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.