Robust Estimation of Regression Models with Potentially Endogenous Outliers via a Modern Optimization Lens

Abstract: This paper addresses the robust estimation of linear regression models in the presence of potentially endogenous outliers. Through Monte Carlo simulations, we demonstrate that existing $L_1$-regularized estimation methods, including the Huber estimator and the least absolute deviation (LAD) estimator, exhibit significant bias when outliers are endogenous. Motivated by this finding, we investigate $L_0$-regularized estimation methods. We propose systematic heuristic algorithms, notably an iterative hard-thresholding algorithm and a local combinatorial search refinement, to solve the combinatorial optimization problem of the (L_0)-regularized estimation efficiently. Our Monte Carlo simulations yield two key results: (i) The local combinatorial search algorithm substantially improves solution quality compared to the initial projection-based hard-thresholding algorithm while offering greater computational efficiency than directly solving the mixed integer optimization problem. (ii) The $L_0$-regularized estimator demonstrates superior performance in terms of bias reduction, estimation accuracy, and out-of-sample prediction errors compared to $L_1$-regularized alternatives. We illustrate the practical value of our method through an empirical application to stock return forecasting.