The Economic Value of Mean Squared Error: Evidence from Portfolio Selection

When designing and evaluating estimators, the mean squared error (MSE) is the most commonly used generic statistical loss function because it captures the bias-variance tradeoff and allows easy analytical and numerical treatment. However, MSE estimators are often applied to decision problems for which the loss function is different, raising questions about how much value there is in using a generic statistical loss function like the MSE rather than a decision loss function. We elucidate this question through the lens of the portfolio selection problem by showing that for several important portfolio rules, there is a positive linear relation between the MSE and a portfolio-decision loss function. Moreover, shrinkage portfolio estimators derived under these two loss functions are typically close to each other. Our findings highlight the economic value of MSE to serve as a general-purpose statistical loss function in portfolio selection.