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A Design-Based Approach to Testing and Inference in (Quasi-)Experiments with Spillovers
\maketitle
\begin{abstract}
Economic policies rarely affect only their direct targets. To study these spillovers, researchers summarize who else was treated with a simple exposure measure, such as the share of treated neighbors within a radius. But for many settings, economic theory provides little guidance on choosing the functional form (e.g., ring) of that measure or its parameters (e.g., radius). We show that the data can inform both choices. Correctly specified exposure measures imply orthogonality conditions that can be used for both estimation and testing. We establish consistency and asymptotic normality of the resulting estimator under spatial and network dependence in a design-based framework, with all randomness arising from treatment assignment. We then characterize the efficient moment conditions. Applied to two large-scale anti-poverty programs, the framework supports some prior radius estimates but rejects others. In the latter case, the revised radius yields substantively different policy-effect estimates.
\bigskip
\noindent{\bf Keywords:} spillovers; exposure mappings; design-based inference; specification testing; spatial and network dependence; general equilibrium effects
\end{abstract}
\newpage
\section{Introduction}
\label{sec:intro}
Economic policies can generate effects far beyond their direct targets. Because economic agents are embedded in spatial, market, and social networks, shocks may propagate through prices, local labor markets, commuting and shopping patterns, and social interactions. Classical analyses of social multipliers in economics emphasize that such spillovers can amplify or attenuate policy impacts in important ways (e.g., \citealp{glaeser2003social,mani2013poverty}). More recently, a growing body of experimental and quasi-experimental work in development and spatial economics has made spillovers the primary object of interest, including studies of large-scale cash transfers, public works, and transport infrastructure \citep[e.g.][]{miguel2004worms,donaldson2016railroads,egger2022general,muralidharan2023generaleq,franklin2024urban,walker2024slack}.
Despite this interest, spillovers pose econometric challenges. Unlike the standard no-interference setting (where each unit's outcome depends only on its own treatment), spillovers mean that unit
$i$'s outcome can depend on many, or even all units' treatments. Whereas under the standard assumption each unit has only a small number of potential outcomes (e.g., two in the binary-treatment case), with spillovers the number of potential outcomes can be very large (e.g., $2^n$ in the binary case), since each of the possible assignments can in principle produce a different outcome for that unit.
Applied work typically makes the problem tractable by replacing the full assignment vector \(W\) with a low-dimensional exposure mapping \(g(W;X_i,\theta)\) \citep{Manski2013TreatmentResponse,AronowSamii2017}\footnote{An alternative is to restrict interference to small, known groups (partial interference; e.g., households or dorm rooms), the special case in which the exposure map reads only own-group assignments. We focus on the general case because our applications feature market- and space-mediated spillovers that may cross such groupings.}. The vector \(W\) collects all units' treatment statuses, \(X_i\) contains baseline covariates such as networks or commuting flows, and \(\theta\) parameterizes features of the mapping, such as a distance radius or elasticity parameter. Examples include ring or neighborhood averages \citep{miguel2004worms,egger2022general} and smooth spatial-decay kernels such as gravity or market-access measures \citep{redding2004economic,donaldson2016railroads,franklin2024urban}.
This approach raises two key challenges. The first is the choice of the map structure \(g\left(W ; X_i, \theta\right)\). Often economic theory does not provide an exact functional form for this map beyond broad guidance, such as the idea that spillovers should decay with distance (in space, in a network, or along trade and commuting links). Reflecting such ambiguity, even in closely related empirical settings, different researchers make very different choices among these constructions. For example, in large-scale anti-poverty programs in low-income countries, researchers studying settings with very similar institutional features choose different exposure map functional forms \citep[e.g.][]{egger2022general,franklin2024urban,walker2024slack}.\footnote{\citet{egger2022general} study large cash transfers in rural Kenya and proxy local spillovers by per-capita transfers within distance bands around each market (``distance buffers''). \citet{franklin2024urban} evaluate an urban public-works program in Addis Ababa and, to capture spillovers across neighborhoods, combine a spatial-equilibrium model with commuting data so that exposure is mediated by the commuting network rather than by geographic distance alone. In related Kenyan data, \citet{walker2024slack} construct gravity-style ``shopping network'' exposure indices that weight village-level activity by baseline shopping flows from villages to markets, and compare the performance of these network-based measures to buffer-style measures of spatial exposure in explaining cross-market inflation \citep[][Figure~A.17]{walker2024slack}.} Even within a single paper, researchers often report multiple exposure constructions and informally compare them \citep{donaldson2016railroads,muralidharan2023generaleq}.\footnote{For instance, \citet{muralidharan2023generaleq} take as their main specification a ring-based exposure measure, given by the treated share of units within a 20 km radius of each location, but also show in their Appendix~B that similar results obtain when exposure is measured using a market-access-style kernel. Likewise, \citet{donaldson2016railroads} summarize the impact of the nineteenth-century U.S.\ railroad network through a market-access index, but Appendix Table~3 reports estimates under a range of alternative exposure maps: they redefine market access using different measures of ``market size'' (population versus county wealth) and restrict the set of trading partners (for example, only counties beyond certain distance thresholds, only urban counties, only major cities, or only New York City).}
Second, even given a particular functional form for the exposure mapping, empirical researchers must still select \(\theta\) and account for the uncertainty this choice introduces into downstream policy estimates. For example, \citet{egger2022general} note that they ``had no a priori knowledge of the relevant distances over which general equilibrium effects might operate\ldots''. In practice, applied papers often report results for several plausible values of \(\theta\), or select a preferred specification using model-selection criteria or informal diagnostics \citep[e.g.][]{currie2015toxicplants,donaldson2016railroads,egger2022general}. Yet standard errors and confidence intervals typically condition on this choice, treating \(\theta\) as fixed rather than as a source of estimation uncertainty.\footnote{In \citet{egger2022general}, local price changes are regressed on exposure measures defined over multiple distance buffers, with the outer radius chosen by a Schwarz/Bayesian information criterion; the selected 0--2 km band is then used as the main exposure definition in the reduced-form analysis \citep[][Section~4.2]{egger2022general}. \citet{currie2015toxicplants} estimate effects within several distance bands around polluting manufacturing plants and report alternative distances, but focus on a preferred near/far comparison, 0--1 versus 1--2 miles, guided by atmospheric dispersion evidence \citep[][pp.~278, 299--312]{currie2015toxicplants}. \citet{donaldson2016railroads} recompute market access under alternative trade elasticities as robustness checks \citep[][pp.~843--844, Appendix Table~3]{donaldson2016railroads}. In each case, variation across \(\theta\) is treated as specification analysis rather than propagated into sampling uncertainty.}
To address these challenges in choosing and calibrating exposure mappings, this paper develops a design-based framework for testing and estimating exposure mappings in settings with spillovers. A key difficulty in such settings is that correctly specifying the outcome model is especially challenging given the complex dependence induced by spillovers. We adopt a design-based approach that sidesteps this by treating potential outcomes as fixed for the experimental sample and letting all randomness come from the known assignment mechanism; this randomization serves as the key assumption needed to test and conduct inference on the exposure mapping. Starting from a proposed exposure map $g(W; X_i, \theta)$, we develop tools to (i) test whether the implications of this map are consistent with observed outcomes under the known assignment mechanism and (ii) estimate the tuning parameters $\theta$. To quantify uncertainty, we develop a design-based law of large numbers and central limit theorem for GMM estimators under spatial and network dependence, characterize an efficiency bound and optimal moments within the class of design-based estimators, and propagate the resulting exposure-map uncertainty into downstream policy estimands.
Formally, we work with a known randomized (or quasi-randomized) assignment design \(\mathcal{D}_n\) and a proposed exposure map \(g(W;X_i,\theta)\). A key observation is that a correctly specified exposure mapping implies an \emph{exposure sufficiency} property: there exists a true parameter \(\theta_0\) such that potential outcomes depend on the full treatment assignment vector only through the exposure mapping \(g(W;X_i,\theta_0)\), so that conditional on \(g(W;X_i,\theta_0)\) the remaining details of \(W\) carry no additional outcome-relevant information.
This property implies a large set of orthogonality conditions between outcomes and functions of \(W\), which we construct via what we call \emph{design-side residuals}. Given any integrable outcome transformation \(\phi\) and design function \(\psi\), we orthogonalize \(\psi(W)\) with respect to the exposure mapping by subtracting its conditional expectation given \(g(W;X_i,\theta)\) \citep[in the spirit of][]{BorusyakHull2023}, and for each unit \(i\) define the design-side residual
\(
R_{i,\theta}{\psi}(W)
:=
\psi(W)
-
\mathbb{E}\big[\psi(W)\mid g(W;X_i,\theta)\big],
\)
where the conditional expectation is taken with respect to \(\mathcal{D}_n\), and thus can be computed by repeatedly sampling from the known design. Under exposure sufficiency and (quasi-)random assignment, we show that at the true parameter these design-side residuals are orthogonal to any function of unit $i$'s outcome:
\(
\mathbb{E}\big[\phi(Y_i)\, R_{i,\theta_0}{\psi}(W)\big] = 0
\)
for every \(\phi\) and \(\psi\). Varying \((\phi,\psi)\) generates a family of moment conditions that form the basis for GMM estimation; when the system is overidentified, the associated \(J\)-test assesses whether the proposed exposure mapping is consistent with the design.
In this way, a single design-side residualization strategy supports both of our key objectives: evaluating the fit of a proposed exposure map and learning about the tuning parameter \(\theta_0\) from the data.
For inference, we establish consistency and asymptotic normality of the design-based GMM estimator under spatial and network dependence. We work in the affinity-set dependence framework of \citet{chandrasekhar2023general}, which nests classical structures such as mixing random fields, m-dependence, and dependency graphs as special cases. Existing results in this framework are pointwise; to our knowledge we are the first to establish uniform consistency and asymptotic normality in it, and we give primitive conditions that cover both smooth exposure maps (gravity and market-access kernels) and non-smooth ones (the ring radius).
We also characterize efficiency within this moment class: there is an asymptotic-variance lower bound for any moment the framework constructs, and a feasible sieve attains it.
Finally, we carry the resulting uncertainty about $g(W; X_i, \theta)$ into downstream policy estimands (e.g., average effects under counterfactual assignment rules) so that inference reflects both the experimental variation and uncertainty about the exposure specification itself.
We illustrate the framework in two applications to large-scale anti-poverty
programs in development economics: the large-scale public-works reform in rural India analyzed
by \citet{muralidharan2023generaleq}, and the GiveDirectly cash-transfer
experiment in rural Kenya studied by \citet{egger2022general} together with the
follow-up structural analysis of \citet{walker2024slack}. Both study large interventions in
low-income economies, are explicitly motivated by general-equilibrium
spillovers, involve spatially linked local markets, and rely primarily on
ring-based exposure mappings.
The two applications yield contrasting conclusions. In the
\citet{muralidharan2023generaleq} application, the design-based
overidentification tests do not reject their ring specification, and the
estimated radii broadly support the original 20 km labor-market scale for the
main income outcomes.
In the \citet{egger2022general} application,
by contrast, the ring specification is rejected for several core outcomes, and our support diagnostics reject the original 2 km radius for all outcomes,
with the smallest non-rejected supports often being 4--6 km or larger. This pattern is consistent with the follow-up analysis of
\citet{walker2024slack}, which argues that very local rings can miss
market-level ambient spillovers that are common to households within local
markets. Consistent with this narrative, we obtain a local fiscal transfer multiplier of 1.57, which is smaller than \citet{egger2022general}'s original 2.5, and much closer to the model-predicted multiplier of 1.54 by \citet{walker2024slack}.
Together, these applications illustrate the value of treating exposure mappings as objects of design-based inference rather than fixed researcher choices: the framework can support an economically motivated map, detect and guide revisions when it does not, and propagate the resulting uncertainty into policy-relevant estimates.
\subsection{Related literature}
\paragraph{Design-based uncertainty.}
We adopt a finite-population, design-based perspective in which $\{Y_i(\cdot),X_i\}_{i=1}^n$ are fixed and randomness arises only from the assignment mechanism. This tradition traces to \citet{Neyman1923} and motivates modern distinctions between design-based and superpopulation uncertainty in regression and experimental analyses \citep{AbadieAtheyImbensWooldridge2020}. Beyond fully randomized designs, \citet{RambachanRoth2025} develop a design-based theory of uncertainty for canonical quasi-experimental estimators when treatment propensities can vary across units, and \citet{LiDing2017} provide finite-population central limit theorems yielding $\sqrt{n}$-Gaussian approximations under randomization-based dependence. Our contribution operates within this design-based framework but targets moment conditions implied by exposure-sufficiency restrictions for parameterized exposure mappings, under the cross-unit dependence induced by interference.
\paragraph{Interference and exposure mappings.}
Foundational potential-outcomes treatments of interference include \citet{Sobel2006} and \citet{HudgensHalloran2008}. Building on this tradition, \citet{AronowSamii2017} formalize \emph{exposure mappings} and develop randomization-based estimators for causal effects under general (but specified) interference structures, while \citet{SavjeAronowHudgens2021} clarify large-sample behavior of conventional estimators under unknown interference and show that standard variance formulas can fail. \citet{Leung2022ANI} allows treatments assigned to increasingly distant units to have smaller but nonzero effects under approximate neighborhood interference. Recent work also studies what can and cannot be learned about interference structure from the design itself: \citet{GaoHarshawSavjeWang2026} show that no specification test can have uniform power against any richer exposure-mapping alternative to an exposure-mapping model\footnote{See Remark \ref{rem:gaocomp} for detailed discussion.}, while \citet{Zhong2025Unconditional} develops finite-sample randomization tests for the existence and extent of interference (for example, whether spillovers vanish beyond a given distance) under minimal assumptions on the network. Complementary work emphasizes robustness to unknown spillover mechanisms within broad classes and studies estimators (and sometimes designs) with minimax or neighborhood-adaptive guarantees; see \citet{belloni2022neighborhood} and \citet{faridani2024linear}. Our approach is complementary to these robust and testing-based methods: rather than remaining fully agnostic about structure or testing a partially sharp no-spillover null, we take the parametric exposure maps used in applied work (e.g.,\ ring, gravity, market-access) as maintained and use the known design to (i) estimate the tuning parameter $\theta_0$ and (ii) test the associated exposure-sufficiency restrictions via overidentifying, design-implied orthogonality conditions.\footnote{A further complementary literature asks how exposure-based estimands should be interpreted when the researcher's chosen exposure mapping is misspecified. \citet{Savje2024Misspecified} separates the use of an exposure mapping to define an estimand from the assumption that the mapping captures the complete causal structure, and establishes conditions under which exposure effects remain estimable under misspecification. \citet{ParkYang2026} instead take the marginal policy effect as primitive and show that a researcher-chosen exposure mapping induces a pseudo-true outcome model and a corresponding decomposition into direct and spillover components.}
\paragraph{Testing models with trusted shocks.}
A line of work going back to \citet{Lucas1980} views structural models as objects to be disciplined by their responses to shocks with well-understood sources and exogeneity properties. In quantitative trade, \citet{AdaoCostinotDonaldson2023} propose IV-based goodness-of-fit statistics that compare a model's predicted response to quasi-experimental tariff shocks with the observed response. We bring this ``trusted shocks'' logic to spillovers and exposure mappings: the parametric exposure family $g(W;X_i,\theta)$ plays the role of the structured model, a structural restriction on how treatments propagate even in otherwise reduced-form designs, the (quasi-)experimental assignment $W\sim\mathcal{D}_n$ (or $W\sim G_n(\cdot\mid X)$) provides the trusted shock, and the design-implied orthogonality conditions deliver both specification tests for the maintained exposure class and design-based estimators of the exposure parameter $\theta_0$.
\paragraph{Orthogonalization and recentering.}
Our design-side residualization builds on a long tradition of orthogonalization in semiparametric and GMM settings, including the partially linear model of \citet{robinson1988root} and the construction of ``orthogonal instruments'' in IO \citep{AckerbergCrawford2009,AckerbergCrawfordHahn2011,AndrewsBarahonaGentzkowRambachanShapiro2025}. Closest in spirit is the recentering approach of \citet{BorusyakHull2023} and \citet{borusyak2025estimating}, who study settings where a known shock design generates a constructed regressor (a formula instrument) and recenter it by subtracting its conditional mean given covariates to obtain orthogonal moments for a downstream causal or structural parameter. Extensions to optimality are provided by \citet{BorusyakHull2026Optimal}. Whereas that work uses recentering to identify and estimate a downstream parameter given a fixed formula or exposure, we apply the same orthogonalization logic upstream to discipline and test the exposure map itself: we treat $g(W;X_i,\theta)$ as the object of interest and use the known (quasi-)experimental design to identify, estimate, and test $\theta_0$. Relatedly, \citet{Ritzwoller2025Spillovers} develops reweighting procedures for proximity-exposure regressions using residualized proximity measures to isolate variation orthogonal to alternative mediating channels.
The rest of the paper is organized as follows.
Sections~\ref{sec:design-orthogonality}--\ref{sec:db-optimal-instruments} develop
the framework: identification via design-based orthogonality
(Section~\ref{sec:design-orthogonality}), a large-sample theory under spatial and
network dependence (Section~\ref{sec:DB-consistency-AN-affinity}), and the
efficient choice of moments (Section~\ref{sec:db-optimal-instruments}). Section~\ref{sec:stage2} propagates
the estimated map into downstream estimands, and
Section~\ref{sec:applications-main} presents the two applications.
\section{Setup and Moment Conditions}
\label{sec:design-orthogonality}
\subsection{Design, outcomes, and exposure maps}
We observe units $i=1,\dots,n$ (regions, households, or network nodes) and a treatment
assignment vector $W = (W_1,\dots,W_n)^\top$ drawn from a known experimental (or
quasi-experimental) design $\mathcal{D}_n$ (e.g., complete, Bernoulli, stratified,
cluster-randomized, shock-based, etc.).
The assignment takes values in a known support $\mathcal{W}$. Binary
treatment, $\mathcal{W}=\{0,1\}^n$, is the leading case, but nothing in what
follows requires it: the individual assignment $W_i$ may be multivalued or continuous, as with the
per-village transfer amounts in our second application.
For each assignment $w \in \mathcal{W}$, unit $i$ has a potential outcome $Y_i(w)$, a
function of the \emph{entire} assignment vector $w$ rather than unit $i$'s own
assignment $w_i$ alone; this is what allows unit $i$'s outcome to depend on other
units' treatments. The realized outcome is $Y_i = Y_i(W)$.
Let $X_i$ denote observed unit-level information, taking values in a set $\mathcal{X}$, that may be relevant for how
assignment affects unit $i$, such as location, network links, strata, or baseline
covariates. An \emph{exposure map} is a function
\[
g : \mathcal{W} \times \mathcal{X} \times \Theta \to \mathbb{R}^k,
\]
with parameter $\theta \in \Theta \subset \mathbb{R}^p$ and $k \ll n$, intended to
summarize the aspects of the assignment that are relevant for $Y_i$ through
\[
g(W;X_i,\theta).
\]
When there is no risk of confusion, we use the shorthand notation
\[
g_i(W;\theta) := g(W;X_i,\theta),
\]
and write $g_i(w;\theta)$ for the exposure induced by a nonrandom assignment
$w \in \mathcal{W}$.
Throughout, we adopt a finite-population, design-based perspective: the potential-outcome
schedule $\{Y_i(\cdot)\}_{i=1}^n$ and covariates $\{X_i\}_{i=1}^n$ for the experimental (or quasi-experimental) sample are treated as fixed (or, equivalently, conditioned upon), and all randomness
comes from $W \sim \mathcal{D}_n$.
We write $\mathbb{E}$ and $\mathbb{P}$ for expectation and probability with respect to $\mathcal{D}_n$,
conditioning implicitly on this fixed schedule of potential outcomes and covariates.
This perspective, as in
\citet{Neyman1923,AbadieAtheyImbensWooldridge2020,LiDing2017}, anchors inference to the specific network, market, or spatial
environment actually exposed to the policy, rather than positing a hypothetical
superpopulation experiment in which entire economies, including their equilibrium
prices, networks, and cross-unit dependence, are repeatedly resampled and subjected
to new assignments $W$. Such an experiment is least credible exactly when spillovers
are system-wide, since it would then require strong assumptions about how the joint
distribution of $\{Y_i(w)\}_{i,w}$ and the broader equilibrium environment vary
coherently across draws. The design-based framework is correspondingly most appealing
here: taking the realized economy as fixed and the known randomization or shock design
as the sole source of uncertainty, it delivers a transparent basis for inference
without additional assumptions on the population-generating process.
\subsection{Examples of exposure maps}
\label{subsec:three-exposure-maps}
As a concrete illustration, we document three families of exposure maps that recur in applied work. Let
\(d(i,j)\) denote a fixed, nonstochastic distance between units \(i\) and \(j\), where
distance may be geographic, travel-time, or network distance.
\begin{example}[Ring exposure]
\label{ex:ring-egger}
A ring exposure map imposes a hard spatial cutoff: only treated units within distance
\(\theta\) of unit \(i\) contribute to exposure. Define the row-normalized weights
\begin{equation}
a_{ij}(\theta)
:=
\frac{\mathbf 1\{d(i,j)\le \theta\}\,\mathbf 1\{j\neq i\}}
{\sum_{k=1}^n \mathbf 1\{d(i,k)\le \theta\}\,\mathbf 1\{k\neq i\}},
\qquad
a_{ii}(\theta):=0,\footnote{
We use the convention $a_{ij}(\theta):=0$ for all $j$ when no other unit lies within distance $\theta$ of $i$ (empty denominator).}
\label{eq:ring-exposure-perunit}
\end{equation}
and let
\[
g^{\mathrm{ring}}(W;X_i,\theta)
:=
\sum_{j=1}^n a_{ij}(\theta) W_j .
\]
Thus \(g^{\mathrm{ring}}(W;X_i,\theta)\) is the average treatment status among units
lying within radius \(\theta\) of \(i\). This is the logic behind the distance-buffer
specifications in \citet{egger2022general}. It is also the main exposure design in
\citet{muralidharan2023generaleq}, where the baseline specification uses the treated share
of locations within a fixed \(20\) km radius.
\end{example}
\begin{example}[Smooth spatial decay]
\label{ex:smooth-spatial-decay}
Smooth spatial-decay maps replace the hard cutoff in a ring design with weights that decline
continuously with distance. Let \(L_j>0\) denote a pre-treatment measure of the economic size
or attractiveness of location \(j\), such as population, employment, or market size. Define
\begin{equation}
a_{ij}(\theta)
:=
\frac{K(d(i,j),L_j;\theta)\,\mathbf 1\{j\neq i\}}
{\sum_{k=1}^n K(d(i,k),L_k;\theta)\,\mathbf 1\{k\neq i\}},
\qquad
a_{ii}(\theta):=0,
\label{eq:smooth-spatial-decay-weights}
\end{equation}
and
\begin{equation}
g^{\mathrm{ssd}}(W;X_i,\theta)
:=
\sum_{j=1}^n a_{ij}(\theta) W_j .
\label{eq:smooth-spatial-decay-exposure}
\end{equation}
This formulation nests several familiar kernels. One example is a gravity-style exponential
kernel,
\[
K_{\mathrm{grav}}(d,L;\theta) := L\,\exp(-\theta d),
\]
under which exposure decays smoothly with distance at rate \(\theta\), used in \citet{franklin2024urban,walker2024slack}. Another important
example is a market-access kernel with power decay,
\[
K_{\mathrm{ma}}(d,L;\theta) := L(1+\alpha d)^{-\theta},
\]
which is closely related to market-access measures in \citet{redding2004economic} and
\citet{donaldson2016railroads}. This is the alternative exposure class considered in
Appendix~B of \citet{muralidharan2023generaleq}, where the authors use the form
\((1+\alpha d)^{-\theta}\) with \(\alpha=1/100\) and fix \(\theta=8\) based on
\citet{donaldson2016railroads}.
\end{example}
\begin{example}[Network exposure]
\label{ex:network-distance}
When spillovers propagate through a known baseline network, distance may be measured in graph
steps rather than kilometers. The ring exposure in Example~\ref{ex:ring-egger} also covers network spillovers as a
special case if distance is interpreted as graph distance rather than geographic distance. Let \(G=(V,E)\) be a fixed graph on nodes
\(V=\{1,\dots,n\}\), and let \(\mathrm{dist}_G(i,j)\) denote shortest-path distance on
\(G\). For \(\theta\in\{1,2,\dots\}\), define the neighborhood
\[
\mathcal N_{\le \theta}(i)
:=
\{\,j\neq i : \mathrm{dist}_G(i,j)\le \theta\,\}.
\]
A natural scalar exposure map is then
\begin{equation}
g^{\mathrm{net}}(W;X_i,\theta)
:=
\frac{1}{|\mathcal N_{\le \theta}(i)|}
\sum_{j\in\mathcal N_{\le \theta}(i)} W_j,
\label{eq:network-hop-exposure}
\end{equation}
with the convention \(g^{\mathrm{net}}(W;X_i,\theta)=0\) if
\(|\mathcal N_{\le \theta}(i)|=0\).
In social-network applications, researchers often further decompose this into the share of
treated friends, the share of treated friends of friends, and so on up to distance
\(\theta\).\footnote{In the network-interference literature this is often described as a
\(K\)-hop restriction. We write the truncation parameter as \(\theta\) here to keep notation
consistent across the three exposure families.} When \(\theta=1\),
\eqref{eq:network-hop-exposure} reduces to the familiar share of treated friends. This type
of restriction is common in empirical work on peer effects and network spillovers; see, for
example, \citet{cai2015social} and \citet{BramoulleDjebbariFortin2009}.
\end{example}
Across these examples, the role of \(\theta\) is always the same: it indexes how far
spillovers reach or how quickly they decay. Table~\ref{tab:exposure-cartoon} illustrates the three families.
\begin{table}[t]
\centering
\caption{Illustration of canonical exposure maps}
\label{tab:exposure-cartoon}
\small
\begin{tabularx}{\textwidth}{@{} l
m{3.0cm}
X
@{}}
\toprule
& Illustration & Informal description \\
\midrule
\multicolumn{3}{l}{} \\[-0.75ex]
Ring
& \raisebox{-0.5\height}{
\begin{tikzpicture}[
scale=0.7,
every node/.style={circle, draw, inner sep=1pt, minimum size=5pt},
treated/.style={circle, draw, fill=black, inner sep=1pt, minimum size=5pt},
untreated/.style={circle, draw, fill=white, inner sep=1pt, minimum size=5pt}
]
\node[treated,label=above:{\scriptsize $i$}] (i) at (0,0) {};
\draw[dashed] (0,0) circle [radius=1.2];
\foreach \angle/\style in {90/treated,150/treated,210/untreated,330/treated}{
\node[\style] at (1.2*cos{\angle},1.2*sin{\angle}) {};
}
\foreach \angle in {30,270}{
\node[untreated] at (2.0*cos{\angle},2.0*sin{\angle}) {};
}
\end{tikzpicture}
}
& Exposure is the average treatment status among units lying within distance
$\theta$ of unit $i$. This is the main specification of
\citet{muralidharan2023generaleq} and the distance-buffer logic used in
\citet{egger2022general}. \\[0.9ex]
Smooth spatial decay
& \raisebox{-0.5\height}{
\begin{tikzpicture}[
scale=0.8,
every node/.style={circle, draw, inner sep=1pt, minimum size=4pt},
center/.style={circle, draw, fill=black, inner sep=1pt, minimum size=5pt},
ring/.style={line width=0.25pt, draw=black!35},
arrowstyle/.style={-stealth}
]
\node[center,label=above:{\scriptsize $i$}] (i) at (-0.1,0) {};
\foreach \r in {0.7,1.2,1.7} {
\draw[ring] (i) circle[radius=\r];
}
\node (n1) at ($(i)+(0.7,0.2)$) {};
\node (n2) at ($(i)+(-1.0,0.7)$) {};
\node (n3) at ($(i)+(1.1,-0.7)$) {};
\node[minimum size=7pt] (n4) at ($(i)+(-1.6,-0.2)$) {};
\draw[arrowstyle, ultra thick] (n1) -- (i);
\draw[arrowstyle, thick] (n2) -- (i);
\draw[arrowstyle, thin] (n3) -- (i);
\draw[arrowstyle, very thin] (n4) -- (i);
\end{tikzpicture}
}
& All locations can affect $i$, but their influence declines smoothly with
distance. Gravity-style exponential kernels and market-access kernels of the
form $(1+\alpha d)^{-\theta}$ are leading examples; the latter is the
alternative exposure class considered in Appendix~B of
\citet{muralidharan2023generaleq}. \\[0.9ex]
Network
& \raisebox{-0.5\height}{
\begin{tikzpicture}[
scale=0.6,
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& Exposure is averaged over nodes within graph distance $\theta$ of $i$. In
social-network applications this often corresponds to the share of treated
friends, or more generally treated friends, friends of friends, and so on,
up to distance $\theta$. \\
\bottomrule
\end{tabularx}
\end{table}
\subsection{Exposure sufficiency}
\label{subsec:randomization-exposure-sufficiency}
We now state the two primitive conditions that generate the design-based moment
restrictions. The first condition is the usual finite-population randomization
condition: the potential-outcome schedule is fixed, and all randomness comes from
the known assignment law.
\begin{assumption}[Randomized assignment]
\label{ass:rand-indep}
The assignment $W$ is drawn from a known law $\mathcal{D}_n$ that does
not depend on the potential outcomes. Formally, for every collection of potential
outcomes $\{Y_i(\cdot)\}_{i=1}^n$,
\[
W \,\big|\, \{Y_i(\cdot),X_i\}_{i=1}^n \sim \mathcal{D}_n .
\]
\end{assumption}
The second condition is the exposure-map hypothesis. It states that, at the
correct value $\theta_0$, the candidate exposure map contains all information in the
assignment vector that is relevant for unit $i$'s outcome.
\begin{hypothesis}[Exposure map]
\label{hyp:exposure-suff}
\label{ass:exposure-suff}
\label{ass:exposure-sufficiency}
The exposure mapping $g(W;X_i,\theta)$ is \emph{well specified} if there exist
$\theta_0\in\Theta$ and measurable functions
$\widetilde Y_i:\mathbb R^k\to\mathbb R$ such that, for every assignment $w$,
\[
Y_i(w)
=
\widetilde Y_i\bigl(g(w;X_i,\theta_0)\bigr),
\qquad i=1,\dots,n .
\]
\end{hypothesis}
Thus Hypothesis~\ref{ass:exposure-suff} makes the exposure \(g(W;X_i,\theta_0)\) a summary of the assignment for unit $i$:
the map is \emph{well specified} when the exposure captures everything in $W$ relevant to
$Y_i$. The goal is to test this restriction and learn $\theta_0$ using the known design
\(\mathcal{D}_n\).
Our main result is that, combined with the known design, well specification has an
observable consequence. Because the assignment law is known and does not depend on the
potential outcomes, conditioning on the exposure removes all dependence between $Y_i$ and
the residual variation in $W$. We call this consequence exposure sufficiency. It is what
makes the map testable: the design leaves assignment variation that a well-specified
exposure must render irrelevant to $Y_i$, and any leftover dependence is evidence against
the map.
\begin{theorem}[Exposure sufficiency]
\label{thm:design-orthogonality}
\label{thm:design-orthogonality-unit}
\label{thm:exposure-sufficiency}
Suppose Assumption~\ref{ass:rand-indep} and Hypothesis~\ref{hyp:exposure-suff} hold.
Then, for every unit \(i\),
\[
Y_i \;\perp\!\!\!\perp\; W
\;\big|\;
g(W;X_i,\theta_0)
\]
under the design distribution.
\end{theorem}
To see what the theorem requires, and why it can be tested, consider the ring map
of Example~\ref{ex:ring-egger}, where $g(W;X_i,\theta)$ is the treated share of units within
radius $\theta$ of unit $i$. Applied researchers already probe this choice informally:
one draws rings of half a mile, a mile, two miles, and looks for the radius at which
estimated spillovers level off. The design-based restriction makes that intuition precise.
Well specification at radius $\theta_0$ says unit $i$'s outcome depends on the assignment
only through the within-$\theta_0$ treated share: two assignments with the same share
produce the same outcome for $i$, no matter which of those neighbors are treated and
regardless of the treatment of any unit beyond $\theta_0$. This is a substantive economic
restriction, and it can fail in two ways: a treated unit just outside the radius still
affects $i$, or the identity rather than the count of treated neighbors matters. The known
randomization turns each failure into something observable. If the ring is correct, then
under the design the treatments of units outside radius $\theta_0$ are uncorrelated with
$Y_i$ once we condition on the within-ring share; leftover correlation is evidence that
spillovers reach past $\theta_0$. This is exactly the equal-to-zero condition that the
moments of the following subsection formalize, both to test a candidate radius and to
select $\theta_0$.
\subsection{Orthogonal moments}
\label{subsec:orthogonal-moments}
The ring discussion tested one exposure map with one natural statistic, the
treatments of units outside the radius. The conditional independence in
Theorem~\ref{thm:exposure-sufficiency} says much more: \emph{any} function of the
assignment, once purged of what the candidate exposure explains, must be unrelated to
\emph{any} function of the outcome. We encode this with a residual that strips from a
design function the part predictable from the exposure.
To construct the moments, let
\(\psi:\mathcal{W}\times\mathcal{X}\to\mathbb R\) be any integrable design function, possibly depending on the unit's covariates \(X_i\).\footnote{The design residual \(R_{i,\theta}\) below evaluates the design function at the unit's covariates \(X_i\), so the unit-centering enters through the same index \(i\) that \(R_{i,\theta}\) already carries. This covers the leading applied case, in which \(\psi\) aggregates the assignment in a neighborhood of \(X_i\) (for example, population-weighted treatment averages over distance bands around unit \(i\)). To avoid clutter we continue to write \(\psi(W)\), and \(\psi_m(W)\) for dictionary elements, each understood to be evaluated at the relevant unit's covariates inside \(R_{i,\theta}\).} For each unit \(i\) and candidate parameter
\(\theta\), define the design residual
\begin{equation}
R_{i,\theta}{\psi}(W)
:=
\psi(W;X_i)
-
\mathbb{E}\!\left[\psi(W;X_i)\mid g(W;X_i,\theta)\right],
\label{eq:Ri-theta-def}
\end{equation}
where the conditional expectation is taken under the known design \(\mathcal{D}_n\).
This residual removes from \(\psi(W)\) the component explained by the candidate
exposure \(g(W;X_i,\theta)\).
\begin{corollary}[Orthogonal moments]
\label{cor:design-moment-unit}
Under the conditions of Theorem~\ref{thm:exposure-sufficiency}, for any outcome
transformation \(\phi:\mathbb R\to\mathbb R\) and design function \(\psi\)
with \(\mathbb{E}[\phi(Y_i)^2]<\infty\) and \(\mathbb{E}[\psi(W)^2]<\infty\),
\begin{equation}
\label{eq:design-moment-unit}
\mathbb{E}\!\left[
\phi(Y_i)\,
R_{i,\theta_0}{\psi}(W)
\right]
=
0 .
\end{equation}
Equivalently,
\[
\mathbb{E}\!\left[
\phi(Y_i)
\left\{
\psi(W)
-
\mathbb{E}[\psi(W)\mid g(W;X_i,\theta_0)]
\right\}
\right]
=
0 .
\]
\end{corollary}
Two features of the corollary matter for what follows. First, it is agnostic about
the map: nothing in \eqref{eq:design-moment-unit} is special to the ring, so the same
construction disciplines gravity, market-access, or network exposures, with only
$g(W;X_i,\theta)$ changing. Second, the equalities hold at $\theta_0$ for \emph{every}
admissible $(\phi,\psi)$, which is what lets a single strategy serve two ends: matching
them identifies and estimates $\theta_0$, while a candidate map that violates them is
detectable through the resulting overidentification.
For estimation, fix a finite dictionary of moment-generating pairs
\[
\{(\phi_m,\psi_m):m=1,\dots,M\},
\]
where each \(\phi_m:\mathbb R\to\mathbb R\) is an outcome transformation and each
\(\psi_m:\mathcal{W}\times\mathcal{X}\to\mathbb R\) is a design function. Define the $i$-specific moment
contribution
\begin{equation}
\eta_{m,i,n}(\theta)
:=
\phi_m(Y_i)\,R_{i,\theta}{\psi_m}(W),
\label{eq:single-unit-score}
\end{equation}
and the corresponding sample moment
\[
\eta_{m,n}(\theta)
:=
\frac{1}{n}\sum_{i=1}^n\eta_{m,i,n}(\theta).
\]
Stacking the \(M\) moments gives
\[
\eta_n(\theta)
:=
\bigl(
\eta_{1,n}(\theta),\dots,\eta_{M,n}(\theta)
\bigr)^\top .
\]
A design-based GMM estimator is then
\begin{equation}
\hat\theta_n
\in
\arg\min_{\theta\in\Theta}
Q_n(\theta),
\qquad
Q_n(\theta)
:=
\eta_n(\theta)^\top \Lambda_n \eta_n(\theta),
\label{eq:design-gmm-criterion}
\end{equation}
where \(\Lambda_n\) is a positive semidefinite weight matrix.
\begin{remark}[Practical implementation]
\label{rem:computing-design-projection}
The conditional expectations
\[
\mathbb{E}[\psi_m(W)\mid g(W;X_i,\theta)]
\]
are design-side objects: because the assignment law $\mathcal{D}_n$ is known
(Assumption~\ref{ass:rand-indep}), each is a functional of the design alone and
involves neither the potential outcomes nor any outcome model, so it can be computed
without additional assumptions. When enumeration of the support of $\mathcal{D}_n$ is
feasible, the projection is evaluated directly. Otherwise, one draws assignments
$W^{(b)}\sim\mathcal{D}_n$, $b=1,\dots,B$, and approximates
$\mathbb{E}[\psi_m(W)\mid g(W;X_i,\theta)=s]$ by regressing $\psi_m(W^{(b)})$ on a flexible
function of the simulated exposure $g(W^{(b)};X_i,\theta)$: for a discrete exposure,
by averaging $\psi_m$ within its realized values; for a continuous exposure, by a
low-order polynomial or kernel smoother in $s$, with $B$ and the smoother chosen so
that the residual approximation error is first-order negligible. In the applications
we use a quadratic in the exposure index across a few hundred placebo draws.
When the conditional distribution $W\mid g(W;X_i,\theta)$ is degenerate, so that the
exposure value pins down $\psi_m(W)$, the design residual $R_{i,\theta}\psi_m(W)$ is
identically zero and unit $i$ contributes nothing to the moment. This reflects a lack
of residual assignment variation left after conditioning, not a failure of the
construction, and it is why we work with the conditional-moment projection, which
pools information across draws and neighboring exposure values, rather than with exact
cell-by-cell conditioning. The design-based identification condition in
Section~\ref{sec:DB-consistency-AN-affinity} makes the amount of such residual variation that the moments require explicit.
\end{remark}
\begin{remark}[Overidentification and testing exposure maps]
\label{rem:overidentification-exposure-testing}
When \(M>\dim(\theta)\), the system is overidentified. The associated design-based
\(J\)-statistic can therefore be used to test the maintained exposure specification or
to compare alternative exposure maps. Section~\ref{sec:DB-consistency-AN-affinity}
derives the large-sample behavior of \(\hat\theta_n\), and
Section~\ref{sec:db-optimal-instruments} studies the efficient choice of
moments.
\end{remark}
\begin{remark}[Multi-unit restrictions]
\label{rem:joint-exposure-future-work}
Hypothesis~\ref{ass:exposure-suff} also implies orthogonality restrictions
for any finite set of units. For any finite
$S\subset\{1,\dots,n\}$, let
\[
Y_S := (Y_i)_{i\in S},
\qquad
g_S(W;\theta) := \bigl(g(W;X_i,\theta)\bigr)_{i\in S}.
\]
If the exposure map is well specified, then
\[
Y_S \;\perp\!\!\!\perp\; W
\;\big|\; g_S(W;\theta_0),
\]
and hence, for any square-integrable $\phi_S$ and $\psi$,
\[
\mathbb{E}\!\left[
\phi_S(Y_S)
\left\{
\psi(W)
-
\mathbb{E}[\psi(W)\mid g_S(W;\theta_0)]
\right\}
\right]
=0.
\]
These restrictions could generate moments based on pairs or larger groups of
units, such as moments involving $(Y_i,Y_j)$ and the joint exposure vector
$(g(W;X_i,\theta_0),g(W;X_j,\theta_0))$. We do not pursue the full multi-unit
moment system here: it greatly expands the class of possible moments, and it is
less clear how to choose among them. Instead, the paper focuses on the unit-level
moments, which nest the moments used in applied work and already deliver tractable
estimators, specification tests, and efficiency analysis. Developing practical
procedures that exploit the broader cross-unit restrictions is left for future work.
\end{remark}
\begin{remark}[Quasi-experimental shock designs and relation to recentering]
\label{rem:quasi-and-BH}
Many applications of exposure mappings are based on \emph{quasi-experimental} variation
rather than literal randomized assignment.
In such settings, the researcher observes a realized shock vector $W$ (e.g.,\ line
openings in a transport network, sectoral demand shocks, or other environmental shocks),
treats the baseline covariates $X=(X_1,\dots,X_n)$ as fixed, and specifies a
\emph{shock design} $\mathcal{D}_n^{\text{shock}}$ that captures the as-good-as-random
component of $W$.
Provided this design is known and satisfies the same structural property as
Assumption~\ref{ass:rand-indep},
\[
W \,\big|\, \{Y_i(\cdot)\}_{i=1}^n \sim \mathcal{D}_n^{\text{shock}}
\quad\text{for all potential-outcome schedules},
\]
all of the constructions above go through after replacing $\mathcal{D}_n$ by
$\mathcal{D}_n^{\text{shock}}$.
In practice, $\mathcal{D}_n^{\text{shock}}$ is implemented via permutations, placebo
networks, or other simulation schemes, and the re-randomization step used to approximate
$\mathbb{E}[\psi_m(W)\mid g(W;X_i,\theta)]$ is carried out by simulating $W$ from this shock
design.
This design-side residualization is analogous in spirit to the recentering approach of
\citet{BorusyakHull2023}: both start from a known shock or assignment distribution and
construct transformed instruments or moments that are orthogonal, by design, to certain
components of the assignment.
However, the goals and maintained structures are different.
\citet{BorusyakHull2023} assume a particular linear homogeneous treatment-effect model
and a given formula instrument, and their objective is to identify and estimate the
resulting coefficient $\beta$.\footnote{The recentering logic is not intrinsically tied to a linear model: recently, \citet{borusyak2025estimating} apply it to nonlinear (nested and mixed logit)
demand estimation.}
By contrast, in the present framework the exposure mapping itself is the primary object
of interest: we use the randomized or quasi-random shock design to learn about
$\theta_0$ in $g(W;X_i,\theta_0)$ and to test whether a proposed exposure class is
consistent with the design-implied orthogonality conditions. That said, we provide a general procedure to conduct formal inference on policy functionals in a downstream stage, discussed in Section~\ref{sec:stage2}.
\end{remark}
\section{Design-Based Consistency and Asymptotic Normality}
\label{sec:DB-consistency-AN-affinity}
Section~\ref{sec:design-orthogonality} derived design-based moment restrictions implied
by exposure sufficiency. We now study the large-sample behavior of GMM estimators and
tests constructed from those moments. Our goal is to establish design-based consistency
and asymptotic normality of the GMM estimator, and to develop the associated
overidentification tests of the exposure-map specification. See
Remark~\ref{rem:exact-randomization-tests} for discussion of exact randomization tests.
To separate the asymptotic argument from any particular choice of moments, we work with a
generic finite-dimensional moment contribution
\[
\Psi_i(\theta)\in\mathbb R^q .
\]
In the exposure-mapping application, this vector is obtained by stacking finitely many
residualized design moments,
\[
\Psi_i(\theta)
=
\bigl(
\eta_{1,i,n}(\theta),\dots,\eta_{q,i,n}(\theta)
\bigr)^\top,
\qquad
\eta_{m,i,n}(\theta)
=
\phi_m(Y_i)R_{i,\theta}{\psi_m}(W).
\]
The generic notation lets us state the LLN, consistency, CLT, and asymptotic-normality
results once, in terms of the dependence structure induced by the assignment design. These
results are then used directly for exposure-map estimation here and for the optimal-moment
analysis in Section~\ref{sec:db-optimal-instruments}.
\subsection{Finite-population setup and GMM criterion}
Following the design-based literature (e.g.,\
\citealp{freedman2008regression,AronowSamii2017}), we work with a triangular array
of experiments $\{(\mathcal{U}_n,\mathcal{D}_n)\}_{n\geq 1}$, where both the population
size $N_n:=|\mathcal{U}_n|$ and the assignment design $\mathcal{D}_n$ are allowed to
change with $n$.
For each $n$, let $\mathcal{U}_n=\{1,\dots,N_n\}$ denote the finite experimental
population. For each unit $i\in\mathcal{U}_n$ and parameter
$\theta\in\Theta\subset\mathbb{R}^p$, let $\Psi_i(\theta)\in\mathbb{R}^q$ be a
$q$-dimensional moment vector, with true parameter value $\theta_0$.
The array $\{\Psi_i(\theta)\}_{i\in\mathcal{U}_n}$ is generated by the known
assignment design $\mathcal{D}_n$, while the potential outcomes and baseline
attributes are treated as fixed. The design fixes the distribution of these moments in any given population; the large-sample results below require, in addition, regularity conditions on the dependence the design induces across units, which we impose through the affinity-set structure introduced next.
The sample moment is the
normalized finite-population average
\[
\bar\Psi_n(\theta)
:=
\frac{1}{N_n}\sum_{i=1}^{N_n}\Psi_i(\theta),
\qquad
\mu_n(\theta)
:=
\frac{1}{N_n}\sum_{i=1}^{N_n}\mathbb{E}[\Psi_i(\theta)].
\]
Let $\Lambda_n$ be a symmetric positive semidefinite $q\times q$ weight matrix
with $\Lambda_n\stackrel{p}{\to} \Lambda\succeq 0$, and define the quadratic GMM criterion
\[
Q_n(\theta)
:=
\bar\Psi_n(\theta)^\top \Lambda_n\,\bar\Psi_n(\theta),
\qquad
\hat\theta_n \in \arg\min_{\theta\in\Theta} Q_n(\theta).
\]
In the limit, write
\begin{equation}
\label{eq:Q-theta-unit}
\begin{aligned}
\mu(\theta)
&:=
\lim_{n\to\infty}\mu_n(\theta),\\
Q(\theta)
&:=
\mu(\theta)^\top \Lambda\,\mu(\theta),
\end{aligned}
\end{equation}
whenever the limit exists. In particular, we assume that the limit is centered
at the true parameter,
$ \mu(\theta_0)=0$.\footnote{
This centering follows directly from the
design-based orthogonality result in
Corollary~\ref{cor:design-moment-unit}.}
\subsection{Affinity sets and consistency}
\label{sec:affset}
To accommodate design-induced spatial or network dependence, for each $i\in\mathcal{U}_n$
we fix an \emph{affinity set} $A_i\subseteq\mathcal{U}_n$ with $i\in A_i$ collecting units
whose assignments may have non-negligible covariance with that of unit $i$.
We write $|A_i|$ for its cardinality and allow arbitrary dependence within $A_i$.
In the spatial examples, $A_i$ can be read as a growing geographic or travel-time
ball around unit $i$; in the market-access and gravity examples, it collects
locations whose assignment shocks receive non-negligible kernel weight for $i$;
and in the network examples, it corresponds to a local graph neighborhood whose
radius may grow slowly with $N_n$.
Outside $A_i$ we do not impose conditional independence; instead, we only assume
that the aggregate covariance contribution from $\{j\notin A_i\}$ is
asymptotically negligible relative to the contribution from within $A_i$.
Define centered variables
\[
Z_{i,n}(\theta)
:= \Psi_i(\theta) - \mathbb{E}[\Psi_i(\theta)],
\qquad
\bar Z_n(\theta)
:= \frac{1}{N_n} \sum_{i=1}^{N_n} Z_{i,n}(\theta)
= \bar\Psi_n(\theta)-\mu_n(\theta),
\]
and the \emph{within-affinity covariance matrix}\footnote{We take the affinity
relation to be symmetric ($j\in A_i\Leftrightarrow i\in A_j$), so that
$\Omega_n(\theta)$ is symmetric; this holds in all our examples (metric or
travel-time balls, symmetric kernels, undirected graph neighborhoods), and any
directed relation may be replaced by its symmetrization $\{j:j\in A_i\ \text{or}\
i\in A_j\}$.}
\[
\Omega_n(\theta)
:= \sum_{i=1}^{N_n} \ \sum_{j \in A_i} \
\operatorname{Cov}\!\big(Z_{i,n}(\theta),\, Z_{j,n}(\theta)\big)
\in \mathbb{R}^{q \times q}.
\]
We first state the high-level ULLN and identification requirements. The
following subsection then explains how the ULLN is verified under primitive
conditions in the two leading cases.
\begin{assumption}[Design-based ULLN and identification]
\label{ass:as-lln}
\mbox{}
\begin{enumerate}[label=\textup{AS--LLN\arabic*}, ref=\textup{AS--LLN\arabic*}, leftmargin=1.6cm]
\item\label{ass:as-lln1}
\textbf{Deterministic stabilization.}
There exists a deterministic $\mu:\Theta\to\mathbb{R}^q$ such that
\[
\sup_{\theta\in\Theta}\,\|\mu_n(\theta)-\mu(\theta)\|\ \to\ 0.
\]
\item\label{ass:as-lln2}
\textbf{Uniform LLN for the design-based moments.}
The sample moments converge uniformly in probability to their limits:
\[
\sup_{\theta\in\Theta}\big\|\bar\Psi_n(\theta)-\mu_n(\theta)\big\|
\ \stackrel{p}{\to}\ 0.
\]
\item\label{ass:as-lln3}
\textbf{Parameter space and identification of the population criterion.}
The parameter space $\Theta\subset\mathbb{R}^p$ is compact and $\theta_0\in\Theta$.
The limit map $\mu:\Theta\to\mathbb{R}^q$ from \ref{ass:as-lln1} is continuous on $\Theta$.
Given \ref{ass:as-lln1}, define
$Q(\theta)=\mu(\theta)^\top \Lambda\,\mu(\theta)$ on $\Theta$, and assume
\[
Q(\theta)=0
\quad\Longleftrightarrow\quad
\theta=\theta_0.
\]
\item\label{ass:as-lln4}
\textbf{Weight convergence.}
$\Lambda_n\stackrel{p}{\to} \Lambda\succeq0$.
\end{enumerate}
\end{assumption}
Assumption~\ref{ass:as-lln} is stated at a high level in terms of the
population map $\mu(\theta)$ and the sample map $\bar\Psi_n(\theta)$.
The next result shows that these high-level conditions are sufficient for
consistency.\footnote{Genuinely discrete exposure parameters (for example, an
integer hop-count) are handled by a separate finite-grid selection argument in
Appendix~\ref{app:finite-grid-ulln-proof}.}
\begin{remark}[Interpretable necessary conditions for identification]
\label{rem:identification-necessary}
We impose identification at the criterion level in \ref{ass:as-lln3}. Two necessary conditions specific to the moments
\eqref{eq:single-unit-score} are worth mentioning. First, the
exposure map must coarsen the assignment: if $w\mapsto g(W;X_i,\theta)$ is
injective at some $\theta$, then conditioning on $g(W;X_i,\theta)$ is equivalent
to conditioning on $W$, so $R_{i,\theta}\psi\equiv0$ for every $\psi$ by
\eqref{eq:Ri-theta-def} and the moments are uninformative about that $\theta$;
identification therefore requires strictly positive residual variation,
$\mathbb{E}[(R_{i,\theta}\psi(W))^2]>0$ for some admissible $\psi$. This is the
degeneracy of a continuous exposure parameter paired with a finely resolved
assignment, such as a smooth market-access decay in (near-)continuous
distances; coarsening the assignment, as the ring map already does, removes it.
Second, the outcomes must respond to the assignment: under the sharp null of no
effect on any unit each $\phi(Y_i)$ is design-nonrandom, so
$\mathbb{E}[\phi(Y_i)R_{i,\theta}\psi(W)]=\phi(Y_i)\,\mathbb{E}[R_{i,\theta}\psi(W)]=0$ for
\emph{every} $\theta$ by the law of iterated expectations, and $\theta_0$ is not
separated.
\end{remark}
\begin{theorem}[Design-based GMM consistency]
\label{thm:as-consistency}
Suppose Assumption~\ref{ass:as-lln} holds. Then
\[
\sup_{\theta\in\Theta}\big|Q_n(\theta)-Q(\theta)\big| \;\stackrel{p}{\to}\; 0,
\]
and any sequence of minimizers
$\hat\theta_n\in\arg\min_{\theta\in\Theta} Q_n(\theta)$ satisfies
$\hat\theta_n\stackrel{p}{\to}\theta_0$.
\end{theorem}
\subsection{Primitive conditions for the uniform law of large numbers}
\label{subsec:primitive-ulln-main}
Consistency was established above under the high-level
uniform law of large numbers assumed in \ref{ass:as-lln2},
\[
\sup_{\theta\in\Theta}\bigl\|\bar\Psi_n(\theta)-\mu_n(\theta)\bigr\|\stackrel{p}{\to} 0 .
\]
This subsection gives primitive conditions under which that uniform law holds. The smooth case follows from a standard argument; the ring needs care, because its
sample path is a step function of the radius $\theta$ and the smooth-GMM argument does
not apply. We verify it for the two exposure maps used in the applications, continuing
Examples~\ref{ex:smooth-spatial-decay} and~\ref{ex:ring-egger};
Appendix~\ref{subsec:primitive-LLN-proof} states the primitive conditions more formally
and gives the proofs.
\par\smallskip
\noindent\textbf{Example~\ref{ex:smooth-spatial-decay} (smooth spatial decay), continued.}\ \ignorespaces
When the weights $a_{ij}(\theta)$ vary smoothly with $\theta$, as in the gravity and
market-access kernels, the moment vector is Lipschitz in $\theta$ and a standard
covering argument applies. The condition that is not automatic concerns the design
projection $m_{i,m,\theta}(s):=\mathbb{E}[\psi_m(W)\mid g_i(W;\theta)=s]$, which must itself vary
regularly in both arguments,
\[
\bigl|m_{i,m,\theta}(s)-m_{i,m,\theta'}(s')\bigr|
\le M_{i,n}\bigl(\|\theta-\theta'\|+\|s-s'\|\bigr),
\qquad
\frac1{N_n}\sum_{i=1}^{N_n}\mathbb{E}\,M_{i,n}=O(1).
\]
Smoothness of the exposure map does not by itself deliver this, so we impose the displayed
regularity condition directly as a primitive assumption in Appendix~\ref{subsec:primitive-LLN-proof}.
\par\smallskip
\noindent\textbf{Example~\ref{ex:ring-egger} (ring exposure), continued.}\ \ignorespaces
Raising the radius $\theta$ changes exposure only through the units whose distance to
$i$ crosses $\theta$, so the sample path $\theta\mapsto\bar\Psi_n(\theta)$ moves in steps
and is not differentiable; in place of a smoothness argument, the uniform law rests on a
condition on the distances. That condition is the spatial counterpart of the bounded-density requirement familiar from regression-discontinuity
and threshold-regression designs, with pairwise distance $d_{ij}$ playing the role of the
running variable and the radius $\theta$ that of the cutoff: as a band of radii shrinks,
the weighted mass of distances inside it must vanish. Formally, for an interval
$I\subset\mathbb{R}_{+}$ of length $|I|$,
\[
\lim_{\delta\downarrow0}\ \limsup_{n\to\infty}\
\sup_{|I|\le\delta}\ \frac1{N_n}\sum_{i=1}^{N_n}\sum_{j\ne i}
a_{ij}\,\mathbbm{1}\{d(i,j)\in I\}=0,
\]
with $a_{ij}$ the exposure weights of Example~\ref{ex:ring-egger}. Moving the radius then
shifts only a vanishing share of the moment, so the empirical criterion converges
uniformly and its population limit is continuous in $\theta$ even though every sample path
jumps.
\begin{remark}[Relation to exact randomization tests]
\label{rem:exact-randomization-tests}
A common inferential approach in experiments with spillovers is the finite-sample exact
randomization test. These tests are cleanest for a \emph{sharp} null: one that pins down
each unit's potential outcome under every assignment, so realized outcomes can be
recomputed for any counterfactual assignment and the statistic has a known permutation
distribution. Exposure sufficiency is not sharp: it restricts how outcomes depend on the
assignment through the exposure map but leaves the potential outcomes otherwise
unspecified. For such non-sharp nulls, exact tests remain available by choosing focal
units and conditioning on the assignment cell within which the focal units' outcomes are
invariant; see, for example, \citet{athey2018exact}.
Our target, however, is not a single fixed null but estimation of the continuous tuning
parameter $\theta$, together with inference on the downstream regression coefficients
through the GMM criterion $Q(\theta)$. Exact randomization inference is already awkward
for the coefficient-based analyses in the applied work cited above, and more so for estimating
$\theta$ or evaluating $Q$. We therefore take the asymptotic conditional-moment route
developed in this section, which draws on the implications of exposure sufficiency
directly.
\end{remark}
\subsection{Asymptotic normality}
We now strengthen the LLN conditions above to obtain a CLT and a GMM asymptotic
normality result.
We retain the finite-population, design-based setup and notation introduced
above, and assume Assumption~\ref{ass:as-lln} holds so that
$\hat\theta_n\stackrel{p}{\to}\theta_0$ by Theorem~\ref{thm:as-consistency}.
Asymptotic normality rests on two further ingredients, both imposed at
the population level. We state them formally as Assumption~\ref{ass:AN-AFF} in
Appendix~\ref{app:DB-AN-affinity} and describe their roles here. The first delivers
a \emph{pointwise} central limit theorem for the moment vector at the truth. Because
the design-based moments are dependent across units through the affinity sets of
Section~\ref{sec:affset}, we invoke the affinity-set central limit theorem of
\citet{chandrasekhar2023general}: under bounded fourth moments, decay of the
within-affinity covariances, and a stabilizing aggregate covariance
$\Omega_n/N_n\to\Omega$ with $\Omega$ positive definite, the centered moments satisfy
$\sqrt{N_n}\,\bar\Psi_n(\theta_0)\Rightarrow\mathcal{N}(0,\Omega)$.
The second ingredient carries this pointwise statement to asymptotic normality of the
GMM estimator $\hat\theta_n$. It has two parts: \emph{mean differentiability} of the
population moment map $\mu(\theta)$ at $\theta_0$, with full-rank Jacobian $G$, and
\emph{stochastic equicontinuity} of the centered empirical process
$\mathbb G_n(\theta)=\sqrt{N_n}\big(\bar\Psi_n(\theta)-\mu(\theta)\big)$ near
$\theta_0$. Both are restrictions on the population map and on the empirical process,
not on the individual sample paths $\Psi_i(\theta)$. This is what lets the argument, a
Z-estimator application of \citet[Theorem~3.3.1]{vaart1996weak} to the projected map
$\theta\mapsto G^\top\Lambda_n\bar\Psi_n(\theta)$, cover the non-differentiable ring
path alongside the smooth gravity and market-access kernels; the affinity-set
conditions enter only through the pointwise CLT and the equicontinuity of
$\mathbb G_n$.
\begin{theorem}[Design-based GMM asymptotic normality]
\label{thm:DB-AN-aff}
Under Assumption~\ref{ass:as-lln} and Assumption~\ref{ass:AN-AFF},
\[
\sqrt{N_n}\,(\hat\theta_n-\theta_0)
\ \Rightarrow\
\mathcal{N}\!\Big(
0,\ (G^\top \Lambda\, G)^{-1} G^\top \Lambda\, \Omega\, \Lambda\, G\, (G^\top \Lambda\, G)^{-1}
\Big),
\]
where $G=G(\theta_0)$ and $\Omega$ is the design-based asymptotic covariance in
Assumption~\ref{ass:AN-AFF}\,\textup{\ref{ass:an-aff3}}.
With the optimal weight $\Lambda=\Omega^{-1}$,
the asymptotic variance simplifies to $(G^\top \Omega^{-1}G)^{-1}$.
Moreover, if $\Lambda_n\stackrel{p}{\to}\Omega^{-1}$, then the
efficient-weight case is obtained.
\end{theorem}
\begin{corollary}[Design-based overidentification test]
\label{cor:db-J-test}
Suppose Assumption~\ref{ass:as-lln} and Assumption~\ref{ass:AN-AFF} hold with
$q>p$, and let $\Lambda_n\stackrel{p}{\to}\Omega^{-1}$ as in Theorem~\ref{thm:DB-AN-aff}. Under
the maintained exposure specification (Hypothesis~\ref{hyp:exposure-suff}),
\[
N_n\,Q_n(\hat\theta_n)
\ \Rightarrow\
\chi^2_{\,q-p}.
\]
\end{corollary}
The statistic $N_n Q_n(\hat\theta_n)$ therefore provides a formal test of the
maintained exposure map, as anticipated in
Remark~\ref{rem:overidentification-exposure-testing}: rejection is evidence that
residualized assignment variation left over after conditioning on
$g(W;X_i,\theta)$ remains predictive of outcomes through the chosen moments.
Different finite moment collections yield different $J$-statistics;
Section~\ref{sec:db-optimal-instruments} studies how to choose moments efficiently.
\begin{remark}[Scope of the specification test]
\label{rem:gaocomp}
Contemporaneously, \citet{GaoHarshawSavjeWang2026} prove that specification
testing of exposure-mapping models is impossible against any richer
exposure-mapping alternative: every testing procedure, design included, has worst-case Type I and Type II errors summing to one, at every sample size. The alternative is simply
too large in that it places no structure across units and so admits
adversarial outcome schedules that are maximally separated from the null yet
generate the same observed-data distribution under every assignment. The implication is that informative tests exist only
against alternatives restricted beyond what exposure mappings alone impose,
as in their consistent test against a linear-in-means model.
Our test naturally embodies the restriction their result requires, in the form applied work already adopts. We maintain a
parametric exposure class $\{g(\cdot;\theta):\theta\in\Theta\}$ and a finite
set of residualized design moments; the null states that some
$\theta_0\in\Theta$ satisfies $\mu(\theta_0)=0$. The $J$-test then has power
against alternatives that keep the moment criterion bounded away from zero
uniformly over $\Theta$, under the convergence conditions of
Assumption~\ref{ass:as-lln} maintained along the alternative sequence.
Rejection therefore signals that assignment variation left over after
conditioning on the proposed exposure map remains predictive of outcomes.
Non-rejection means the maintained class is consistent with the design through
these moments, and nothing more: it does not certify exposure sufficiency
against the unrestricted alternative, a guarantee no procedure can provide.
\end{remark}
\begin{remark}[Nonconservative design-based variance]
\label{rem:nonconservative-variance}
The variance in Theorem~\ref{thm:DB-AN-aff} is design-based but not a
Neyman-style conservative bound. Neyman-type bounds arise because the exact
variance of a treatment-effect estimator involves co-moments of the same unit's
potential outcomes across assignments, which are never jointly observed
\citep{Neyman1923, AbadieAtheyImbensWooldridge2020}. No such term arises
here: by Corollary~\ref{cor:design-moment-unit}, the score
\[
\phi(Y_i)\left\{\psi(W)-\mathbb{E}[\psi(W)\mid g(W;X_i,\theta_0)]\right\}
\]
is mean zero unit by unit, whatever the outcome functions $\widetilde{Y}_i$ may
be, so the centered score is observed for the realized assignment and $\Omega$
is identified as a second-moment functional of observed scores under the known
design. Given the graph-HAC condition of Assumption~\ref{ass:graph-hac}, the
sandwich therefore estimates the exact large-sample variance for the maintained
exposure model.\end{remark}
It remains to estimate the asymptotic design covariance matrix \(\Omega\).
In our setting $\Omega$ captures spatial and network dependence through the
affinity sets, so a natural object is a spatial/graph HAC estimator built from
$\{Z_{i,n}\}_{i\le N_n}$.
Rather than spell out primitive conditions for a particular estimator, we impose
a high-level consistency requirement in the spirit of spatial and network HAC
methods, and refer to the existing literature for sufficient conditions.
For spatial dependence, see \citet{Conley1999,KimSun2011}.
For general network dependence and $\psi$-dependent processes on graphs,
\citet{KojevnikovMarmerSong2021,sasaki2025gmm} provide LLN, CLT, and consistency results for a
closely related network HAC estimator.
We formalize our requirement as follows.
\begin{assumption}[Graph-HAC estimation of the asymptotic covariance]
\label{ass:graph-hac}
Let $\Omega$ be the design-based asymptotic covariance in
Assumption~\ref{ass:AN-AFF}\,\textup{\ref{ass:an-aff3}}.
There exists a sequence of graph-HAC (Conley-type) estimators $\hat\Omega_n$
constructed from $\{Z_{i,n}\}_{i\le N_n}$ and the affinity sets
$\{A_i\}_{i\le N_n}$ such that
$\hat\Omega_n$ is consistent in operator norm for the normalized
asymptotic covariance:
\[
\big\|\hat\Omega_n - \Omega_n/N_n\big\|_{\mathrm{op}} \ \stackrel{p}{\to}\ 0.
\]
Evaluated at a preliminary consistent estimator $\hat\theta_n$ (Algorithm~\ref{alg:twostep-practical-scores}), the resulting plug-in error is $o_p(1)$: the design centering is computed from the known design rather than estimated, and $\hat\theta_n\stackrel{p}{\to}\theta_0$ under Assumption~\ref{ass:as-lln} (Theorem~\ref{thm:as-consistency}).
\end{assumption}
\section{Efficient Moments}
\label{sec:db-optimal-instruments}
Sections~\ref{sec:design-orthogonality}--\ref{sec:DB-consistency-AN-affinity}
introduced a class of design-based moments parameterized by an outcome
transformation $\phi$ and a design function $\psi$,
\[
\eta_{\phi,\psi,n}(\theta)
:=
\frac1{N_n}\sum_{i\in\mathcal{U}_n}
\phi(Y_i)\,R_{i,\theta}\psi(W).
\] Two facts delimit what optimality within this class means.
First, the class is exhaustive at the unit level:
Appendix~\ref{app:conditional-poirier} shows that the collection of these
moments over all bounded measurable $(\phi,\psi)$ is equivalent to the
exposure sufficiency $Y_i \perp\!\!\!\perp W \mid g(W;X_i,\theta_0)$. Second,
a correctly-specified exposure map also implies cross-unit restrictions, involving pairs or
larger sets of units (Remark~\ref{rem:joint-exposure-future-work}), which lie
outside this class and are not exploited. The bound below is therefore an
efficiency bound within the unit-level moment class, the class containing the
moments used in applied work and in our applications, not a semiparametric
bound over all implications of an exposure map hypothesis.
\begin{theorem}[Efficiency bound and sieve attainment; informal]
\label{thm:efficiency-informal}
Under the regularity conditions of Appendix~\ref{app:db-efficiency}:
\begin{enumerate}[label=\textup{(\roman*)}, leftmargin=1.2cm]
\item There is a finite $V^\star>0$ such that every GMM estimator of $\theta_0$
built from finitely many moments in the unit-level class has design-based
asymptotic variance at least $V^\star$
(Theorem~\ref{thm:oracle_variance_bound}).
\item The bound is feasibly attained: for a dictionary
$\{(\phi_m,\psi_m):m\ge1\}$ whose span is dense in the class, two-step GMM on
the first $M_n$ moments attains $V^\star$ asymptotically, provided
$M_n\to\infty$ with $M_n^2 b_n/N_n\to0$, where $b_n:=\max_i|A_i|$ is the
maximal affinity-set size (Theorem~\ref{thm:sieve_oracle}).
\end{enumerate}
\end{theorem}
For empirical work the implication is that fixing a finite dictionary and
running two-step GMM with an estimated covariance matrix is both standard
and, as the dictionary grows, asymptotically efficient within the class.
\begin{algorithm}[Two-step GMM]
\label{alg:twostep-practical-scores}
Given a dictionary $\{(\phi_m,\psi_m):m=1,\ldots,M\}$, stack
$\eta_n^{(M)}(\theta):=\bigl(\eta_{1,n}(\theta),\ldots,\eta_{M,n}(\theta)\bigr)^\top$,
$\eta_{m,n}(\theta):=\frac1{N_n}\sum_{i\in\mathcal{U}_n} \phi_m(Y_i)R_{i,\theta}\psi_m(W)$.
\begin{enumerate}[label=\textup{(\alph*)}, leftmargin=1.2cm]
\item Compute $\hat\theta_n^{(1)}\in\arg\min_{\theta\in\Theta}
\eta_n^{(M)}(\theta)^\top\eta_n^{(M)}(\theta)$.
\item Estimate the covariance of $\sqrt{N_n}\,\eta_n^{(M)}(\theta_0)$ by a
graph-HAC estimator $\hat\Omega_{M,n}$ (Assumption~\ref{ass:graph-hac})
evaluated at $\hat\theta_n^{(1)}$.
\item Compute $\hat\theta_n^{(2)}\in\arg\min_{\theta\in\Theta}
\eta_n^{(M)}(\theta)^\top\hat\Omega_{M,n}^{-1}\eta_n^{(M)}(\theta)$; then
\[
\sqrt{N_n}(\hat\theta_n^{(2)}-\theta_0)\Rightarrow
\mathcal{N}\bigl(0,(G^{(M)\top}\Omega^{(M)-1}G^{(M)})^{-1}\bigr)
\]
as in Theorem~\ref{thm:DB-AN-aff}, where $G^{(M)}$ and $\Omega^{(M)}$ are the
limiting Jacobian and covariance of the stacked moments
(Appendix~\ref{app:sieve_gmm}).
\end{enumerate}
\end{algorithm}
\section{Stage 2: Outcome regression given the exposure map}
\label{sec:stage2}
Stage~1 learns the exposure map $g(W;X_i,\hat\theta_n)$. In most applications,
the object of ultimate interest is a downstream exposure--response or policy parameter, such as
the coefficient from an outcome regression on the learned exposure index. This
section states the main implication for such second-step inference.
The main case is a regular exposure-map parameter $\theta\in\Theta\subset\mathbb{R}^p$
estimated in Stage~1. Since $\hat\theta_n$ is an input into the downstream
exposure--response equation, the sampling uncertainty in $\hat\theta_n$ must be
propagated into second-step inference.
For concreteness, consider the linear exposure--response projection. Define
\[
Z_i(\theta):=
\begin{pmatrix}
1\\
g(W;X_i,\theta)
\end{pmatrix}.
\]
The target $(\alpha_0,\beta_0)$ is the design-based projection coefficient
satisfying
\begin{equation}
\label{eq:linear-main}
\lim_{n\to\infty}
\frac{1}{N_n}\sum_{i\in\mathcal{U}_n}
\mathbb{E}\!\left[
Z_i(\theta_0)
\{Y_i-\alpha_0-\beta_0^\top g(W;X_i,\theta_0)\}
\right]
=0.
\end{equation}
A correctly specified linear conditional mean is sufficient for
\eqref{eq:linear-main}, but the projection interpretation does not require it.
Collect the regular parameters as
\[
\zeta=(\theta^\top,\alpha,\beta^\top)^\top .
\]
The two-step estimator is characterized by stacking the projected Stage~1 GMM
equation with the Stage~2 exposure--response equation:
\[
G_1^\top\Lambda_{1,n}\bar\Psi_{1,n}(\hat\theta_n)=o_p(N_n^{-1/2}),
\qquad
\bar\Psi_{2,n}(\hat\theta_n,\hat\alpha_n,\hat\beta_n)=o_p(N_n^{-1/2}).
\]
Under a joint affinity-set CLT, mean differentiability of the stacked population map,
and stochastic equicontinuity, Appendix~\ref{app:stage2} shows that
\[
\sqrt{N_n}(\hat\zeta_n-\zeta_0)
\Rightarrow
\mathcal{N}(0,V_\zeta),
\]
where $V_\zeta$ is the corresponding design-based sandwich covariance matrix.
Inference for the exposure--response coefficient $\beta_0$ uses the
$(\beta,\beta)$ block of $V_\zeta$. More general low-dimensional Stage~2
moments, including nonlinear regressions or policy-functional estimating
equations, are handled by the same stacked-system argument.
\begin{remark}[Design-based interpretation of the Stage-2 standard errors]
\label{rem:stage2-conservative}
The Stage-2 projection moments, unlike the Stage-1 moments
(Remark~\ref{rem:nonconservative-variance}), are mean zero only in aggregate:
\eqref{eq:linear-main} fixes the average moment at zero but leaves the
unit-level means nonzero, and those means are not identified from the realized
assignment (each involves $\widetilde{Y}_i$ at unrealized exposures). The
feasible sandwich must therefore center at the sample mean, so it need not
reproduce the exact design variance of $\hat\beta$. When the affinity sets are
singletons or block-diagonal the discrepancy is positive semidefinite and the
reported standard errors are conservative, with no assumption of correct
specification; it vanishes when the linear exposure--response holds unit by
unit; and it is of ambiguous sign under general overlapping dependence absent a
local-alignment condition. Appendix~\ref{subsec:beta-inference-app} makes this
precise and states the condition.
\end{remark}
\section{Applications}
\label{sec:applications-main}
This section applies the framework to two large-scale anti-poverty programs in
development economics that explicitly study general-equilibrium (GE) effects:
(i) the Smartcards reform of India's National Rural Employment Guarantee Scheme
analyzed by \citet{muralidharan2023generaleq}; and (ii) the GiveDirectly
cash-transfer experiment in rural Kenya studied by \citet{egger2022general},
together with the follow-up structural analysis of \citet{walker2024slack}.
The two settings are chosen to be similar ex ante, but we find that they yield contrasting conclusions.
Both applications follow the same two-stage structure: Stage~1 estimates the exposure
map from the design-based moments and tests it against the randomization; Stage~2 re-estimates
the authors' original outcome equations with the estimated map in place of their fixed one,
propagating the Stage-1 uncertainty.
\subsection{\texorpdfstring{Revisiting \citet{muralidharan2023generaleq}}{Revisiting Muralidharan et al. (2023)}}
\label{subsec:mns-main}
We begin with the general-equilibrium effects of India's
National Rural Employment Guarantee Scheme (NREGS) studied by
\citet{muralidharan2023generaleq}. The program guarantees up to 100 days of
public employment per year to rural households. \citet{muralidharan2023generaleq}
exploit a large randomized rollout of biometric ``Smartcards'' that improved
the implementation of NREGS at the mandal level, and combine direct effects on
treated mandals with spillovers to nearby untreated areas.\footnote{Administrative units nest as district $\supset$ mandal $\supset$
Gram Panchayat (GP): a mandal is a sub-district (average population roughly
62,500 in the study sample), and a GP is a village-cluster government
comprising one or more census villages. Treatment was randomized at the
mandal level, outcomes are measured at the GP (or household) level, and
spatial exposure is computed from geocoded 2001 Census village locations
\citep{muralidharan2023generaleq}. Our Stage~1 below implements the exposure
measure at the census-village level; see Appendix~\ref{app:mns-stage1}.}
\paragraph{Ring-based exposure measure.}
In their main specification (their equation~(1)),
\citet{muralidharan2023generaleq} estimate regressions of the form
\begin{equation}
Y_{pmd}
\;=\;
\alpha
\;+\;\beta_0 T_m
\;+\;\beta_1 \mathrm{NR}_{pmd}^{20}
\;+\; X_{pmd}'\gamma
\;+\;\varepsilon_{pmd},
\label{eq:mns-main-eq}
\end{equation}
where $Y_{pmd}$ is an outcome such as NREGS earnings, wage-labor income, or
total income for Gram Panchayat (GP) $p$ in mandal $m$ and district $d$,
$T_m$ is the Smartcard treatment indicator for mandal $m$, and $X_{pmd}$ are
controls. The key spillover variable $\mathrm{NR}_{pmd}^{20}$ is the share of
GPs in other mandals within 20 km of $p$ that were assigned to treatment; it is a
ring-based neighborhood-treatment measure constructed at the GP level using
geographic distance, an instance of the ring exposure map of
Example~\ref{ex:ring-egger} with radius $\theta = 20$ km.
The headline finding of \citet{muralidharan2023generaleq} is that 86\% of beneficiary
income gains came from non-program earnings: from higher private-sector wages and employment
rather than from NREGS payments themselves, with statistically significant spatial spillovers
of wages and employment onto nearby areas.
The authors fix the radius defining $\mathrm{NR}^{20}_{pmd}$
a priori at 20 km, motivated by commuting distances and the geographic scale of
local labor markets, rather than estimate it or test it against the design.
\paragraph{Stage~1: letting the data choose the radius.}
We apply the Stage~1 design-based GMM procedure of
Sections~\ref{sec:design-orthogonality} and~\ref{sec:DB-consistency-AN-affinity},
taking the ring radius as the tuning parameter $\theta$. The exposure measure
follows \citet{muralidharan2023generaleq}'s construction: for each GP we compute the
population-weighted fraction of non-same-mandal treated census villages within
radius $\theta$, using the same 6,662-village geometry as the original paper.
The design functions are population-weighted annulus averages of the mandal
treatment indicator in distance bands around GP $i$, built from the same
village-level distance structure as the exposure.
Because physical interaction distances vary continuously in space, we treat the
radius as a continuous parameter and search over a fine 0.1 km grid. We report
Wald confidence intervals justified by the asymptotic normality result and the design-based overidentification
($J$) test of Section~\ref{sec:DB-consistency-AN-affinity}. Full technical details (the
two-step criterion, Wald interval construction, and overidentification test) are in
Appendix~\ref{app:mns-stage1}.
Table~\ref{tab:mns-stage1-main} reports the Stage-1 estimates: the selected radii are
23.7 km for total income, 14.1 km for NREGS earnings, and 25.8 km for wage-labor income. For
each outcome the table also gives the Wald standard error and 95\% confidence interval for
$\hat\theta$, and the overidentification statistic $J(\hat\theta)$ with its $p$-value. The $J$
statistics are small and none is significant at conventional levels ($p = 0.42$, $0.54$, and
$0.40$), so at the selected radii the design-based moments give no evidence against the ring
specification.
NREGS earnings and wage-labor income decompose total income into a more local and a
broader-reaching component (rows two and three of Table~\ref{tab:mns-stage1-main}), and they
separate in the direction the authors' own mechanism predicts: NREGS earnings, the direct
public-employment margin, localize to 14.1 km, whereas wage-labor income, which operates
through private labor markets, reaches farther, to 25.8 km.\footnote{\citet{muralidharan2023generaleq}
emphasize that jobcard holders ``could only do NREGS work in their own villages,'' so the
public-employment margin should track local program access; the 14.1 km estimate has a tight
95\% confidence interval of [12.8, 15.4] km. In their mechanism for wage-labor income, better
NREGS implementation raises reservation wages and propagates through markets they describe as
``spatially integrated beyond individual GPs or even mandals.''}
The total-income radius, 23.7 km, falls between them.
Figure~\ref{fig:mns-objective-main} plots the Stage-1 criterion against the candidate
radius for each outcome, with the dashed vertical line marking the selected radius
$\hat\theta$; the criterion is small where the ring's implied design-based moments are close to
zero. In all three panels the criterion is high at short radii, falls to a clear interior
minimum at the selected radius, and rises beyond it, so the design identifies a finite
interaction radius for every outcome. The total-income criterion also has local minima near the
two component radii, consistent with total income aggregating a more local and a
broader-reaching margin.
\begin{table}[htbp]
\centering
\caption{Stage-1 radius estimates for \citet{muralidharan2023generaleq} outcomes.}
\label{tab:mns-stage1-main}
\begin{tabular}{lcccc}
\toprule
Outcome $Y$ & $\hat\theta$ (km) & Wald SE (km) & 95\% Wald CI & $J(\hat\theta)$ ($p$-value) \\
\midrule
Total income & 23.7 & 1.22 & [21.30,\;26.10] & 6.01 (0.422) \\
NREGS earnings & 14.1 & 0.64 & [12.84,\;15.36] & 5.05 (0.538) \\
Wage-labor income & 25.8 & 1.05 & [23.75,\;27.85] & 6.26 (0.395) \\
\bottomrule
\end{tabular}
\begin{flushleft}
\footnotesize Notes: Population-weighted village-level exposure and
instruments (3 km bands, 0.1 km radius grid).
Wald SE and CI treat the radius as a continuous parameter; see
Appendix~\ref{app:mns-stage1} for details.
The $J$-statistic is the overidentification test, with $6$ degrees of freedom; $p$-values in parentheses.
\end{flushleft}
\end{table}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.48\textwidth]{figures/mns_discrete_criterion_total_w3_rmax20_bands.png}
\vspace{0.5em}
\includegraphics[width=0.48\textwidth]{figures/mns_discrete_criterion_nrega_w3_rmax20_bands.png}
\includegraphics[width=0.48\textwidth]{figures/mns_discrete_criterion_wage_labor_w3_rmax20_bands.png}
\caption{Stage-1 objective functions for total income (top),
NREGS earnings (bottom left), and wage-labor income (bottom right).
Dashed line: selected radius $\hat\theta$. Population-weighted village-level
exposure and instruments (3 km bands), strict placebo design.}
\label{fig:mns-objective-main}
\end{figure}
Taken together, the exhibits broadly support the exposure choice that
\citet{muralidharan2023generaleq} make on institutional grounds: the ring specification is not
rejected for any outcome, and the estimated radius for total income,
the headline outcome, is 23.7 km,
close to the 20 km they adopt a priori (first row of
Table~\ref{tab:mns-stage1-main}).
The 20 km choice also lies inside the range spanned by the two component radii, 14.1
to 25.8 km.
\paragraph{Stage~2: the income-source decomposition.}
Re-estimating \citet{muralidharan2023generaleq}'s outcome equations at the outcome-specific radii leaves their conclusions essentially intact (Table~\ref{tab:mns-stage2-main}). Total income
($9344.2$) and wage-labor income ($7627.9$) are approximately unchanged from the 20 km
specification; only NREGS earnings move materially, falling from $1294.6$ to
$758.7$. \citet{muralidharan2023generaleq} attribute 14\% of the income gain to
program earnings and 86\% to non-program earnings; the corresponding share computed
from the 20 km column of Table~\ref{tab:mns-stage2-main} is 13.5\%. At the design-based radii the
program share falls to 8.1\% ($758.7$ of $9344$), so non-program
earnings account for roughly 91.9\% of the gain (bottom panel of
Table~\ref{tab:mns-stage2-main}).\footnote{Estimating sources at different radii need not yield shares that sum to one. As an alternative accounting exercise, restricting to the two sources with outcome-specific radii (NREGS and wage labor) gives a program share of 9.0\%, of the same small magnitude as the 8.1\% in the text.} Their central finding, that the
income gains come predominantly from non-program (general-equilibrium) earnings
rather than direct program payments, is thus robust to estimating the radius.
The design-based standard errors are close to their fixed-radius
counterparts despite additionally propagating the sampling uncertainty in
$\hat\theta$, so treating the radius as estimated rather than known does
not materially inflate the uncertainty around the decomposition.
\begin{table}[htbp]
\centering
\caption{Treatment effects and implied income-source decomposition for
\citet{muralidharan2023generaleq} outcomes: original 20\,km specification vs.\
design-consistent radii.}
\label{tab:mns-stage2-main}
\resizebox{0.98\linewidth}{!}{
\begin{tabular}{lcccc}
\toprule
Outcome & 20\,km spec.\ (MNS) & Design-based ($\hat\theta$) & $\hat\theta$ (km) & $\Delta$ vs.\ 20\,km \\
\midrule
Total income & 9579.5 (4538.7) & 9344.2 (4969.1) & 23.7 & $-235.3$ \\
NREGS earnings & 1294.6 (1061.0) & \phantom{0}758.7 \phantom{0}(624.6) & 14.1 & $-535.9$ \\
Wage-labor income & 7607.2 (2720.7) & 7627.9 (2996.3) & 25.8 & $+20.7$ \\
\midrule
\multicolumn{5}{l}{\emph{Implied income-source decomposition (share of total-income gain)}} \\
Program (NREGS earnings) & 13.5\% & 8.1\% & -- & $-5.4$\,pp \\
Non-program & 86.5\% & 91.9\% & -- & $+5.4$\,pp \\
\bottomrule
\end{tabular}
}
\begin{flushleft}
\footnotesize
\emph{Notes:} Entries are adjusted treatment effects with standard errors in
parentheses. The ``20\,km spec.'' column is \citet{muralidharan2023generaleq}'s
original fixed-radius specification; the ``Design-based'' column re-estimates the
same Equation~\eqref{eq:mns-main-eq}-style regression (treatment, neighborhood
exposure, district fixed effects, strata controls; baseline total income
included only for total income) at the outcome-specific radius $\hat\theta$
selected in Stage~1. Standard errors are clustered at the mandal level; the
design-based standard error additionally propagates the Stage-1 Wald variance of
$\hat\theta$ via the delta method.
The bottom panel reports the implied income-source decomposition: within each
column, the program share divides the NREGS-earnings effect by the total-income effect
($1294.6/9579.5 = 13.5\%$ at 20\,km; $758.7/9344.2 = 8.1\%$ at the design-based radii), and
the non-program share is its complement. \citet{muralidharan2023generaleq}'s published
accounting reports a program share of 14\%. Because the design-based column estimates each
source at its own radius, see the text footnote for an alternative accounting.
\end{flushleft}
\end{table}
\subsection{Revisiting Egger et al.\ (2022)}
\label{subsec:egger-main}
\citet{egger2022general} study the GiveDirectly cash-transfer experiment in rural
Kenya, in which treatment was randomized both across villages and, through a second-tier
saturation design, across sublocations. The two-tier design generates experimental variation in
indirect exposure to transfers at the village and local-market level; this is the variation our
Stage~1 exploits below.\footnote{See
\citet{egger2022general} for details on the experimental design and data
collection.}
Their main specification relates household outcomes to the amount transferred to the
household's own village and to the amount transferred to other villages within 2 km. The 2 km
outer radius, the distance beyond which transfers are assumed to have no effect, is selected
by a Bayesian information criterion over concentric 2 km distance bands.
The headline finding of \citet{egger2022general} is a local transfer multiplier of
roughly 2.5 (their Table~V), together with large positive spillovers onto non-recipient
households. The spillover component of that accounting is estimated from the $0$--$2$ km
exposure, so the multiplier inherits the support choice; and the support is selected by
in-sample predictive fit rather than estimated from, or tested against, the experimental
design.
\paragraph{Stage~1: estimating the weights and selecting the support.}
Stage~1 here estimates a richer object than the scalar radius of the previous
application: the annulus weights of a linear spillover index, together with its support. Let
\[
A_v(W)
=
\left(
T_{v,0-2}^{\neg v}(W),
T_{v,2-4}^{\neg v}(W),
\ldots,
T_{v,18-20}^{\neg v}(W)
\right)'
\]
denote neighboring-village transfer amounts in the ten annuli up to 20 km.
The first of these, $T_{v,0-2}^{\neg v}(W)$, is \citet{egger2022general}'s own
$0$--$2$ km spillover variable. For a
candidate \emph{support} $R_m=2m$, the radius beyond which transfers are
assumed to have no causal effect on a household, Stage~1 forms the linear
spillover exposure index
\[
g_v^{(m)}(W;\theta_m)
=
\sum_{j=1}^{m}
\theta_{m,j}T_{v,2(j-1)-2j}^{\neg v}(W),
\qquad
\mathbf 1'\theta_m=1,
\]
and estimates $\theta_m$ from the design-based exposure-sufficiency moments.\footnote{The scale of $\theta_m$ is immaterial for
exposure sufficiency: replacing $\theta_m$ by $c\theta_m$, with $c\neq0$, only
rescales the downstream coefficient on the index. We impose $\mathbf 1'\theta_m=1$
and let the Stage~2 coefficient absorb scale. When $m=1$, this normalization
implies $\theta_{1,1}=1$. Thus the 2 km case coincides with the original
Egger et al.\ one-ring exposure measure, and Stage~1 has no nontrivial shape
parameter to estimate.} In
this notation, the exposure map used in Stage~1 is
\[
\left(
T_v^{\mathrm{own}}(W),
g_v^{(m)}(W;\theta_m)
\right).
\]
The support $R_m$ itself is also selected by the design. The 2 km support is an
untested restriction, and the fully flexible 20 km index turns out to estimate the
pure-spillover effects too imprecisely to be informative
(Figure~\ref{fig:egger-radius-path-core}); we therefore select the smallest support the design
does not reject. Exposure sufficiency makes the support a testable restriction: if the annuli
beyond $R_m$ carry no outcome-relevant assignment variation, zeroing their coefficients leaves
the design-based moments satisfied, which is the restriction the overidentification test of
Section~\ref{sec:DB-consistency-AN-affinity} (Corollary~\ref{cor:db-J-test}) evaluates. We test
the unrestricted 20 km index against the restricted index that zeroes all annuli beyond $R_m$,
implemented as a likelihood-ratio statistic at each candidate support (reported in
the lower panels of Figure~\ref{fig:egger-radius-path-core}), and select the smallest
non-rejected support.\footnote{As a check on post-selection and spatial-dependence concerns, the diamonds in
Figure~\ref{fig:egger-radius-path-core} report a sublocation-level sample-splitting
exercise. We split at the level of \emph{sublocations} (the higher administrative
tier at which the saturation design randomizes treatment intensity, each containing
many villages and a local market) rather than at the village or household level:
because cash-transfer spillovers operate through the local market
\citep{walker2024slack}, dependence is contained within a sublocation and is
approximately negligible across them, so assigning whole sublocations to the
selection and estimation subsamples makes the two approximately independent while
leaving the within-sublocation spillovers the exposure map captures intact. We aggregate across repeated splits
following the median procedure of \citet{chernozhukov2025fisher}: we report the median point estimate across splits and form the band
from the median lower and median upper confidence limits, yielding an approximately
$90\%$ interval.
The split estimates are noisier but point the same way: $2$ km is too local and the
selected supports are modestly larger.} This is
the same parsimony goal behind the Bayesian information criterion \citet{egger2022general} use
to set their outer radius; the difference is that the BIC scores cross-sectional predictive
fit, whereas the design-based test scores consistency with the randomization, the identifying
assumption the downstream estimates rely on.\footnote{\citet{egger2022general} choose the radius $r$ by minimizing
$\mathrm{BIC}(r)=N\log\hat\sigma_r^2+k(r)\log N$, where $\hat\sigma_r^2$ is the residual
variance from regressing $Y_i$ on the radius-$r$ ring exposures and controls, $k(r)$ is the
number of parameters, and $N$ is the number of villages: the radius is scored by how much the
rings reduce residual variance, traded off against the penalty.}
The two criteria can disagree: a spillover component that is approximately common to
households within a local market is largely absorbed by the intercept and controls, so outer
annuli that mainly carry such a component barely reduce residual variance, and a fit criterion
prefers short supports even when the beyond-2 km transfers generating that component move the
total response.
Figure~\ref{fig:egger-radius-path-core} traces \citet{egger2022general}'s recipient
and non-recipient effects, re-estimated at each candidate support from their 2 km to the fully
flexible 20 km: the recipient
point estimates are stable across the entire path, while the non-recipient effects attenuate
and lose precision until, by 20 km, they are generally no longer distinguishable from
zero. The
two endpoints anchor the comparison. At $R=2$ km the normalized index collapses to their single
$0$--$2$ km ring ($\theta_{1,1}=1$), so the leftmost point replicates the original
\citet{egger2022general} specification; at $R=20$ km the index imposes no support
restriction across the ten annuli; each interior point imposes, ex ante, that transfers beyond
$R$ km do not affect the household. The contrast between the two paths is what one would
expect: recipients respond mainly to the direct, own-village transfer, which does not depend on
the support, whereas non-recipients respond only through spillovers, which do.
The confidence intervals do not shrink monotonically as the support widens;
Appendix~\ref{app:egger-extra}, part~D, takes up this feature separately.
The lower panel of each plot reports the $p$-value of the support test described
above at each candidate support.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.48\textwidth]{figures/figure_radius_path_p2_consumption_wins_PPP.pdf}
\includegraphics[width=0.48\textwidth]{figures/figure_radius_path_p1_assets_wins_PPP.pdf}
\vspace{0.5em}
\includegraphics[width=0.52\textwidth]{figures/figure_radius_path_p3_totincome_wins_PPP.pdf}
\caption{Recipient and non-recipient effects across candidate supports,
\citet{egger2022general} application}
\label{fig:egger-radius-path-core}
\begin{flushleft}
\footnotesize
\emph{Notes:} The figure plots recipient total effects and non-recipient pooled
spillover effects across candidate supports $R=2,\ldots,20$ km. The shaded band
marks the selected support. The lower panel reports the likelihood-ratio test $p$-value
path, and the dashed horizontal line marks $\alpha=0.05$. Diamonds report
sublocation-level split-sample medians with their median intervals (dotted
whiskers). The appendix reports additional outcome diagnostics.
\end{flushleft}
\end{figure}
The design-based support test rejects the original 2 km support for all three
outcomes: in every lower panel of Figure~\ref{fig:egger-radius-path-core}, the $p$-value lies
below $0.05$ at 2 km and crosses above it by 4 km. The smallest non-rejected support is $4$ km
for household expenditure, assets, and household income; across the broader set of outcomes in
Appendix~\ref{app:egger-extra} the selected supports are $4$--$6$ km, occasionally larger. The
$2$ km exposure is thus statistically inconsistent with the design. At the selected supports,
the overidentification ($J$) test of the estimated linear-index specification (last column of Table~\ref{tab:egger-core-IV-linear-index}) rejects for
expenditure and assets ($p = 0.002$ and $0.001$) and does not reject for income ($p = 0.129$);
the rejection is consistent with a spillover component that no annular index spans at any
support, a point we return to below.
\paragraph{Stage~2.}
We next estimate \citet{egger2022general}'s original IV specification, given by
\[
Y_{iv}
=
\alpha
+
\beta_{\mathrm{own},m}T_v^{\mathrm{own}}(W)
+
\gamma_m g_v^{(m)}(W;\widehat\theta_m)
+
X_{iv}'\delta
+
\varepsilon_{iv},
\]
replacing their $0$--$2$ km spillover measure with the estimated index
$g_v^{(m)}(W;\widehat\theta_m)$ at the selected support, and keeping the same controls and
instrumenting logic as the original.\footnote{Standard errors account for the generated weights
$\widehat\theta_m$ by the delta method.}
Table~\ref{tab:egger-core-IV-linear-index} compares \citet{egger2022general}'s
original estimates with the design-based estimates at the selected supports. The exposure
measure is the only difference between the two columns. The change matters only for the
non-recipient effects. The recipient total effects survive the change of
exposure map: all three point estimates move by less than half of \citet{egger2022general}'s
own standard error. Every pooled spillover, by contrast, attenuates by about one to one and a half of
\citet{egger2022general}'s standard errors. Expenditure falls from $334.7$ to $128.9$ and
remains significant; income falls from $225.0$ to $81.4$ and is no longer distinguishable from
zero at the $5\%$ level; assets, not significant in the original specification, falls from
$135.4$ to $41.9$.\footnote{The ``Egger IV'' column re-estimates the original 2 km
specification on our replication sample; the estimates closely match the published Table~1
values (see the table notes).} Relative to the \citet{muralidharan2023generaleq} application,
then, the exposure-map choice does matter here, and it matters for the spillovers rather than
for the direct effects.
\begin{table}[htbp]
\centering
\caption{Core IV components: Egger et al.\ vs. design-based estimates}
\label{tab:egger-core-IV-linear-index}
\footnotesize
\setlength{\tabcolsep}{5pt}
\begin{tabular}{lcccc}
\toprule
& Egger IV & Design-based IV & $\Delta\beta,\ \Delta$SE & $J$ p-val \\
\midrule
\multicolumn{5}{l}{\emph{Household expenditure, annualized}} \\
\quad Recipients & 338.6 (100.6) & 296.1 (58.7) & $-42.5,\ -41.9$ & 0.002 \\
\quad Non-recipients & 334.7 (131.1) & 128.9 (56.0) & $-205.8,\ -75.1$ & 0.002 \\[0.3em]
\multicolumn{5}{l}{\emph{Assets (non-land, non-house), net borrowing}} \\
\quad Recipients & 183.4 (55.2) & 177.3 (31.2) & $-6.1,\ -23.9$ & 0.001 \\
\quad Non-recipients & 135.4 (99.3) & 41.9 (45.9) & $-93.5,\ -53.4$ & 0.001 \\[0.3em]
\multicolumn{5}{l}{\emph{Household income, annualized}} \\
\quad Recipients & 135.7 (82.1) & 97.3 (37.0) & $-38.4,\ -45.1$ & 0.129 \\
\quad Non-recipients & 225.0 (100.1) & 81.4 (42.7) & $-143.6,\ -57.3$ & 0.129 \\
\bottomrule
\end{tabular}
\begin{flushleft}
\footnotesize\emph{Notes:} Entries are IV point estimates with standard errors in parentheses. Egger IV values re-estimate the original 2~km Table~1 specification on our replication sample (cluster standard errors); they closely match the published Table~I values. Design-based IV values use each outcome's selected design-based support ($\hat R = 4.0$~km for all three core outcomes); the design-side conditional expectations entering the index are estimated by least squares on a quadratic polynomial in the exposure index, using $200$ placebo draws of the assignment. Recipient entries report total effects; non-recipient entries report pooled spillover effects, both with joint delta-method standard errors. $J$ p-val is the design-GMM overidentification p-value for the selected support; the support-selection rule is the likelihood-ratio test.
\end{flushleft}
\end{table}
Recomputing \citet{egger2022general}'s Table~V multiplier accounting with the
design-selected indexes gives a mean local transfer multiplier of about $1.57$, well below the
original headline value near $2.5$ and close to the $1.54$ implied by
\citet{walker2024slack}'s calibrated general-equilibrium model
(Table~\ref{tab:egger-multiplier-results}). The expenditure and income components are about
$1.60$ and $1.54$.
With only $84$ sublocation clusters
these multipliers are imprecisely estimated, with standard errors comparable to
\citet{egger2022general}'s own; the mean multiplier carries a standard error of
$1.12$, so although its point estimate lies well below $2.5$, it is not
statistically distinguishable from one.\footnote{Standard errors are delta-method
values that propagate both the Stage-1 uncertainty in the index weights and the Table~V
coefficient uncertainty; Appendix~\ref{app:egger-extra}, part~E details the reparameterization
and the inference.}
The \citet{walker2024slack} benchmark is on the same real (deflated) basis as
\citet{egger2022general}'s Table~V.
\begin{table}[htbp]
\centering
\caption{Local transfer multiplier: design-based estimates and benchmarks}
\label{tab:egger-multiplier-results}
\begin{tabular}{lccc}
\toprule
& Design-based & \citet{egger2022general} & \citet{walker2024slack} \\
& & Table~V & model \\
\midrule
Expenditure & 1.60 (1.49) & 2.58 (1.44) & -- \\
Income & 1.54 (0.99) & 2.47 (1.71) & -- \\
Mean & 1.57 (1.12) & 2.52 (1.39) & 1.54 \\
\bottomrule
\end{tabular}
\begin{flushleft}
\footnotesize \emph{Notes:} Entries are multiplier point estimates with
standard errors in parentheses, all on a real basis: transfers and
outcomes are deflated to January~2015 USD, matching \citet{egger2022general}'s
Table~V. Our design-based estimates recompute that deflated accounting after
replacing each ring exposure with the design-selected index; their standard
errors are delta-method values that propagate the Stage-1 uncertainty in the
index weights and the Table~V coefficient uncertainty (Appendix~\ref{app:egger-extra}, part~E). The \citet{egger2022general} column reproduces their Table~V
estimates, whose standard errors come from a wild clustered bootstrap. The \citet{walker2024slack} figure is a
model-implied real multiplier from a calibrated general-equilibrium model, with
no sampling standard error and reported only in aggregate (their nominal
counterpart is $1.83$).
\end{flushleft}
\end{table}
The economics of the setting supports spillovers that reach beyond 2 km.
\citet{walker2024slack} argue that local general-equilibrium responses to the transfers include
a market-level ``slack'' component: part of the response is common to all households trading in
the same local market, rather than declining smoothly with distance from the transferred
villages. Such a component is consistent with both Stage-1 findings. It gives beyond-2 km
transfers real effects on a household's outcomes, which is why the support test extends the
support past 2 km; and it is not spanned by an annular index at any support, which is why the
$J$ test still rejects at the selected supports.
The second application thus illustrates the revision use of the framework: when the
design rejects the original exposure map, both the map and the headline policy estimate change
materially.
\section{Conclusion}
\label{sec:conclusion}
Estimates of spillover effects depend on an exposure map that turns the realized
assignment into an index of each unit's exposure. Both the functional form of
that map (e.g., a ring) and its parameter (e.g., the radius) are usually fixed
before estimation, with little guidance for either choice. This paper shows that
the same randomization that identifies treatment effects can guide both choices. Specifically, a correctly specified exposure map implies
orthogonality conditions that can be used for both estimation and testing:
design-based GMM can estimate the mapping parameter, and the same conditions, when
overidentified, can test the map itself.
We establish consistency and asymptotic normality of the resulting estimator for
the maps used in applied work, with all randomness arising from the assignment,
and we characterize the efficient moments. Finally, we carry the resulting
uncertainty about $g(W;X_i,\theta)$ into downstream policy estimands (e.g.,
average effects under alternative assignment rules), so that inference reflects
both the experimental variation and the uncertainty about the exposure
specification itself.
We apply the method to two large-scale anti-poverty experiments. For the NREGS
reform studied by \citet{muralidharan2023generaleq}, the 20~km radius chosen on
institutional grounds is not rejected by the design, and the program's headline
decomposition is robust: even at the data-chosen radius, the income gains come
predominantly from non-program earnings. For the cash-transfer experiment of
\citet{egger2022general}, the 2~km support is rejected for every core outcome,
and replacing it with the smallest support the design does not reject lowers the
estimated local transfer multiplier (mean) from about $2.5$ to $1.57$, close to the
value implied by the independent structural model of \citet{walker2024slack}.
\bibliographystyle{apalike}
\bibliography{references}
\newpage