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

Abstract

where μ1 and μ2 are the means, and σ1 and σ2 are the standard errors of the dependent variable y and the predictor x, respectively, and sgn(ρ) is the sign of Pearson correlation ρ of these variables. In contrast to the best linear prediction (1), the slope of the best linear predictor (2), obtained with the restriction that the variance of the predictor of y equals the variance of y itself, is expressed by the sgn(ρ) replacing the actual value of ρ in (1). The formula (1) corresponds to the simple regression, while the formula (2) coincides with the so-called diagonal regression. The diagonal regression was proposed by Ragnar Frisch (1934), one of the founders of modern economics and the first economics Nobel laureate, who coined such terms as econometrics and collinearity. Up to the variables centering, the formula (2) defines the slope as the signed quotient of the standard deviations of the dependent and independent variables, and the diagonal regression for one and two predictors was considered in Cobb (1939, 1943). The model of the form (2) for one predictor is identical to the so-called geometric mean regression, standard (reduced) major axis regression, and some others, reviewed in the work by Xe (2014), with an extensive list of many researchers independently proposed and developed these models. Derivation of the diagonal regression (2) for the models with errors in measurement by both variables via the maximum likelihood criterion is described in Leser (1974, Chapt. 2). More references on diagonal regres-