Invariance Properties of Uncorrelated Residual Transformations

Abstract

Summary 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), Brown et 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.

Algorithm AS 194: An Algorithm for Testing Goodness of Fit of ARMA (P,Q) Models

“…where C. is upper triangular, the "mirror image" of the transformation of Brown, Durbin and Evans (1975). The latter image is given by formula (4.2) of Godolphin and de Tullio (1978).…”

A Method for Testing the Order of an Autoregressive-Moving Average Process

“…But an invariance theorem of Godolphin & de Tullio (1978) shows that nw'w = nr'r which is the Box-Pierce statistic. Thus if Tk = m then w* = w from (3-1) and the statistic (3.2) is given by nQ0 = nw'Q 11w which is nw'w under Ho.…”

“…The elements of w given by (2.5) can be obtained from the residual serial autocorrelations rl, ..., rT by using an expression for N* (Godolphin & de Tullio, 1978). The elements of w given by (2.5) can be obtained from the residual serial autocorrelations rl, ..., rT by using an expression for N* (Godolphin & de Tullio, 1978).…”

“…de Tullio (1978) have derived the following result. Given the partition (2-1), let the (T-lk) x (T-lk) and (T-lk) x T matrices N* and N…”

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“…Observe that (2.15), (2.17) and (2.18) are the general conditions for LUSresiduals Godolphin & De Tullio (1978). proved that (2.17) and (2.18) hold if and only if W'W = M ,…”