Combining Return Distributions with Time-Series Models
Summary
The document addresses a mistaken either-or choice between modeling returns with a parametric probability distribution and using an autoregressive model. These describe different parts of a model: a distribution specifies the form of uncertainty, while time-series dynamics describe how parameters or conditional moments evolve. They can therefore be combined rather than treated as competing alternatives.
For example, an ARMA process can describe changes in a distribution’s location, while a GARCH process models its changing scale; a parametric distribution can then describe the remaining return innovations. The discussion offers this modeling framework but no model comparison, dataset, or selection procedure. It emphasizes that choosing a suitable specification for stock returns is a broad empirical problem, and that research has not established a clear, universally best model. Model choice therefore depends on the question, data, and assumptions under study.
Key ideas
- A return distribution and an autoregressive process describe different components of a statistical model.
- A parametric distribution can describe return innovations while time-series models govern its parameters.
- ARMA models can represent evolving location, while GARCH models can represent changing scale.
- The document offers no universal model-selection rule or clear winning specification for stock returns.
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Full text
# Should stock return series be modeled with a parametric distribution, or an autoregressive function? # Should stock return series be modeled with a parametric distribution, or an autoregressive function? 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? ## Answer by Richard Hardy (score 1) https://quant.stackexchange.com/a/59891 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.
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