Why is the Mean Squared Error (MSE) often favored for its mathematical properties, particularly convexity, in linear regression?

MSE is favored because its objective function is convex. A convex function has a single global minimum, meaning that any local minimum found during optimization is guaranteed to be the global minimum. This property simplifies the optimization process considerably, as algorithms like Gradient Descent are guaranteed to converge to the optimal set of parameters without getting stuck in local minima, leading to a unique and stable solution for the linear regression coefficients.