Error Distributions in ARIMA and ARIMA-GARCH Estimation

Summary

The document compares error assumptions in simple ARIMA models with those used in ARIMA-GARCH estimation. It explains that some ARIMA or ARMA models can be fitted with ordinary or iterative least squares, while maximum likelihood estimation is often used when conditional variance changes over time, as in GARCH. Maximum likelihood requires specifying an error distribution for the likelihood calculation.

The answers also caution that the apparent difference depends on the estimation software. An ARIMA fitting routine may use an implicit Gaussian assumption, or may let users choose among distributions such as Gaussian and Student t. Thus a distribution assumption may be present even when a simple ARIMA workflow does not ask the user to state it. The discussion is a high-level explanation rather than a derivation, and it does not establish that every package or estimator follows the same procedure.

Key ideas

• Simple ARIMA models may be estimated with least squares methods.
• GARCH models represent time-varying conditional variance, which standard least squares does not capture directly.
• Maximum likelihood estimation for GARCH requires a specified error distribution.
• Software may impose an implicit distribution assumption even when it is not exposed as a setting.
• Distribution choices and estimation details vary by package.

Tags

# Error distribution assumption in a simple ARIMA model


# Error distribution assumption in a simple ARIMA model












why in an ARIMA-GARCH structure I have to assume an error distribution to run the estimation while in a simple ARIMA model it is not required?

Thank you

## Answer by Neeraj (score 0, accepted)

https://quant.stackexchange.com/a/41387

Simple ARIMA model can be estimated using OLS methods. MA models (or ARMA model) can be estimated using iterative OLS, which provide similar results if MLE is used, assuming that error terms follows normal distribution with constant variance.

Since, GARCH model assumes that conditional variance is not constant. Such dynamic behaviour in volatility can not be accommodated into OLS method. Therefore, we use MLE for GARCH models, which mandatorily require assumption about distribution of error terms.

## Answer by Tim Wilding (score 0)

https://quant.stackexchange.com/a/41386

I would imagine this depends on the package you are using to estimate the ARIMA model and the GARCH model. Often, there is an implicit error distribution used in the fitting process. Many packages will not allow different error distributions, and will assume a Normal error distribution.

So, for example, if you look at the MATLAB ARIMA time series modelling, you can see that you can specify an error distribution (see halfway down https://uk.mathworks.com/help/econ/arima-class.html where you can choose 'Gaussian' or 'T').

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