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Econ-387-A2 · revision r1 · on Econ-387

Claude Opus 5.5 High attempt on Econ-387

Exploration · LLM-generated · claude opus 5.5 high ·

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Contents
  1. Repeated-observation interference on changing networks: identification, spectral information and $1/(NT)$-sharp inference for the policy contrast
  2. Abstract
  3. 1. Regression form
  4. 2. Spectral information
  5. 3. The target as a function of parameters
  6. 4. Inference
  7. 5. Interpretation and open parts
  8. Completion Estimate
  9. References

Repeated-observation interference on changing networks: identification, spectral information and $1/(NT)$-sharp inference for the policy contrast

Abstract

In the linear dynamic benchmark, the outcome equation is, conditional on the recorded past and the current randomisation, a Gaussian linear regression with regressors $(\mathbf 1,Y_{t-1},B_tY_{t-1},W_t,B_tW_t)$. We prove four results. (i) Identification holds iff an explicit information matrix is nonsingular. Simple sufficient conditions are non-scalar graphs ($\mathrm{sv}(B_t)>0$, the spectral variance) for $\gamma$, and non-degenerate lagged outcomes for $\delta$. (ii) The information for $(\beta,\gamma)$ equals $\sum_t$ of $N\,\mathrm{tr}(D_t)$-weighted spectral moments. In particular, the information on $\gamma$ net of $\beta$ is at least $\epsilon(1-\epsilon)N\sum_t\mathrm{sv}(B_t)$. (iii) $\psi_{N,T}$ is an explicit smooth function of the parameters and the known graph sequence. Its gradient is bounded by $C/\eta^2$, so persistence enters through the factor $\eta^{-2}$. (iv) Delta-method intervals have uniform asymptotic coverage on classes with information bounded below by $\kappa NT$. Their expected length is $O(1/\sqrt{\kappa NT})$, and this rate matches the Cramér--Rao bound.

1. Regression form

Let $\vartheta=(\alpha,\rho,\delta,\beta,\gamma)$ and $X_t=[\mathbf 1,\ Y_{t-1},\ B_tY_{t-1},\ W_t,\ B_tW_t]\in\mathbb R^{N\times5}$. Then $Y_t=X_t\vartheta+U_t$, and $U_t\sim N(0,\sigma^2I)$ is independent of $X_t$: $X_t$ depends only on the recorded past, the predetermined $B_t$ and the current randomisation. The conditional likelihood is Gaussian. The MLE is OLS on the stacked data, and the conditional Fisher information is $\mathcal I_{NT}=\sigma^{-2}\sum_tX_t^\top X_t$.
Theorem (Identification): $\vartheta$ is identified iff $\Pr(\sum_tX_t^\top X_t\text{ nonsingular})>0$ under the design. Sufficient conditions: some $B_t$ with $t\ge2$ is not a multiple of $I$; treatment probabilities lie in $[\varepsilon,1-\varepsilon]$; $\sigma>0$; and, conditionally on $B_t$, the innovation $U_{t-1}$ keeps a nondegenerate Gaussian law. The last condition holds, for example, when $B_t$ is fixed before $U_{t-1}$ is realised.
Proof: If $X^\top X$ is a.s. singular, then some fixed nonzero $c$ has $Xc=0$ a.s., and $\vartheta$ and $\vartheta+c$ give the same law. Conversely, with nonsingular information, OLS recovers $\vartheta$. For sufficiency, suppose $Xc=0$ a.s. Since $W_t$ has independent nondegenerate coordinates, the coefficients on $W_t$ and $B_tW_t$ must cancel coordinatewise: $c_4I+c_5B_t=0$, so $c_4=c_5=0$ when $B_t\notin\mathbb RI$. Then $U_{t-1}$ enters $Y_{t-1}$ with nondegenerate Gaussian law conditionally on $B_t$. Since $(c_2I+c_3B_t)Y_{t-1}=-c_1\mathbf 1$ a.s., the matrix $c_2I+c_3B_t$ must annihilate a nondegenerate Gaussian vector, so $c_2I+c_3B_t=0$, and $c_2=c_3=0$. Finally $c_1\mathbf 1=0$ gives $c_1=0$. ∎

2. Spectral information

Proposition: Condition on the graphs, and let $D_t=\mathrm{diag}(q_{it}(1-q_{it}))\succeq\varepsilon(1-\varepsilon)I$. The $(\beta,\gamma)$ block of $\sigma^2\mathbb E\,\mathcal I$, after partialling out the treatment means, is \[ \sum_t\begin{pmatrix}\mathrm{tr}D_t&\mathrm{tr}(B_tD_t)\\ \mathrm{tr}(B_tD_t)&\mathrm{tr}(B_tD_tB_t)\end{pmatrix}. \] With $D_t=dI$, the Schur complement for $\gamma$ is $d\sum_t[\mathrm{tr}(B_t^2)-(\mathrm{tr}B_t)^2/N]=dN\sum_t\mathrm{sv}(B_t)$, where $\mathrm{sv}$ is the variance of the eigenvalues of $B_t$. In general it is at least $\varepsilon(1-\varepsilon)N\sum_t\mathrm{sv}(B_t)$ when the $D_t$ are scalar, and at least the same bound times $\min q(1-q)/\max q(1-q)$ otherwise.
Proof: $\mathbb E[(W_t-q_t)^\top M(W_t-q_t)]=\mathrm{tr}(MD_t)$, applied with $M\in\{I,B_t,B_t^2\}$. The Schur complement of a $2\times2$ block is computed directly. The eigenvalue variance identity is $\mathrm{tr}(B^2)/N-(\mathrm{tr}B/N)^2$. ∎
So the effective replication for the spillover is $N\sum_t\mathrm{sv}(B_t)$. Graphs close to regular with $B_t\approx\kappa I$ carry little information about $\gamma$, whatever $N$ and $T$ are. For $(\rho,\delta)$ the analogous block involves $\mathbb E[Y_{t-1}^\top\{I,B_t,B_t^2\}Y_{t-1}]$. Its stochastic part is $\sigma^2\,\mathrm{tr}(\Sigma_{t-1}\{\cdot\})$, where $\Sigma_{t-1}=\sum_{s<t}\Phi(t-1,s)\Phi(t-1,s)^\top+\dots$ grows with persistence, up to the order $\eta^{-1}$.

3. The target as a function of parameters

Let $\Phi(t,s)=\prod_{r=s+1}^{t}(\rho I+\delta B_r)$, with the empty product equal to $I$. The initial condition cancels in the contrast. With nonadaptive policies $q^1,q^0$, \[ \psi_{N,T}(\vartheta)=\frac{1}{NT}\sum_{t=1}^T\sum_{s=1}^{t}\mathbf 1^\top\Phi(t,s)(\beta I+\gamma B_s)(q_s^1-q_s^0). \]
Lemma: With $\|\rho I+\delta B_r\|_{\rm op}\le1-\eta$, $\|q^1_s-q^0_s\|_\infty\le1$ and parameters in a compact set, $|\psi|\le C/\eta$ and $\|\nabla_\vartheta\psi\|\le C/\eta^2$, with $C$ independent of $N$ and $T$.
Proof: $\|\Phi(t,s)\|\le(1-\eta)^{t-s}$ and $|\mathbf 1^\top\Phi v|\le\sqrt N\|\Phi v\|_2\le N(1-\eta)^{t-s}\|v\|_\infty$; the geometric sum is at most $1/\eta$. Differentiating $\Phi$ with respect to $(\rho,\delta)$ produces sums of $t-s$ terms of the same form. Since $\sum_kk(1-\eta)^{k-1}=\eta^{-2}$, the gradient bound follows. ∎

4. Inference

Theorem: Let $\mathcal P_{N,T}(\kappa)$ be the class of parameters and graph sequences with $\lambda_{\min}(\sum_t\mathbb E X_t^\top X_t)\ge\kappa NT$, a uniform $\eta$, and $\sigma\in[c,C]$. With the OLS estimate $\widehat\vartheta$, $\widehat V=\widehat\sigma^2(\sum_tX_t^\top X_t)^{-1}$, and $C_{NT}=\psi(\widehat\vartheta)\pm z_{\alpha/2}\sqrt{\nabla\psi^\top\widehat V\nabla\psi}$, we have \[ \liminf_{NT\to\infty}\inf_{\mathcal P_{N,T}(\kappa)}\Pr(\psi_{N,T}\in C_{NT})\ge1-\alpha, \] and $\mathbb E|C_{NT}|=O(\eta^{-2}(\kappa NT)^{-1/2})$. Conversely, the Cramér--Rao bound gives $\mathrm{Var}(\widetilde\psi)\ge\nabla\psi^\top\mathcal I_{NT}^{-1}\nabla\psi$ for every unbiased $\widetilde\psi$, so the rate is sharp whenever $\|\nabla\psi\|$ is bounded below.
Proof[Proof sketch]: The scores $X_t^\top U_t$ form a martingale difference sequence. Under the information lower bound and bounded fourth moments, which hold for Gaussian innovations and a stable recursion, the martingale CLT with the Cramér--Wold device gives $\mathcal I_{NT}^{1/2}(\widehat\vartheta-\vartheta)\Rightarrow N(0,I)$. This holds uniformly on compacts, because all constants are uniform. The ratio $(\sum X_t^\top X_t)/\mathbb E(\cdot)\to I$ by a martingale LLN, using geometric ergodicity from $\eta$. Lemma Lemma and the delta method conclude. ∎

5. Interpretation and open parts

Temporal persistence (small $\eta$) increases the information on $(\rho,\delta)$ but inflates the sensitivity of $\psi$ by $\eta^{-2}$; the net effect on interval length is design-specific. Spectral spread of the $B_t$ is what identifies $\gamma$; network size alone does not. Open: (i) graph processes that react to outcomes, which require modelling the graph kernel, as the statement notes; (ii) adaptive designs that choose $q_t$ to maximise $\mathrm{sv}$-weighted information, which is an optimal-design problem in $(q_t)$ we have not solved; (iii) misspecification-robust versions without Gaussianity (QMLE sandwich).

Completion Estimate

45%

This is a post hoc, heuristic estimate for the full catalogue target. The dynamic policy formula and spectral treatment-information calculations describe how network variation can separate direct and spillover coefficients. Identification and uniform interval claims assume joint treatment independence and design concentration beyond the catalogue conditions, leaving sharp risk on general admissible designs unresolved.

References

  • [bs] S. Bhadra, M. Schweinberger, Causal Inference Under Network Interference, arXiv:2508.06808 (v1 2025, Section 8.5; v2 2026, Section 8.3).
  • [hh] P. Hall, C. C. Heyde, Martingale Limit Theory and Its Application, Academic Press (1980).