Improving test statistics for the weighted chi-square distribution

Abstract

Many econometric test statistics—F, likelihood‑ratio (LR), and the GMM overidentification J—share a common large‑sample limit under heteroskedasticity, clustering, or serial dependence: a weighted sum of chi‑squares. In this talk I first review some of this framework and how p‑values are computed numerically. I then bring in recent psychometric advances, ie. penalized smoothing of the sorted plug‑in eigenvalues pEBA. These procedures target the well‑documented finite‑sample “sorting bias” (large eigenvalues over‑estimated, small under‑estimated) that can make weighted‑chi‑square tests conservative at realistic sample sizes. I propose pEBA/pOLS as drop‑in refinements to Hansen‑style tests: compute the same matrices and eigenvalues, stabilization with pEBA/pOLS, then evaluate the weighted chi‑square distribution. Monte Carlo experiments for several settings show that pEBA delivers closer‑to‑nominal size with comparable or better power than unadjusted plug‑in weighted‑chi‑square p‑values, without resorting to bootstrap. This positions pEBA as a simple, fast way to improve finite‑sample inference for a broad class of econometric tests.