Unbiased Estimation of Central Moments in Unbalanced Two- and Three-Level Models

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.

💡 Research Summary

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The paper develops closed‑form unbiased estimators for central moments of latent components in hierarchical random‑effects models when the panel is unbalanced, i.e., group sizes differ across clusters. The authors treat two‑level and three‑level linear mixed models and consider two natural weighting schemes: (i) group‑level averaging, where each cluster receives equal weight, and (ii) observation‑level averaging, where each individual observation receives equal weight.

In the two‑level setting the observed outcome is written as yᵢⱼ = uᵢ + vᵢⱼ, with uᵢ (group‑level random effect) and vᵢⱼ (within‑group error) mutually independent, identically distributed within each level, and with zero means. Sample means are defined for each cluster (ȳᵢ), for the whole sample using group‑level weights (ȳ_g), and using observation‑level weights (ȳ_o). Lemma A.1, proved in the appendix, gives the expectations of powers of a weighted sum of i.i.d. zero‑mean variables. Applying this lemma to the within‑group deviations yᵢⱼ − ȳᵢ yields unbiased estimators for the second, third, and fourth central moments of v (µ₂ᵥ, µ₃ᵥ, µ₄ᵥ). The formulas involve simple sums of powers of the deviations divided by combinatorial factors that depend on the cluster size Jᵢ (e.g., Σ (Jᵢ−1) for the variance, Σ Jᵢ(Jᵢ−1)(Jᵢ−2) for the third moment).

Having obtained unbiased estimates of the v‑moments, the authors turn to the group‑level random effect u. By expressing the between‑group deviation ȳᵢ − ȳ_g (or ȳᵢ − ȳ_o) as a weighted sum of uᵢ and the already‑estimated v‑averages, they derive linear systems that link the unknown moments of u (µ₂ᵤ, µ₃ᵤ, µ₄ᵤ) to observable quantities and the previously estimated v‑moments. The systems are 2 × 2 for the variance and fourth‑order moment, and 3 × 3 for the third‑order moment. The coefficients a_{·} in these systems are explicit functions of the cluster sizes (Jᵢ) and the number of clusters (n). Solving the systems yields closed‑form unbiased estimators for µ₂ᵤ, µ₃ᵤ, and µ₄ᵤ under both weighting schemes.

The three‑level model extends the framework to yᵢⱼₖ = uᵢ + vᵢⱼ + wᵢⱼₖ, adding a lower‑level random component w. Analogous definitions of cluster‑level, group‑level, and observation‑level means are introduced, now involving the additional dimensions Kᵢⱼ (size of the lowest‑level sub‑clusters). Lemma A.1 is applied first to the within‑sub‑cluster deviations to obtain unbiased estimators of µ₂_w and µ₃_w. These are then used to adjust the between‑sub‑cluster and between‑group deviations, leading to unbiased estimators of µ₂_v, µ₃_v, µ₂_u, and µ₃_u under both averaging schemes. The resulting formulas again involve sums of powers of centered observations divided by combinatorial factors that depend on both Jᵢ and Kᵢⱼ.

The paper’s contribution is threefold. First, it provides the first known closed‑form unbiased estimators for third‑order and fourth‑order central moments in unbalanced hierarchical models. Second, it treats both group‑level and observation‑level averaging within a unified algebraic framework, allowing researchers to match the estimator to their sampling design. Third, it demonstrates that even for the fourth moment—where cross‑product terms appear—the problem can be reduced to solving a small linear system, making the estimators computationally tractable.

Practical implications are highlighted through examples where skewness and kurtosis of latent shocks matter, such as earnings dynamics, firm growth distributions, and insurance risk. The authors note that while the fourth‑order formulas are manageable, extending the approach to fifth‑order and higher moments would lead to increasingly complex systems, suggesting case‑by‑case derivations or alternative methods (e.g., Bayesian priors or shrinkage) for future work. The paper thus equips applied econometricians with a rigorous toolkit for unbiased high‑order moment estimation in realistic, unbalanced multilevel data settings.