# Should stock return series be modeled with a parametric distribution, or an autoregressive function? [closed]

If I have prior knowledg that a stock return series follows a parametric distribution, such as a Student t-distribution with 4 degrees of freedom, without actively looking for prior knowledge of functions outside of parametric pdf's such as autoregressive functions (which are not parametric pdf's), is there anything in financial theory that can help with the dilemma of deciding whether the parametric distribution or some autoregressive function (i.e. AR(1), ARMA(1,2), GARCH(1,1), etc) would be more appropriate for modeling the stock returns?

• @develarist, virtually every article on ARMA, GARCH or ARMA-GARCH and their applications on modeling stock returns is about combining parametric distributions with autoregressive models. The parametric distributions are usually kept in the background, but they are there. A GARCH(1,1) with Student-$t$ standardized innovations sounds like a reasonable benchmark. You could add something for skewness, e.g. GJR-GARCH or log-GARCH or replace the vanilla Student-$t$ distribution with its skewed counterpart. – Richard Hardy Dec 11 '20 at 7:32
• @develarist, estimating models as basic as ARMA(1,1) or GARCH(1,1) by maximum likelihood simultaneously optimizes across all the parameters you have listed. You may look up the likelihood function of any of these models in an advanced time series textbook or perhaps in some software documentation (for example, I would check rugarch in R). – Richard Hardy Dec 11 '20 at 19:11