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

The use of residuals for detecting departures from the assumptions of the linear model with full-rank covariance, whether the design matrix is full rank or not, has long been recognized as an important diagnostic tool. Once it became feasible to compute different kinds of residual in a straight forward way, various methods have focused 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 for models having uncorrelated errors with equal variances are given in terms of the observation vector or the usual least-squares 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.