Application of the Moore-Penrose inverse of a data matrix in multiple regression

Consider an adjoined n x p matrix Z = (Y:X) relating to the regression of a dependent variable Y on a set of predictors X. It is shown that the Moore-Penrose inverse of Z contains a useful summary of information about multiple regressions between any column of Z and all other columns, as well as a set of case diagnostics that may be used to identify outliers and influential points. Z and the inverse are dual, so that Z is itself a diagnostic indicator of multiple regressions in the inverse. It is shown how the inverse may be used as a case diagnostic for both leverage and outlyingness, and also provides information about the dependence of subset regressions on particular cases.