Kernel Conditional Moment Test via Maximum Moment Restriction

TL;DR

This paper introduces kernel conditional moment (KCM) tests that leverage a novel RKHS representation to efficiently test conditional moment restrictions, offering consistency and strong finite-sample performance.

Contribution

The paper develops a new KCM testing framework using conditional moment embedding in RKHS, providing an analytically tractable, consistent, and computationally efficient method.

Findings

KCM tests outperform existing methods in finite-sample scenarios.

The proposed test fully characterizes conditional moment restrictions.

Analytic expressions enable easy computation and asymptotic analysis.

Abstract

We propose a new family of specification tests called kernel conditional moment (KCM) tests. Our tests are built on a novel representation of conditional moment restrictions in a reproducing kernel Hilbert space (RKHS) called conditional moment embedding (CMME). After transforming the conditional moment restrictions into a continuum of unconditional counterparts, the test statistic is defined as the maximum moment restriction (MMR) within the unit ball of the RKHS. We show that the MMR not only fully characterizes the original conditional moment restrictions, leading to consistency in both hypothesis testing and parameter estimation, but also has an analytic expression that is easy to compute as well as closed-form asymptotic distributions. Our empirical studies show that the KCM test has a promising finite-sample performance compared to existing tests.