Kernel Minimum Distance Estimation and Testing with Conditional Moment Restrictions: A Unified Framework

2026-07-17

AI Summary

This study addresses the computational instability and complexity associated with nonparametric orthogonalization in conditional moment models by proposing a kernel minimum distance framework based on reproducing kernel Hilbert space (RKHS) embeddings. The approach constructs a closed-form V-statistic objective function without requiring auxiliary nonparametric estimation, thereby unifying parameter estimation and model specification testing. By incorporating an endogenous treatment effect through a projected kernel structure and employing a multiplier bootstrap procedure, the method achieves both statistical consistency and computational tractability. Theoretical analysis establishes that the estimator is √n-consistent and asymptotically normal, while the test statistic exhibits desirable properties under the null, alternative, and local alternatives. Simulations and an empirical application to UK household expenditure data confirm the robustness of the method in finite samples.

Abstract

We propose a unified Kernel Minimum Distance (KMD) framework for estimating and testing models defined by conditional moment restrictions. By embedding conditional moments into a Reproducing Kernel Hilbert Space (RKHS), we construct a closed-form $V$-statistic objective function that quantifies the distance from the restrictions. We establish the $\sqrt{n}$-consistency and asymptotic normality of the associated minimum distance estimator. Within this framework, the minimized objective function naturally yields a consistent omnibus specification test. Unlike projection-based methods that require auxiliary nonparametric estimation for Neyman orthogonalization, our test inherently captures the estimation effect via a projected kernel structure. We derive asymptotic properties of the test statistics under the null hypothesis, the alternative hypothesis, and a sequence of local alternatives converging to the null at the parametric rate $n^{-1/2}$. The validity of a computationally simple multiplier bootstrap is established to facilitate inference. Simulation results demonstrate robust finite-sample performance, and the framework is illustrated by analyzing Engel curves using UK Family Expenditure Survey data.

Problem

• conditional moment restrictions
• minimum distance estimation
• specification testing
• kernel methods
• omnibus test

Innovation

• Kernel Minimum Distance
• Conditional Moment Restrictions
• Reproducing Kernel Hilbert Space
• Omnibus Specification Test
• Multiplier Bootstrap