  Heteroscedasticity is common in real world applications and is often handled
by incorporating case weights into a modeling procedure. Intuitively, models
fitted with different weight schemes would have a different level of complexity
depending on how well the weights match the inverse of error variances.
However, existing statistical theories on model complexity, also known as model
degrees of freedom, were primarily established under the assumption of equal
error variances. In this work, we focus on linear regression procedures and
seek to extend the existing measures to a heteroscedastic setting. Our analysis
of the weighted least squares method reveals some interesting properties of the
extended measures. In particular, we find that they depend on both the weights
used for model fitting and those for model evaluation. Moreover, modeling
heteroscedastic data with optimal weights generally results in fewer degrees of
freedom than with equal weights, and the size of reduction depends on the
unevenness of error variance. This provides additional insights into weighted
modeling procedures that are useful in risk estimation and model selection.
