The paper gives a self-contained account of linear least squares regression when the errors have an arbitrary error covariance matrix. The finite sample size case is treated algebraically by methods which are entirely analogous to those used for the asymptotic study of the same problem by spectral analysis when the errors are generated by a covariance stationary process. The algebraic methods and results are of interest in themselves and may also be useful as an introduction to the difficult analysis involved in the asymptotic treatment.