Second-Order Least Squares as a Special Case of the Polynomial Maximization Method
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 e2−σ2, share one influence function, and attain the slope variance c2g2/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.
Related
- Optimal Experimental Design for the Polynomial Maximization Method: a Higher-Order Reserve Beyond Second-Order Design Theory
- Second-Order Least Squares Method for Statistical Inference in Linear Models with Asymmetric Errors
- Concave Roofs and Downward Recovery for DR-Submodular Polynomial Maximization
- Polynomial area growth for special Kähler metrics
- Quaternionic Sums of Squares, Moments, and Quaternionic Polynomial Optimization
BibTeX
@misc{zabolotnii2026secondorder,
title = {{Second-Order Least Squares as a Special Case of the Polynomial Maximization Method}},
author = {Serhii Zabolotnii},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.11421},
eprint = {2606.11421},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.11421},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Second-Order Least Squares as a Special Case of the Polynomial Maximization Method
AU - Serhii Zabolotnii
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2606.11421
UR - https://arxiv.org/abs/2606.11421
AB - 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.
ER -