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?

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    $\begingroup$ Hi: ARMA models are a class of "time-series" models. GLM's, generally speaking, have nothing to do with time series. They are a different class of models that are more general ( sort of ) . GLM's are an extension of regression models where the response and it's distribution isn't required to be normally distributed. So, I wouldn't think of them as at all related. $\endgroup$ – mlofton Dec 30 '18 at 16:38
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    $\begingroup$ Thank you, mlofton. Why do time series have nothing to do with GLMs? I formulated the question in terms of ARMA (a type of time series) so as to be able to be more specific, given the great variety of different type series models. I edited the title of question to broaden to all time series. GLMs include models that have a continuous predictor, which we assume to be normally distributed. Then why aren't time series GLMs? What exactly is it? Is it the lack of independence between observations? GLMMs also have non independent observations. Then why aren't Time Series an extension of GLMs? $\endgroup$ – ColorStatistics Dec 30 '18 at 17:09
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    $\begingroup$ One can use GLMs for time series analysis. However, ARMA models formulate a very particular way in which observations may be related: in particular, they involve additive terms that are unsuited for (and cannot even be accommodated by) most GLMs. Where people have been successful in combining the approaches they have supposed there is some underlying ARMA (or ARIMA or whatever) model for an unobserved process that describes the GLM link function. As such, neither ARMA nor GLM can be viewed as "extensions" of each other, but both offer useful techniques that can be combined. $\endgroup$ – whuber Dec 30 '18 at 17:18
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    $\begingroup$ @whuber: Thank you. Am I to conclude that a time series is not a type of GLM because the observations are not independent? A time series is not an extension of GLM because unlike in GLMM, where some generic/unspecified dependence among observations within a cluster is allowed for, in a time series, we stipulate the precise way in which observations are correlated? And while we can describe the random component, linear predictor, and link function in a time series, it is the stipulation of the exact way in which observations are dependent that place time series in a category of their own? $\endgroup$ – ColorStatistics Dec 30 '18 at 20:57
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    $\begingroup$ ColorStatistics: I'm honestly at a loss for how to explain it but they are pretty unrelated. GLMs ( I'm not familar with GLMM's ) assume that the response is based on the predictors ( exogenous ) in some parametric way. ARIMA models only have two kind of predictors: the previous values of the series and error terms. I imagine that there must be those who somehow use GLM's for time series prediction but this was not the original intention of Mcullogh and Nelder. $\endgroup$ – mlofton Dec 31 '18 at 19:43

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