Invariance Properties of Uncorrelated Residual Transformations
Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 1978 · Volume 40 · Issue 3
DOI: 10.1111/j.2517-6161.1978.tb01044.x
Abstract
The problem of obtaining an uncorrelated residual in the general linear model of full rank which is orthogonal to the design matrix has been discussed by Theil (1965), Brownet al.(1975) and others. In this paper it is shown that the problem is equivalent to stipulating an invariance requirement for the corresponding residual sum of squares, thus implying that alternative uncorrelated residuals are necessarily related by an orthogonal transformation. The set of uncorrelated residuals satisfying a given partitioning of the design matrix is derived. It is also shown that each member of this set enjoys an additional invariance property which enables it to be expressed as the same transformation of the vectors of observations, disturbances or least-squares residuals. Several implications of the results are discussed.
Keywords
blus residual, least-squares residual, recursive residual, scalar covariance matrix
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