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arXiv · 2606.11421

Second-Order Least Squares as a Special Case of the Polynomial Maximization Method

Serhii Zabolotnii
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Abstract

For linear regression with i.i.d. homoskedastic errors, optimally weighted second-order least squares (SLS) is the degree-two polynomial maximization method (PMM) estimating equation: both select the same optimal combination of $e$ and $e^2-σ^2$, share one influence function, and attain the slope variance $c_2g_2/N$. The identity is one of estimating equations and stops where the optimal combination depends on the design. Within the polynomial moment class, degree three holds an efficiency reserve: on four right-skewed laws the cubic direction adds between $30\%$ and $50\%$ asymptotic efficiency, and under symmetric platykurtic errors it is the only informative direction. The reserve belongs to the degree-three span, which an efficient generalized method of moments estimator also attains. We give the asymptotic law of the feasible plug-in estimator, a rule for choosing the degree, Lean 4 checks of the algebra, and simulation evidence.

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Serhii Zabolotnii. 2026-09-16. Second-Order Least Squares as a Special Case of the Polynomial Maximization Method. https://arxiv.org/abs/2606.11421

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