Efficient difference-in-differences estimation under partial interference with incremental propensity score policies

Efficient difference-in-differences estimation under partial interference with incremental propensity score policies

Introduction

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 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 lee2025efficient (hereafter LZH).

First, we adopt LZH's data structure. Clusters are independent and identically distributed draws from a super-population, and the cluster size $N_i$ is a 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 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 kennedy2019nonparametric to clustered interference, instead 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 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 hudgens2008toward,tchetgen2012causal,liu2019doubly,park2022efficient,lee2025efficient from cross-sectional designs to the difference-in-differences setting, building on the doubly robust DID estimator of sant2020doubly and the incremental propensity score interventions of kennedy2019nonparametric,munoz2012population,wen2023intervention. On the DID side, it complements recent work on identification and estimation under interference xu2023difference,sun2025difference and on the interpretation of conventional estimators when interference is misspecified savje2021average, by delivering policy-relevant, efficient estimands with explicit influence-function-based inference.

The remainder of the paper is organized as follows. Section (ref) sets up the model, the CIPS policy, and the estimands. Section (ref) states the identifying assumptions and identification result. Section (ref) develops the global and local efficiency theory (Theorems (ref) and (ref)). Section (ref) reports a simulation study and Section (ref) presents the empirical application; Section (ref) concludes and the appendix collects all proofs.

Set-up

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 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 $\Delta Y_{ij}$.

Potential outcomes

We use the potential outcome framework for partial interference 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.

The cluster incremental propensity score policy

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 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 cluster incremental propensity score (CIPS) policy proposed by lee2025efficient, which extends the incremental propensity score framework of 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

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

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 $q_{j,\delta}(x,n)<q_{j',\delta}(x,n)$.

Causal estimands

The direct and spillover effects for the policy are defined by

where

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 (ref)--(ref). 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,

The construction in Equations (ref) and (ref) parallels the causal estimands of 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 (ref)--(ref) 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 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 sant2020doubly for every $\delta$, and $\tau^{SATT}(\delta)\equiv0$.

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 $a_{-j}$ \[ r_{j}(a_{-j},n)=\mathbb{P}(A_{ij}=1,A_{i(-j)}=a_{-j}\mid N_i=n)=\mathbb{E}_n[q_{j}(X_i,n)\,e_{1,j}(a_{-j}\mid X_i,n)]. \]

Assumption (ref) is used to identify the potential outcomes from the observed data.

Semiparametric efficiency under partial interference

Global efficiency

Let $\mathcal{M}_{NP}$ denote the nonparametric model implied by the i.i.d.\ cluster-sampling structure of Section (ref). Throughout this section starred quantities denote evaluation at the truth. Our first main result characterizes the efficient influence function (EIF) of $\tau^{\bullet}(\delta)$.

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

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

where the first term is the covariate variation of the peer law and the second is the propensity-estimation correction, with

the last factor being $\partial q_{j',\delta}/\partial q_{j'}$ obtained from Equation (ref). The policy-estimation piece $\theta_{j}^{\bullet*}\psi_{j,a_{-j}}^{\delta*}$ in (ref) is the DID analogue of the weight-estimation term $\phi_Q$ in Corollary 1 of 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) reduces to Proposition 1(a) of 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) is given in (ref).

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Proposed Estimators

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 (ref), the peer law $\pi_\delta$ of (ref), and the derivative $g_{j,a_{-j}}'$ of (ref) are explicit functionals of $(q_{j},e_{a,j})$ and are evaluated by substitution. Define the uncentered efficient influence function:

where each component is defined as in Section (ref) 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). 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 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:

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.

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 lee2025efficient, whose condition $r_\pi=o(M^{-1/4})$ plays the same role.

Simulation

To assess the finite-sample performance of the proposed estimators (ref), 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 (ref), 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).

From Table (ref), 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). (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 $M=2000$.

Application: social pensions and intra-household spillovers

We illustrate the estimator in the setting of 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 huang2021power. We re-cast the setting into the framework of Sections (ref)--(ref), 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 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 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 (ref) 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), $\tau^{DATT}(\delta)$ 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 huang2021power document but avoid estimating as a causal contrast.

Figure (ref) reports $\widehat\tau^{DATT}(\delta)$ and $\widehat\tau^{SATT}(\delta)$ on a grid $\delta\in[0.5,2]$ with $95\%$ confidence intervals. The 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 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.”

The spillover effect is cleaner and, relative to 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 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.

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.