This work proposes deep nonparametric Instrumental variable quantile regression (IVQR), a two-stage estimator that combines conditional diffusion modeling with a kernel-smoothed conditional moment formulation. In the first stage, we estimate the joint conditional distribution of the outcome and endogenous covariates given the instrument using a variance-preserving conditional diffusion model. In the second stage, we approximate the conditional moment operator through Monte Carlo sampling and a kernel-smoothed surrogate for the indicator function, and then estimate the structural quantile function by empirical risk minimization over deep neural networks. We establish an excess-risk bound for the proposed estimator and derive end-to-end total variation guarantees for the conditional diffusion model under unbounded support, explicitly accounting for score estimation, early stopping, and discretization errors. Our theory is developed under a polynomial-tail envelope on the data distribution and degenerates continuously to the exponential setting: as the tail index grows, the obtained excess-risk rate converges to the minimax-optimal rate of nonparametric regression, thus our heavy-tailed theory covers the classical light-tailed nonparametric guarantees as a limiting case. Simulation studies and a real-data application demonstrate that the proposed method outperforms existing nonparametric IVQR approaches, with gains that become increasingly pronounced as the dimensionality of the covariates and instruments increases.
Many two-stage estimators assess the first-stage learner by prediction error, even when the next stage uses its residual. In control-function instrumental variables, that residual must preserve the latent control direction without removing the treatment variation that identifies the structural response. A scalar prediction score does not reveal how the learner allocates this variation. Under piecewise-smooth graph geometry, interpolation can suppress the control, whereas isotropic smoothing can leak systematic variation across boundaries. We formulate this as a variation-allocation problem and introduce Adaptive Anisotropic Instrumental Heat Flow (A-IHF). The method uses pilot treatment contrasts to adapt edge conductance, takes the complement of a sparse graph resolvent as the generated control, and selects candidates without consulting outcomes. For a linear control-function regression, the generated control is identified only by its span. Working in that projective geometry, we derive an exact finite-sample fidelity--relevance frontier, spectral identities for remaining treatment variation and coefficient distortion, and a lower bound for monotone fixed-graph residual filters. A connected construction proves that adapting conductance can remove the corresponding fixed-graph obstruction. In a 54-cell benchmark, the A-IHF family wins 32 cells; its guarded observational variant lowers mean nonlinear response error by 8.3%, with the largest gains in fractured designs. Controlled rewiring explains when the graph should be used, replaced by a fallback, or rejected. The resulting lesson is task-specific: a first stage for generated controls should be judged by control fidelity, downstream relevance, and graph compatibility together.
We consider debiased inference on least-squares solutions to inverse problems as a way to avoid having to assume exact solutions exist. Such assumptions are substantive and not innocuous and their failure may well imperil inference when we impose them on the statistical model. Our approach instead allows us to conduct inference on a quantity that is defined regardless of solutions existing and coincides with the usual estimands when they do. For the case of instrumental variables, this means we can motivate the analysis with structural models but these do not need to hold exactly for the inferential procedure to remain valid.