  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.
