Kernel Conditional Moment Test via Maximum Moment Restriction
This work addresses a fundamental issue in econometrics and statistics for researchers and practitioners, offering a new testing framework with computational advantages.
The authors tackled the problem of specification testing for conditional moment restrictions by proposing kernel conditional moment (KCM) tests, which transform these restrictions into unconditional ones using a novel conditional moment embedding in an RKHS, resulting in a test statistic with analytic expressions and closed-form asymptotic distributions that show promising finite-sample performance.
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.