Finite-Sample Risk Approximation and Risk-Consistent Tuning for Generalized Ridge Estimation in Nonlinear Models: Controlling Extreme Realizations

Masamune Iwasawa

Papers from arXiv.org

Abstract: Maximum likelihood estimation in nonlinear models can exhibit substantial instability in finite samples when the data provide limited information about certain parameters. Such instability is driven by rare but extreme realizations of the estimator, which can dominate mean squared error (MSE) and lead to poor performance of conventional estimators. To address this issue, we consider ridge estimators that directly target MSE through regularization and thereby control extreme realizations. Developing this approach raises several challenges, including characterizing finite-sample MSE, selecting the penalty parameter, and achieving oracle risk performance. We address these challenges using a unified framework based on a finite-sample approximation to the MSE. Building on higher-order expansions, we derive an explicit first-order approximation to the finite-sample MSE of generalized ridge estimators in a broad class of nonlinear models. This approximation reveals an explicit bias--variance trade-off and shows that generalized ridge estimators can improve upon the MLE in terms of MSE at the first-order level, even under target misspecification. It also provides a tractable foundation for analyzing data-driven tuning, enabling us to show that the proposed MSE-based selection rule achieves oracle risk consistency. Simulation results demonstrate that the proposed method substantially reduces the frequency and impact of extreme realizations, leading to large improvements in finite-sample risk relative to both the maximum likelihood estimator and cross-validation-based methods. An empirical illustration shows that the proposed MSE-based tuning approach can stabilize first-stage propensity score estimation and reveal sensitivity in subsequent treatment effect estimates that remains hidden under conventional estimators.

Date: 2025-04, Revised 2026-04
New Economics Papers: this item is included in nep-dcm, nep-ecm, nep-ets and nep-inv
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