Some properties of, and relationships among, several uncorrelated and homoscedastic residual vectors

Abstract

In this paper we examine the properties of four types of residual vectors, arising from fitting a linear regression model to a set of data by least squares. The four types of residuals are (i) the Stepwise residuals (Hedayat and Robson, 1970), (ii) the Recursive residuals (Brown, Durbin, and Evans, 1975), (iii) the Sequentially Adjusted residuals (to be defined herein), and (iv) the BLUS residuals (Theil, 1965, 1971). We also study the relationships among the four residual vectors. It is found that, for any given sequence of observations, (i) the first three sets of residuals are identical, (ii) each of the first three sets, being identical, is a member of Thei’rs (1965, 1971) family of residuals; specifically, they are Linear Unbiased with a Scalar covariance matrix (LUS) but not Best Linear Unbiased with a Scalar covariance matrix (BLUS). We find the explicit form of the transformation matrix and show that the first three sets of residual vectors can be written as an orthogonal transformation of the BLUS residual vector. These and other properties may prove to be useful in the statistical analysis of residuals.

Keywords:

• Adjusted residuals
• BLUS Residuals
• Cholesky decomposition
• LUS residuals
• Recursive residuals
• Sequentially Adjusted residuals
• Stepwise residuals