Comment on “On Optimal Correlation-Based Prediction,” by Bottai et al. (2022)

Abstract

Bottai et al. (2022) examine the best predictors that maximize two correlation criteria and in particular examine predictors that are restricted to have the same mean and variance as what they are trying to predict. We give a brief demonstration that their best correlation predictor, subject to the mean and variance conditions, also minimizes the expected squared error prediction loss subject to those constraints on the predictors. , , , to predict y from the values of x 1 , . . . , x p 1 . the vector x x 1 , . . . , x p 1 ) (cid:2) A reasonable criterion for choosing a predictor of y to pick a predictor h ( x ) that minimizes the mean squared E [ y − h ( x ) ] 2 . The expected value is taken over the joint distribution of y and x . It is well known that the best predictor is (essentially), using notation from both Christensen (2020, sec. 6.3) and Bottai et al. (2022),