_{kj}]$,
and we have shown that $E[\eta_{ki}]=\tau_{ki}-\tau_{k}$. Hence, for a
counterexample where $V_{k}^{CGM}-v_{k}<0$, it suffices that
$\sum_{i}\sum_{j\in\mathcal{N}_{i}}(\tau_{ki}-\tau_{k})(\tau_{kj}-\tau_{k})<0$.

Let $k_{1}$ denote an odd number, and $(g,h)$ describe a cluster intersection
that is in cluster $g$ on the $G$ dimension and in cluster $h$ on the $H$
dimension. Suppose there are $M$ $G$ clusters and $M$ $H$ clusters, where $M$
is even. Let $M_{0}$ be a fixed number. In the population, individuals can
only belong to cluster intersections of the form $(k_{1},k_{1})$, $(k_{1},k\pm
1)$ and $(k_{1}\pm 1,k_{1})$. This assumption implies that there cannot be
individuals in cluster intersections where both cluster indexes are even, or
if their difference is more than one. There are $4M_{0}$ observations in
$(k_{1},k_{1})$ clusters and $M_{0}$ observations in other cluster
intersections that are nonempty. Hence, the total number of observations is
$n_{k}=4M_{0}M$.

Table 2: Distribution of population with bounded cluster sizes Type | Proportion | (g,h) | $\tau_{ki}-\tau_{k}$
---|---|---|---
1 | 1/2 | $(k_{1},k_{1})$ | 1
2 | 1/4 | $(k_{1},k_{1}\pm 1)$ | -1
3 | 1/4 | $(k_{1}\pm 1,k_{1})$ | -1

The $\tau_{ki}-\tau_{k}$ values for observations belonging to the various
cluster intersections are given by Table 2. Then, I show how this particular
construction results in
$\sum_{i}\sum_{j\in\mathcal{N}_{i}}(\tau_{ki}-\tau_{k})(\tau_{kj}-\tau_{k})<0$.

First, consider an observation $i$ in $(k_{1},k_{1})$ — they account for half
the observations. Then, $|\mathcal{N}_{i}|=8M_{0}$, where $4M_{0}$ of those
observations are in $(k_{1},k_{1})$ and the remaining $4M_{0}$ are either in
$(k_{1}\pm 1,k_{1})$ or $(k_{1},k_{1}\pm 1)$. Hence,
$\sum_{j\in\mathcal{N}_{i}}(\tau_{ki}-\tau_{k})(\tau_{kj}-\tau_{k})=(1)(4-1-1-1-1)=0$.

Next, consider an observation $i$ in $(k_{1},k_{1}\pm 1)$. The treatment of
$(k_{1}\pm 1,k_{1})$ is identical. The units in either $(k_{1},k_{1}\pm 1)$ or
$(k_{1}\pm 1,k_{1})$ account for the other half of the units. Here,
$|\mathcal{N}_{i}|=7M_{0}$. For instance, for some $(k_{1},k_{1}+1)$, $4M_{0}$
of those are in $(k_{1},k_{1})$ intersections, $M_{0}$ are in
$(k_{1},k_{1}+1)$, $M_{0}$ are in $(k_{1},k_{1}-1)$ and $M_{0}$ are in
$(k_{1}+2,k_{1}+1)$. Then,
$\sum_{j\in\mathcal{N}_{i}}(\tau_{ki}-\tau_{k})(\tau_{kj}-\tau_{k})=(1)(4-1-1-1)=-1$.

Combining these results,

$\displaystyle\frac{1}{N}\sum_{i}\sum_{j\in\mathcal{N}_{i}}(\tau_{ki}-\tau_{k})(\tau_{kj}-\tau_{k})$
$\displaystyle=\frac{1}{2}(1)+\frac{1}{2}(-1)=-1/2<0$

##  Appendix B Proofs

###  B.1 Proofs for Section 3.1

###### Proof of Theorem 1.

Observe that $\sum_{i=1}^{n_{k}}u_{ki}(w)=0$. Hence,

$\displaystyle\hat{\tau}_{k}-\tau_{k}$
$\displaystyle=\frac{1}{N_{k1}}\sum_{i=1}^{n_{k}}R_{ki}W_{ki}Y_{ki}-\frac{1}{N_{k0}}\sum_{i=1}^{n_{k}}R_{ki}\left(1-W_{ki}\right)Y_{ki}-\frac{1}{n_{k}}\sum_{i=1}^{n_{k}}\left(y_{ki}(1)-y_{ki}(0)\right)$
$\displaystyle=\frac{1}{N_{k1}}\sum_{i=1}^{n_{k}}R_{ki}W_{ki}y_{ki}(1)-\frac{1}{N_{k0}}\sum_{i=1}^{n_{k}}R_{ki}\left(1-W_{ki}\right)y_{ki}(0)-\frac{1}{n_{k}}\sum_{i=1}^{n_{k}}\left(y_{ki}(1)-y_{ki}(0)\right)$
$\displaystyle=\frac{1}{N_{k1}}\sum_{i=1}^{n_{k}}R_{ki}W_{ki}u_{ki}(1)-\frac{1}{N_{k0}}\sum_{i=1}^{n_{k}}R_{ki}\left(1-W_{ki}\right)u_{ki}(0)$
$\displaystyle=\frac{1}{N_{k1}}\sum_{i=1}^{n_{k}}R_{ki}W_{ki}u_{ki}(1)-\frac{b_{k1}}{N_{k1}}\sum_{i=1}^{n_{k}}u_{ki}(1)-\frac{1}{N_{k0}}\sum_{i=1}^{n_{k}}R_{ki}\left(1-W_{ki}\right)u_{ki}(0)+\frac{b_{k0}}{N_{k0}}\sum_{i=1}^{n_{k}}u_{ki}(0)$
$\displaystyle=\frac{1}{N_{k1}}\sum_{i=1}^{n_{k}}\left(R_{ki}W_{ki}-b_{k1}\right)u_{ki}(1)-\frac{1}{N_{k0}}\sum_{i=1}^{n_{k}}\left(R_{ki}\left(1-W_{ki}\right)-b_{k0}\right)u_{ki}(0)$
$\displaystyle=\frac{n_{k}}{N_{k1}}\frac{E\left[N_{k1}\right]}{n_{k}}\frac{1}{b_{k1}n_{k}}\sum_{i=1}^{n_{k}}\left(R_{ki}W_{ki}-b_{k1}\right)u_{ki}(1)-\frac{n_{k}}{N_{k0}}\frac{E\left[N_{k0}\right]}{n_{k}}\frac{1}{b_{k0}n_{k}}\sum_{i=1}^{n_{k}}\left(R_{ki}\left(1-W_{ki}\right)-b_{k0}\right)u_{ki}(0)$
$\displaystyle=\frac{b_{k1}}{\hat{b}_{k1}}\hat{a}_{k1}-\frac{b_{k0}}{\hat{b}_{k0}}\hat{a}_{k0}$

where

$\displaystyle\hat{b}_{k1}$
$\displaystyle=\frac{N_{k1}}{n_{k}}\qquad\hat{b}_{k0}=\frac{N_{k0}}{n_{k}}$
$\displaystyle\hat{a}_{k1}$
$\displaystyle=\frac{1}{b_{k1}n_{k}}\sum_{i=1}^{n_{k}}\left(R_{ki}W_{ki}-b_{k1}\right)u_{ki}(1)$
$\displaystyle\hat{a}_{k0}$
$\displaystyle=\frac{1}{b_{k0}n_{k}}\sum_{i=1}^{n_{k}}\left(R_{ki}\left(1-W_{ki}\right)-b_{k0}\right)u_{ki}(0)$

Since $R_{ki},W_{ki}$ are multiway clustered and well-behaved, using the law
of large numbers implied by Yap (2023), it is immediate that
$\frac{b_{k1}}{\hat{b}_{k1}}=1+o_{P}(1)$ and
$\frac{b_{k0}}{\hat{b}_{k0}}=1+o_{P}(1)$. Then,

$\displaystyle\hat{\tau}_{k}-\tau_{k}$
$\displaystyle=\left(1+o_{P}(1)\right)\hat{a}_{k1}-\left(1+o_{P}(1)\right)\hat{a}_{k0}$
$\displaystyle=\hat{a}_{k1}-\hat{a}_{k0}+o_{P}(1)\left(\hat{a}_{k1}-\hat{a}_{k0}\right)$

It is sufficient to show that
$(\sqrt{N_{k}}/\sqrt{v_{k}})(\hat{a}_{k1}-\hat{a}_{k0})\xrightarrow{d}N(0,1)$.
If that result holds, then

$\displaystyle\frac{\sqrt{N_{k}}}{\sqrt{v_{k}}}(\hat{\tau}_{k}-\tau_{k})$
$\displaystyle=\frac{\sqrt{N_{k}}}{\sqrt{v_{k}}}(\hat{a}_{k1}-\hat{a}_{k0})+o_{P}(1)\left(\hat{a}_{k1}-\hat{a}_{k0}\right)\frac{\sqrt{N_{k}}}{\sqrt{v_{k}}}$
$\displaystyle=\frac{\sqrt{N_{k}}}{\sqrt{v_{k}}}(\hat{a}_{k1}-\hat{a}_{k0})+o_{P}(1)O_{P}(1)\xrightarrow{d}N(0,1)$

Hence,

$\displaystyle\hat{a}_{k1}-\hat{a}_{k0}$
$\displaystyle=\frac{1}{n_{k}}\sum_{i=1}^{n_{k}}\left(\frac{1}{b_{k1}}\left(R_{ki}W_{ki}-b_{k1}\right)u_{ki}(1)-\frac{1}{b_{k0}}\left(R_{ki}\left(1-W_{ki}\right)-b_{k0}\right)u_{ki}(0)\right)$
$\displaystyle=\frac{1}{n_{k}}\sum_{i=1}^{n_{k}}\xi_{ki}=\hat{\tau}_{k}-\tau_{k}$

It is obvious that $E[\xi_{ki}]=0$. Since the conditions of the central limit
theorem (CLT) of Yap (2023) are satisfied,

$\displaystyle\frac{\sum_{i=1}^{n_{k}}\xi_{ki}}{\sqrt{\sum_{i=1}^{n_{k}}\sum_{j\in\mathcal{N}_{i}}E[\xi_{ki}\xi_{kj}]}}\xrightarrow{d}N(0,1)$

The expression for $v_{k}$ is obtained by solving:

$\displaystyle\frac{1}{n_{k}^{2}}\sum_{i=1}^{n_{k}}\sum_{j\in\mathcal{N}_{i}}E[\xi_{ki}\xi_{kj}]=\frac{v_{k}}{N_{k}}$

Finally, it remains to find an expression for $E[\xi_{ki}\xi_{kj}]$.

$\displaystyle E\left[\xi_{ki}\xi_{kj}\right]$
$\displaystyle=E[\left(\frac{1}{b_{k1}}\left(R_{ki}W_{ki}-b_{k1}\right)u_{ki}(1)-\frac{1}{b_{k0}}\left(R_{ki}\left(1-W_{ki}\right)-b_{k0}\right)u_{ki}(0)\right)$
$\displaystyle\qquad\left(\frac{1}{b_{k1}}\left(R_{kj}W_{kj}-b_{k1}\right)u_{kj}(1)-\frac{1}{b_{k0}}\left(R_{kj}\left(1-W_{kj}\right)-b_{k0}\right)u_{kj}(0)\right)]$
$\displaystyle=E\left[\frac{1}{b_{k1}^{2}}\left(R_{ki}W_{ki}-b_{k1}\right)u_{ki}(1)\left(R_{kj}W_{kj}-b_{k1}\right)u_{kj}(1)-\frac{1}{b_{k1}b_{k0}}\left(R_{ki}W_{ki}-b_{k1}\right)u_{ki}(1)\left(R_{kj}\left(1-W_{kj}\right)-b_{k0}\right)u_{kj}(0)\right]$
$\displaystyle\qquad+E\left[\frac{1}{b_{k0}^{2}}\left(R_{ki}\left(1-W_{ki}\right)-b_{k0}\right)u_{ki}(0)\left(R_{kj}\left(1-W_{kj}\right)-b_{k0}\right)u_{kj}(0)\right]$
$\displaystyle\qquad-E\left[\frac{1}{b_{k1}b_{k0}}\left(R_{kj}W_{kj}-b_{k1}\right)u_{kj}(1)u_{ki}(1)\left(R_{ki}\left(1-W_{ki}\right)-b_{k0}\right)u_{ki}(0)\right]$

Expand the individual terms.

$\displaystyle E$
$\displaystyle\left[\frac{1}{b_{k1}^{2}}\left(R_{ki}W_{ki}-b_{k1}\right)u_{ki}(1)\left(R_{kj}W_{kj}-b_{k1}\right)u_{kj}(1)\right]$
$\displaystyle=E\left[\frac{1}{b_{k1}^{2}}\left(R_{ki}W_{ki}R_{kj}W_{kj}-b_{k1}R_{kj}W_{kj}-b_{k1}R_{ki}W_{ki}+b_{k1}^{2}\right)u_{ki}(1)u_{kj}(1)\right]$
$\displaystyle=E\left[\frac{1}{b_{k1}^{2}}\left(R_{ki}W_{ki}R_{kj}W_{kj}\right)u_{ki}(1)u_{kj}(1)\right]-2\frac{E\left[R_{ki}W_{ki}\right]b_{k1}}{b_{k1}^{2}}u_{ki}(1)u_{kj}(1)+u_{ki}(1)u_{kj}(1)$
$\displaystyle=\frac{1}{b_{k1}^{2}}E\left[R_{ki}W_{ki}R_{kj}W_{kj}\right]u_{ki}(1)u_{kj}(1)-u_{ki}(1)u_{kj}(1)$
$\displaystyle=\left(\frac{1}{b_{k1}^{2}}E\left[R_{ki}W_{ki}R_{kj}W_{kj}\right]-1\right)u_{ki}(1)u_{kj}(1)$

Since sampling and assignment are independent,

$E\left[R_{ki}W_{ki}R_{kj}W_{kj}\right]=E\left[R_{ki}R_{kj}\right]E\left[W_{ki}W_{kj}\right]$

The remaining terms are:

$\displaystyle E$
$\displaystyle\left[\frac{1}{b_{k1}b_{k0}}\left(R_{ki}W_{ki}-b_{k1}\right)u_{ki}(1)\left(R_{kj}\left(1-W_{kj}\right)-b_{k0}\right)u_{kj}(0)\right]$
$\displaystyle=\frac{1}{b_{k1}b_{k0}}E\left[R_{ki}W_{ki}R_{kj}\left(1-W_{kj}\right)-b_{k1}b_{k0}\right]u_{ki}(1)u_{kj}(0)$
$\displaystyle=\left(\frac{1}{b_{k1}b_{k0}}E\left[R_{ki}W_{ki}R_{kj}\left(1-W_{kj}\right)\right]-1\right)u_{ki}(1)u_{kj}(0)$

$\displaystyle E$
$\displaystyle\left[\frac{1}{b_{k0}^{2}}\left(R_{ki}\left(1-W_{ki}\right)-b_{k0}\right)u_{ki}(0)\left(R_{kj}\left(1-W_{kj}\right)-b_{k0}\right)u_{kj}(0)\right]$
$\displaystyle=\frac{1}{b_{k0}^{2}}E\left[\left(R_{ki}\left(1-W_{ki}\right)-b_{k0}\right)\left(R_{kj}\left(1-W_{kj}\right)-b_{k0}\right)\right]u_{ki}(0)u_{kj}(0)$
$\displaystyle=\frac{1}{b_{k0}^{2}}E\left[\left(R_{ki}\left(1-W_{ki}\right)R_{kj}\left(1-W_{kj}\right)\right)-b_{k0}^{2}\right]u_{ki}(0)u_{kj}(0)$
$\displaystyle=\left(\frac{1}{b_{k0}^{2}}E\left[R_{ki}\left(1-W_{ki}\right)R_{kj}\left(1-W_{kj}\right)\right]-1\right)u_{ki}(0)u_{kj}(0)$

with

$\displaystyle E\left[R_{ki}W_{ki}R_{kj}\left(1-W_{kj}\right)\right]$
$\displaystyle=E\left[R_{ki}R_{kj}\right]E\left[W_{ki}\left(1-W_{kj}\right)\right]$
$\displaystyle
E\left[R_{ki}\left(1-W_{ki}\right)R_{kj}\left(1-W_{kj}\right)\right]$
$\displaystyle=E\left[R_{ki}R_{kj}\right]E\left[\left(1-W_{ki}\right)\left(1-W_{kj}\right)\right]$

The result is then obtained by plugging in these expressions. ∎

###### Proof of Lemma 1.

If $i$ and $j$ do not share a cluster on the respective dimensions,
$E\left[R_{ki}R_{kj}\right]=E[R_{ki}]E[R_{kj}]=p_{k}^{2}q_{k}^{2}$, and
$E\left[W_{ki}W_{kj}\right]=\mu^{2}$. If they share the assignment dimension,
then

$\displaystyle E\left[W_{ki}W_{kj}\right]$
$\displaystyle=E\left[E\left[W_{ki}W_{kj}|A_{h(i)}\right]\right]$
$\displaystyle=E\left[A_{h}^{2}\right]$ $\displaystyle=\sigma^{2}+\mu^{2}$

All other derivations are analogous. ∎

###### Proof of Lemma 2.

$\displaystyle E\left[W_{ki}W_{kj}\right]$
$\displaystyle=E\left[E\left[W_{ki}W_{kj}|A_{g},B_{h}\right]\right]$
$\displaystyle=E\left[A_{g}^{2}B_{h}^{2}\right]$
$\displaystyle=E\left[A_{g}^{2}\right]E\left[B_{h}^{2}\right]$
$\displaystyle=\left(\mu_{A}^{2}+\sigma_{A}^{2}\right)\left(\mu_{B}^{2}+\sigma_{B}^{2}\right)$

Further,

$\displaystyle E\left[W_{ki}W_{kj}\right]$
$\displaystyle=E\left[E\left[W_{ki}W_{kj}|A_{g},B_{h},B_{h^{\prime}},g(i)=g(j)=g,h(i)=h,h(j)=h^{\prime}\right]\right]$
$\displaystyle=E\left[A_{g}^{2}B_{h}B_{h^{\prime}}\right]$
$\displaystyle=E\left[A_{g}^{2}\right]E\left[B_{h}\right]^{2}$
$\displaystyle=\left(\mu_{A}^{2}+\sigma_{A}^{2}\right)\mu_{B}^{2}$

∎

###  B.2 Proofs for Section 3.2

###### Proof of Theorem 2.

Due to Theorem 1, and applying a similar argument to show that
$\hat{\alpha}_{k}/{\alpha_{k}}\xrightarrow{p}1$,
$\hat{U}_{ki}=U_{ki}(1+o_{P}(1))$. Hence,
$\hat{\eta}_{ki}=\eta_{ki}(1+o_{P}(1))$, so it suffices to show the result for
$\eta_{ki}$ directly.

First consider the EHW result. We want to show that:

$\displaystyle P$
$\displaystyle\left(\frac{1}{\sum_{i}E[\eta_{ki}^{2}]}\sum_{i}\left(\eta_{ki}^{2}-E\left[\eta_{ki}^{2}\right]\right)>\epsilon\right)\leq\frac{1}{\epsilon^{2}}\frac{1}{\left(\sum_{i}E[\eta_{ki}^{2}]\right)^{2}}E\left[\left(\sum_{i}\left(\eta_{ki}^{2}-E\left[\eta_{ki}^{2}\right]\right)\right)^{2}\right]$
$\displaystyle=\frac{1}{\epsilon^{2}}\frac{1}{\left(\sum_{i}E[\eta_{ki}^{2}]\right)^{2}}\sum_{i}\sum_{j}\left(E\left[\eta_{ki}^{2}\eta_{kj}^{2}\right]-E\left[\eta_{ki}^{2}\right]E\left[\eta_{kj}^{2}\right]\right)\rightarrow
0$

Let $D_{ij}$ denote dependency between $i$ and $j$, so these units are
correlated. Under our regularity conditions, for some arbitrary constant $K$,
the probability above is:

$\displaystyle\frac{K}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{i}}\left(E\left[\eta_{ki}^{2}\eta_{kj}^{2}\right]-E\left[\eta_{ki}^{2}\right]E\left[\eta_{kj}^{2}\right]\right)$
$\displaystyle=\frac{K}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{i}}D_{ij}$
$\displaystyle\leq\frac{K}{n_{k}^{2}}\left(\sum_{g}(N^{G}_{g})^{2}+\sum_{h}(N^{H}_{h})^{2}\right)$

To obtain convergence, simply observe that:

$\displaystyle\frac{1}{n_{k}^{2}}\sum_{g}(N^{G}_{g})^{2}\leq\frac{\operatornamewithlimits{max}_{g}N^{G}_{g}}{n_{k}}\sum_{g}(N^{G}_{g})\frac{1}{n_{k}^{3/2}}=o(1)\frac{n_{k}}{n_{k}}=o(1)$

Turning to the LZ estimator, we have
$\lambda_{k}^{G}=\sum_{i}\sum_{j\in\mathcal{N}^{G}_{g_{ki}}}E[\eta_{ki}\eta_{kj}]$.
We want to show that the following is $o_{P}(1)$, even under multi-way
correlation.

$\displaystyle P$
$\displaystyle\left(\frac{1}{\lambda_{n}^{G}}\sum_{i}\sum_{j\in\mathcal{N}_{g(i)}^{G}}\left(\eta_{ki}\eta_{kj}-E\left[\eta_{ki}\eta_{kj}\right]\right)>\epsilon\right)\leq\frac{1}{\epsilon^{2}}\frac{1}{\left(\lambda_{n}^{G}\right)^{2}}E\left[\left(\sum_{i}\sum_{j\in\mathcal{N}_{g(i)}^{G}}\left(\eta_{ki}\eta_{kj}-E\left[\eta_{ki}\eta_{kj}\right]\right)\right)^{2}\right]$
$\displaystyle=\frac{1}{\epsilon^{2}}\frac{1}{\left(\lambda_{n}^{G}\right)^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{g(i)}^{G}}\sum_{s}\sum_{l\in\mathcal{N}_{g(k)}^{G}}\left(E\left[\eta_{ki}\eta_{kj}\eta_{ks}\eta_{kl}\right]-E\left[\eta_{ki}\eta_{kj}\right]E\left[\eta_{ks}\eta_{kl}\right]\right)$

$E\left[\eta_{ki}\eta_{kj}\eta_{ks}\eta_{kl}\right]=E\left[\eta_{ki}\eta_{kj}\right]E\left[\eta_{ks}\eta_{kl}\right]$
whenever $(i,j)$ and $(s,l)$ do not share any cluster, even if $i,j$ share a
cluster with each other. Hence, we will need to count the moment when
$D_{is}=1$ or $D_{il}=1$ or $D_{js}=1$ or $D_{jl}=1$. Since moments are
bounded by Assumption 2,

$P\left(\frac{1}{\lambda_{n}^{G}}\sum_{i}\sum_{j\in\mathcal{N}_{g(i)}^{G}}\left(\eta_{ki}\eta_{kj}-E\left[\eta_{ki}\eta_{kj}\right]\right)>\epsilon\right)\leq\frac{K_{0}}{\epsilon^{2}}\frac{1}{\left(\lambda_{n}^{G}\right)^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{g(i)}^{G}}\sum_{s}\sum_{l\in\mathcal{N}_{g(s)}^{G}}\left(D_{is}+D_{il}+D_{js}+D_{jl}\right)$

It suffices to consider the sum on $D_{is}$ because everything else is
analogous.

$\displaystyle\frac{1}{\left(\lambda_{n}^{G}\right)^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{g(i)}^{G}}\sum_{s}\sum_{l\in\mathcal{N}_{g(s)}^{G}}D_{is}$
$\displaystyle\leq\frac{1}{\left(\lambda_{n}^{G}\right)^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{g(i)}^{G}}\left(\sum_{s\in\mathcal{N}_{g(i)}^{G}}+\sum_{s\in\mathcal{N}_{h(i)}^{H}}\right)\sum_{l\in\mathcal{N}_{g(s)}^{G}}D_{is}$
$\displaystyle\leq\frac{1}{\left(\lambda_{n}^{G}\right)^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{g(i)}^{G}}\left(\sum_{s\in\mathcal{N}_{g(i)}^{G}}+\sum_{s\in\mathcal{N}_{h(i)}^{H}}\right)\operatornamewithlimits{max}_{g}N_{g}^{G}$
$\displaystyle=\frac{1}{\left(\lambda_{n}^{G}\right)^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{g(i)}^{G}}\left(N_{g(i)}^{G}+N_{h(i)}^{H}\right)\operatornamewithlimits{max}_{g}N_{g}^{G}$

Taking the first term,

$\displaystyle\frac{1}{\left(\lambda_{n}^{G}\right)^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{g(i)}^{G}}N_{g(i)}^{G}\operatornamewithlimits{max}_{g}N_{g}^{G}$
$\displaystyle=\frac{\operatornamewithlimits{max}_{g}N_{g}^{G}}{\left(\lambda_{n}^{G}\right)^{2}}\sum_{i}\left(N_{g(i)}^{G}\right)^{2}$
$\displaystyle=\frac{\operatornamewithlimits{max}_{g}N_{g}^{G}}{\left(\lambda_{n}^{G}\right)^{2}}\sum_{g}\left(N_{g}^{G}\right)^{3}$

Under Assumption 3,
$\frac{\operatornamewithlimits{max}_{g}\left(N_{g}^{G}\right)^{2}}{\lambda_{n}^{G}}\rightarrow
0$, and $\frac{1}{\lambda_{n}}\sum_{g}\left(N_{g}^{G}\right)^{2}$ is finite,
so

$\frac{\operatornamewithlimits{max}_{g}N_{g}^{G}}{\left(\lambda_{n}^{G}\right)^{2}}\sum_{g}\left(N_{g}^{G}\right)^{3}\leq\frac{\operatornamewithlimits{max}_{g}\left(N_{g}^{G}\right)^{2}}{\lambda_{n}^{G}}\frac{1}{\lambda_{n}^{G}}\sum_{g}\left(N_{g}^{G}\right)^{2}\rightarrow
0$

For the other term,

$\displaystyle\frac{1}{\left(\lambda_{n}^{G}\right)^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{g(i)}^{G}}N_{h(i)}^{H}\operatornamewithlimits{max}_{g}N_{g}^{G}$
$\displaystyle=\frac{\operatornamewithlimits{max}_{g}N_{g}^{G}}{\left(\lambda_{n}^{G}\right)^{2}}\sum_{i}N_{g(i)}^{G}N_{h(i)}^{H}$
$\displaystyle\leq\frac{\operatornamewithlimits{max}_{g}N_{g}^{G}\operatornamewithlimits{max}_{h}N_{h}^{H}}{\lambda_{n}^{G}}\frac{1}{\lambda_{n}^{G}}\sum_{i}N_{g(i)}^{G}$
$\displaystyle=\frac{\operatornamewithlimits{max}_{g}N_{g}^{G}\operatornamewithlimits{max}_{h}N_{h}^{H}}{\lambda_{n}^{G}}\frac{1}{\lambda_{n}^{G}}\sum_{g}\left(N_{g}^{G}\right)^{2}\rightarrow
0$

Convergence in the last line occurs due to
$\operatornamewithlimits{max}_{g}N^{G}_{g}/(\lambda_{n}^{G})^{1/2}=(\operatornamewithlimits{max}_{g}(N^{G}_{g})^{2}/\lambda_{n}^{G})^{1/2}=o(1)$.

A similar argument works for the CGM estimator. We now want to show that the
following is $o(1)$:

$\displaystyle\frac{K_{0}}{\epsilon^{2}}\frac{1}{\left(\sum_{i}\sum_{j\in\mathcal{N}_{i}}E[\eta_{ki}\eta_{kj}]\right)^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{i}}\sum_{r}\sum_{l\in\mathcal{N}_{s}}\left(D_{is}+D_{il}+D_{js}+D_{jl}\right)$

As before, it suffices to consider
$\sum_{i}\sum_{j\in\mathcal{N}_{i}}\sum_{s}\sum_{l\in\mathcal{N}_{r}}D_{is}$.
Let $\lambda_{k}:=\sum_{i}\sum_{j\in\mathcal{N}_{i}}E[\eta_{ki}\eta_{kj}]$.

$\displaystyle\frac{1}{\lambda_{k}^{2}}$
$\displaystyle\sum_{i}\sum_{j\in\mathcal{N}_{i}}\sum_{s}\sum_{l\in\mathcal{N}_{r}}D_{is}\leq\frac{1}{\lambda_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{i}}\left(N_{g(i)}^{G}+N_{h(i)}^{H}\right)\left(\operatornamewithlimits{max}_{g}N_{g}^{G}+\operatornamewithlimits{max}_{h}N_{h}^{H}\right)$
$\displaystyle\leq\frac{1}{\lambda_{k}^{2}}\sum_{i}\left(N_{g(i)}^{G}+N_{h(i)}^{H}\right)^{2}\left(\operatornamewithlimits{max}_{g}N_{g}^{G}+\operatornamewithlimits{max}_{h}N_{h}^{H}\right)$

It suffices to consider the leading term as the other terms are analogous. Due
to Assumption 2,

$\displaystyle\frac{\operatornamewithlimits{max}_{g}N_{g}^{G}}{\lambda_{k}^{2}}\sum_{i}(N_{g(i)}^{G})^{2}=\frac{\operatornamewithlimits{max}_{g}(N_{g}^{G})^{2}}{\lambda_{k}}\frac{1}{\lambda_{k}}\sum_{g}(N_{g}^{G})^{2}=o(1)O(1)=o(1)$

Hence, we can get asymptotic convergence to the expectation.

Since the CGM2 estimator is built from LZ estimators on the G and H
dimensions, the established convergence of the LZ estimators implies the
convergence of CGM2. ∎

###### Proof of Corollary 1.

Observe that:

$\displaystyle V_{k}^{CGM}-v_{k}$
$\displaystyle=\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{i}}E\left[\eta_{ki}\right]E\left[\eta_{kj}\right]$
$\displaystyle=\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{i}}\left(u_{ki}(1)-u_{ki}(0)\right)\left(u_{kj}(1)-u_{kj}(0)\right)$
$\displaystyle V_{k}^{EHW}-v_{k}$
$\displaystyle=\frac{N_{k}}{n_{k}^{2}}\sum_{i}E\left[\eta_{ki}^{2}\right]+\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{i}}E\left[\xi_{ki}\xi_{kj}\right]$
$\displaystyle=\frac{N_{k}}{n_{k}^{2}}\sum_{i}E\left[\xi_{ki}^{2}\right]+\frac{N_{k}}{n_{k}^{2}}\sum_{i}E\left[\eta_{ki}\right]^{2}-\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{i}}E\left[\xi_{ki}\xi_{kj}\right]$
$\displaystyle=\frac{N_{k}}{n_{k}^{2}}\sum_{i}E\left[\eta_{ki}\right]^{2}-\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{i}\backslash\left\\{i\right\\}}E\left[\xi_{ki}\xi_{kj}\right]$
$\displaystyle V_{k}^{LZG}-v_{k}$
$\displaystyle=\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{\mathcal{N}}_{g(i)}^{G}}E\left[\eta_{ki}\eta_{kj}\right]-\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{i}}E\left[\xi_{ki}\xi_{kj}\right]$
$\displaystyle=\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{\mathcal{N}}_{g(i)}^{G}}E\left[\xi_{ki}\xi_{kj}\right]+\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{\mathcal{N}}_{g(i)}^{G}}E\left[\eta_{ki}\right]E\left[\eta_{kj}\right]-\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{i}}E\left[\xi_{ki}\xi_{kj}\right]$
$\displaystyle=-\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{i}\backslash\mathcal{\mathcal{N}}_{g(i)}^{G}}E\left[\xi_{ki}\xi_{kj}\right]+\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{\mathcal{N}}_{g(i)}^{G}}E\left[\eta_{ki}\right]E\left[\eta_{kj}\right]$
$\displaystyle V_{k}^{CGM2}-v_{k}$
$\displaystyle=\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{\mathcal{N}}_{g(i)}^{G}}E\left[\xi_{ki}\xi_{kj}\right]+\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{\mathcal{N}}_{g(i)}^{G}}E\left[\eta_{ki}\right]E\left[\eta_{kj}\right]-\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{N}_{i}}E\left[\xi_{ki}\xi_{kj}\right]$
$\displaystyle\qquad+\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{\mathcal{N}}_{h(i)}^{H}}E\left[\xi_{ki}\xi_{kj}\right]+\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{\mathcal{N}}_{h(i)}^{H}}E\left[\eta_{ki}\right]E\left[\eta_{kj}\right]$
$\displaystyle=\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{\mathcal{N}}_{m(i)}^{M}}E\left[\xi_{ki}\xi_{kj}\right]+\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{\mathcal{N}}_{h(i)}^{H}}E\left[\eta_{ki}\right]E\left[\eta_{kj}\right]+\frac{N_{k}}{n_{k}^{2}}\sum_{i}\sum_{j\in\mathcal{\mathcal{N}}_{g(i)}^{G}}E\left[\eta_{ki}\right]E\left[\eta_{kj}\right]\geq
0$

∎

To prove Corollary 2, I first prove Lemma 3.

######  Lemma 3.

Consider any scalar random variable $W_{i}\in\mathbb{R}$ and define
$Q_{n}:=Var(\sum_{i}W_{i})=\sum_{i}\sum_{j\in\mathcal{N}_{i}}E[(W_{i}-E[W_{i}])(W_{j}-E[W_{j}])]$
and
$\hat{V}^{w}_{CGM2}:=\sum_{g}\sum_{i,j\in\mathcal{N}^{G}_{g}}W_{i}W_{j}+\sum_{h}\sum_{i,j\in\mathcal{N}^{H}_{h}}W_{i}W_{j}$.
Suppose $W_{i}\perp\\!\\!\\!\perp W_{j}$ when $g(i)\neq g(j)$ and $h(i)\neq
h(j)$, and all the second moments exist. Then,
$E[Q_{n}^{-1}\hat{V}^{w}_{CGM2}]\geq 1$.

###### Proof of Lemma 3.

Observe that:

$\displaystyle\hat{V}^{w}_{CGM2}$
$\displaystyle:=\sum_{g}\sum_{i,j\in\mathcal{N}^{G}_{g}}W_{i}W_{j}+\sum_{h}\sum_{i,j\in\mathcal{N}^{H}_{h}}W_{i}W_{j}$
$\displaystyle=\sum_{i}\sum_{j\in\mathcal{N}_{i}}W_{i}W_{j}+\sum_{m}\sum_{i,j\in\mathcal{N}^{M}_{m}}W_{i}W_{j}$

Define:

$\displaystyle Q^{*}_{n}:=\sum_{i}\sum_{j\in\mathcal{N}_{i}}E[W_{i}]E[W_{j}]$

Then,

$\displaystyle E[Q_{n}^{-1}\hat{V}^{w}_{CGM2}]$
$\displaystyle=Q_{n}^{-1}E\left(\sum_{i}\sum_{j\in\mathcal{N}_{i}}(W_{i}-E[W_{i}])(W_{j}-E[W_{j}])+\sum_{i}\sum_{j\in\mathcal{N}_{i}}(2W_{i}E[W_{j}]-E[W_{i}]E[W_{j}])+\sum_{m}\sum_{i,j\in\mathcal{N}^{M}_{m}}W_{i}W_{j}\right)$
$\displaystyle=1+Q_{n}^{-1}E\left(Q^{*}_{n}+\sum_{m}\sum_{i,j\in\mathcal{N}^{M}_{m}}W_{i}W_{j}\right)$

Then, it remains to show that
$\left(Q^{*}_{n}+\sum_{m}\sum_{i,j\in\mathcal{N}^{M}_{m}}E[W_{i}W_{j}]\right)$
is weakly positive.

$\displaystyle Q^{*}_{n}$
$\displaystyle+\sum_{m}\sum_{i,j\in\mathcal{N}^{M}_{m}}E[W_{i}W_{j}]=\sum_{i}\sum_{j\in\mathcal{N}_{i}}E[W_{i}]E[W_{j}]+\sum_{m}\sum_{i,j\in\mathcal{N}^{M}_{m}}E[W_{i}W_{j}]$
$\displaystyle=\sum_{g}\sum_{i,j\in\mathcal{N}^{G}_{g}}E[W_{i}]E[W_{j}]+\sum_{h}\sum_{i,j\in\mathcal{N}^{H}_{h}}E[W_{i}]E[W_{j}]+\sum_{m}\sum_{i,j\in\mathcal{N}^{M}_{m}}(E[W_{i}W_{j}]-E[W_{i}]E[W_{j}])$

Terms like $\sum_{g}\sum_{i,j\in\mathcal{N}^{G}_{g}}E[W_{i}]E[W_{j}]$ are
positive definite. Let $W_{g}\in\mathbb{R}^{N^{G}_{g}}$ be the vector that
stacks the $W_{i}$’s in cluster $g$. Then,

$\displaystyle\sum_{g}\sum_{i,j\in\mathcal{N}^{G}_{g}}E[W_{i}]E[W_{j}]$
$\displaystyle=\sum_{g}1_{N^{G}_{g}}^{\prime}E[W_{g}]E[W_{g}]^{\prime}1_{N^{G}_{g}}$

Since $E[W_{g}]E[W_{g}]^{\prime}$ is positive definite by construction of an
outer product,
$1_{N^{G}_{g}}^{\prime}E[W_{g}]E[W_{g}]^{\prime}1_{N^{G}_{g}}\geq 0$. Thus, it
comes down to
$\sum_{m}\sum_{i,j\in\mathcal{N}^{M}_{m}}(E[W_{i}W_{j}]-E[W_{i}]E[W_{j}])$.
Observe that:

$\displaystyle E[(W_{i}-E[W_{i}])(W_{j}-E[W_{j}])]$
$\displaystyle=E[W_{i}W_{j}]-2E[W_{i}]E[W_{j}]+E[W_{i}]E[W_{j}]$
$\displaystyle=E[W_{i}W_{j}]-E[W_{i}]E[W_{j}]$

Thus,

$\displaystyle\sum_{m}\sum_{i,j\in\mathcal{N}^{M}_{m}}(E[W_{i}W_{j}]-E[W_{i}]E[W_{j}])$
$\displaystyle=\sum_{m}\sum_{i,j\in\mathcal{N}^{M}_{m}}E[(W_{i}-E[W_{i}])(W_{j}-E[W_{j}])]$

Applying a similar argument,

$\displaystyle\sum_{m}\sum_{i,j\in\mathcal{N}^{M}_{m}}E[(W_{i}-E[W_{i}])(W_{j}-E[W_{j}])]$
$\displaystyle=\sum_{m}E[1_{N_{m}}^{\prime}(W_{m}-E[W_{m}])(W_{m}-E[W_{m}])^{\prime}1_{N_{m}}]\geq
0$

Thus,
$\left(Q^{*}_{n}+\sum_{m}\sum_{i,j\in\mathcal{N}^{M}_{m}}E[W_{i}W_{j}]\right)\geq
0$, yielding the result.

∎

###### Proof of Corollary 2.

Following the proof of Theorem 2, $\hat{\eta}_{ki}=\eta_{ki}(1+o_{P}(1))$, so
it suffices to work with $\eta_{ki}$. By using $\eta_{ki}$ in place of $W_{i}$
in Lemma 3, $V_{k}^{CGM2}/v_{k}\geq 1$, so $V_{k}^{CGM2}\geq v_{k}$. ∎

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