Exactly what is being modelled by the systematic component in a heteroscedastic linear regression

Abstract

The distribution of the stochastic component of semi- and non-parametric models is often assumed to belong to a large class of distributions. In such models, the identifiability of the structural component of the model becomes important. For example, in the location problem, the class is restricted to symmetric distributions so that the parameter is always identifiable (as the center of symmetry). In linear regression problems, the slope parameters are identifiable even if the distributions are asymmetric. However, if in addition the errors in the regression model are not identically distributed, the slope parameters are not identifiable. This means that in practice large biases (which do not necessarily vanish with increasing sample size) occur. These biases arise from the difference between the distribution functional (e.g., the mean or median) which is being modelled by structural linearity and the functional being estimated by the statistical procedure used. The possible extent of this bias is illustrated here. The conclusion: it does matter what functional of the distribution is being modelled because this determines which estimator should be used.