Second-Order Least Squares and the Polynomial Maximization Method: An Estimating-Equation Identity and a Higher-Degree Efficiency Reserve

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 the residual e and of e^2 - sigma^2, share one influence function, and attain the slope variance c_2 g_2 / N, where g_2 = 1 - gamma_3^2 / (2 + gamma_4) depends only on the skewness and kurtosis of the error law. The identity is one of estimating equations, not merely of asymptotic variances: the Kunchenko centered-correlation matrix F_2 is the SLS weighting operator after a change of variables, and the design-dependent factors of the SLS residual cancel in the slope score. The identity stops where the optimal combination depends on the design; outside that scope conditional SLS is the more general object. 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, since g_2 = 1 there. The reserve belongs to the degree-three span, which an efficient generalized method of moments estimator also attains; the PMM normal system is the low-dimensional route to it, and the difference is visible at n <= 200. The talk gives the asymptotic law of the feasible plug-in estimator, a bootstrap pretest for choosing the degree, Lean 4 checks of the degree-specific algebra, simulation evidence, and a residual-calibrated study on drilling innovations from the Utah FORGE 58-32 well. Preprint: arXiv:2606.11421

probabilitystatistics theorydata analysis, statistics and probability

Audience: researchers in the topic

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Asymptotic Methods in Statistics

Series comments: One can find video files and slides of talks at the seminar on Asymptotic Methods in Statistics here: www.youtube.com/@SeminarforAsymptoticMethods

A link to a slide is below the description of the video, namely shorturl.at/jUAhL and shorturl.at/6wjUA shorturl.at/rdpqs

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