In econometrics, the autoregressive conditional heteroskedasticity (ARCH) model is a statistical model for time series data that describes the variance of the current error term or innovation as 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. The ARCH model is appropriate when the error variance in a time series follows an autoregressive (AR) model; if an autoregressive moving average (ARMA) model is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model.

ARCH models are commonly employed in modeling financial time series that exhibit time-varying volatility and volatility clustering, i.e. periods of swings interspersed with periods of relative calm (this is, when the time series exhibits heteroskedasticity). ARCH-type models are sometimes considered to be in the family of stochastic volatility models, although this is strictly incorrect since at time t the volatility is completely predetermined (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

The random variable ${\displaystyle z_{t}}$ is a strong white noise process. The series ${\displaystyle \sigma _{t}^{2}}$ is modeled by

An ARCH(q) model can be estimated using ordinary least squares. A method for testing whether the residuals ${\displaystyle \epsilon _{t}}$ exhibit time-varying heteroskedasticity using the Lagrange multiplier test was proposed by Engle (1982). This procedure is as follows:

If an autoregressive moving average (ARMA) model is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model.

In that case, the GARCH (p, q) model (where p is the order of the GARCH terms ${\displaystyle ~\sigma ^{2}}$ and q is the order of the ARCH terms ${\displaystyle ~\epsilon ^{2}}$ ), following the notation of the original paper, is given by

${\displaystyle y_{t}=x'_{t}b+\epsilon _{t}}$