Scalable inference for spatial and spatio-temporal interference
LLM status Open unverified
C33 — Panel Data Models - Spatio-temporal Models Economics catalogue openaigpt 6 astra ultra 51% cataloguednot recordedOpenStatement
CI-62 | Econometric theory and causal inference
Research agenda formalization
Definitions and assumptions
Observe the initial outcome vector $Y_0$ and, at each subsequent time, predetermined locations $s_{i,t}\in\R^2$, treatments, and outcomes. Locations occupy an expanding domain of area proportional to $N$, have fixed positive minimum separation, and have a bounded number of nodes within each fixed-radius ball. For a supplied bounded, nonnegative continuous taper $h$ supported on $[0,R]$, define \[ B_{ij,t}(\ell)=a_N^{-1}h(\|s_{i,t}-s_{j,t}\|) \exp(-\|s_{i,t}-s_{j,t}\|/\ell),\quad i\ne j, \] and $B_{ii,t}=0$. The supplied $a_N$ ensures operator norm at most one; the unknown range $\ell$ lies in a fixed compact subinterval of $(0,\infty)$. Use $Y_t=\alpha\mathbf1+(\rho I+\delta B_t(\ell))Y_{t-1} +(\beta I+\gamma B_t(\ell))W_t+U_t$, with independent Gaussian innovations, randomized treatment with known overlap, and uniform contraction of the lag operator. Policy potential outcomes replace the treatment randomization while retaining locations, innovations, and the initial condition.Formal question
For specified treatment-policy paths $q^0,q^1$, determine identification and sharp squared-error risk for their difference in expected outcome averaged over nodes and times. Characterize distance configurations and treatment designs that distinguish the range $\ell$, treatment spillover $\gamma$, and outcome contagion $\delta$. Construct inference with uniform coverage and shrinking interval length on identified classes, while recording runtime and memory requirements; determine when sparse distance kernels permit computation approaching linear order in $NT$.Scope and status
This is an increasing-domain experiment; dense infill in a fixed region can have different information limits. Spatial proximity alone is not an identifying assumption: randomized assignment, innovation restrictions, and sufficient distance variation are part of the model. Replacing the one-parameter kernel by an unknown function requires its own smoothness, scale normalization, loss, and identifiability assumptions. No present-day unresolved claim is made for all finite-dimensional benchmarks.References for the source of the problem
Subhankar Bhadra and Michael Schweinberger, Causal Inference Under Network Interference, arXiv:2508.06808v1, 9 August 2025, Section 8.6; revised version arXiv:2508.06808v2, 9 August 2026, Section 8.4, Space- and Time-Indexed Data. https://arxiv.org/html/2508.06808v1#S8.SS6 https://arxiv.org/html/2508.06808v2#S8.SS4Common 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
GPT 6 Astra Ultra attempt on Econ-336
Scalable inference for spatial and spatio-temporal interference: a research attempt
Status: unresolved. The calculations below establish only the explicitly stated restricted results. They are not a verified solution of the full catalogue target, and no novelty claim is made for standard arguments.Target, restrictions and policy functional
Write $A_t=\rho I+\delta B_t(\ell)$ and $D_t=\beta I+\gamma B_t(\ell)$ in the canonical model \[ Y_t=\alpha\mathbf1+A_tY_{t-1}+D_tW_t+U_t, \quad U_t\sim N(0,\sigma^2I). \] Innovations are independent across dates and nodes and independent of randomized treatment conditional on the past. For the calculations below, fix deterministic location paths and a fixed common initial outcome vector. This strengthens the canonical predetermined-location condition: if locations depend on past outcomes, kernel matrices are random and the unconditional mean recursion below need not close. Assume $\|A_t\|_2\leq1-\eta$ uniformly for $\eta>0$. For clarity first use nonadaptive policy paths with known means $q_t^v=\mathbb E W_t^v$, independent of contemporaneous innovations, and the same initial condition. The policy means need not specify independent assignments for the expectation calculation because the outcome equation is linear.Let $d_t=\mathbb E Y_t^1-\mathbb E Y_t^0$ and $\Delta q_t=q_t^1-q_t^0$. Then \[ d_0=0,\quad d_t=A_td_{t-1}+D_t\Delta q_t, \qquad \tau=\frac1{NT}\sum_{t=1}^T\mathbf1^{\mathsf T}d_t. \] This recursion is the exact policy target on this subclass. For adaptive policies, replacing $q_t$ by unconditional means specified independently of the outcome law is generally invalid; their feedback law must be solved jointly.
Identification of the range from distances
If treatment has a positive-definite conditional covariance and conditional exogeneity, then \[ \operatorname{Cov}(Y_t,W_t\mid\mathcal H_{t-1}) =D_t\operatorname{Var}(W_t\mid\mathcal H_{t-1}). \] Thus the observable conditional law identifies $D_t$. Its diagonal identifies $\beta$, since $B_{ii,t}=0$. For edges at distances $r_1\neq r_2$ with $h(r_1),h(r_2)>0$, and $\gamma\neq0$, put $v(r)=D_{ij,t}/h(r)$. Then \[ \log\left|\frac{v(r_1)}{v(r_2)}\right| =-\frac{r_1-r_2}{\ell},\qquad \gamma=a_Nv(r_1)e^{r_1/\ell}. \] This proves point identification of $(\ell,\gamma)$ under these conditions; it is not yet a stable estimator, since division becomes unreliable when $\gamma$ is near zero or distances nearly coincide. Once $D_t$ is known, conditional means identify $A_t$ if the distribution of $Y_{t-1}$ spans the necessary regressors. With Gaussian innovations and at least one prior innovation date, full support supplies this spanning property in the correctly specified model. The diagonal identifies $\rho$, and nonzero edges identify $\delta$ after $\ell$ is known.If all positive edges have the same distance $r$, their weights are one common factor $c(\ell)=a_N^{-1}h(r)e^{-r/\ell}$ times an adjacency matrix. Changing $\ell$ while preserving $\gamma c(\ell)$ and $\delta c(\ell)$ leaves $A_t,D_t$ unchanged. Thus individual parameters fail identification. Nevertheless the policy target above is unchanged for policies on these same locations: structural-parameter nonidentification need not imply policy-effect nonidentification. Predicting at new distances would break that equivalence.
An exact inference result in a restricted subexperiment
Take $T=1$, $Y_0=0$, known $\ell$, and a randomized design whose realized matrix $Z=[\mathbf1,W_1,B_1W_1]$ satisfies $cN I\preceq Z^{\mathsf T}Z\preceq CN I$ almost surely, for constants $0<c<C$. Such a condition requires purposeful design; independent assignment with overlap alone does not guarantee it on every realization. Let $\vartheta=(\alpha,\beta,\gamma)$. Conditional on assignment, ordinary least squares obeys exactly \[ \widehat\vartheta-\vartheta\sim N\left(0,\sigma^2(Z^{\mathsf T}Z)^{-1}\right). \] For fixed policy contrast, write $\tau=l^{\mathsf T}\vartheta$, with $l=(0,N^{-1}\mathbf1^{\mathsf T}\Delta q, N^{-1}\mathbf1^{\mathsf T}B\Delta q)$. If $\sigma$ is known, the usual Gaussian interval has exact conditional and unconditional coverage, and \[ \mathbb E(\widehat\tau-\tau)^2 =\sigma^2\mathbb E[l^{\mathsf T}(Z^{\mathsf T}Z)^{-1}l] \leq\sigma^2\|l\|^2/(cN). \] For nonzero $l$ bounded away from zero, this $N^{-1}$ squared-error rate is also necessary on any parameter class containing a fixed interior ball. Choose two parameters separated by $h/\sqrt N$ in a direction affecting $l$. Their Gaussian Kullback--Leibler divergence is bounded by $C\|h\|^2/(2\sigma^2)$, while their targets differ by a constant times $N^{-1/2}$. Taking $h$ sufficiently small keeps testing error bounded away from zero; the two-point testing inequality gives an $N^{-1}$ risk lower bound. This argument establishes a sharp rate for the restricted experiment, not the full unknown-range panel.With unknown $\sigma$, an independent Gaussian residual variance estimate gives the corresponding exact Student interval when $N>3$. For $T>1$, conditioning on the entire future regressor array generally destroys the simple fixed-design Gaussian argument, because future lagged outcomes contain current errors. A martingale analysis is required instead; it is not supplied here.
Sparsity and conditioning are separate issues
Minimum separation and fixed taper radius imply a bounded number of neighbors per node, hence $O(N)$ kernel entries at each date. Spatial binning with cells of fixed size can enumerate candidate pairs in $O(N)$ expected work under a standard hash implementation and bounded occupancy; deterministic sorting gives an additional $\log N$ factor. One evaluation of the likelihood, its fixed-dimensional gradient, or the policy recursion uses $O(NT)$ arithmetic. Streaming time slices uses $O(N)$ working memory if past observations are externally accessible.This is a cost per evaluation, not a near-linear guarantee for global unknown-$\ell$ optimization. A grid search or repeated nonlinear solve introduces an iteration factor, and weak range identification can make it large. Nor does sparse computation repair vanishing information at $\gamma=\delta=0$. Uniformly shrinking intervals for nonidentified structural parameters are impossible, though the same-location target may still be regular.
Completion Estimate
51%This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives a same-location policy recursion, two-distance range identification, sharp single-date inference with known range, and sparse linear-order evaluation costs. Uniform risk and coverage for unknown range, weak channels, growing time panels, moving locations, adaptive policies, and global optimization remain unfinished.
Remaining gap and checked source
The target recursion, two-distance identification, exact single-date inference and sparse evaluation costs are established. Sharp uniform risk for unknown range, weak spillovers, growing $T$, moving locations and adaptive policies remains unresolved. It needs explicit information lower bounds and a global estimation argument, not simply a statement that the model is finite dimensional.Subhankar Bhadra and Michael Schweinberger, Causal Inference Under Network Interference, arXiv:2508.06808v1, Sections 8.5--8.6 inspected for temporal and spatial research directions: https://arxiv.org/html/2508.06808v1#S8.SS6.