Graph Diffusion Residuals for Control-Function Instrumental Variables

• URL: http://arxiv.org/abs/2606.14636v1
• Date: Fri, 12 Jun 2026 17:02:53 GMT
• Title: Graph Diffusion Residuals for Control-Function Instrumental Variables
• Abstract summary: High-capacity first stages can interpolate treatment and leave too little residual information for the outcome equation.<n>We study Adaptive Anisotropic Instrumental Heat Flow (A-IHF), a deterministic graph-diffusion residual extractor for flexible control functions.<n>A-IHF treats treatment as a signal on a graph of first-stage features, uses pilot diffusion to detect large treatment jumps, attenuates conductance across those jumps, and computes the generated control with a sparse graph resolvent.
• Score: 82.39671985893028
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Control-function instrumental variable estimators need a first-stage residual, not merely a first-stage prediction. High-capacity first stages can interpolate treatment and leave too little residual information for the outcome equation. We study Adaptive Anisotropic Instrumental Heat Flow (A-IHF), a deterministic graph-diffusion residual extractor for flexible control functions. A-IHF treats treatment as a signal on a graph of first-stage features, uses pilot diffusion to detect large treatment jumps, attenuates conductance across those jumps, and computes the generated control with a sparse graph resolvent. Its observational selection rule uses only $(Z,X)$, combining graph generalized cross-validation, roughness, residualized-treatment relevance, and graph-admissibility filtering. The analysis decomposes error into structural leakage, residual attenuation, and residualized treatment variation, yielding finite-sample bounds, graph-admissibility rates under latent piecewise-smooth geometry, and finite-path selection calibration. Across 54 synthetic benchmark cells with tuned graph, kernel, tree, boosting, series, and neural control-function baselines, guarded observational A-IHF has the lowest average structural-response MSE; the A-IHF family beats the best non-A-IHF baseline in 32 cells. Performance is strongest when the graph captures piecewise-smooth first-stage structure.

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