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Modelling Skewed Innovations in ARMA Time Series

Article Quant Q&A · Author: Hans-Peter Schrei
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Summary

The document explains how to model constant conditional skewness alongside an ARMA conditional mean. Its central point is that ARMA specifies the structure of the conditional mean, while the innovation distribution can be chosen to have skewness and kurtosis. This allows the model to represent skewed errors without changing the basic ARMA mean specification.

For practical estimation, the answer suggests using a skewed innovation distribution in a broader ARFIMA-GARCH framework and names an R package that supports several such distributions. ARMA can be represented as a restricted case within that framework. The discussion distinguishes constant skewness from research on time-varying conditional skewness, but it does not provide a fitted example, model-selection advice, or diagnostic procedures. Its proposed software route is a practical pointer rather than evidence that a particular distribution or specification will suit every series.

Key ideas

  • ARMA models specify the conditional mean structure, while the innovation distribution can determine skewness and kurtosis.
  • A skewed error distribution can represent constant conditional skewness in an ARMA model.
  • The answer proposes fitting an ARMA specification through an ARFIMA-GARCH framework with skewed innovations.
  • The suggested approach does not address how to choose among skewed distributions or test whether skewness changes over time.

Tags

StatisticsVolatility
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# Modelling Skew when using ARMA Time Series


# Modelling Skew when using ARMA Time Series












I am currently modelling financial time series via ARMA processes, but I have reason to believe that in addition to significant autocorrelation, the time series also exhibit skewness. Is there a way to estimate them jointly?

I am aware of Simulation of Non-normal Autocorrelated Variables, but it only talks about how to combine AR and MA models to achieve a desired skew and kurtosis. There is also this paper Looking for skewness in financial time series analyzing time series to show that they exhibit time-varying conditional skewness instead of unconditional skewness.

There is also this paper Time series models based on the unrestricted skew-normal process, which models skew innovations.

Does a general approach for modelling skewness with ARMA models exist that I am overlooking here? Do heuristics exist?

## Answer by Richard Hardy (score 2, accepted)

https://quant.stackexchange.com/a/61019

Conceptually, if you want constant conditional skewness, you could simply choose an error distribution that is skewed for your ARMA model. ARMA only restricts the conditional mean of the time series to vary in a certain way, but all the other parameters such as skewness or kurtosis can be chosen freely.

In practice, you need a way to estimate such a model. If I needed to do this myself, I would use the `rugarch` package in R. It has a wide variety of distributions, including multiple skewed ones, to be used in AR(FI)MA-GARCH models. ARMA is a restricted version of ARFIMA-GARCH, and vis made feasible in `rugarch`. You can specify and fit an ARMA model with unconditional skenwness by using functions `arfimaspec` and `arfimafit`, respectively.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.