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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 denote the error terms (return residuals, with respect to a mean process), i.e. the series terms. These are split into a stochastic piece and a time-dependent standard deviation characterizing the typical size of the terms so that


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