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Econ-71 · economics · Economics

Causal effects with endogenous evolving interference networks

LLM status Open unverified

Statement

CI-13 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

Observe $m$ independent trajectories of a directed network with $N$ vertices through times $1,\ldots,T$. Each history $H_t$ includes the current graph $G_t$, current pretreatment covariates, and all past assignments and outcomes. Counterfactuals under a regime $g=(g_t)$ replace only the assignment kernels; network and outcome transition mechanisms continue responding to the counterfactual history. The total-effect target is \[ \theta_{N,T}(g,g')=\frac1N\sum_i \E[Y_{T,i}^{g}-Y_{T,i}^{g'}]. \] Impose consistency and sequential exchangeability given $H_t$, either by randomization or as a stated causal assumption. Require $g_t(\cdot\mid h)\ll q_t(\cdot\mid h)$ and similarly for $g'$, with declared density-ratio bounds.

One quantitative class has $G_t$ contained in a fixed candidate graph of maximum degree $D$, local transition kernels, and influence coefficients $L_{ij}$ whose row sums are at most $\lambda<1$ in total-variation distance. Conditional on the past and assignments, node transitions use independent noise.

Formal question

Characterize sufficient histories for identifying the total effect and determine minimax risk and honest interval length in $(m,N,T,D,\lambda)$ and the policy density-ratio bounds. Establish when increasing network size yields additional information beyond independent trajectory replication, and when temporal dependence or poor support prevents consistency. Give valid estimators that propagate endogenous graph changes rather than condition on the final graph as a pretreatment variable.

Scope and status

The ordinary longitudinal identification formula addresses fully observed exchangeable histories; the refinement asks for sharp guarantees in a specified growing-network class. Conditioning on treatment-induced network features can change the target or induce selection bias. Graph-mediated decompositions would require additional counterfactual definitions and assumptions and are not implicit in this total-effect question.

References for the source of the problem

Carlos Cinelli, Avi Feller, Guido Imbens, Edward Kennedy, Sara Magliacane, and Jose Zubizarreta, Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference, arXiv:2508.17099v1, 23 August 2025, Section 3.2.3, paragraph Time evolution. https://arxiv.org/html/2508.17099v1.

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-71-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-71

Causal effects with endogenous evolving interference networks: Identification and the limits of contraction as an effective-sample-size claim

Target

With $m$ independent network trajectories, the effect averages terminal outcomes over vertices while allowing the graph to evolve under policy: \[ \theta_{N,T}(g,g')=\frac1N\sum_{i=1}^N \mathbb E[Y_{T,i}^g-Y_{T,i}^{g'}]. \] The evolving-network agenda is in [agenda, Section 3.2.3]. For the quantitative results in this note impose $|Y_{T,i}|\le1$; this is an explicit additional condition because the canonical statement does not itself give a terminal-outcome moment bound. We prove an identified known-assignment estimator, a counterexample to unqualified $mN$ information scaling, and a sufficient local-independence condition under which network size does help.

Keep the graph transition in the intervention law

Let $Z_{t+1}$ include the next graph, covariates and outcomes. Factor the observed trajectory law as \[ p_0(dH_1)\prod_{t=1}^T q_t(dA_t\mid H_t) K_t(dZ_{t+1}\mid H_t,A_t). \] Every $H_t$ is measured before its assignment. Sequential exchangeability and consistency identify the intervention law by replacing $q_t$ by $g_t$ and retaining $K_t$. Put \[ R_g=\prod_{t=1}^T\frac{dg_t(\cdot\mid H_t)}{dq_t(\cdot\mid H_t)}(A_t), \qquad \bar Y=N^{-1}\sum_iY_{T,i}. \]
Proposition: Suppose the assignment kernels and both policy ratios are supplied and $R_g,R_{g'}\le B$ almost surely. Then \[ \widehat\theta=\frac1m\sum_{j=1}^m(R_{g,j}-R_{g',j})\bar Y_j \] is unbiased and has mean-squared error at most $4B/m$. The interval $[\widehat\theta\pm2\sqrt{B/(m\alpha)}]$ intersected with $[-2,2]$ has coverage at least $1-\alpha$.
Proof: Cancel assignment factors in the joint density to obtain $\mathbb E[R_g\bar Y]=\mathbb E_g\bar Y$ and $\mathbb E R_g=1$. All graph-transition factors remain in the resulting integral, so it is the total policy effect. Since $R_g^2\le BR_g$, \[ \mathbb E[(R_g-R_{g'})\bar Y]^2 \le2\mathbb E(R_g^2+R_{g'}^2)\le4B. \] Independence of trajectories gives the risk bound. Chebyshev's inequality proves coverage, and intersection with the known target range preserves it. ∎
This formula also states what a sufficient history must contain: all measured variables needed for the asserted exchangeability of each assignment and its invariant subsequent transition law. Conditioning on the final graph instead would replace the intervention law by a different conditional population.

Row contraction alone does not deliver $mN$ information

Theorem: Fix $0<a<1/4$. For all $N$ there is a one-step model with an empty candidate graph, independent node transition noises conditional on the observed past and assignments, and influence row sums at most $a<1$, in which minimax risk for the total effect is at least $c_a/m$ uniformly in $N$. Both policy density ratios are bounded by two.
Proof: In each trajectory draw an observed baseline $Z\sim\operatorname{Ber}(p)$, where $p\in[1/4,3/4]$, and put that same observed value in every node's baseline covariate. Draw a common randomized assignment $A\sim \operatorname{Ber}(1/2)$ and assign $A_i=A$ to all vertices. Given these variables, generate independent \[ Y_i\sim\operatorname{Ber}(1/2+a A_i Z_i),\qquad Z_i=Z. \] The two policies assign all ones or all zeros. Their likelihood ratios against the common assignment are $2\mathbf1\{A=1\}$ and $2\mathbf1\{A=0\}$. The effect is $\theta=ap$. Each transition uses only its own baseline and assignment, and changing its baseline changes its Bernoulli law in total variation by at most $a$; there is no influence from other nodes. Thus its influence matrix has row sum at most $a$, and the conditional node noises are independent. Nevertheless the conditional distribution of every assignment and outcome given $Z$ is independent of $p$. The likelihood for $p$ consists of exactly $m$ Bernoulli baseline draws, regardless of $N$. For $p_\pm=1/2\pm b/\sqrt m$ the sample KL is bounded by $Cb^2$. Choosing small $b$ and testing the two targets gives risk at least $c a^2/m$. All assumptions in the theorem, including sequential exchangeability, hold by construction. ∎
The missing restriction in any $mN$ claim here is on the initial spatial law or shared observed random inputs. The contraction condition addresses transition sensitivity, not how many independent draws of that initial law have been observed. In this counterexample $N$ repeated baseline copies contain only one draw of $Z$.

A positive result from independent causal cones

The following stronger submodel makes the information gained from $N$ explicit. Let a fixed candidate graph have maximum undirected degree $D$. Initial node variables and all node-time innovations, including assignment randomization, are mutually independent. Graph changes may be generated locally as part of the state. Suppose the terminal outcome at vertex $i$ and every assignment ancestor needed for its policy intervention depend only on innovations in a deterministic space-time cone $\mathcal C_i$. All vertices in that cone are within graph distance $R$ of $i$. Assume the observational assignment law factorizes into known local kernels and that the policy kernels have the same local ancestry. Define $R_{i,g}$ as the product of policy-to-observed ratios for assignment nodes ancestral to $Y_{T,i}$. Suppose $R_{i,g},R_{i,g'}\le B_{\rm loc}$. Let $H$ be the maximum number of cones, including itself, intersecting any given cone. In particular \[ H\le \min\left\{N,\ 1+\sum_{k=1}^{2R}D^k\right\}. \tag{1} \]
Theorem: Under precisely these cone and independence conditions, \[ \widehat\theta_{\rm loc}= \frac1{mN}\sum_{j=1}^m\sum_{i=1}^N (R_{i,g,j}-R_{i,g',j})Y_{T,i,j} \] is unbiased, has risk at most $4B_{\rm loc}^2H/(mN)$, and gives an honest interval of radius $2B_{\rm loc}\sqrt{H/(mN\alpha)}$ by Chebyshev's inequality.
Proof: Marginalize all nodes outside the ancestral cone in reverse topological order; each removed conditional density integrates to one. Inside the cone, weighting cancels precisely the ancestral assignment factors. This proves each summand's expectation is the required vertex effect, including the local graph transitions. Write the summand as $Z_i$. It is a function of the innovations in $\mathcal C_i$ and has $|Z_i|\le2B_{\rm loc}$. Disjoint cones have independent innovations and hence zero covariance. For an intersecting pair, Cauchy--Schwarz bounds covariance in absolute value by $4B_{\rm loc}^2$. At most $NH$ ordered pairs intersect, so $\operatorname{Var}(N^{-1}\sum_iZ_i)\le4B_{\rm loc}^2H/N$. Divide by $m$ for independent trajectories and apply Chebyshev. If cones intersect, their center vertices are at distance at most $2R$; counting graph walks gives (1). ∎

Unresolved dependence on $T,D,\lambda$

For nearest-neighbor updates a radius proportional to $T$ gives a conservative bound through (1), often exponential in $T$. It does not exploit contraction $\lambda$ to sharpen that dependence. Doing so needs control of initial correlations, assignment dependence, and how likelihood ratios interact with influence decay. The counterexample shows why those conditions cannot be omitted. Unknown kernels, unbounded outcomes without moment assumptions, and a sharp joint minimax rate in the full evolving- network class remain unresolved.

References

  • [agenda] C. Cinelli, A. Feller, G. Imbens, E. Kennedy, S. Magliacane and J. Zubizarreta. Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference (2025), arXiv:2508.17099v1. https://arxiv.org/html/2508.17099v1.

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