# 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?

What are the advantages and disadvantages of these two competing approaches?

## 1 Answer

Parametric distributions and autoregressive functions live in different dimensions. You cannot contrast them as you cannot contrast, say, a person's race with gender. But you can combine them, letting the parameters of a distribution follow an autoregressive pattern. This is what ARMA, GARCH and ARMA-GARCH models do. You can have a parametric distribution with its location parameter evolving according to ARMA and scale parameter evolving according to GARCH. For details, see "What is the difference between GARCH and ARMA?".

Regarding how to select an appropriate model for stock returns, this is a very broad question. Or perhaps the question is concrete enough, but there is no easy answer. Thousands of studies have been done trying to find good models, thousands have been published, but it does not seem a clear winner has emerged.

• would t-distribution plus GARCH be good for modeling a stock return series? what are good articles on combining parametric distributions with autoregressive models? Dec 10 '20 at 23:19
• @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. Dec 11 '20 at 7:32
• Most articles tend to only focus on one or the other, not both at the same time (like your link). To fit a parametric distribution, we use maximum likelihood estimation of its parameters. but how does this fitting procedure change, if we have to also find optimal parameters for the autoregressive model on top of that? Any sources discussing the technicals of fitting both the parameters of the parametric distribution and the parameters of the autoregressive model would help me understand how the combined approach can actually be implemented. double maximum-likelihood estimation? Dec 11 '20 at 17:32
• @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). Dec 11 '20 at 19:11
• I hope this has become clearer now once "Reference request for t-distribution GARCH maximum-likelihood estimation" has been answered (and after the discussion there in the comments). Dec 11 '20 at 20:14