Individual value of residuals from a least squares regression equation can be analyzed as a set of originally observed quantities. If the analysis of residuals is in the form of another regression analysis, the total analysis of the data can be described as a two-stage regression. Such a two-stage approach is not equivalent to the usual, one-stage analysis, and therefore the resulting estimates and tests are biased. It is the purpose of this paper to obtain expressions for these biases.
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Freund et al. (1961) studied this question.