Unbiased Estimation of Central Moments in Unbalanced Two- and Three-Level Models
Dan Ben-Moshe, David Genesove
TL;DR
The paper tackles unbiased finite-sample estimation of higher central moments in unbalanced multilevel (two- and three-level) designs, focusing on latent components. It develops closed-form estimators for the second, third, and fourth central moments in the two-level case and the second and third moments in the three-level case, under both group-level and observation-level averaging. The method proceeds by expressing centered sample averages as weighted sums of latent components, applying moment expansions to relate expectations to latent moments, and solving the resulting linear systems with sample analogs and lower-order estimators. The results provide practical, finite-sample corrections for skewness and kurtosis in hierarchical data, with explicit formulas and Appendix-provided coefficients, enabling improved inference in economics applications where unbalanced designs are common.
Abstract
This paper derives closed-form unbiased estimators of central moments in multilevel random-effects models with unbalanced group sizes. In a two-level model, we provide unbiased estimators for the second, third, and fourth central moments under both group-level and observation-level averaging. In a three-level model, we provide unbiased estimators for the second and third central moments.