A particularly important area of application for univariate time series models is the prediction of volatility. The volatility of financial time series is usually not constant over time but changes, with bouts of volatility clustering together. Changes in variance create challenges for time series forecasting using the classical ARIMA models. To address this challenge, we will now model volatility so that we can predict changes in variance.

Heteroskedasticity is the technical term for changes in a variable's variance. The autoregressive conditional heteroskedasticity (ARCH) model expresses the variance of the error term as a function of the errors in previous periods. More specifically, it assumes that the error variance follows an AR(p) model.

The generalized autoregressive conditional heteroskedasticity (GARCH) model broadens the scope to ARMA models. Time series forecasting often combines ARIMA models for the expected mean and ARCH/GARCH models for the expected variance of a time series. The 2003 Nobel Prize in Economics was awarded to Robert Engle and Clive Granger for developing this class of models. The former also runs the Volatility Lab at New York University's Stern School (see GitHub references) with numerous online examples and tools concerning the models we will discuss and their numerous extensions.