Difference Between ARIMA and ARCH Models for Time Series Analysis:
Auto Regressive Integrated Moving Average (ARIMA) models and a similar concept known as Auto Regressive Conditional Heteroskedasticity (ARCH) models will our focus here. The distinction and different purposes of the “MA” or the “CH” respectively offers an interesting map for when to choose between the two.
ARIMA:
ARIMA models are able to measure relationships on our Time Series data that have both long term trends (AR) and sudden disruptions (MA). An ARIMA model is essentially two different models added together. The Auto Regressive aspect models the predicted value on previous values of itself earlier in time. The Moving Average aspect does the same thing with respect to previous values in time, but with the residuals. The Moving Average in this way can capture variation in the data or “sudden shocks”.
Here we are dealing with only one variable. Previous values of that variable become our predictors. It is useful to have a model that is robust to shocks, but what about when the variation itself has patterns that we want to take account of?
ARCH:
An ARCH model is very similar to ARIMA, but its CH component models the previous squared resisduals at each previous point in time. We are using the term “volatility” here as a proxy for variance or squared residuals. The CH model is predicting a future squared residual as part of the wider ARCH model. Below we can illustrate the two components slightly differently.
Conditional Heteroskedasticity:
From our studies of the necessary assumptions of OLS regression, we will review the concept of heteroskedasticity. Heteroskedicity and be definied as non-constant error variance or non-constant volatility in the data. When it is present an OLS model cannot be trusted becasue the assumption of the residuals being normally distributed will not hold. Conditional Heteroskedasticity is when this non-constant variance itself is correlated at each lagged point in time in a Time Series model. More exactly, CH is heteroskedasticity that is time dependent.
The identification of Conditional Heteroskedasticity as being useful for characterizing changing periods of volatiity led to a nobel prize in Economics for the associated work by Robert F. Engle and Clive Granger.
General Context: When is Each Model Type Appropriate?
ARIMA models can be used for measuring trends that have occasional interruptions. One example of a trend below relating to temperature data:
We can see that there is a clear trend but the volatility is relatively constant throughout. This data would be a good candidate for an ARIMA model if we wanted to make predictions.
An ARCH Model becomes appropriate when there are clear changes in volatility regimes or different “volatility clusters.” We can probably expect to have more trustworthy predictions from an ARCH model with the type of data represented below.
This is exactly when we benefit from the regressed squared residual aspect of the ARCH model to capture the change in variation. There could be some kind of pattern contributing to the wider and narrower bands of variation. If we were relying on ARIMA model to model this data we might not be able to clearly capture this aspect. With ARCH, we have a model component that delivers coefficients that can offer insight and clarity to the different volatility clusters.
Summary:
Our OLS based Time Series models have another tool now when we see patterns in the variation of our data. Fortunately, the ARCH model itself is a slight variation on the ARIMA models we are already familiar with. There is even a standalone API for working with ARCH models (conveniently called ARCH) that can help us get started exploring what these models can tell us.
Background on Engle and Granger Nobel Prize:
https://www.nobelprize.org/prizes/economic-sciences/2003/press-release/
ARCH API Documentation:
https://arch.readthedocs.io/en/latest/#
