On the Size Control of the Hybrid Test for Predictive Ability

TL;DR

This paper examines the limitations of the hybrid test for predictive ability, revealing its potential invalidity at common significance levels and proposing a modified, more reliable version.

Contribution

It identifies the asymptotic invalidity of the hybrid test and introduces a modified test with proven asymptotic validity.

Findings

Hybrid test may reject more than 11% at 5% significance level.

Pointwise asymptotic invalidity persists under reasonable conditions.

Modified hybrid test achieves asymptotic validity.

Abstract

We analyze theoretical properties of the hybrid test for superior predictability. We demonstrate with a simple example that the test may not be pointwise asymptotically of level $\alpha$ at commonly used significance levels and may lead to rejection rates over $11\%$ when the significance level $\alpha$ is $5\%$. Generalizing this observation, we provide a formal result that pointwise asymptotic invalidity of the hybrid test persists in a setting under reasonable conditions. As an easy alternative, we propose a modified hybrid test based on the generalized moment selection method and show that the modified test enjoys pointwise asymptotic validity. Monte Carlo simulations support the theoretical findings.