Are log-linear models exponential models?

Question

What is usually referred to as "log-linear models"? Is a log-linear model an exponential model where the normalization constant is 1 (since its logarithm needs to be a linear function)? Or is there very little difference between the use of the two terms, and they are in fact interchangeable?

Answer 1

It is hard to say what is "usually referred to" without more context, as terminology is not well standardized across fields.

In the most common statistical context, I would say "log-linear model" refers to a Poisson GLM that is applied to a multi-way contingency table and presented in a special form. This is the way Agresti uses the term in his Categorical Data Anaysis, for example (Google book search). I have an R-based example of using a log-linear model here: $χ^2$ of multidimensional data.

Answer 2

They are different, but it's a bit ambiguous without extra context.

Log-linear models usually refer to an OLS linear model with logged response, or sometimes a GLM with a Normal family, log link function. The Normal distribution is in the exponential family.

If you actually have an exponential response, you would use a GLM with Gamma family, but this typically uses the inverse link, not a log. Also in the exponential family.