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Conformal Prediction for Nonparametric Instrumental Regression

Masahiro Kato

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.

Conformal Prediction for Nonparametric Instrumental Regression

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 . 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.

Paper Structure

This paper contains 74 sections, 9 theorems, 136 equations, 1 figure, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution
  3. Related Work
  4. Setup
  5. Variables and Observations
  6. Variables
  7. Observations
  8. Prediction under Endogeneity
  9. Recap of identification through conditional moment restrictions
  10. Reformulation by conditional prediction given IVs
  11. Ideal (but Infeasible) Goal: Prediction Interval with Conditional Coverage Guarantee
  12. Target of Interest and IV Shift Relaxation
  13. Exact Conditional Coverage
  14. Relaxation via IV Shifts
  15. Prediction Intervals and Radius Classes
  16. ...and 59 more sections

Key Result

Proposition 3.1

Equation (eq:exact_conditional) holds if and only if where ${\mathcal{M}}$ denotes the set of all bounded measurable functions $f\colon {\mathcal{Z}}\to{\mathbb{R}}$.

Figures (1)

  • 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 $z=0$.

Theorems & Definitions (22)

  • Proposition 3.1: Moment characterization of exact conditional coverage
  • Proposition 3.2: Radius-class nesting
  • Proposition 4.1: Importance-weighted identity for the $X$-indexed class
  • Remark
  • Theorem 6.1: Shift-robust coverage over joint tilts
  • Theorem 6.2: Shift-robust coverage over IV tilts
  • Remark
  • Proposition 6.3: Stability to NPIV estimation error
  • Theorem 6.4: Length inflation bound
  • Remark
  • ...and 12 more