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
A generalized linear model (GLM) is a regression model of the form
where denotes the expected value of the dependent variable
, is a vector of explanatory variables, an
unknown parameter vector and
a known link function.
An essential feature of the GLM is that the expectation
is directly dependent on a function of
Additionally, one assumes that
which relates and is called the link
function. (Note that
McCullagh and Nelder (1989) actually denote as the link
It is easy to see that GLM covers a range of widely used models, e.g.
- Linear regression (OLS)
Hence the classical linear model falls into the GLM framework
with an identity link function and a variance function .
- Binary response models (Logit, Probit)
The probability for Bernoulli distributed is
identical to the expectation . Hence Logit
or Probit models
is the Logistic or Gaussian
distribution function, can be estimated within the GLM framework
as well. The variance function is
in this case.