A Note on Alternative Regressions - Samuelson, Paul A. (Econometrica, 1942)
IN THE JANUARY issue of this journal Mr. Elliott B. Woolley presented a method of determining a straight-line regression by the summed absolute values of the areas of rectangles formed by the projections of each observation upon the regression line.' The resulting line possesses the usual property of passing through the point of means, and its slope is a simple average of the elementary regression slopes derived by in each direction; it is the geometric mean of the elementary regression coefficients, each referred to the same axis, and has their algebraic sign. It should be pointed out that this is nothing other than Frisch's regression (cf. Statistical Confluence Analysis . . .), and a statistical parameter which has long appeared in the literature. In terms of a correlation surface it represents the major axis of the concentric ellipses of equal frequency. While Mr. Woolley has made an interesting contribution in proving this minimizing property of the diagonal regression,2 his further argument that it is to be preferred in any sense as a method of determining regression lines seems to require brief comment. (a) The lack of consistency between the elementary regressions is a necessary property of a linear multivariate frequency surface. It is expressed in the purely formal statistical law of regression towards the average. The elementary regressions are not thereby illogical. (b) If the aim of the investigation is not simply a characterization of the properties of the multivariate distribution, but rather the search for a hypothetical true (in some sense) linear relationship, upon which has been superimposed a distribution of errors, then no definite method of determining the regression equation can be specified until some assumptions have been made concerning the nature of the disturbing causes. These assumptions must be in the nature of postulates; by no possible method can they be determined inductively from an examination of the data, even in an infinitely large sample. This last statement must be emphasized since some of the recent literature seems at first sight to suggest otherwise. This is because seemingly innocent, but in fact highly restrictive and often arbitrary, assumptions of noncorrela-