The exact contents of citations.db main_text.text for this paper — one flattened LaTeX string, title through conclusion, appendix excluded, unmodified except for removing email addresses. This is what our citation measures are computed over.
47,717 characters
Efficient difference-in-differences estimation under partial interference with incremental propensity score policies
\begin{frontmatter}
\title{Efficient difference-in-differences estimation under partial interference with incremental propensity score policies}
\author[hit]{Junjie Li}
\ead{[email removed]}
\author[hit]{Yukitoshi Matsushita\fnref{fund}}
\ead{[email removed]}
\address[hit]{Graduate School of Economics, Hitotsubashi University,
2-1 Naka, Kunitachi, Tokyo 186-8601, Japan.}
\fntext[fund]{Matsushita acknowledges financial support from the JSPS KAKENHI (23K01331).}
\begin{abstract}
This paper develops efficient difference-in-differences (DID) estimation under partial interference with a cluster incremental propensity score (CIPS) policy. We define direct and spillover average treatment effects on the treated, establish their identification, and derive their efficient influence functions, from which we construct a cross-fitted estimator. Simulations confirm its finite-sample validity, and an application to China's New Rural Pension Scheme uncovers a significantly negative within-household spillover of pension participation on co-residents' labour income.
\end{abstract}
\begin{keyword}
Difference-in-differences \sep Partial interference \sep Incremental propensity score \sep Stochastic policy \sep Semiparametric efficiency \sep Spillover effects
\end{keyword}
\end{frontmatter}
\section{Introduction}
\label{sec:intro}
Difference-in-differences (DID) is among the most widely used research designs for evaluating policies with panel or repeated cross-section data. Its standard justification rests on a no-interference (SUTVA) assumption: a unit's potential outcomes depend only on its own treatment. In many settings this assumption is untenable. When a place-based subsidy is granted to some firms, neighbouring firms may benefit from agglomeration spillovers; when some children in a household are encouraged to attend school, their siblings may be affected through shared resources; when some villages in a county enter a special economic zone, surrounding villages may gain or lose through factor reallocation and local-market linkages. In such cases a unit's outcome responds both to its own treatment (a direct effect) and to the treatment of its peers (a spillover effect), and an analysis that ignores interference both misstates the estimand and discards the spillover, which is often of independent policy interest.
This paper studies efficient DID estimation under \emph{partial} (clustered) interference: units are grouped into clusters, interference is unrestricted within a cluster but absent across clusters, and the number of clusters grows while cluster sizes stay bounded.
Two modelling choices set the present paper apart from Park and Kang (2022), and both are imported from \citet{lee2025efficient} (hereafter LZH).
First, we adopt LZH's \emph{data structure}. Clusters are independent and identically distributed draws from a super-population, and the cluster size $N_i$ is a \emph{bounded random variable} that is observed and carried through the analysis.
There is no latent cluster type $L_i$, no type-frequency parameters $p_k$, and no device that fixes the cluster size at a type-specific constant $M_k$.
Heterogeneity across clusters of different sizes is captured simply by conditioning the nuisance functions on the observed size $N_i$; arbitrary within-cluster dependence remains permitted. This is both more transparent and closer to applications, in which cluster sizes vary continuously rather than falling into a handful of discrete types.
Second, we replace the type-B counterfactual---each peer independently assigned to treatment with the same probability $\alpha$---by a \emph{cluster incremental propensity score} (CIPS) policy. As LZH emphasize, the type-B policy can lack real-world relevance because realistic counterfactual scenarios allow the probability of treatment to vary across units according to their covariates, and because, away from the observed treatment saturation, the type-B estimand averages over peer configurations that are essentially never observed. The CIPS policy, extending the incremental propensity score intervention of \citet{kennedy2019nonparametric} to clustered interference, instead \emph{modulates the observed propensities}: it multiplies the odds of each unit's treatment by a user-chosen factor $\delta\in(0,\infty)$. The choice $\delta=1$ reproduces the factual allocation, so the policy never overwrites the propensity with an exogenous constant, the unit ranking by treatment probability is preserved, and moderate values of $\delta$ keep all probability mass inside the supported region. No parametric model for the propensity is required to \emph{define} the estimand.
We make three contributions. First, under a nonparametric model we define direct and
spillover average treatment effects on the treated under the CIPS policy, $\tau_{DATT}(\delta)$ and $\tau_{SATT}(\delta)$,
and establish their identification under conditional parallel trends, no anticipation, overlap, and
bounded cluster size (Proposition 1). Second, we derive the efficient influence functions (EIFs)
of both estimands, characterizing the semiparametric efficiency bounds under partial interference
(Theorem 1). Because the CIPS allocation law is itself a functional of the propensity score, the
EIF carries an additional policy-estimation term---the DID analogue of the weight-estimation term
$\phi_Q$ in Corollary 1 of LZH. This term vanishes identically under the type-B policy, in which case
our EIF reduces to the partial-interference efficiency theory of Park and Kang (2022). Third, we propose
a cross-fitted estimator that averages the uncentered EIF over held-out folds, with the nuisance
functions estimated by arbitrary machine-learning or parametric methods, and show that it is
asymptotically normal and attains the efficiency bound under standard rate conditions (Theorem 2).
Unlike the type-B case, the required convergence rate for the propensity score cannot be traded off
against outcome-model accuracy: the policy-estimation term depends on the assignment mechanism
alone and is unaffected by how well the outcome regression is estimated, so an inaccurate propensity
estimate cannot be compensated by a more accurate outcome model. This one-sided robustness is
the exact DID counterpart of the behaviour reported by LZH for their CIPS estimator.
This work connects two strands of literature. On the causal-inference side, it extends the
semiparametric efficiency and stochastic-policy machinery for partial interference
\citep{hudgens2008toward,tchetgen2012causal,liu2019doubly,park2022efficient,lee2025efficient}
from cross-sectional designs to the difference-in-differences setting, building on the doubly
robust DID estimator of \citet{sant2020doubly} and the incremental propensity score interventions
of \citet{kennedy2019nonparametric,munoz2012population,wen2023intervention}. On the DID side, it
complements recent work on identification and estimation under interference
\citep{xu2023difference,sun2025difference} and on the interpretation of conventional estimators
when interference is misspecified \citep{savje2021average}, by delivering policy-relevant, efficient
estimands with explicit influence-function-based inference.
The remainder of the paper is organized as follows. Section~\ref{sec:setup} sets up the model, the CIPS policy, and the estimands. Section~\ref{sec:identification} states the identifying assumptions and identification result. Section~\ref{sec:efficiency} develops the global and local efficiency theory (Theorems~\ref{thm:eif} and~\ref{thm:dr}). Section~\ref{sec:simulation} reports a simulation study and Section~\ref{sec:application} presents the empirical application; Section~\ref{sec:discussion} concludes and the appendix collects all proofs.
\section{Set-up}
\label{sec:setup}
\subsection{Data structure}
We observe $M$ clusters drawn independently from a super-population $\mathbb{P}$. Cluster $i$ has a random size $N_i\in\mathbb{N}$ and contains units $j=1,\dots,N_i$. We focus on two time periods $t=1,2$, with treatment realized only in period $2$; following the DID convention we write the period-2 treatment as $A_{ij}\equiv A_{ij2}\in\{0,1\}$. For unit $j$ in cluster $i$, let $Y_{ijt}\in\mathbb{R}$ be the outcome at time $t$ and $X_{ij}\in\mathbb{R}^p$ a vector of pre-treatment covariates. Stacking units within a cluster,
\[
Y_{it}=(Y_{i1t},\dots,Y_{iN_it})^\top,\quad
A_i=(A_{i1},\dots,A_{iN_i})^\top\in\mathcal{A}(N_i),\quad
X_i=(X_{i1}^\top,\dots,X_{iN_i}^\top)^\top,
\]
where $\mathcal{A}(s)$ denotes the set of all length-$s$ binary vectors. For unit $j$ in cluster $i$, let
$A_{i(-j)}=(A_{ik})_{k\ne j,\,1\le k\le N_i}$ denote the treatment vector of $j$'s peers, so that
$A_i=(A_{ij},A_{i(-j)})$ after a relabeling of coordinates. The observed data for cluster $i$ are
\[
O_i=(Y_{i1},Y_{i2},A_i,X_i,N_i),
\]
and we assume $(O_1,\dots,O_M)$ are independent and identically distributed draws from $\mathbb{P}$, and that there exists a fixed $n_{\max}\in\mathbb{N}$ with $\mathbb{P}(N_i\le n_{\max})=1$. Asymptotics are driven by $M\to\infty$ with the cluster size bounded, which follows the model of \citet{liu2019doubly,park2022efficient,lee2025efficient}.
For ease of notation we write
\[
\nu_n=\mathbb{P}(N_i=n),\qquad \mathbb{E}_n[\cdot]=\mathbb{E}[\cdot\mid N_i=n],\qquad n\le n_{\max}.
\]
For unit $j$ define the observed first difference $\Delta Y_{ij}=Y_{ij2}-Y_{ij1}$ and the reduced data
\[
W_{ij}=(\Delta Y_{ij},A_{ij},A_{i(-j)},X_i,N_i);
\]
the analysis depends on the outcomes only through $\Delta Y_{ij}$.
\subsection{Potential outcomes}
We use the potential outcome framework for partial interference \citep{hudgens2008toward,tchetgen2012causal,liu2019doubly,park2022efficient,lee2025efficient}. Let $Y_{ijt}(a_i)=Y_{ijt}(a_{ij},a_{i(-j)})$ be the potential outcome of unit $j$ at time $t$ when cluster $i$ receives $a_i$, where $a_i=(a_{ij},a_{i(-j)})$ is a realized cluster treatment vector. Clustered interference is the assumption, embedded in this notation, that a unit's potential outcome may depend on the treatments of others in the same cluster but not on those in other clusters.
\subsection{The cluster incremental propensity score policy}
\label{sec:cips}
A stochastic policy assigns, for a focal unit $j$ in a cluster of size $n$ with covariates $x$, a counterfactual conditional distribution over the treatment assignments of the remaining $n-1$ units, denoted by $\mathcal{A}(n-1)$. In the presence of interference, such stochastic policies provide a natural framework for defining causal estimands that account for treatment spillovers.
Following \citet{tchetgen2012causal,liu2019doubly,park2022efficient}, we first consider the commonly used type-B policy, which is defined as
\[
\pi_B(a_{-j}\mid n;\alpha)=\prod_{j'\neq j}\alpha^{a_{j'}}(1-\alpha)^{1-a_{j'}},
\qquad a_{-j}\in\mathcal{A}(n-1).
\]
Under this policy, the treatments of all peers are assigned independently with a common probability ($\alpha$), regardless of their individual characteristics. The type-B policy has been widely adopted as a benchmark for constructing causal estimands under interference.
However, this specification imposes a restrictive homogeneity assumption by ignoring potential heterogeneity in treatment assignment across units. In many empirical settings, treatment adoption may depend on observed characteristics, implying that a more realistic stochastic policy should allow treatment probabilities to vary with covariates. Motivated by this consideration, we adopt the \emph{cluster incremental propensity score} (CIPS) policy proposed by \citet{lee2025efficient}, which extends the incremental propensity score framework of \citet{kennedy2019nonparametric} to settings with clustered interference. Unlike the type-B policy, the CIPS policy accommodates covariate-dependent treatment probabilities and therefore provides a more flexible framework for defining causal effects under interference.
Let
\[
q_{j}(x,n)=\mathbb{P}(A_{ij}=1\mid X_i=x,N_i=n)
\]
denote the individual propensity score of unit $j$. The CIPS policy shifts the odds of treatment by a user-chosen factor $\delta\in(0,\infty)$, producing the shifted propensity
\begin{equation}\label{eq:qdelta}
q_{j,\delta}(x,n)=\frac{\delta\,q_{j}(x,n)}{\delta\,q_{j}(x,n)+1-q_{j}(x,n)},
\qquad\text{equivalently}\qquad
\frac{q_{j,\delta}(x,n)}{1-q_{j,\delta}(x,n)}=\delta\,\frac{q_{j}(x,n)}{1-q_{j}(x,n)}.
\end{equation}
Assuming that, given $(X_i,N_i)$, the treatments of distinct units within a cluster are conditionally independent, so that $\mathbb{P}(A_{i(-j)}=a_{-j}\mid X_i=x,N_i=n)=\prod_{j'\neq j}q_{j'}(x,n)^{a_{j'}}\{1-q_{j'}(x,n)\}^{1-a_{j'}}$, shifting each peer's odds by $\delta$ yields the conditional CIPS policy distribution
\begin{equation}\label{eq:pidelta}
\pi_\delta(a_{-j}\mid x,n)
=\prod_{j'\neq j}q_{j',\delta}(x,n)^{a_{j'}}\bigl(1-q_{j',\delta}(x,n)\bigr)^{1-a_{j'}},
\qquad a_{-j}\in\mathcal{A}(n-1).
\end{equation}
The CIPS policy has three properties that the type-B policy lacks. (i) At $\delta=1$, $q_{j,1}=q_{j}$, so $\pi_1(\cdot\mid x,n)$ is the peer law with the factual propensity score, guaranteed to be supported by the data. (ii) As $\delta\downarrow0$ all peers are sent to control and as $\delta\uparrow\infty$ all peers are sent to treatment, so $\delta$ smoothly indexes a one-parameter family of interventions around the allocation strategy. (iii) The ranking of units by treatment probability is preserved: $q_{j}(x,n)<q_{j'}(x,n)$ implies $q_{j,\delta}(x,n)<q_{j',\delta}(x,n)$.
\subsection{Causal estimands}
\label{sec:estimands}
The direct and spillover effects for the policy are defined by
\begin{equation}\label{eq:tau}
\tau^{\bullet}(\delta)
=\mathbb{E}\!\left[N_i^{-1}\sum_{j=1}^{N_i}\tau_{j}^{\bullet}(\delta,N_i)\right],
\qquad \bullet\in\{DATT,SATT\},
\end{equation}
where
\begin{equation}\label{eq:tauj}
\tau_{j}^{\bullet}(\delta,n)
=\sum_{a_{-j}\in\mathcal{A}(n-1)}\bar\pi_\delta(a_{-j};n)\,\theta_{j}^{\bullet}(a_{-j},n),
\qquad \bullet\in\{DATT,SATT\}
\end{equation}
denotes the unit-level effect under the policy, with the average CIPS policy $\bar\pi_\delta(a_{-j};n)=\mathbb{E}_n\bigl[\pi_\delta(a_{-j}\mid X_i,n)\bigr]$
and the configuration-specific effects $\theta_j^{DATT}$ and $\theta_j^{SATT}$ defined below in \eqref{eq:thetaDATT}--\eqref{eq:thetaSATT}.
Fix a focal unit $j$ in a cluster of size $n$. For a peer configuration $a_{-j}\in\mathcal{A}(n-1)$, define the configuration-specific direct and spillover average treatment effects on the treated,
\begin{align}
\theta_{j}^{DATT}(a_{-j},n)
&=\mathbb{E}\!\left[Y_{ij2}(1,a_{-j})-Y_{ij2}(0,a_{-j})\mid A_{ij}=1,A_{i(-j)}=a_{-j},N_i=n\right],
\label{eq:thetaDATT}\\
\theta_{j}^{SATT}(a_{-j},n)
&=\mathbb{E}\!\left[Y_{ij2}(0,a_{-j})-Y_{ij2}(0,0)\mid A_{ij}=1,A_{i(-j)}=a_{-j},N_i=n\right].
\label{eq:thetaSATT}
\end{align}
The construction in Equations~\eqref{eq:tau} and~\eqref{eq:tauj} parallels the causal estimands of \citet{park2022efficient,lee2025efficient}, extended here to the DID design. Note that the policy enters through its covariate average $\bar\pi_\delta(a_{-j};n)$ rather than through the conditional peer law $\pi_\delta(a_{-j}\mid X_i,N_i)$ itself.
The two estimands admit an exact decomposition of a total effect.
Defining the configuration-specific total effect on the treated
$\theta_{j}^{TATT}(a_{-j},n)=\mathbb{E}[Y_{ij2}(1,a_{-j})-Y_{ij2}(0,0)\mid
A_{ij}=1,A_{i(-j)}=a_{-j},N_i=n]
=\theta_{j}^{DATT}(a_{-j},n)+\theta_{j}^{SATT}(a_{-j},n)$,
aggregation with the common weights in
\eqref{eq:tau}--\eqref{eq:tauj} yields
$\tau^{TATT}(\delta)=\tau^{DATT}(\delta)+\tau^{SATT}(\delta)$, for every $\delta\in(0,\infty)$.
This is the CIPS-policy analogue of the overall ATT decomposition in
Equation~(14) of \citet{sun2025difference}: there the direct and spillover
effects are averaged over the observed neighbourhood treatment distribution
among the treated, whereas here the weights are the counterfactual policy
$\bar\pi_\delta$, so the decomposition holds along the entire family of
interventions indexed by $\delta$.
In the no-interference special case $N_i\equiv1$, every cluster contains a single unit with no peers, $\mathcal{A}(0)=\{\emptyset\}$, $\bar\pi_\delta(\emptyset;1)=1$, and $Y_{ij2}(a_{ij},a_{i(-j)})=Y_{ij2}(a_{ij})$; then $\tau^{DATT}(\delta)$ reduces to the standard average treatment effect on the treated (ATT) of \citet{sant2020doubly} for every $\delta$, and $\tau^{SATT}(\delta)\equiv0$.
\section{Identification}
\label{sec:identification}
To identify the configuration-specific treatment effects, we introduce the following outcome regression and treatment assignment nuisance functions.
For $a\in\{0,1\}$,
\[
m_{a,j}(a_{-j},x,n) = \mathbb{E}[\Delta Y_{ij}\mid A_{ij}=a,A_{i(-j)}=a_{-j},X_i=x,N_i=n],
\]
\[
q_{j}(x,n)=\mathbb{P}(A_{ij}=1\mid X_i=x,N_i=n),\qquad
e_{a,j}(a_{-j}\mid x,n)=\mathbb{P}(A_{i(-j)}=a_{-j}\mid A_{ij}=a,X_i=x,N_i=n),
\]
and the joint probability of observing unit $j$ treated together with peer configuration $a_{-j}$
\[
r_{j}(a_{-j},n)=\mathbb{P}(A_{ij}=1,A_{i(-j)}=a_{-j}\mid N_i=n)=\mathbb{E}_n[q_{j}(X_i,n)\,e_{1,j}(a_{-j}\mid X_i,n)].
\]
\begin{asm}[Identification assumptions]\label{asm:1}
For all $a_{-j}\in\mathcal{A}(n-1)$, $x$ in the support of $X_i\mid N_i=n$, and $n\le n_{\max}$, the following hold.
\begin{enumerate}[label=(\roman*)]
\item (Consistency)
$Y_{ijt}=\sum_{a_i\in\mathcal{A}(N_i)}\mathbf{1}(A_i=a_i)\,Y_{ijt}(a_i)$ for all $i,j,t$.
\item (No anticipation) Treatment occurs only in period $2$, and
$Y_{ij1}(a_{ij},a_{i(-j)})=Y_{ij1}(0,0)$ for all $i,j$.
\item (Conditional parallel trends) For the identification of $\tau^{DATT}(\delta)$,
\begin{align*}
&\mathbb{E}[Y_{ij2}(0,a_{-j})-Y_{ij1}(0,0)\mid A_{ij}=1,A_{i(-j)}=a_{-j},X_i=x,N_i=n]\\
&\quad=\mathbb{E}[Y_{ij2}(0,a_{-j})-Y_{ij1}(0,0)\mid A_{ij}=0,A_{i(-j)}=a_{-j},X_i=x,N_i=n].
\end{align*}
For the identification of $\tau^{SATT}(\delta)$, the following two conditions hold:
\noindent (CPT-S1, standard parallel trends for $Y(0,a_{-j})$)
\begin{align*}
&\mathbb{E}[Y_{ij2}(0,a_{-j})-Y_{ij1}(0,a_{-j})\mid A_{ij}=1,A_{i(-j)}=a_{-j},X_i=x,N_i=n]\\
&\quad=\mathbb{E}[Y_{ij2}(0,a_{-j})-Y_{ij1}(0,a_{-j})\mid A_{ij}=0,A_{i(-j)}=a_{-j},X_i=x,N_i=n].
\end{align*}
\noindent (CPT-S2, cross-configuration trend for $Y(0,0)$)
\begin{align*}
&\mathbb{E}[Y_{ij2}(0,0)-Y_{ij1}(0,0)\mid A_{ij}=1,A_{i(-j)}=a_{-j},X_i=x,N_i=n]\\
&\quad=\mathbb{E}[Y_{ij2}(0,0)-Y_{ij1}(0,0)\mid A_{ij}=0,A_{i(-j)}=0,X_i=x,N_i=n].
\end{align*}
CPT-S1 is the usual conditional parallel trends assumption within peer configuration $a_{-j}$; CPT-S2 is a cross-configuration restriction stating that treated units with peer configuration $a_{-j}$ exhibit the same $Y(0,0)$ trend with control units that have no treated peers.
\item (Overlap) There exists a constant $c>0$ such that, for all $n\le n_{\max}$, all $j$, all $a_{-j}\in\mathcal{A}(n-1)$, and all $x$ in the support of $X_i\mid N_i=n$,
\[
c\le q_{j}(x,n)\le 1-c,\qquad
e_{a,j}(a_{-j}\mid x,n)\ge c\ \ (a\in\{0,1\}).
\]
These conditions imply $r_{j}(a_{-j},n)=\mathbb{E}_n[q_{j}e_{1,j}]\ge c^{2}$ for \emph{every} $a_{-j}\in\mathcal{A}(n-1)$, so each configuration cell carries positive treated mass. The requirement is imposed over all of $\mathcal{A}(n-1)$, rather than only over the realized support of $A_{i(-j)}\mid N_i=n$, because the aggregation \eqref{eq:tauj} runs over all configurations and $\bar\pi_\delta(a_{-j};n)>0$ for each of them.
\item (Moments) For every $n\le n_{\max}$ and $j$,
\[
\mathbb{E}_n[\Delta Y_{ij}^{2}]<\infty,\qquad
\mathbb{E}_n\!\left[\max_{a\in\{0,1\},\,a_{-j}\in\mathcal{A}(n-1)}|m_{a,j}(a_{-j},X_i,n)|^{2}\right]<\infty.
\]
\end{enumerate}
\end{asm}
Assumption~\ref{asm:1} is used to identify the potential outcomes from the observed data.
\begin{prop}[Identification]\label{prop:id}
Under Assumption~\ref{asm:1}, for every $\delta\in(0,\infty)$ the configuration-specific direct and spillover average treatment effects on the treated are identified as
\begin{align}
\theta_{j}^{DATT}(a_{-j},n)
&=\mathbb{E}\!\left[m_{1,j}(a_{-j},X_i,n)-m_{0,j}(a_{-j},X_i,n)\mid A_{ij}=1,A_{i(-j)}=a_{-j},N_i=n\right],
\label{eq:idDATT}\\
\theta_{j}^{SATT}(a_{-j},n)
&=\mathbb{E}\!\left[m_{0,j}(a_{-j},X_i,n)-m_{0,j}(0,X_i,n)\mid A_{ij}=1,A_{i(-j)}=a_{-j},N_i=n\right].
\label{eq:idSATT}
\end{align}
Consequently the average CIPS policy weights $\bar\pi_\delta(a_{-j};n)=\mathbb{E}_n[\pi_\delta(a_{-j}\mid X_i,n)]$ are identified through the shifted propensity \eqref{eq:qdelta} together with the conditional distribution of $X_i$ given $N_i=n$. Hence $\tau^{DATT}(\delta)$ and $\tau^{SATT}(\delta)$ in \eqref{eq:tau} are identified. A proof is provided in \ref{app:id}.
\end{prop}
\section{Semiparametric efficiency under partial interference}
\label{sec:efficiency}
\subsection{Global efficiency}
\label{sec:global}
Let $\mathcal{M}_{NP}$ denote the nonparametric model implied by the i.i.d.\ cluster-sampling structure of Section~\ref{sec:setup}. Throughout this section starred quantities denote evaluation at the truth. Our first main result characterizes the efficient influence function (EIF) of $\tau^{\bullet}(\delta)$.
\begin{thm}[Global efficiency under the CIPS policy]\label{thm:eif}
Under Assumption~\ref{asm:1}, for $\bullet\in\{DATT,SATT\}$ the efficient influence function of $\tau^{\bullet}(\delta)$ under $\mathcal{M}_{NP}$ is
\begin{equation}\label{eq:eif}
\varphi^{\bullet*}(O_i;\delta)
=\bar\varphi^{\bullet*}(O_i;\delta)+\Bigl\{\bar\tau^{\bullet}(\delta,N_i)-\tau^{\bullet}(\delta)\Bigr\},
\qquad
\bar\tau^{\bullet}(\delta,n)=n^{-1}\sum_{j=1}^{n}\tau_{j}^{\bullet}(\delta,n),
\end{equation}
and
\begin{equation}\label{eq:barphi}
\bar\varphi^{\bullet*}(O_i;\delta)
=N_i^{-1}\sum_{j=1}^{N_i}\sum_{a_{-j}\in\mathcal{A}(N_i-1)}
\Bigl\{\,\bar\pi_\delta^*(a_{-j};N_i)\,\phi_{j,a_{-j}}^{\bullet*}(W_{ij})
+\theta_{j}^{\bullet*}(a_{-j},N_i)\,\psi_{j,a_{-j}}^{\delta*}(O_i)\Bigr\},
\end{equation}
where $\phi_{j,a_{-j}}^{\bullet*}$ is the EIF of $\theta_{j}^{\bullet}(a_{-j},n)$ and $\psi_{j,a_{-j}}^{\delta*}$ is the EIF of $\bar\pi_\delta(a_{-j};n)$, given in \eqref{eq:phiDATT}--\eqref{eq:phiSATT} and \eqref{eq:psi} below, respectively. The semiparametric efficiency bound for $\tau^{\bullet}(\delta)$ under $\mathcal{M}_{NP}$ is $\mathbb{E}[\{\varphi^{\bullet*}(O_i;\delta)\}^{2}]$.
\end{thm}
\medskip
Define the importance weights
\[
\omega_{j}^*(a_{-j},x,n)=\frac{q_{j}^*(x,n)\,e_{1,j}^*(a_{-j}\mid x,n)}{(1-q_{j}^*(x,n))\,e_{0,j}^*(a_{-j}\mid x,n)},\qquad
\omega_{j}^{0*}(a_{-j},x,n)=\frac{q_{j}^*(x,n)\,e_{1,j}^*(a_{-j}\mid x,n)}{(1-q_{j}^*(x,n))\,e_{0,j}^*(0\mid x,n)},
\]
and recall $r_{j}^*(a_{-j},n)=\mathbb{P}(A_{ij}=1,A_{i(-j)}=a_{-j}\mid N_i=n)$. The efficient influence functions of the configuration-specific direct and spillover average treatment effects on the treated are
\begin{align}
\phi_{j,a_{-j}}^{DATT*}(W_{ij})
=&\;\frac{A_{ij}\mathbf{1}\{A_{i(-j)}=a_{-j}\}}{r_{j}^*(a_{-j},N_i)}\bigl\{\Delta Y_{ij}-m_{1,j}^*(a_{-j},X_i,N_i)\bigr\}\nonumber\\
&-\frac{(1-A_{ij})\mathbf{1}\{A_{i(-j)}=a_{-j}\}}{r_{j}^*(a_{-j},N_i)}\omega_{j}^*(a_{-j},X_i,N_i)\bigl\{\Delta Y_{ij}-m_{0,j}^*(a_{-j},X_i,N_i)\bigr\}\nonumber\\
&+\frac{A_{ij}\mathbf{1}\{A_{i(-j)}=a_{-j}\}}{r_{j}^*(a_{-j},N_i)}\bigl\{m_{1,j}^*(a_{-j},X_i,N_i)-m_{0,j}^*(a_{-j},X_i,N_i)-\theta_{j}^{DATT*}(a_{-j},N_i)\bigr\},
\label{eq:phiDATT}
\end{align}
\begin{align}
\phi_{j,a_{-j}}^{SATT*}(W_{ij})
=&\;\frac{(1-A_{ij})\mathbf{1}\{A_{i(-j)}=a_{-j}\}}{r_{j}^*(a_{-j},N_i)}\omega_{j}^*(a_{-j},X_i,N_i)\bigl\{\Delta Y_{ij}-m_{0,j}^*(a_{-j},X_i,N_i)\bigr\}\nonumber\\
&-\frac{(1-A_{ij})\mathbf{1}\{A_{i(-j)}=0\}}{r_{j}^*(a_{-j},N_i)}\omega_{j}^{0*}(a_{-j},X_i,N_i)\bigl\{\Delta Y_{ij}-m_{0,j}^*(0,X_i,N_i)\bigr\}\nonumber\\
&+\frac{A_{ij}\mathbf{1}\{A_{i(-j)}=a_{-j}\}}{r_{j}^*(a_{-j},N_i)}\bigl\{m_{0,j}^*(a_{-j},X_i,N_i)-m_{0,j}^*(0,X_i,N_i)-\theta_{j}^{SATT*}(a_{-j},N_i)\bigr\}.
\label{eq:phiSATT}
\end{align}
Each $\phi_{j,a_{-j}}^{\bullet*}$ is the efficient influence function of the configuration-specific effect $\theta_{j}^{\bullet}(a_{-j},n)$ given $N_i=n$, and has size-conditional mean zero. The EIF of the CIPS policy weight is
\begin{equation}\label{eq:psi}
\psi_{j,a_{-j}}^{\delta*}(O_i)
=\bigl\{\pi_\delta^*(a_{-j}\mid X_i,N_i)-\bar\pi_\delta^*(a_{-j};N_i)\bigr\}
+\sum_{j'\neq j}g_{j,a_{-j}}'(X_i,N_i)\,\bigl(A_{ij'}-q_{j'}^*(X_i,N_i)\bigr),
\end{equation}
where the first term is the covariate variation of the peer law and the second is the propensity-estimation correction, with
\begin{equation}\label{eq:gprime}
g_{j,a_{-j}}'(x,n)
=\frac{\partial \pi_\delta^*(a_{-j}\mid x,n)}{\partial q_{j'}(x,n)}
=\pi_\delta^*(a_{-j}\mid x,n)\,\frac{a_{j'}-q_{j',\delta}^*(x,n)}{q_{j',\delta}^*(x,n)\{1-q_{j',\delta}^*(x,n)\}}\,
\frac{\delta}{\{\delta q_{j'}^*(x,n)+1-q_{j'}^*(x,n)\}^{2}},
\end{equation}
the last factor being $\partial q_{j',\delta}/\partial q_{j'}$ obtained from Equation~\eqref{eq:qdelta}. The policy-estimation piece $\theta_{j}^{\bullet*}\psi_{j,a_{-j}}^{\delta*}$ in \eqref{eq:barphi} is the DID analogue of the weight-estimation term $\phi_Q$ in Corollary~1 of \citet{lee2025efficient}.
{In addition, if $N_i\equiv1$, so that every cluster contains a single unit with no peers (i.e. $\tau^{SATT}(\delta)\equiv0$), the EIF of DATT stated in Theorem~\ref{thm:eif} reduces to Proposition~1(a) of \citet{sant2020doubly}. Specifically, in this case $\mathcal{A}(0)=\{\emptyset\}$, and the CIPS policy and the size law are both degenerate, so that only $\phi_{1,\emptyset}^{DATT*}$ remains. Moreover $e_{a,1}^*(\emptyset\mid x,1)=1$, $r_{1}^*(\emptyset,1)=\mathbb{E}[q_{1}^*(X_i,1)]=\mathbb{P}(A_{i1}=1)$, and $\omega_{1}^*(\emptyset,x,1)=q_{1}^*(x,1)/\{1-q_{1}^*(x,1)\}$, and $\theta_{1}^{DATT*}$ reduces to the standard ATT.}
The proof of Theorem~\ref{thm:eif} is given in \ref{app:eif}.
{
\begin{rem}[Efficiency under the type-B policy]\label{rem:typeB}
Suppose the CIPS policy is replaced by the type-B policy $\pi_B(a_{-j}\mid n;\alpha)=\prod_{j'\neq j}\alpha^{a_{j'}}(1-\alpha)^{1-a_{j'}}$, i.e.\ the estimands are defined by \eqref{eq:tau}--\eqref{eq:tauj} with the fixed weight $\pi_B(a_{-j}\mid n;\alpha)$ in place of $\bar\pi_\delta(a_{-j};n)$; denote them by $\tau_B^{\bullet}(\alpha)$ and $\bar\tau_B^{\bullet}(\alpha,n)$. Because $\pi_B$ is a known constant that does not depend on the observed-data distribution, the policy-weight influence function vanishes, $\psi_{j,a_{-j}}^{\delta*}\equiv0$, and the efficient influence function of $\tau_B^{\bullet}(\alpha)$ under $\mathcal{M}_{NP}$ reduces to
\begin{equation}\label{eq:eifB}
\varphi_B^{\bullet*}(O_i;\alpha)
=N_i^{-1}\sum_{j=1}^{N_i}\sum_{a_{-j}\in\mathcal{A}(N_i-1)}\pi_B(a_{-j}\mid N_i;\alpha)\,\phi_{j,a_{-j}}^{\bullet*}(W_{ij})
+\Bigl\{\bar\tau_B^{\bullet}(\alpha,N_i)-\tau_B^{\bullet}(\alpha)\Bigr\},
\end{equation}
in which only the $\pi_B$-weighted configuration-specific EIFs $\phi_{j,a_{-j}}^{\bullet*}$ and the size-mean centring survive. This is the difference-in-differences analogue of the type-B efficiency bound of \citet{park2022efficient}. The proof is given at the end of \ref{app:eif}.
\end{rem}
}
\subsection{Proposed Estimators}
\label{sec:proposed est}
We collect the nuisances as $\eta=\bigl(q_{j},\,e_{a,j},\,m_{a,j},\,r_{j},\,\theta_{j}^{\bullet},\,\bar\pi_\delta\bigr)$, based on the EIFs given in the previous section, comprising the regression components $(q_{j},e_{a,j},m_{a,j})$ and the derived cell-level components $(r_{j},\theta_{j}^{\bullet},\bar\pi_\delta)$. The weights $(\omega_{j},\omega_{j}^{0})$, the shifted propensity $q_{j,\delta}$ of \eqref{eq:qdelta}, the peer law $\pi_\delta$ of \eqref{eq:pidelta}, and the derivative $g_{j,a_{-j}}'$ of \eqref{eq:gprime} are explicit functionals of $(q_{j},e_{a,j})$ and are evaluated by substitution. Define the uncentered efficient influence function:
\begin{equation}\label{eq:momentfn}
\Gamma^{\bullet}(O_i;\delta,\eta)
=\varphi^{\bullet}(O_i;\delta,\eta)
+\tau^{\bullet}(\delta;\eta)
=\bar\tau^{\bullet}(\delta,N_i;\eta)+\bar\varphi^{\bullet}(O_i;\delta,\eta),
\end{equation}
where each component is defined as in Section~\ref{sec:global} with every starred quantity replaced by the corresponding component of $\eta$. Note that $\mathbb{E}[\Gamma^{\bullet}(O_i;\delta,\eta^*)]=\tau^{\bullet}(\delta)$, which follows from Theorem~\ref{thm:eif}. Then, the estimator of $\tau^{\bullet}(\delta)$ is constructed as follows. First, cluster-level data $(O_1,\dots,O_M)$ are randomly partitioned into $K$ folds of equal size, $\mathcal{I}_1,\dots,\mathcal{I}_K$, with $M_k= |\mathcal{I}_k|=M/K$, for each $k \in \{1,\dots,K\}$, so that $\frac{1}{K}\frac{1}{M_k}=\frac{1}{M}$ and the average over folds of the within-fold means equals the overall sample mean, $\frac{1}{K}\sum_{k=1}^{K}\frac{1}{M_k}\sum_{i\in\mathcal{I}_k}=\frac{1}{M}\sum_{i=1}^{M}$. Then, for each fold $k$, the nuisance functions are estimated on the training complement $\mathcal{I}_k^{c}$, and the moment function is evaluated on the held-out fold $\mathcal{I}_k$, following the sample-splitting strategy of \citet{chernozhukov2018double}. {The training complement $\mathcal{I}_k^{c}$ is itself split into two halves: the regression nuisances $(\widehat q_{j},\widehat e_{a,j},\widehat m_{a,j})$ are fitted on the first half, while the derived cell-level components $(\widehat r_{j},\widehat\theta_{j}^{\bullet},\widehat{\bar\pi}_\delta)$ are formed as empirical means on the second half, so that they are independent of the regression fits. It is a proof device that costs nothing asymptotically; in practice using the full complement for both steps behaves similarly.} Finally, the estimator averages over folds:
\begin{align}
\widehat\tau^{\bullet}(\delta)
&=\frac{1}{K}\sum_{k=1}^{K}\frac{1}{M_k}\sum_{i\in\mathcal{I}_k}\Gamma^{\bullet}\bigl(O_i;\delta,\widehat\eta^{(-k)}\bigr)\nonumber\\
&=\frac{1}{K}\sum_{k=1}^{K}\frac{1}{M_k}\sum_{i\in\mathcal{I}_k}\frac{1}{N_i}\sum_{j=1}^{N_i}\sum_{a_{-j}\in\mathcal{A}(N_i-1)}
\Bigl\{\widehat{\bar\pi}_\delta(a_{-j};N_i)\bigl[\widehat\theta_{j}^{\bullet}(a_{-j},N_i)+\widehat\phi_{j,a_{-j}}^{\bullet}(W_{ij})\bigr]
+\widehat\theta_{j}^{\bullet}(a_{-j},N_i)\,\widehat\psi_{j,a_{-j}}^{\delta}(O_i)\Bigr\},
\label{eq:tauhat}
\end{align}
where every starred quantity is replaced by the corresponding estimated nuisance $\widehat\eta^{(-k)}$ trained on $\mathcal{I}_k^{c}$. Next we discuss the large-sample properties of the estimator $\widehat\tau^{\bullet}(\delta)$, and we impose the following conditions on the nuisance estimators.
\begin{asm}[Nuisance estimation]\label{asm:2}
For $f=f(O_i)$ write $\|f\|=\{\mathbb{E} f(O_i)^2\}^{1/2}$, the expectation taken over a new observation with the training-fold nuisances held fixed, and set
$\varepsilon_q=\max_{j,n}\|\widehat q_{j}-q_{j}^*\|$,
$\varepsilon_e=\max_{a,j,a_{-j},n}\|\widehat e_{a,j}-e_{a,j}^*\|$,
$\varepsilon_m=\max_{a,j,a_{-j},n}\|\widehat m_{a,j}-m_{a,j}^*\|$.
For every fold $k$:
\begin{enumerate}[label=(\roman*)]
\item (Boundedness) There are constants $c'>0$, $C<\infty$ with $c'\le\widehat q_{j}\le 1-c'$, $\widehat e_{a,j}\ge c'$, $\widehat r_{j}\ge c'$, $|\widehat m_{a,j}|\le C$, and $\mathbb{E}[\Delta Y_{ij}^{2}\mid A_i,X_i,N_i]\le C$ almost surely.
\item (Consistency) $\varepsilon_q,\varepsilon_e,\varepsilon_m=o_p(1)$.
\item (Rates) $\varepsilon_q^{2}+(\varepsilon_q+\varepsilon_e)\,\varepsilon_m=o_p(M^{-1/2})$.
\end{enumerate}
\end{asm}
Condition (iii) holds in particular when every nuisance converges faster than $M^{-1/4}$, the rate attained by many machine-learning estimators under structure (sparsity, smoothness) and, trivially at $M^{-1/2}$, by correctly specified parametric models. Note the asymmetry within (iii): the terms $\varepsilon_q\varepsilon_m$ and $\varepsilon_e\varepsilon_m$ admit the usual double-robust trade-off between assignment and outcome accuracy, but $\varepsilon_q^{2}$ stands alone---the propensity must itself converge faster than $M^{-1/4}$, and no accuracy in the outcome model can compensate. This is the estimation-level footprint of the policy-estimation term: the CIPS allocation law is a functional of $q_{j}$ alone, exactly as in \citet[\S4.2]{lee2025efficient}, whose condition $r_\pi=o(M^{-1/4})$ plays the same role.
\begin{thm}[Asymptotic normality and local efficiency]\label{thm:dr}
Suppose Assumptions~\ref{asm:1} and~\ref{asm:2} hold and $K$ is fixed. Then, as $M\to\infty$, for $\bullet\in\{DATT,SATT\}$,
\[
\sqrt M\bigl\{\widehat\tau^{\bullet}(\delta)-\tau^{\bullet}(\delta)\bigr\}
=\frac{1}{\sqrt M}\sum_{i=1}^M\varphi^{\bullet*}(O_i;\delta)+o_p(1)
\xrightarrow{d}\mathcal{N}\!\left(0,\mathbb{E}[\{\varphi^{\bullet*}(O_i;\delta)\}^{2}]\right),
\]
so $\widehat\tau^{\bullet}(\delta)$ attains the semiparametric efficiency bound of Theorem~\ref{thm:eif}. Moreover, the cross-fitted variance estimator
\begin{equation}\label{eq:Vhat}
\widehat V^{\bullet}(\delta)=\frac1M\sum_{k=1}^{K}\sum_{i\in\mathcal{I}_k}\bigl\{\Gamma^{\bullet}(O_i;\delta,\widehat\eta^{(-k)})-\widehat\tau^{\bullet}(\delta)\bigr\}^{2}
\end{equation}
satisfies $\widehat V^{\bullet}(\delta)\to_p\mathbb{E}[\{\varphi^{\bullet*}(O_i;\delta)\}^{2}]$, and
$\widehat\tau^{\bullet}(\delta)\pm z_{1-\gamma/2}\sqrt{\widehat V^{\bullet}(\delta)/M}$
is, for any $\gamma\in(0,1)$, an asymptotically valid $(1-\gamma)$ confidence interval for $\tau^{\bullet}(\delta)$. The second-order remainder derived in the proof consists of the product terms of Assumption~\ref{asm:2}(iii) and nothing else; in particular it contains $\varepsilon_q^{2}$ with no compensating factor. Consequently, consistent estimation of the propensity is necessary: if $\widehat q_{j}$ is inconsistent (for example, a misspecified parametric model), the estimated weights converge to a wrong allocation law and $\widehat\tau^{\bullet}(\delta)$ is in general inconsistent, no matter how well the outcome regression is estimated. There is thus no model-type double robustness for the CIPS policy. This property of the CIPS estimator is the exact DID counterpart of the behavior reported in \citet{lee2025efficient}. The proof of Theorem~\ref{thm:dr} is given in \ref{app:dr}.
\end{thm}
\begin{rem}[Double robustness under the type-B policy]\label{rem:typeBdr}
Under the type-B policy the estimator \eqref{eq:tauhat} \emph{is} doubly robust: because $\pi_B(a_{-j}\mid n;\alpha)$ is a known constant, the weight $\widehat{\bar\pi}_\delta(a_{-j};N_i)$ requires no estimation, and the policy-estimation term $\widehat\theta_{j}^{\bullet}(a_{-j},N_i)\,\widehat\psi_{j,a_{-j}}^{\delta}(O_i)$ vanishes since $\psi_{j,a_{-j}}^{\delta*}\equiv0$ (Remark~\ref{rem:typeB}). The only surviving nuisance error is then the configuration-specific augmented inverse-probability-weighting remainder $\mathbb{E}_n[(1-q^*)e_0^*\{\widehat\omega-\omega^*\}\{\widehat m_0-m_0^*\}]$, which vanishes whenever \emph{either} the assignment or the outcome model is correctly specified; the configuration-specific double robustness thus passes to the aggregate, giving the difference-in-differences analogue of the doubly robust estimator of \citet{park2022efficient}.
\end{rem}
\section{Simulation}
\label{sec:simulation}
To assess the finite-sample performance of the proposed estimators \eqref{eq:tauhat}, we generate
$M = \{1000,\ 2000\}$ independent clusters. The cluster size $N_i\in\{1,2,3\}$ is drawn from $\nu=(\nu_1,\nu_2,\nu_3)=(0.30,0.40,0.30)$, and each unit carries a scalar covariate $X_{ij}\sim N(0,1)$ drawn independently across units. The propensity score is logistic,
$q(x)=\operatorname{expit}(a_0+a_1 x),\ a_0=0,\ a_1=0.8$,
and treatments are assigned independently within a cluster, $A_{ij}\mid X_{ij}\sim\text{Bernoulli}\bigl(q(X_{ij})\bigr)$. Let $s_{ij}=\sum_{k\neq j}A_{ik}$, the outcome evolution is generated from
$\Delta Y_{ij}=b_0+b_A A_{ij}+b_s s_{ij}+b_{As}A_{ij}s_{ij}+g\,X_{ij}+\varepsilon_{ij}$,
with $(b_0,b_A,b_s,b_{As},g,\sigma)=(0,\,0.30,\,0.15,\,0.10,\,0.50,\,1)$, and $\varepsilon_{ij}\sim N(0,\sigma^2)$.
The peers' CIPS policy exposure law is $\text{Binomial}\bigl(n-1,\bar q_\delta\bigr)$ with $\bar q_\delta=\mathbb{E}[q_\delta(X)]$, with constant $\delta\in\{0.5,1,2\}$, and the configuration-specific effects are $\theta^{DATT}(s)=b_A+b_{As}s$ and $\theta^{SATT}(s)=b_s s$. The causal estimands therefore admit the closed forms
$\tau^{DATT}(\delta)=\sum_{n}\nu_n\bigl(b_A+b_{As}(n-1)\bar q_\delta\bigr)$, and
$\tau^{SATT}(\delta)=\sum_{n}\nu_n\,b_s(n-1)\bar q_\delta$. We estimate the nuisances by generalized linear models (a logistic regression for the propensity and a linear regression with an $A\times s$ interaction for the outcome, both correctly specified here) with $K=5$ cross-fitting folds, and report results over $D=2{,}000$ Monte-Carlo replications.
Denote the estimator of an estimand $\tau=\tau^{\bullet}(\delta)$ and its standard-error estimator from \eqref{eq:Vhat}, computed on the $d$th of $D$ simulated datasets, by $\widehat\tau_d$ and $\widehat\sigma_d$. We compute the bias $\bigl(\mathrm{Bias}=\sum_{d=1}^{D}(\widehat\tau_d-\tau)/D\bigr)$, the root-mean-squared error $\bigl(\mathrm{RMSE}=\{\sum_{d=1}^{D}(\widehat\tau_d-\tau)^2/D\}^{1/2}\bigr)$, the average standard error $\bigl(\mathrm{SE}=\sum_{d=1}^{D}\widehat\sigma_d/D\bigr)$, the empirical standard deviation of the estimates $\bigl(\mathrm{SD}=\operatorname{sd}\{\widehat\tau_d\}_{d=1}^{D}\bigr)$, and the coverage of the nominal $95\%$ confidence interval (Cov.). These quantities are reported in Table~\ref{tab:sim}.
\begin{table}[ht]
\centering
\small
\setlength{\tabcolsep}{4.5pt}
\caption{Simulation results for CIPS policy with constant $\delta$.}
\label{tab:sim}
\begin{tabular}{cc ccccc ccccc}
\toprule
& & \multicolumn{5}{c}{$M=1000$} & \multicolumn{5}{c}{$M=2000$} \\
\cmidrule(lr){3-7}\cmidrule(lr){8-12}
$\delta$ & Truth & Bias & RMSE & SD & $\mathrm{SE}$ & Cov. & Bias & RMSE & SD & $\mathrm{SE}$ & Cov. \\
\midrule
\multicolumn{12}{l}{\emph{Panel A. Direct effect $\tau^{DATT}(\delta)$}}\\
\addlinespace
$0.5$ & $0.335$ & $-0.001$ & $0.060$ & $0.060$ & $0.060$ & $0.950$ & $\phantom{-}0.000$ & $0.042$ & $0.042$ & $0.042$ & $0.950$ \\
$1.0$ & $0.350$ & $-0.001$ & $0.059$ & $0.059$ & $0.058$ & $0.950$ & $\phantom{-}0.000$ & $0.041$ & $0.041$ & $0.041$ & $0.951$ \\
$2.0$ & $0.365$ & $-0.001$ & $0.061$ & $0.061$ & $0.060$ & $0.948$ & $\phantom{-}0.000$ & $0.042$ & $0.042$ & $0.042$ & $0.954$ \\
\addlinespace
\multicolumn{12}{l}{\emph{Panel B. Spillover effect $\tau^{SATT}(\delta)$}}\\
\addlinespace
$0.5$ & $0.053$ & $-0.001$ & $0.042$ & $0.042$ & $0.038$ & $0.957$ & $\phantom{-}0.000$ & $0.027$ & $0.027$ & $0.026$ & $0.955$ \\
$1.0$ & $0.075$ & $-0.001$ & $0.057$ & $0.057$ & $0.052$ & $0.955$ & $\phantom{-}0.000$ & $0.037$ & $0.037$ & $0.036$ & $0.954$ \\
$2.0$ & $0.097$ & $-0.002$ & $0.073$ & $0.073$ & $0.065$ & $0.955$ & $\phantom{-}0.000$ & $0.047$ & $0.047$ & $0.045$ & $0.958$ \\
\bottomrule
\end{tabular}
\vspace{2pt}
\begin{minipage}{0.95\textwidth}
\footnotesize
\emph{Notes:} Results are for the cross-fitted CIPS estimators of the direct effect $\tau^{DATT}(\delta)$ (Panel~A) and the spillover effect $\tau^{SATT}(\delta)$ (Panel~B), with $M=1000$ and $M=2000$ reported side by side. ``Bias'' is the average of $\widehat\tau-\tau$ across replications; ``RMSE'' is the root-mean-squared error $\{D^{-1}\sum_{d=1}^{D}(\widehat\tau_d-\tau)^2\}^{1/2}$; SD is the Monte-Carlo standard deviation of the estimates and $\mathrm{SE}$ is the average standard error; ``Cov.'' is the coverage of the nominal $95\%$ confidence interval. Results are based on $D=2{,}000$ Monte-Carlo replications with $K=5$ cross-fitting folds.
\end{minipage}
\end{table}
From Table~\ref{tab:sim}, two conclusions emerge. (i) The proposed estimator has small bias and RMSE and coverage close to the nominal $95\%$, and all three properties improve as the sample size grows. This confirms the asymptotic normality and efficiency of the proposed estimator established in Theorem~\ref{thm:dr}. (ii) For the direct effect $\tau^{DATT}(\delta)$, the empirical standard deviation (SD) and the analytic standard error ($\mathrm{SE}$) are very close, so the plug-in variance estimator is well calibrated. For the spillover effect $\tau^{SATT}(\delta)$, SD and $\mathrm{SE}$ differ somewhat at $M=1000$, but this discrepancy is substantially reduced once the sample size increases to $M=2000$.
\section{Application: social pensions and intra-household spillovers}
\label{sec:application}
We illustrate the estimator in the setting of \citet{huang2021power}, who study the effect of China's New Rural Pension Scheme (NRPS), a large noncontributory social pension, on the rural elderly. The setting is a difference-in-differences problem with within-household partial interference: a member's pension may affect that member directly and spillover onto co-resident relatives through shared resources and a reallocation of labour, as pointed out by \citet{huang2021power}.
We re-cast the setting into the framework of Sections~\ref{sec:setup}--\ref{sec:efficiency}, treating the household as the cluster so that the within-household spillover becomes the object of interest, indexed by the CIPS odds multiplier $\delta$. The NRPS rolled out county-by-county from September 2009 and was near-universal by the end of 2012, and this staggered timing is the identifying variation \citep{huang2021power}: once a county was covered, any rural resident aged $16+$ could voluntarily enrol, with a basic pension from age $60$, so participation varies across units and we take NRPS participation as the unit-level treatment. Following \citet{huang2021power}, we use the China Family Panel Studies (CFPS), waves $2010$ (pre) and $2012$ (post), so treatment is realized only in the second period, and we drop the $55$ of $162$ CFPS counties whose rollout year is $2010$ or earlier, leaving $107$ counties untreated at the $2010$ baseline so that no-anticipation and conditional parallel trends hold cleanly; the treatment $A_{ij}=1$ indicates that member $j$ participates for the first time by $2012$, and the outcome is individual log labour income. Each household is a cluster ($i=1,\dots,M$) and each co-resident rural-hukou member aged $60$ and over is a unit ($j=1,\dots,N_i$), with households drawn i.i.d., within-household interference unrestricted, and cross-household (village) interference treated as second order. The cluster size $N_i$ (the observed number of co-resident sample members) is a bounded random variable entering \eqref{eq:tauhat} directly, and the size law $\nu_n$ is its empirical distribution; the analysis sample has $2{,}392$ members in $1{,}821$ households. The covariate vector $X_i$ stacks the baseline ($2010$) covariates: gender, age, age$^2$, education. Under Assumption~\ref{asm:1}, $\tau^{DATT}(\delta)$ is then the effect of a member's own participation on that member, and $\tau^{SATT}(\delta)$ the effect on a member's own labour income of a co-resident's participation, holding the member's own participation fixed at non-participation, the intra-household spillover \citet{huang2021power} document but avoid estimating as a causal contrast.
Figure~\ref{fig:income} reports $\widehat\tau^{DATT}(\delta)$ and $\widehat\tau^{SATT}(\delta)$ on a grid $\delta\in[0.5,2]$ with $95\%$ confidence intervals. The \emph{direct} effect on a participant's own labour income is negative at every $\delta$ but statistically insignificant throughout, with wide intervals ($[-0.87,\,0.63]$ at $\delta=1$). The sign (participants earn less) matches the labour-supply reduction that \citet{huang2021power} document for the age-eligible elderly, while the insignificance is consistent with their observation (their footnote~15) that the reduction is concentrated in farmwork, which brings in little cash, so the labour-supply response ``may not end up in lower income.''
\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth]{application_cfps/cips_did_figure1_income.pdf}
\caption{Cross-fitted CIPS estimates of the direct ($\tau^{DATT}(\delta)$, left) and spillover ($\tau^{SATT}(\delta)$, right) effects on individual \emph{log labour income}, over the CIPS odds multiplier $\delta$. Red dots are point estimates and vertical bars are $95\%$ confidence intervals; the dashed line marks the null. The direct effect is negative but insignificant; the spillover effect is significantly negative throughout and strengthens with $\delta$.}
\label{fig:income}
\end{figure}
The \emph{spillover} effect is cleaner and, relative to \citet{huang2021power}, new. A co-resident member's NRPS participation significantly lowers a member's own labour income at every policy strength, with the interval excluding zero across the whole range and the effect strengthening as $\delta$ rises. This is the within-household income effect that \citet{huang2021power} could only describe qualitatively: they show that age-ineligible co-residents reallocate labour (farm to non-farm) but stop short of an individual-level causal contrast because of such spillovers. Our CIPS DID estimator identifies and quantifies it: a household member's pension enrolment relaxes the shared budget constraint and induces other members to withdraw labour income, a negative intra-household labour-income spillover that grows as more members are drawn into enrolement.
\section{Discussion}
\label{sec:discussion}
This paper extended semiparametric efficiency theory for partial interference to the difference-in-differences design and to incremental-propensity-score counterfactuals. The CIPS policy indexes a one-parameter family of interventions around the factual allocation, so the estimands stay inside the support of the data; because the allocation law is a functional of the propensity score, the efficient influence function acquires a policy-estimation term, and the cross-fitted estimator attains the semiparametric efficiency bound while remaining one-sided robust---the propensity rate cannot be traded against outcome-model accuracy. The simulation confirms that the estimator is nearly unbiased with well-calibrated inference at sample sizes typical of applications, and the empirical study delivers a policy-relevant finding that the original analysis could only conjecture: a co-resident's pension participation significantly lowers a member's own labour income, a negative within-household spillover that strengthens as take-up is scaled up.
Several limitations point to future work. The partial interference assumption restricts spillovers to within-household pairs, and extending the framework to more general network structures would broaden its applicability. The two-period DiD design also abstracts from staggered adoption and time-varying effects, so adapting the CIPS estimand to multi-period settings is a natural next step. Further, one-sided robustness does not protect against outcome-model misspecification the way a doubly robust estimator would; whether a doubly robust policy-estimation term exists without sacrificing efficiency remains open.
Finally, the policy parameter---how far to shift the propensity score from the factual allocation---is treated here as fixed rather than chosen; casting it instead as a decision variable to be optimized against a stated welfare or budget criterion would connect the framework to the policy learning literature and yield a data-driven recommendation rather than an estimate at an analyst-specified shift.
\clearpage
\bibliographystyle{elsarticle-harv}
\bibliography{references}
\newpage{}