Graph Diffusion Residuals for Control-Function Instrumental Variables
English summary
The paper proposes Adaptive Anisotropic Instrumental Heat Flow (A-IHF), a deterministic graph-diffusion residual extractor for control-function instrumental variable estimators. A-IHF treats treatment as a signal on a feature graph, uses pilot diffusion to detect treatment jumps, attenuates conductance across those jumps, and computes control via a sparse graph resolvent. The observational selection rule combines graph generalized cross-validation, roughness, residualized-treatment relevance, and graph-admissibility filtering without outcome data. The analysis decomposes error into structural leakage, residual attenuation, and residualized treatment variation, yielding finite‑sample bounds and graph‑admissibility rates under latent piecewise‑smooth geometry. Across 54 synthetic benchmark cells, guarded observational A‑IHF achieves the lowest average structural‑response MSE and the A‑IHF family beats the best non‑A‑IHF baseline in 32 cells, excelling when the graph captures piecewise‑smooth first‑stage structure.
Chinese summary
论文提出自适应各向异性工具热流（A‑IHF），一种用于控制函数工具变量估计的确定性图扩散残差提取器。A‑IHF 将处理变量视为特征图上的信号，利用先导扩散检测处理跳跃，跨越跳跃处衰减电导，并通过稀疏图求解器计算生成控制。观测选择规则仅使用工具变量和协变量，结合图广义交叉验证、粗糙度、残差化处理相关性和图可容许性过滤。误差分解为结构泄漏、残差衰减和残差化处理变差，给出了潜在分片光滑几何下的有限样本界和图可容许速率。在54个合成基准单元中，受防护的观测式A‑IHF取得了最低的平均结构响应MSE，A‑IHF族在32个单元中击败最佳非‑A‑IHF基线，尤其在图形捕捉到分片光滑第一阶段结构时表现最强。
Key points
Introduces Adaptive Anisotropic Instrumental Heat Flow (A‑IHF), a deterministic graph‑diffusion method that extracts first‑stage residuals for control‑function IV estimation.
提出自适应各向异性工具热流（A‑IHF），一种用于控制函数IV估计的确定性图扩散残差提取方法。
A‑IHF operates by modeling treatment as a graph signal, using pilot diffusion to detect jumps and attenuate conductance, then computing control via a sparse graph resolvent.
A‑IHF 通过将处理变量建模为图信号、使用先导扩散检测跳跃并衰减电导，再通过稀疏图求解器计算控制。
The observational selection rule uses only (Z,X) data and combines graph generalized cross‑validation, roughness, residualized‑treatment relevance, and graph‑admissibility filtering.
观测选择规则仅使用工具变量和协变量，结合图广义交叉验证、粗糙度、残差化处理相关性和图可容许性过滤。
Theoretical analysis provides finite‑sample error bounds and graph‑admissibility rates under a latent piecewise‑smooth geometry assumption.
理论分析在潜在分片光滑几何假设下给出了有限样本误差界和图可容许速率。
Empirically, guarded observational A‑IHF achieves lowest average structural‑response MSE across 54 synthetic benchmarks, outperforming the best non‑A‑IHF baseline in 32 cells, especially when the graph captures piecewise‑smooth first‑stage structure.
实证上，受防护的观测式A‑IHF在54个合成基准中取得最低平均结构响应MSE，在32个单元中击败最佳非‑A‑IHF基线，尤其在图形捕捉分片光滑第一阶段结构时表现突出。