A general analysis of bias in the estimated standard errors of least squares coefficients

It is well known that when errors in the usual regression model are not independently distributed with equal variances, the application of ordinary least squares leads to calculated variances of the coefficient estimates which are biased and inconsistent. The nature of this bias has been investigated extensively, but the existing literature is limited in two significant ways. First, derivations of exact expressions for the bias have been restricted to special cases and, except for the simplest of these, the expressions derived are almost unmanageably complex. Second, for general error specifications, attention has been focused exclusively on deriving bounds for the bias, which are usually wide and do not allow even the probable direction of any bias to be determined. This paper derives an asymptotic expression for the bias which allows both its sign and approximate magnitude to be described easily in most regression problems. This expression is then used to investigate the bias in the cases of serial correlation of an arbitrary degree, variance components models and approximation of a non-linear relationship with a linear specification.