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 $e^2-{\sigma }^2$, share one influence function, and attain the slope variance $c_2{g}_2/N$. 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.

34 pages, 3 figures, 20 tables; supplement included as an appendix. v2: revision for Ann. Inst. Statist. Math. Corrects the kurtosis condition in the degree-three proof; the efficiency reserve is restated as a property of the degree-three span, not a limit of SLS. Adds feasible-estimator theory, a GMM competitor and real data. Code: https://github.com/SZabolotnii/PMM-SLS-BRIDGE-code-supplement