I am trying to understand the relationship between ARMA Time Series models and the GLM (Generalized Linear Model) family of models. As far I know, all GLMs have the following 3 components: 1) random component, 2) linear predictor, and 3) link function. Also, as far I know all GLMs are estimable by some flavor of Maximum Likelihood (e.g. "Conditional", "Partial").
Do ARMA Time Series models fall under the GLM family of models?
If so, please specify the random component, linear predictor, link function that qualify it to be a GLM and what sort of likelihood method is used to estimate it.
If not, please specify why it isn't in terms of the above 3 components and please comment on challenges of likelihood estimation, if relevant. Finally, if it borderline misses out on inclusion in the GLM family of models, why aren't ARMA models considered an extension of GLMs, the way GLMMs and GAMs are? Is it that much more of stretch from GLMs to ARMA models than from GLMs to GLMMs or from GLMs to GAMs?