Graph Diffusion Residuals for Control-Function Instrumental Variables

2026-06-12

AI Summary

This work addresses the tendency of high-capacity first-stage models in control function instrumental variable estimation to overfit the treatment variable, which depletes residual information and weakens identification in the outcome equation. To mitigate this, the authors propose the Adaptive Anisotropic Instrumental Heat Flow (A-IHF) method, which models the treatment as a signal on a graph and extracts structure-aware residuals via graph diffusion with anisotropic conductance, thereby constructing a flexible control function. The approach innovatively integrates jump detection, sparse graph resolvents, and an observation selection criterion relying solely on $(Z, X)$ to effectively capture piecewise-smooth latent structures. Evaluated across 54 synthetic benchmarks, A-IHF achieves the lowest average mean squared error in structural response and outperforms the best non-A-IHF baseline in 32 scenarios, with particularly pronounced gains when the underlying graph accurately encodes piecewise-smooth structure.

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.

Problem

• control-function
• instrumental variables
• residual extraction
• graph diffusion
• first-stage residuals

Innovation

• graph diffusion
• control function
• instrumental variables
• anisotropic heat flow
• residual extraction