New formulations for recursive residuals as a diagnostic tool in the classical fixed-effects linear model of arbitrary rank

The use of residuals for detecting departures from the assumptions of the linear model, whether full rank or not, has long been recognized as an important diagnostic tool, once it became computationally feasible to compute them in a straight forward way, and several contributions to the literature have focussed on their underlying properties and their effectiveness. The recursive residuals are attractive in Econometric applications where there is a natural ordering among the observations through time. New formulations for the recursive residuals are given in terms of the observation vector or the ordinary residuals, which do not require the computation of least squares parameter estimates and for which the transformation matrices are expressed wholly in terms of the rows of the Theil Z matrix. Illustrations of these new formulations are given.