7. Generalized Linear Models

Marlene Müller
28 July 2004

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

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 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

implies that

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

where is the Logistic or Gaussian distribution function, can be estimated within the GLM framework as well. The variance function is in this case.