The standard linear regression model is analysed using a method called functional least squares which yields a family of estimators for the slope parameter indexed by a real variable t, |t| ≦ T. The choice t = 0 corresponds to ordinary least squares, non-zero values being appropriate if the error distribution is long-tailed, and it is argued that the approach is a natural extension of least squares methodology. It emerges that the asymptotic normal distribution of these estimators has a covariance matrix characterised by a scalar function of t, called the variance function, which is determined by the error distribution. Properties of this variance function suggest graphical criteria for detecting departures from normality.