GLM Invariants

Those familiar with classic linear regression, as many actuaries are, are aware that, for any regression including an intercept term, there is an exact balance (equality) between (weighted) fitted and (weighted)observed values in aggregate over the whole dataset. Many are also aware that this balance also holds in aggregate over any level of any classification variable appearing in the regression as a main effect. What many may not be aware of is where these balances come from or the fact that they sometimes, but not always, extend to the GLM setting. This paper will discuss the source of the balance conditions in the so-called GLM "Normal Equations". In those cases where balance does not hold, the Normal Equations imply another invariance. The paper will also discuss some applications of these invariants.

Keywords: Generalized Linear Models (GLMs), balance conditions, invariants.