7. Generalized Linear Models

McCullagh and Nelder (1989) summarized many approaches
to relax the distributional assumptions of the classical
linear model under the common term **Generalized Linear
Models** (GLM).
A generalized linear model (GLM) is a regression model of the form

An essential feature of the GLM is that the expectation
is directly dependent on a function of
the index
.
Additionally, one assumes that
.
The function
which relates and is called the **link
function**. (Note that
McCullagh and Nelder (1989) actually denote as the link
function.)

It is easy to see that GLM covers a range of widely used models, e.g.

**Linear regression (OLS)**

The model**Binary response models (Logit, Probit)**

The probability for Bernoulli distributed is identical to the expectation . Hence Logit or Probit models