Conformal Prediction for Nonparametric Instrumental Regression

Masahiro Kato

econ.EMMar 26, 2026

Conformal Prediction for Nonparametric Instrumental Regression

Authors: Masahiro Kato

Organizations: The University of Tokyo

Abstract

We propose a method for constructing distribution-free prediction intervals in nonparametric instrumental variable regression (NPIV), with finite-sample coverage guarantees. Building on the conditional guarantee framework in conformal inference, we reformulate conditional coverage as marginal coverage over a class of IV shifts $\mathcal{F}$ . Our method can be combined with any NPIV estimator, including sieve 2SLS and other machine-learning-based NPIV methods such as neural networks minimax approaches. Our theoretical analysis establishes distribution-free, finite-sample coverage over a practitioner-chosen class of IV shifts.

Figures & tables

Radius	IV Shift	Radius Model	Test Dist. of IVs	Cov. Ratio		Interval Length
Radius	IV Shift	Radius Model	Test Dist. of IVs	Mean	SD	Mean	SD
XZ-indexed	Bins		Observed	0.902	0.020	4.456	9.446
			Linear Tilt	0.901	0.024	4.288	7.181
			Local Tilt	0.900	0.027	4.335	6.623
			Step Tilt	0.901	0.022	4.329	7.742
	Linear		Observed	0.908	0.021	4.080	7.179

Table 1: Results of the experiment using Dataset 1.

Radius	IV Shift	Radius Model	Test Dist. of IVs	Cov. Ratio		Interval Length
Radius	IV Shift	Radius Model	Test Dist. of IVs	Mean	SD	Mean	SD
XZ-indexed	Bins		Observed	0.901	0.020	2.751	0.210
			Linear Tilt	0.898	0.024	2.800	0.246
			Local Tilt	0.889	0.028	2.822	0.274
			Step Tilt	0.900	0.021	2.786	0.235
	Linear		Observed	0.914	0.023	2.858	0.239

Table 2: Results of the experiment using Dataset 2.

Radius	IV Shift	Radius Model	Test Dist. of IVs	Cov. Ratio		Interval Length
Radius	IV Shift	Radius Model	Test Dist. of IVs	Mean	SD	Mean	SD
XZ-indexed	Bins		Observed	0.899	0.022	34.507	141.721
			Linear Tilt	0.898	0.023	34.532	141.043
			Local Tilt	0.899	0.025	34.710	142.243
			Step Tilt	0.898	0.024	34.487	140.992
	Linear		Observed	0.916	0.021	37.134	149.733

Table 3: Results of the experiment using Dataset 3.

Figure 1: Visualization of prediction intervals for Dataset 1 under the observed IV distribution. The $(X,Z)$ -indexed and $Z$ -indexed procedures use the Linear shift family, and the $X$ -indexed procedure uses the Linear radius model. The top row shows the interval surfaces over $(x,z)$ , the middle row shows $Y$ – $Z$ slices at $x=0$ , and the bottom row shows $Y$ – $X$ slices at $z=0$ .

Radius	IV Shift	Radius Model	Test Dist. of IVs	Cov. Ratio		Interval Length
Radius	IV Shift	Radius Model	Test Dist. of IVs	Mean	SD	Mean	SD
XZ-indexed	Bins		Observed	0.911	0.055	1.489	2.392
			Linear Tilt	0.918	0.060	1.282	1.511
			Local Tilt	0.918	0.070	1.335	1.621
			Step Tilt	0.919	0.057	1.360	1.816
	Linear		Observed	0.928	0.051	inf	-

Table 4: Results of the experiment using the CigarettesSW dataset.

Radius	IV Shift	Radius Model	Test Dist. of IVs	Cov. Ratio		Interval Length
Radius	IV Shift	Radius Model	Test Dist. of IVs	Mean	SD	Mean	SD
XZ-indexed	Bins		Observed	0.901	0.017	2.638	4.845
			Linear Tilt	0.893	0.020	2.521	4.446
			Local Tilt	0.882	0.028	2.386	4.043
			Step Tilt	0.891	0.021	2.502	4.402
	Linear		Observed	0.903	0.017	2.668	4.744

Table 5: Results of the experiment using the CollegeDistance dataset.

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Perturbative methods for non-parametric instrumental variable

Wei Bu, Arthur Gretton

Harvard University, Northeastern University, Physics Department · University College London.

Kernel Ridge Regression Instrumental Variable Estimation Causal Inference Inverse Problems

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We introduce a perturbative approach for nonparametric instrumental variable (NPIV) estimation. By drawing inspiration from perturbation theory in physics, we extend standard kernel ridge methods with systematic higher perturbation order corrections that significantly improve estimation accuracy. Spectrally, the perturbation introduces mixing between different eigenmodes of the expectation integral operator, which becomes especially useful when the integral equation is ill-defined. One source for such ill-definedness can be the curse of dimensionality. Our method performs across various dimensionality regimes, particularly when the dimensionality parameter

β

which is defined through the number of samples

n

and dimension

d

n^β= d

, becomes large. Experimental results show that our first-order perturbative corrections can reduce prediction error by up to 99% in high-dimensional ill-defined cases (

β> 0.7

) compared to standard ridge regression approaches. The performance improvement is maintained across a wide range of dimensions, with the advantage becoming more pronounced as dimensionality increases.

Abstract

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