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Randomized Experiments with Ambiguous Record Linkage
\maketitle
\begin{abstract}
Randomized experiments are often linked to administrative data to measure outcomes, but coarse identifiers can leave multiple administrative records consistent with each experimental unit. We show that unbiased estimation of the sample average treatment effect is possible without resolving these ambiguous links or specifying an outcome model. Our key linkage requirement is that the true administrative outcome is contained among the candidate outcomes for each experimental unit. We characterize the precision cost of linkage ambiguity and study two complementary ways to mitigate it: regression adjustment and reweighting based on observed linkage characteristics. We establish finite-sample and asymptotic properties of these estimators and develop valid randomization-based and large-sample inference.
\end{abstract}
\section{Introduction}
\label{sec:intro}
Administrative records provide an attractive way to measure outcomes in randomized experiments. They can reduce survey costs, avoid attrition from follow-up surveys, and measure outcomes that are difficult to collect directly. A practical obstacle is that the experimental file and the administrative file may not share a unique identifier. If the available identifiers are coarse---for example, names shared by multiple individuals or only a subset of the digits of a national identification number---one experimental unit may be compatible with several administrative records. These compatible records form the unit's \emph{candidate set}.
We show that the sample average treatment effect (SATE) can be identified without resolving these ambiguous links. For candidate sets defined by fixed coarse identifiers, we require that the true administrative outcome of each experimental unit be contained among its candidate outcomes. Rather than selecting or imputing the true record, we construct an alternative outcome by summing the outcomes of all administrative records consistent with the observed identifier. This sum contains the true outcome together with nuisance components contributed by false candidates. We exploit the known treatment-assignment mechanism to account for these components and construct unbiased estimators of the SATE without identifying the true record or assigning probabilities to candidate identities.
Our setting is particularly relevant when the administrative data contain the outcome of interest for a population that is broader than the experimental sample, but individual records can only be linked to experimental units through coarse identifiers. Examples include earnings from tax records, hospitalizations from health records, and birth outcomes from vital-statistics records. When absence from the administrative outcome file implies that the true outcome is zero, the database need only contain records for units with nonzero outcomes. For mortality or participation in a social program, for example, absence from the administrative outcome file can then be interpreted as a zero outcome, provided all nonzero outcomes are recorded.
We establish both finite-sample and large-sample results in a finite-population framework. For expositional clarity, our main analysis focuses on independent Bernoulli treatment assignment. The baseline estimator applies the Horvitz--Thompson contrast to the candidate-sum outcomes and is exactly unbiased for the SATE. When the experimental sample is drawn representatively from a well-defined target subpopulation, it is also unbiased for the corresponding subpopulation average treatment effect. We also characterize the precision cost associated with ambiguous linkage and develop valid randomization-based and large-sample inference that accounts for the dependence induced by shared coarse identifiers.
The precision loss from ambiguous linkage can be mitigated in two complementary ways. First, reweighting assigns less weight to linkage cells expected to contribute more to the estimator’s variance. For example, cells with more candidate records may generate noisier candidate sums. Under heterogeneous treatment effects, however, reweighting generally changes the target from the SATE to a different weighted average of treatment effects. Second, regression adjustment can remove predictable differences in candidate-sum levels. When larger linkage cells have larger expected candidate sums, adjusting for cell size can reduce variation caused by chance imbalances between treatment and control. Sample-based adjustment can introduce finite-sample bias but preserves consistency for the SATE. The two strategies can also be combined.
Finally, we extend the analysis to stratified randomization with fixed treatment counts, including complete randomization as a special case. Under these designs, dependence between assignments can generate bias when experimental units share both a coarse identifier and a randomization stratum. We characterize this bias and provide a design-based correction that restores exact unbiasedness under suitable conditions. We establish asymptotic theory and variance estimators both for a fixed number of growing strata and for a growing number of small strata. The analysis also accommodates reweighting and regression adjustment.
There is a large literature on the analysis of imperfectly linked data
(see \cite{kamatgutman2026} for a recent survey). However, existing methods need not consistently estimate the
SATE in randomized experiments. For instance, methods that rely on
unambiguous matches can fail when those matches disproportionately
represent individuals with different treatment effects, while pooled
corrections for linkage error can fail when treatment effects vary with
linkage ambiguity. Appendix~\ref{app:alternative-methods} analyzes a
range of methods, providing examples of settings in which they may fail to recover the SATE
\citep{lahiri2005,chambers2009,kimchambers2012a,kimchambers2012b,hofzwinderman2012,goldstein2012,hofzwinderman2015,steorts2015,sadinle2017,wortman2018,hanlahiri2019,chipperfield2019,chamberssilva2020,shan2021,zhangtuoto2021,zhangtuoto2024,slawskietal2025,bukkeslawski2025,hureiter2027}.
Other approaches use joint Bayesian models for record linkage and relationships among variables in the
linked files, including models for causal inference
\citep{gutman2013,guha2022,kamatshan2023}. Their validity for SATE
estimation, though, depends on the specification and identification of the joint
model. In contrast, under the stated assignment and candidate-set
conditions, we propose estimators that recover the SATE without
specifying an outcome-distribution model, estimating correct-link
probabilities, or restricting how treatment effects vary with linkage
ambiguity. We also develop a theory for precision-motivated weighted
estimators, making explicit the trade-off between precision and retaining
the SATE as the target estimand in settings with ambiguous linkage.
Our candidate-sum approach also accommodates outcomes measured using administrative files containing only nonzero records, whenever absence from the file implies that the outcome is zero, a data structure not directly covered by the implementations compared above. \citet{tahamont2021} study linkage errors in this setting for binary outcomes. Correcting the attenuation characterized in their analysis, however, requires knowledge of true-positive and false-positive rates, which are not generally identified from the positive-record file and treatment assignments alone, even under random assignment. In contrast, our candidate-sum SATE estimators exploit positive-record counts without requiring knowledge of these rates. Finally, other approaches use additional information, such as verified links from clerical review or an audit sample \citep{dasylva2018,gutman2016}. Our candidate-sum SATE estimators do not require such information.
\section{Setup and oracle benchmark}
\label{sec:setup}
\subsection{Finite administrative population and experimental sample}
Consider a fixed finite administrative population
\[
\mathcal U=\{1,\ldots,M\}.
\]
We assume a well-defined treatment and no interference across units in $\mathcal U$ (SUTVA), so each unit's outcome depends only on its own treatment assignment. Unit $i\in\mathcal U$ has fixed potential outcomes $Y_i(0)$ and $Y_i(1)$ and treatment effect
\[
\tau_i=Y_i(1)-Y_i(0).
\]
The realized experimental sample is a fixed subset $\mathcal S\subset\mathcal U$ of size $N$. We do not require $\mathcal S$ to be a random sample from $\mathcal U$. This reflects the setting in which the administrative file may cover a broad population while the experiment is conducted in a particular region or among a selected set of participants. We take the SATE as our target.
\begin{equation}
\tau_{\mathcal S}=\frac{1}{N}\sum_{i\in\mathcal S}\tau_i.
\label{eq:sample-ate}
\end{equation}
{If the experimental sample is randomly drawn from a well-defined subpopulation, the corresponding subpopulation average treatment effect can instead be taken as the target by incorporating the sampling design.}
\subsection{Treatment assignment and administrative outcomes}
\label{sec:assignment}
For simplicity, the main analysis uses independent Bernoulli treatment assignment. For every $i\in\mathcal S$,
\begin{equation}
D_i\sim\operatorname{Bernoulli}(p),
\qquad 0<p<1,
\label{eq:bern-design}
\end{equation}
independently across experimental units. Let $\mathbf D=(D_i)_{i\in\mathcal S}$. For notational convenience set $D_i=0$ for $i\notin\mathcal S$, because treatment is not available outside the experiment. We consider complete and stratified randomization with fixed treatment counts in Section~\ref{sec:other-designs}.
In the full-register setting, the administrative file contains one outcome record per unit, reporting $Y_i^{\mathrm{adm}}$.
The administrative outcome observed after assignment is
\begin{equation}
Y_i^{\mathrm{adm}}
=
Y_i(D_i)
=
Y_i(0)+D_i\tau_i,
\qquad i\in\mathcal U.
\label{eq:admin-y}
\end{equation}
\subsection{Coarse identifiers and candidate outcomes}
The observed coarse identifier partitions the administrative population into linkage cells $\mathcal C_1,\ldots,\mathcal C_{G}$. Let
\[
M_g=|\mathcal C_g|
\]
be the number of administrative records in linkage cell $g$. For each experimental unit $i$, let $g(i)$ denote the index of the administrative linkage cell to which the coarse identifier recorded in the experimental file maps that unit. Our coverage requirement is that $i\in\mathcal C_{g(i)}$ for every $i\in\mathcal S$, so that each unit's true administrative record is included among its candidates.
The partition and the linkage-cell sizes may be arbitrarily related to potential outcomes and treatment effects. We impose no restriction on the number, composition, or outcomes of the false candidates. However, we assume that both the administrative partition and the mapping $g(i)$ are unaffected by treatment assignment.
For each linkage cell define the number of experimental units and treated experimental units,
\begin{equation}
N_g^{\mathrm{cell}}=|\mathcal C_g\cap\mathcal S|,
\qquad
T_g^{\mathrm{cell}}=\sum_{i\in\mathcal C_g\cap\mathcal S}D_i,
\label{eq:ng-tg}
\end{equation}
and define the untreated administrative total in linkage cell $g$
\begin{equation}
B_g=\sum_{j\in\mathcal C_g}Y_j(0).
\label{eq:bg}
\end{equation}
The total observed administrative outcome in linkage cell $g$ is
\begin{equation}
\widetilde Y_g(\mathbf D)
=
\sum_{j\in\mathcal C_g}Y_j^{\mathrm{adm}}
=
B_g+
\sum_{j\in\mathcal C_g\cap\mathcal S}D_j\tau_j.
\label{eq:linked-cell-outcome}
\end{equation}
Every experimental unit $i\in\mathcal C_g$ is assigned the same candidate-sum outcome,
\begin{equation}
\widetilde Y_i(\mathbf D)=\widetilde Y_{g(i)}(\mathbf D),
\label{eq:linked-unit-outcome}
\end{equation}
where this observed outcome may depend on the treatment assignments of other experimental units sharing the same coarse identifier.
\begin{remark}[Administrative files omitting zero outcomes]
Suppose the administrative outcome file contains only records for which $Y_i^{\mathrm{adm}}\neq 0$, with absence from the file implying $Y_i^{\mathrm{adm}}=0$. The sum in \eqref{eq:linked-cell-outcome} is still observed because records with zero outcomes contribute nothing to it. The relevant coverage condition is that every realized nonzero outcome is recorded and retained by the candidate rule.
The set of observed nonzero-outcome records may change with assignment, but the underlying linkage partition and the mapping $g(i)$ remain fixed.
\end{remark}
\subsection{Oracle benchmark}
If exact linkage were available, we would observe each experimental unit's own outcome $Y_i(D_i)$. As a benchmark, consider the infeasible Horvitz--Thompson estimator \citep{horvitz1952} under Bernoulli assignment:\footnote{Compared with the usual difference-in-means estimator, the Horvitz--Thompson estimator uses the known assignment probabilities $p$ and $1-p$, rather than the realized treatment shares $N_1/N$ and $1-N_1/N$, where $N_1=\sum_{i\in\mathcal S}D_i$. Under Bernoulli assignment, both estimators are unbiased for the oracle estimand, conditional on nondegenerate realized treatment counts in the case of difference-in-means. We use the Horvitz--Thompson form here because its extension to the candidate-sum outcomes introduced below remains exactly unbiased in the presence of ambiguous linkage, which simplifies the analysis. Section~\ref{sec:adjustment} studies a broader class of candidate-sum estimators that includes the corresponding difference-in-means estimator as a special case.}
\begin{equation}
\widehat\tau^{\mathrm O}
=
\frac{1}{N}\sum_{i\in\mathcal S}
\left\{
\frac{D_iY_i(D_i)}{p}
-
\frac{(1-D_i)Y_i(D_i)}{1-p}
\right\}.
\label{eq:oracle}
\end{equation}
This estimator satisfies
\begin{equation}
\mathbb{E}_D[\widehat\tau^{\mathrm O}]=\tau_{\mathcal S},
\label{eq:oracle-unbiased}
\end{equation}
and independence across units gives
\begin{equation}
\operatorname{Var}_D(\widehat\tau^{\mathrm O})
=
\frac{1}{N^2}\sum_{i\in\mathcal S}
\left\{
\frac{Y_i(1)^2}{p}
+
\frac{Y_i(0)^2}{1-p}
-
\tau_i^2
\right\},
\label{eq:oracle-var}
\end{equation}
where expectations and variances indexed by $D$ are taken over the treatment assignment, conditional on the fixed administrative population, experimental sample, potential outcomes, and linkage cells.
The oracle will serve as a precision benchmark. Our objective is to preserve the causal identification that exact linkage would deliver while quantifying the additional randomization variance created by ambiguous linkage and mitigating the resulting loss of precision.
\section{Finite-sample results under Bernoulli assignment}
\label{sec:finite}
\subsection{Baseline estimator}
\label{sec:baseline}
We now construct an estimator that is exactly unbiased for the SATE under independent Bernoulli assignment, even when individual links are unresolved and several experimental units share the same candidate set. Define the baseline estimator
\begin{align}
\widehat\tau^{\mathrm B}
&=
\frac{1}{N}\sum_{i\in\mathcal S}
\left\{
\frac{D_i\widetilde Y_i(\mathbf D)}{p}
-
\frac{(1-D_i)\widetilde Y_i(\mathbf D)}{1-p}
\right\}
\notag\\
&=
\frac{1}{N}\sum_{g=1}^{G}
\frac{T_g^{\mathrm{cell}}-pN_g^{\mathrm{cell}}}{p(1-p)}\,\widetilde Y_g(\mathbf D).
\label{eq:baseline}
\end{align}
The first expression compares candidate-sum outcomes across treatment arms, dividing the treated and control totals by the expected numbers of treated and control units, $Np$ and $N(1-p)$, respectively. The second expression writes the same estimator in terms of linkage-cell contributions. Each linkage cell's administrative outcome total is weighted by $T_g^{\mathrm{cell}}-pN_g^{\mathrm{cell}}$, the difference between its realized and expected numbers of treated experimental units, and the sum is normalized by $Np(1-p)$. Thus, linkage cells with more treated units than expected receive positive weights, whereas linkage cells with fewer treated units than expected receive negative weights. Note that linkage cells containing no experimental units have $N_g^{\mathrm{cell}}=T_g^{\mathrm{cell}}=0$ and therefore make no contribution to the estimator.
We next characterize the expectation and variance of the baseline estimator.
\begin{proposition}[Exact unbiasedness and variance]
\label{prop:baseline}
Under independent Bernoulli assignment,
\begin{equation}
\mathbb{E}_D[\widehat\tau^{\mathrm B}]=\tau_{\mathcal S}.
\label{eq:baseline-unbiased}
\end{equation}
Moreover,
\begin{equation}
\operatorname{Var}_D(\widehat\tau^{\mathrm B})
=
\frac{1}{N^2}\sum_{g=1}^{G}V_g,
\label{eq:baseline-var}
\end{equation}
where
\begin{equation}
V_g
=
\theta_g^2
+
\frac{
N_g^{\mathrm{cell}}(B_g+p\theta_g)^2
+2(1-2p)(B_g+p\theta_g)\theta_g
+\{1+(N_g^{\mathrm{cell}}-6)p(1-p)\}Q_g
}{p(1-p)},
\label{eq:vg}
\end{equation}
with
\[
\theta_g=\sum_{i\in\mathcal C_g\cap\mathcal S}\tau_i,
\qquad
Q_g=\sum_{i\in\mathcal C_g\cap\mathcal S}\tau_i^2.
\]
\end{proposition}
Proofs of all propositions and corollaries are provided in Appendix~\ref{app:proofs}.
The expectation result is the central design-based property. Linkage-cell size can be highly informative about potential outcomes, false candidates can have systematically different outcomes from true candidates, and ambiguity can vary arbitrarily across experimental units. None of these features creates bias under independent treatment assignment.
A key potential cost of ambiguous linkage is reduced precision. To make the comparison with exact linkage transparent, consider the case in which no linkage cell contains more than one experimental unit.
\begin{corollary}[Variance relative to exact linkage when $N_g^{\mathrm{cell}}\leq1$]
\label{cor:variance-gap}
Suppose $N_g^{\mathrm{cell}}\leq1$ for every linkage cell. Then
\begin{align}
&\operatorname{Var}_D(\widehat\tau^{\mathrm B})
-
\operatorname{Var}_D(\widehat\tau^{\mathrm O})
\notag\\
&\quad=
\frac{1}{N^2p(1-p)}\sum_{i\in\mathcal S}
\left[
\{B_{g(i)}+(1-p)\tau_i\}^2
-
\{Y_i(0)+(1-p)\tau_i\}^2
\right].
\label{eq:variance-gap}
\end{align}
If \(Y_i(0),Y_i(1)\geq0\) for all units, then the variance difference in \eqref{eq:variance-gap} is nonnegative.
\end{corollary}
For unrestricted real-valued outcomes there is no universal pointwise variance ordering: false-candidate outcomes could, by chance, generate sufficiently strong negative cancellation with the true outcome. Such a variance reduction requires systematic cancellation within linkage cells. In the absence of such cancelation, ambiguous linkage increases variance, which motivates the efficiency methods below.
\subsection{Precision weighting}
\label{sec:weighting}
The precision loss from ambiguous linkage need not be uniform across experimental units. The number and composition of administrative candidates can help predict how much each linkage cell contributes to the variance of the baseline estimator. We investigate whether precision can be improved by reweighting units based on these pre-treatment linkage characteristics. Because reweighting can also change the treatment-effect target, we characterize both the variance and the estimand of the weighted estimator. Let $w_g>0$ be a weight determined only by pre-treatment linkage information, such as $M_g$, name prevalence, or other features of the coarse identifier, and define
\[
W_{\mathcal S}=\sum_g N_g^{\mathrm{cell}}w_g.
\]
The weighted estimator is
\begin{equation}
\widehat\tau^{\mathrm W}
=
\frac{1}{W_{\mathcal S}}
\sum_g
w_g\frac{T_g^{\mathrm{cell}}-pN_g^{\mathrm{cell}}}{p(1-p)}\widetilde Y_g(\mathbf D).
\label{eq:weighted}
\end{equation}
Its estimand is the weighted sample effect
\begin{equation}
\tau_{w,\mathcal S}
=
\frac{1}{W_{\mathcal S}}
\sum_g w_g\theta_g.
\label{eq:weighted-target}
\end{equation}
\begin{proposition}[Fixed precision weights]
\label{prop:weighted}
If the weights are fixed with respect to treatment assignment, then
\begin{equation}
\mathbb{E}_D[\widehat\tau^{\mathrm W}]=\tau_{w,\mathcal S}
\label{eq:weighted-unbiased}
\end{equation}
and
\begin{equation}
\operatorname{Var}_D(\widehat\tau^{\mathrm W})
=
\frac{1}{W_{\mathcal S}^2}\sum_gw_g^2V_g,
\label{eq:weighted-var}
\end{equation}
with $V_g$ given by \eqref{eq:vg}. If treatment effects are homogeneous, $\tau_i=\tau$ for every $i\in\mathcal S$, then $\tau_{w,\mathcal S}=\tau_{\mathcal S}=\tau$ for every positive weighting scheme.
\end{proposition}
The proposition makes the tradeoff explicit. Under heterogeneous effects, precision weighting changes the target. Under homogeneous effects, all positive weighting schemes have the same target, so weights can be chosen solely to reduce variance.
A simple outcome-free choice is
\begin{equation}
w_g=\frac{1}{M_g}.
\label{eq:inverse-k}
\end{equation}
Since candidate-sum outcomes aggregate $M_g$ administrative
outcomes, larger linkage cells may contribute more to the
estimator's variance. Inverse-size weighting downweights these
cells without using observed outcomes.
Section~\ref{sec:efficiency} characterizes asymptotically optimal
weights and discusses conditions under which inverse-size
weighting is optimal when combined with regression adjustment.
\begin{remark}[Linkage-cell-size weighting when zero outcomes are omitted]
When the administrative outcome file omits records with $Y=0$, the full linkage-cell size $M_g$ is generally not observed. Using instead the number of candidates with $Y\neq0$ may be problematic, since this quantity may itself be affected by treatment. An alternative in this case is to use other pre-treatment proxies for $M_g$ or for the degree of linkage ambiguity. For example, one could use the prevalence of the relevant coarse identifier in an auxiliary population register or other pre-treatment data.
\end{remark}
\subsection{Regression adjustment}
\label{sec:adjustment}
We can also seek to improve precision by adjusting for pre-treatment characteristics that predict the level of the candidate-sum outcome. Because this outcome aggregates administrative records, characteristics such as $M_g$ can generate substantial differences in its level across linkage cells. Random treatment assignment may then produce chance imbalances in the distribution of such characteristics between treated and control units, adding variance to the estimator. Regression adjustment can reduce this source of variation by removing the component of the candidate-sum outcome that is predictable from pre-treatment characteristics. Let $x_g\in\mathbb R^L$ denote the characteristics used for adjustment. For example, $x_g$ may include $M_g$ and $N_g^{\mathrm{cell}}$. Other pre-treatment covariates can also be included, provided they are defined at, or aggregated to, the linkage-cell level.
For the theory it is useful first to consider a fixed adjustment $h_g$. Define the general weighted-and-adjusted estimator
\begin{equation}
\widehat\tau(w,h)
=
\frac{1}{W_{\mathcal S}}
\sum_g
w_g\frac{T_g^{\mathrm{cell}}-pN_g^{\mathrm{cell}}}{p(1-p)}
\{\widetilde Y_g(\mathbf D)-h_g\}.
\label{eq:generic-estimator}
\end{equation}
The baseline estimator is $\widehat\tau(1,0)$ and the weighted estimator is $\widehat\tau(w,0)$. Let $V_g(h)$ denote the expression in \eqref{eq:vg} with $B_g$ replaced by $B_g-h_g$. That is,
\[
V_g(h)
=
\theta_g^2
+
\frac{
N_g^{\mathrm{cell}}(B_g-h_g+p\theta_g)^2
+2(1-2p)(B_g-h_g+p\theta_g)\theta_g
+\{1+(N_g^{\mathrm{cell}}-6)p(1-p)\}Q_g
}{p(1-p)}.
\]
\begin{proposition}[Fixed adjustment]
\label{prop:fixed-adjustment}
For any fixed adjustment $h_g$,
\begin{equation}
\mathbb{E}_D[\widehat\tau(w,h)]=\tau_{w,\mathcal S},
\qquad
\operatorname{Var}_D\{\widehat\tau(w,h)\}
=
\frac{1}{W_{\mathcal S}^2}\sum_gw_g^2V_g(h).
\label{eq:fixed-adjustment-result}
\end{equation}
\end{proposition}
In particular, with equal weights, $\widehat\tau(1,h)$ remains exactly unbiased for the SATE for any fixed adjustment $h_g$.
In practice, the adjustment function must be estimated.
For linear adjustment, include an intercept in $x_g$
and write $x_g=(1,z_g')'$.
Let $\widehat\beta_1$ and $\widehat\beta_0$ denote the
coefficients from separate weighted least-squares
regressions of the candidate-sum outcome on $x_{g(i)}$
in the treatment and control arms, respectively,
using weights $w_{g(i)}$. We then set
\[
\widehat h_g
=
(1-p)x_g'\widehat\beta_1
+
p x_g'\widehat\beta_0.
\]
The resulting estimator can equivalently be computed
from a single fully interacted weighted regression.
Define the weighted covariate mean and centered covariates
\[
\overline z_w
=
\frac{1}{W_{\mathcal S}}
\sum_{i\in\mathcal S}w_{g(i)}z_{g(i)},
\qquad z_g^c=z_g-\overline z_w,
\]
and let
\begin{equation}
\begin{aligned}
(\widehat a,\widehat t,\widehat b,\widehat c)
&=
\underset{a,t,b,c}{\arg\min}
\sum_{i\in\mathcal S}w_{g(i)}\\
&\quad\times
\left[
\widetilde Y_{g(i)}(\mathbf D)
-a-tD_i
-(z_{g(i)}^c)'b
-D_i(z_{g(i)}^c)'c
\right]^2.
\end{aligned}
\label{eq:interacted-wls}
\end{equation}
Whenever these regressions are identified, the coefficient
on treatment satisfies
$\widehat t=\widehat\tau(w,\widehat h)$ exactly.
Thus, the adjusted estimator is the treatment coefficient
from weighted least squares of the candidate-sum outcome
on an intercept, treatment, centered covariates, and
their interactions with treatment. With equal weights,
this is the fully interacted regression adjustment of
\citet{lin2013}, applied to the candidate-sum outcome.
If \(x_g\) contains only an intercept, the adjusted estimator reduces exactly to the difference in weighted mean candidate-sum outcomes between treated and control units, using weights $w_{g(i)}$. With equal weights, this is the usual difference in means. More generally, \(x_g\) can include pre-treatment characteristics that predict the level of the candidate-sum outcome, such as \(M_g\), and flexible functions of these characteristics. For example, indicators for different values or ranges of \(M_g\) allow the relationship between linkage-cell size and the candidate-sum outcome to be nonlinear. Sparsely represented categories can be pooled or modeled more parsimoniously. Other pre-treatment covariates can also be included in \(x_g\).
Because \(\widehat h_g\) depends on the realized treatment assignment, the exact finite-sample unbiasedness result in Proposition~\ref{prop:fixed-adjustment} no longer applies. The asymptotic analysis below shows that estimating the adjustment does not affect consistency or the first-order limiting distribution.
\subsection{Randomization inference}
\label{sec:ri}
Consider the sharp null
\begin{equation}
H_0^{\mathrm{sharp}}:
Y_i(1)=Y_i(0)
\quad\text{for every }i\in\mathcal S,
\label{eq:sharp-null}
\end{equation}
and a test statistic
$\mathcal T(\widetilde{\mathbf Y},\mathbf D)$, where
$\widetilde{\mathbf Y}
=(\widetilde Y_g(\mathbf D))_{g=1}^G$
is the vector of observed candidate-sum outcomes and
$\mathbf D=(D_i)_{i\in\mathcal S}$ is the treatment-assignment vector.
We adopt the convention that larger values of $\mathcal T$
provide stronger evidence against the null.
Under \eqref{eq:sharp-null}, every candidate-sum outcome is
fixed with respect to treatment assignment:
\[
\widetilde Y_g(\mathbf d)=B_g
\quad\text{for every }g
\text{ and every }\mathbf d\in\{0,1\}^N.
\]
Consequently, the null distribution of the statistic can be
obtained by holding $\widetilde{\mathbf Y}$ fixed and redrawing
treatment assignments according to the known design.
For two-sided tests, possible choices include
$|\widehat\tau(w,\widehat h)|$ or its studentized version
using the asymptotic variance estimator developed in
Section~\ref{sec:variance-estimation}.
These choices encompass the baseline, weighted, adjusted,
and weighted-and-adjusted estimators.
\begin{proposition}[Exact Fisher randomization inference]
\label{prop:ri}
Let
\[
\mathcal A(\widetilde{\mathbf Y})
=
\left\{
\mathbf d\in\{0,1\}^N:
\mathcal T(\widetilde{\mathbf Y},\mathbf d)
\text{ is defined}
\right\},
\]
and suppose this set has positive probability under the
Bernoulli design in \eqref{eq:bern-design}.
For an observed assignment
$\mathbf D\in\mathcal A(\widetilde{\mathbf Y})$,
define the randomization $p$-value
\begin{equation}
p_{\mathrm{RI}}
=
\Pr_{\mathbf D^*}\!\left\{
\mathcal T(\widetilde{\mathbf Y},\mathbf D^*)
\geq
\mathcal T(\widetilde{\mathbf Y},\mathbf D)
\,\middle|\,
\widetilde{\mathbf Y},\mathbf D,\,
\mathbf D^*\in\mathcal A(\widetilde{\mathbf Y})
\right\},
\label{eq:ri-pvalue}
\end{equation}
where $\mathbf D^*$ is an independent draw from the
Bernoulli design. Under \eqref{eq:sharp-null},
\begin{equation} \label{eq:ri-level}
\Pr_D\!\left\{
p_{\mathrm{RI}}\leq\alpha
\,\middle|\,
\mathbf D\in\mathcal A(\widetilde{\mathbf Y})
\right\}
\leq\alpha
\qquad\text{for every }\alpha\in(0,1).
\end{equation}
Thus, rejecting when $p_{\mathrm{RI}}\leq\alpha$
gives a finite-sample level-$\alpha$ test conditional
on the statistic being defined. Conditional rejection
probability exactly equal to $\alpha$ can be obtained
by randomization at the boundary.
\end{proposition}
The conditioning on
$\mathbf D\in\mathcal A(\widetilde{\mathbf Y})$
in \eqref{eq:ri-level} means that the level guarantee
applies to realizations of the treatment assignment
for which the test statistic can be computed.
For example, for statistics that require observations
in both treatment arms, this excludes assignments
in which all units are treated or all units are controls.
To compute the randomization $p$-value in
\eqref{eq:ri-pvalue}, draws of $\mathbf D^*$ outside
$\mathcal A(\widetilde{\mathbf Y})$ are discarded
and redrawn. When the statistic uses
$\widehat\tau(w,\widehat h)$ with an estimated adjustment,
$\widehat h$ must be re-estimated under each reassignment.
Any variance estimate used for studentization must
likewise be recomputed.
Randomization inference conditions on the fixed outcomes
under the sharp null and reproduces the known assignment
mechanism. The same argument applies to the stratified and
complete randomization designs considered in
Appendix~\ref{app:fixed-count}, provided treatment assignments
are redrawn according to the corresponding randomization
design. Its limitation is that it tests the sharp null that
every individual treatment effect is zero. In many
applications the object of interest is instead a hypothesis
about the SATE while allowing treatment effects to be
heterogeneous. The large-sample theory below provides
inference for such weak null hypotheses.
\section{Asymptotic theory}
\label{sec:asymptotics}
We next study the large-sample behavior of the estimators along sequences of fixed administrative populations and experimental samples, with the experimental sample size $N$ tending to infinity. The asymptotic analysis provides inference for weak null hypotheses, characterizes the first-order behavior of the baseline, weighted, adjusted, and weighted-and-adjusted estimators, and clarifies the role of ambiguous linkage in precision.
\subsection{Asymptotic framework}
For each $N$, the administrative population, experimental sample $\mathcal S$, potential outcomes, and linkage structure are treated as fixed, and randomness arises only from the treatment-assignment mechanism. The coarse identifiers partition the administrative population into linkage cells $\mathcal C_g$ of size $M_g$, while
\[
N_g^{\mathrm{cell}}=|\mathcal C_g\cap\mathcal S|
\]
denotes the number of experimental units in linkage cell $g$. Let
\[
\mathcal G_N=\{g:N_g^{\mathrm{cell}}>0\},
\qquad
G_N=|\mathcal G_N|
\]
denote the represented linkage cells and their number.
\begin{assumption}[Bounded linkage cells]
\label{ass:bounded-cells}
There exists a finite constant $\overline M$ such that
\[
\max_g M_g\leq \overline M
\]
along the sequence.
\end{assumption}
Assumption~\ref{ass:bounded-cells} requires linkage cells to remain uniformly bounded in size as the administrative population grows. It therefore rules out coarse identifiers whose number of matches grows without bound, as could occur, for example, with very common names.\footnote{If some linkage cells grow unusually large, their candidate sums can also grow with linkage-cell size and may dominate the asymptotic variance. In applications, such cases would typically need to be handled separately, for example by excluding extremely large linkage cells or assigning them substantially smaller weights.} The assumption still allows persistent linkage ambiguity: a linkage cell may contain multiple candidate records even asymptotically.
Because $N_g^{\mathrm{cell}}\leq M_g$, Assumption~\ref{ass:bounded-cells} also implies that the number of experimental units in any linkage cell is uniformly bounded. Moreover,
\[
G_N\geq \frac{N}{\overline M},
\]
so the number of linkage cells represented in the experimental sample necessarily diverges as $N\to\infty$. The large-sample approximation is therefore driven by an increasing number of non-dominating linkage cells.
We impose no sampling model for the experimental sample and no additional restriction on the realized degree of overlap among experimental units within linkage cells. Some sequences may have $N_g^{\mathrm{cell}}\leq1$ for every represented linkage cell, while in others multiple experimental units may share linkage cells even asymptotically. The results below cover both cases.
\subsection{A general estimator class}
Let $w_g=w(x_g)>0$ be a bounded pre-treatment weight and let $h_g=h(x_g)$ be a fixed adjustment. Define the linkage-cell contribution
\begin{equation}
H_g(w,h)
=
w_g\frac{T_g^{\mathrm{cell}}-pN_g^{\mathrm{cell}}}{p(1-p)}
\{\widetilde Y_g(\mathbf D)-h_g\}.
\label{eq:Hg}
\end{equation}
Then
\begin{equation}
\widehat\tau(w,h)
=
\frac{\sum_gH_g(w,h)}{W_{\mathcal S}},
\qquad
W_{\mathcal S}=\sum_gw_gN_g^{\mathrm{cell}}.
\label{eq:generic-again}
\end{equation}
The corresponding target is $\tau_{w,\mathcal S}$ in \eqref{eq:weighted-target}. The four estimators considered in the paper are nested as follows:
\begin{center}
\begin{tabular}{lcc}
\toprule
Estimator & Weight & Adjustment \\
\midrule
Baseline $\widehat\tau^{\mathrm B}$ & $w_g=1$ & $h_g=0$ \\
Weighted $\widehat\tau^{\mathrm W}$ & $w_g=w(x_g)$ & $h_g=0$ \\
Adjusted $\widehat\tau^{\mathrm A}$ & $w_g=1$ & $h_g=\widehat h_g$ \\
Weighted + adjusted $\widehat\tau^{\mathrm{WA}}$ & $w_g=w(x_g)$ & $h_g=\widehat h_g$ \\
\bottomrule
\end{tabular}
\end{center}
\begin{assumption}[Regularity]
\label{ass:cells}
Along the sequence:
\begin{enumerate}[label=(\roman*)]
\item $W_{\mathcal S}/N\to w^\ast\in(0,\infty)$ and $0<\underline w\leq w_g\leq\overline w<\infty$;
\item the candidate sums and fixed adjustments have bounded fourth moments on average,
\[
\frac1N\sum_{g\in\mathcal G_N}
\left\{
\mathbb{E}_D\left[\left|\widetilde Y_g(\mathbf D)\right|^4\right]
+|h_g|^4
\right\}=O(1);
\]
\item the exact design variance satisfies
\begin{equation}
V_N(w,h)
\equiv
\frac{N}{W_{\mathcal S}^2}\sum_{g\in\mathcal G_N}w_g^2V_g(h)
\longrightarrow
V(w,h)\in(0,\infty).
\label{eq:VN}
\end{equation}
\end{enumerate}
\end{assumption}
Assumption~\ref{ass:cells} bounds weights and the average fourth moments of candidate sums and fixed adjustments, and requires the root-$N$ design variance to converge to a nondegenerate limit. It does not require uniformly bounded outcomes or a uniform moment bound for every linkage cell. With bounded linkage-cell sizes, part (ii) also bounds the average fourth moments of $H_g(w,h)$, both centered and noncentered, and the average second moments of the candidate sums. Together with Assumption~\ref{ass:bounded-cells}, it gives a triangular array of independent, non-dominating linkage-cell contributions under Bernoulli assignment.
For the adjusted estimators, let
\[
\widehat h_g=h(x_g;\widehat\beta),
\qquad
h_g^*=h(x_g;\beta^*),
\]
where $\beta^*$ is a nonrandom finite-population target. Dependence of
$\beta^*$ on $N$ is left implicit.
\begin{assumption}[Estimated adjustment]
\label{ass:adjustment}
The dimension of $\beta$ is fixed, and:
\begin{enumerate}[label=(\roman*)]
\item
\[
\widehat\beta-\beta^*=O_p(N^{-1/2});
\]
\item $h(x;\beta)$ is twice continuously differentiable in $\beta$ in a
neighborhood of $\beta^*$, with first and second derivatives uniformly
bounded over the realized values of $x_g$ along the sequence.
\end{enumerate}
\end{assumption}
Assumption~\ref{ass:adjustment} accommodates both interacted and
non-interacted regression adjustment. The separate treatment- and
control-arm regressions described above correspond to one particular
parameterization of $h(x_g;\widehat\beta)$.
We can now state a common asymptotic result for this estimator class, allowing both weighting and estimated adjustment.
\begin{proposition}[General design-based central limit theorem]
\label{thm:general-clt}
Suppose Assumptions~\ref{ass:bounded-cells}, \ref{ass:cells}, and
\ref{ass:adjustment} hold, with Assumption~\ref{ass:cells} evaluated at
$h=h^*$. Then
\[
\sqrt N
\{\widehat\tau(w,\widehat h)-\tau_{w,\mathcal S}\}
\Rightarrow
\mathcal N\{0,V(w,h^*)\}.
\]
\end{proposition}
The proposition covers all four estimators introduced above. The baseline and adjusted estimators are consistent and asymptotically normal for the SATE. This includes the usual difference in means of candidate-sum outcomes between treated and control experimental units, which is obtained with equal weights and intercept-only treatment- and control-arm regressions for the feasible adjustment in Section~\ref{sec:adjustment}. In particular, the finite-sample departure from exact unbiasedness induced by estimating the adjustment does not affect consistency or the first-order limiting distribution. The weighted and weighted-and-adjusted estimators are consistent and asymptotically normal for the weighted target in \eqref{eq:weighted-target}. Under homogeneous treatment effects this target coincides with the SATE.
The result does not depend on the degree of experimental overlap within linkage cells or on the limiting behavior of \(N/M\). In particular, it applies both when \(N_g^{\mathrm{cell}}\leq1\) for every represented linkage cell and when multiple experimental units may share a linkage cell.
\subsection{Adjustment and precision weighting}
\label{sec:efficiency}
\subsubsection{Optimal regression adjustment based on $x_g$}
We first characterize the variance-minimizing adjustment for a given fixed weighting rule. Suppose $x_g$ has finite support and let $w_g>0$ denote fixed pre-treatment weights. For each value $x$, define
\[
W_{2,x}
=
\sum_{i\in\mathcal S:x_{g(i)}=x} w_{g(i)}^2
=
\sum_{g:x_g=x} w_g^2 N_g^{\mathrm{cell}}.
\]
For $d\in\{0,1\}$, define the finite-population weighted arm-specific mean
\begin{equation}
\mu_{d,w,N}(x)
=
\frac{1}{W_{2,x}}
\sum_{i\in\mathcal S:x_{g(i)}=x}
w_{g(i)}^2
\mathbb{E}_D\!\left[
\widetilde Y_{g(i)}(\mathbf D)\mid D_i=d
\right].
\label{eq:oracle-arm-mean-weighted}
\end{equation}
The dependence on $N$ emphasizes that these are finite-population quantities and need not converge to fixed limits.
Define
\begin{equation}
h_{w,N}^*(x)
=
(1-p)\mu_{1,w,N}(x)
+
p\mu_{0,w,N}(x).
\label{eq:optimal-h-weighted}
\end{equation}
\begin{proposition}[Optimal adjustment based on $x_g$]
\label{prop:optimal-adjustment}
Fix any positive pre-treatment weighting rule $\{w_g\}$. Among all fixed adjustments of the form $h_g=h(x_g)$, the choice
\[
h_g=h_{w,N}^*(x_g)
\]
minimizes the exact design variance of $\widehat\tau(w,h)$.
\end{proposition}
The oracle adjustment in \eqref{eq:optimal-h-weighted} is generally infeasible because the conditional expectations in \eqref{eq:oracle-arm-mean-weighted} are not observed. A natural saturated sample analogue is obtained by estimating the weighted arm-specific means separately within each category of $x_g$. Define
\begin{equation}
\widehat\mu_{d,w}(x)
=
\frac{
\sum_{i\in\mathcal S}
w_{g(i)}^2
\mathbf{1}\{x_{g(i)}=x,D_i=d\}
\widetilde Y_{g(i)}(\mathbf D)
}{
\sum_{i\in\mathcal S}
w_{g(i)}^2
\mathbf{1}\{x_{g(i)}=x,D_i=d\}
},
\qquad d\in\{0,1\},
\label{eq:muhat-saturated-weighted}
\end{equation}
whenever the denominator is positive, and set
\begin{equation}
\widehat h_w(x)
=
(1-p)\widehat\mu_{1,w}(x)
+
p\widehat\mu_{0,w}(x).
\label{eq:hhat-saturated-weighted}
\end{equation}
Equivalently, this adjustment can be obtained from separate saturated treatment- and control-arm regressions using $w_g^2$ as regression weights.
The following result verifies that this feasible saturated adjustment satisfies Assumption~\ref{ass:adjustment}.
\begin{corollary}[Feasible saturated adjustment]
\label{cor:saturated-adjustment}
Suppose Assumption~\ref{ass:bounded-cells} holds and the weight bounds and
moment condition in Assumption~\ref{ass:cells}(i)--(ii) hold with $h=0$.
Suppose $x_g$ has fixed finite support and, for every retained category $x$,
\begin{equation}
\frac{N_x}{N}\longrightarrow\pi_x>0,
\qquad
N_x=\sum_{i\in\mathcal S}\mathbf{1}\{x_{g(i)}=x\}.
\label{eq:positive-category-share}
\end{equation}
Then
\[
\widehat\mu_{d,w}(x)-\mu_{d,w,N}(x)
=
O_p(N^{-1/2}),
\qquad d\in\{0,1\},
\]
for every retained category $x$. Hence
\[
\widehat h_w(x)-h_{w,N}^*(x)
=
O_p(N^{-1/2}),
\]
and the saturated adjustment satisfies Assumption~\ref{ass:adjustment} with finite-population target $h_{w,N}^*$. Also, Assumption~\ref{ass:cells}(ii) holds with
$h_g=h_{w,N}^*(x_g)$.
\end{corollary}
Condition~\eqref{eq:positive-category-share} also implies that each retained category is represented in both treatment arms with probability approaching one under Bernoulli assignment. Categories that are too sparse for a saturated specification can instead be pooled or modeled more parsimoniously.
Combining Proposition~\ref{prop:optimal-adjustment}, Corollary~\ref{cor:saturated-adjustment}, and Proposition~\ref{thm:general-clt}, the feasible saturated adjustment has the same first-order asymptotic distribution as the infeasible variance-minimizing adjustment $h_{w,N}^*$. Thus, for any given fixed weighting rule, saturated regression adjustment weakly reduces first-order asymptotic variance relative to using the same weights without adjustment.
When the weights are themselves functions only of $x_g$, $w_g^2$ is constant within each category and cancels from \eqref{eq:oracle-arm-mean-weighted} and \eqref{eq:muhat-saturated-weighted}. In that case, the oracle and feasible adjustments reduce to the corresponding unweighted arm-specific means within each category.
\subsubsection{Optimal weights}
Weighting addresses a different source of imprecision. Suppose $x_g$ has finite support and, for each value $x$ in its support,
\begin{equation}
\frac{N_x}{N}
\longrightarrow
\pi_x>0.
\label{eq:Nx-pix}
\end{equation}
For a given fixed adjustment $h$, define the asymptotic design-variance contribution per experimental unit in category $x$ by
\begin{equation}
\overline V_h(x)
=
\lim
\frac{1}{N_x}
\sum_{g\in\mathcal G_N:x_g=x}V_g(h),
\label{eq:Vbarx}
\end{equation}
whenever the limit exists and lies in $(0,\infty)$.
\begin{proposition}[First-order precision weighting]
\label{prop:efficiency}
Suppose the conditions of Proposition~\ref{thm:general-clt} hold and the limits in \eqref{eq:Nx-pix}--\eqref{eq:Vbarx} exist. For a given fixed adjustment $h$, among positive weights that are functions of $x_g$, asymptotic variance is minimized by
\begin{equation}
w^*(x)
\propto
\frac{1}{\overline V_h(x)}.
\label{eq:optimal-weight}
\end{equation}
In particular, let $x_g=M_g$ and suppose that there exists a constant $c>0$
such that
\begin{equation}
\frac{1}{N_m}
\sum_{g\in\mathcal G_N:M_g=m}V_g(h)
\longrightarrow
cm
\label{eq:homoskedastic-benchmark}
\end{equation}
for each $m$ in the support of $M_g$, where
\[
N_m=\sum_{g:M_g=m}N_g^{\mathrm{cell}}.
\]
Then
\[
w^*(m)\propto\frac{1}{m}.
\]
\end{proposition}
Appendix~\ref{app:weighting-primitives} provides primitive stochastic conditions under which the asymptotic proportionality condition in \eqref{eq:homoskedastic-benchmark} holds almost surely. While those conditions are relatively strong, they illustrate the intuition that linkage cells with larger \(M_g\) may contribute more to the design-based variance, providing a rationale for downweighting such cells.
Under homogeneous treatment effects, reweighting does not
change the causal target, so the variance gains described
above correspond to efficiency improvements for the same
target. With heterogeneous treatment effects, reweighting
may involve a trade-off: downweighting cells to improve
precision also changes their contribution to the causal
target, yielding a weighted average of treatment effects
that need not coincide with the SATE.
\subsection{Feasible asymptotic variance estimation}
\label{sec:variance-estimation}
Proposition~\ref{thm:general-clt} establishes root-\(N\) asymptotic normality of the estimators considered above. Therefore, a feasible estimator of the asymptotic variance, or an asymptotically conservative upper bound, is sufficient for asymptotic inference on \(\tau_{w,\mathcal S}\).
Because experimental units that share a linkage cell also share the same candidate-sum outcome, the natural approach is to cluster the variance calculation at the linkage-cell level. Let
\begin{equation}
\widehat H_g
=
w_g\frac{T_g^{\mathrm{cell}}-pN_g^{\mathrm{cell}}}{p(1-p)}
\{\widetilde Y_g(\mathbf D)-\widehat h_g\},
\label{eq:Hhat}
\end{equation}
where $\widehat h_g=0$ for estimators without adjustment.
Let $\widehat\delta_g$ be a centering term for the treatment effect
in cell $g$. A simple choice is $\widehat\delta_g=\widehat\tau$
for every cell; cell-specific values can instead account for
predictable variation in treatment effects. Define
\begin{equation}
\widehat V_{\mathrm{cl}}
=
\frac{N}{W_{\mathcal S}^2}
\sum_{g\in\mathcal G_N}
\left(\widehat H_g-w_gN_g^{\mathrm{cell}}\widehat\delta_g\right)^2.
\label{eq:cluster-vhat}
\end{equation}
For fixed centering targets $\delta_g^*$, define
\begin{equation}
\Delta_N(w,\delta^*)
=
\frac{N}{W_{\mathcal S}^2}
\sum_{g\in\mathcal G_N}
w_g^2
\{\theta_g-N_g^{\mathrm{cell}}\delta_g^*\}^2.
\label{eq:cluster-gap-N}
\end{equation}
\begin{proposition}[Cluster variance and treatment-effect heterogeneity]
\label{prop:cluster-se}
Suppose the conditions of Proposition~\ref{thm:general-clt} hold.
Suppose there exist fixed targets $\delta_g^*$ such that
\[
\frac1N\sum_{g\in\mathcal G_N}|\delta_g^*|^4=O(1),
\qquad
\frac1N\sum_{g\in\mathcal G_N}
(\widehat\delta_g-\delta_g^*)^2=o_p(1),
\]
and $\Delta_N(w,\delta^*)\longrightarrow\Delta(w,\delta^*)$.
Then
\begin{equation}
\widehat V_{\mathrm{cl}}
\overset{p}{\longrightarrow}
V(w,h^*)+\Delta(w,\delta^*),
\label{eq:cluster-limit}
\end{equation}
where $\Delta(w,\delta^*)\geq0$. Thus
$\widehat V_{\mathrm{cl}}$ is asymptotically conservative for the
design variance and is consistent whenever $\Delta(w,\delta^*)=0$.
Under homogeneous treatment effects, the gap vanishes if
$\delta_g^*$ equals the common effect for every represented cell.
\end{proposition}
The common centering $\widehat\delta_g=\widehat\tau$ satisfies these
conditions with $\delta_g^*=\tau_{w,\mathcal S}$, provided the corresponding
gap has a limit. It therefore gives a valid variance estimator for
all four estimators considered above, including the baseline and
weighted estimators with $\widehat h_g=0$, and is consistent under
homogeneous treatment effects. More generally, the gap reflects
variation in linkage-cell average treatment effects around
$\delta_g^*$ and vanishes whenever
$\theta_g=N_g^{\mathrm{cell}}\delta_g^*$ for every represented cell.
For the WLS implementation in \eqref{eq:interacted-wls}, take
\[
\widehat\delta_g=x_g'(\widehat\beta_1-\widehat\beta_0).
\]
Let $\beta_d^*$ denote the finite-population weighted projection
coefficients of $\mathbb{E}_D[\widetilde Y_{g(i)}(\mathbf D)\mid D_i=d]$
on $x_{g(i)}$ over all $i\in\mathcal S$, using weights $w_{g(i)}$.
The corresponding targets are
$h_g^*=(1-p)x_g'\beta_1^*+p x_g'\beta_0^*$ and
$\delta_g^*=x_g'(\beta_1^*-\beta_0^*)$.
Suppose the conditions of Proposition~\ref{thm:general-clt}
hold, with Assumption~\ref{ass:adjustment} applied jointly to the
coefficients of the two arm-specific regressions and their
finite-population targets defined above. In particular,
$\widehat\beta_d-\beta_d^*=O_p(N^{-1/2})$ for $d\in\{0,1\}$.
Suppose also that $x_g$ includes an intercept, has fixed dimension,
and is uniformly bounded, and
\[
\mathsf A_N
=\frac1N\sum_{g\in\mathcal G_N}
w_gN_g^{\mathrm{cell}}x_gx_g'
\longrightarrow\mathsf A\succ0.
\]
Then this centering satisfies the moment and approximation conditions
of Proposition~\ref{prop:cluster-se}. Moreover,
\[
N\widehat{\operatorname{Var}}_{\mathrm{CR0}}(\widehat t)
-\widehat V_{\mathrm{cl}}=o_p(1),
\]
where $\widehat{\operatorname{Var}}_{\mathrm{CR0}}(\widehat t)$ is the
usual cluster-robust variance estimator for the treatment coefficient
in \eqref{eq:interacted-wls}, clustering by linkage cell and without
a degrees-of-freedom correction. Thus, when the corresponding gap
converges, the regression CRVE is also asymptotically conservative.
With an intercept, homogeneous treatment effects imply
$\delta_g^*=\tau_{w,\mathcal S}$, so both variance estimators are consistent
on their respective scales.
This equivalence provides a direct implementation:
run the regression in \eqref{eq:interacted-wls} at the experimental-unit
level and use the standard error clustered by linkage cell.
The result covers both intercept-only WLS adjustment and the fully
interacted adjustment with additional covariates. In the first case,
the regression includes an intercept and treatment; in the second,
it also includes the centered covariates and their interactions with
treatment.
When $N_g^{\mathrm{cell}}\leq1$ for every represented linkage cell,
CR0 coincides with the individual-level heteroskedasticity-robust
estimator HC0 applied to the same regression. When a cell contains
multiple experimental units, clustering accounts for the dependence
induced by their shared candidate-sum outcome.
Usual degrees-of-freedom corrections that tend to one, such as CR1,
leave these asymptotic conclusions unchanged.
Proposition~\ref{prop:cluster-se} conditions on the realized
experimental sample and therefore concerns inference for the SATE,
or for the corresponding weighted treatment-effect target. If the
experimental sample is itself treated as random and the target is
an average treatment effect in a broader population or subpopulation,
sampling variation would provide an additional source of uncertainty.
\section{Other assignment mechanisms}
\label{sec:other-designs}
Many experiments fix the number of treated units, either overall or within randomization strata. Appendix~\ref{app:fixed-count} extends our finite-sample and asymptotic results to these designs.
We show that a generalization of the baseline estimator allowing treatment probabilities to vary across strata, denoted $\widehat\tau^{\mathrm{SR}}$, can be biased for the SATE when experimental units share both a linkage cell and a randomization stratum.\footnote{When treatment probabilities are constant across strata, $\widehat\tau^{\mathrm{SR}}$ coincides numerically with $\widehat\tau^{\mathrm B}$.} Under fixed treatment counts, treating one unit reduces the probability that another unit in the same stratum is treated. When these units share a linkage cell, this dependence prevents the other unit's treatment-effect contribution from canceling in expectation. We derive a corrected estimator, $\widehat\tau^{\mathrm{SR,corr}}$, that is exactly unbiased provided no stratum is entirely contained in one linkage cell. Fisher randomization inference for the sharp null of no individual treatment effects remains exact when assignments follow the actual design and the full statistic is recomputed after each reassignment.
We also study two asymptotic regimes. With a fixed number of growing strata and bounded linkage-cell sizes, the bias of $\widehat\tau^{\mathrm{SR}}$ vanishes and the corrected and uncorrected estimators are first-order asymptotically equivalent. With a growing number of small strata, the uncorrected estimator can remain inconsistent for the SATE, whereas the corrected estimator is consistent and asymptotically normal under the stated moment and variance conditions.
For each regime, we provide a nonnegative variance estimator that is asymptotically conservative in general and consistent under homogeneous treatment effects. With many small strata, linkage cells crossing stratum boundaries can generate first-order cross-stratum covariances, which our variance estimator explicitly accounts for.
Regression adjustment and weighting extend to these designs, subject to the same caveats discussed under Bernoulli assignment.
\section{Discussion}
\label{sec:discussion}
Ambiguous record linkage is usually treated as a problem of recovering individual identities or individual outcomes. For treatment-effect estimation in a randomized experiment, that objective can be unnecessarily demanding. We show that the SATE can be recovered under much weaker conditions on the linkage process: the true outcome must be retained among the candidate outcomes, but the relative probabilities of candidate identities can remain unknown and the ambiguity can be arbitrarily related to potential outcomes.
The baseline estimator trades an identification problem for an efficiency problem. Retaining all candidate outcomes avoids attenuation from selecting or averaging ambiguous links, but can add substantial noise. The linkage structure itself provides useful information about that noise. Regression adjustment can remove predictable level differences while retaining the SATE as the first-order target. Precision-motivated weighting can further reduce variance, but generally changes the target from the SATE to a weighted average when treatment effects are heterogeneous. Together, these approaches characterize how precision can be recovered without returning to a probabilistic model of individual identity and when precision gains come at the cost of changing the estimand.
Several extensions are natural. We focus on candidate sets generated by a coarse identifier and therefore forming a partition. More general overlapping candidate sets can be represented by a known incidence matrix, suggesting extensions of the same design-based logic beyond partitions. A second direction is to study whether candidate-sum outcomes can be used with nonexperimental methods, such as matching, instrumental variables, difference-in-differences, or regression discontinuity.