Linear regression estimators are known to be sensitive to outliers, and one alternative to obtain a robust and efficient estimator of the regression parameter is to model the error with Student's $t$ distribution. In this article, we compare estimators of the degrees of freedom parameter in the $t$ distribution using frequentist and Bayesian maximum a posteriori methods, and study properties of the corresponding estimated regression coefficients. We include a comparison with some recommended approaches in the literature, including fixing the degrees of freedom and robust regression using the Huber loss function. Our results suggest that estimating the degrees of freedom by maximizing the adjusted profile log-likelihood function often yields accurate estimates of the regression coefficients, with performance comparable to that obtained by fixing the degrees of freedom at their true values. These findings provide a detailed synthesis of $t$-based robust regression and underscore a key insight: the proper calibration of the degrees of freedom is as crucial as the choice of the robust distribution itself for achieving optimal performance. The R package that implements our method is available at https://github.com/amanda-ng518/RobustTRegression

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