Autoregressive conditional heteroskedasticity

Autoregressive conditional heteroskedasticity (ARCH) is the condition that one or more data points in a series for which the variance of the current error term or innovation is a function of the actual sizes of the previous time periods' error terms: often the variance is related to the squares of the previous innovations. In econometrics, ARCH models are used to characterize and model time series. A variety of other acronyms are applied to particular structures that have a similar basis.

ARCH models are commonly employed in modeling financial time series that exhibit time-varying volatility clustering, i.e. periods of swings interspersed with periods of relative calm. ARCH-type models are sometimes considered to be in the family of models, although this is strictly incorrect since at time t the volatility is completely pre-determined (deterministic) given previous values.

To model a time series using an ARCH process, let ${\displaystyle ~\epsilon _{t}~}$denote the error terms (return residuals, with respect to a mean process), i.e. the series terms. These ${\displaystyle ~\epsilon _{t}~}$ are split into a stochastic piece ${\displaystyle z_{t}}$ and a time-dependent standard deviation ${\displaystyle \sigma _{t}}$ characterizing the typical size of the terms so that

...
Wikipedia