For linear models, ordinary least squares (OLS) estimators are unbiased regardless of the underlying covariance structure of the residuals, but their variances can be large due to many reasons, such as covariance misspecification. Penalized estimators are typically biased, but can have much smaller variance than these OLS estimators. Much of the theory on the estimation performance of LASSO-type estimators, such as their MSEs, has concentrated on the case of iid normally distributed errors. Little attention has been given to studying the robustness of the LASSO or related estimators to different covariance structures. This talk investigates the robustness of a few LASSO-type estimators, both analytically and through simulation, to deviations of the error correlation structure from iid. Considering both low- and high-dimensional settings, we find that the LASSO is surprisingly robust to covariance misspecification as long as the design matrix satisfies certain properties.