ARCH and GARCH model are used to model stochastic process with time dependant variance (like financial serie). They are simply, AR and ARMA models applied to the volatility term of a stochastic process.
ARCH for autoregressive conditional heteroskedasticity 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 (an AR model for the variance).
Let \(\varepsilon_t = \sigma_t z_t\)
Where:
In the ARCH framework, the serie of the variances \(\sigma_t^2\) is model by an \(AR_p\) process:
\[\sigma_{t}^{2} = \varphi_0 + \sum_{k=1}^{p} \varphi_k \varepsilon_{t-k}^2\]Here, the variance \(\sigma_{t}^{2}\) is autoregressed by the error term values squared.
With certain constraints imposed on the coefficients, the \(\varepsilon_t\) series squared will theoretically be \(AR_p\).
GARCH for generalized autoregressive conditional heteroskedasticity model is a statistical model for time series data that describes the variance of the current error term or innovation as an ARMA model.
Let \(\varepsilon_t = \sigma_t z_t\)
Where:
In the ARCH framework, the serie of the variances \(\sigma_t^2\) is model by an \(ARMA_{p,q}\) process:
\[\sigma_{t}^{2} = \varphi_0 + \sum_{k=1}^{p} \varphi_k \varepsilon_{t-k}^2 + \sum_{k=1}^{q}\theta_k \sigma_{t-k}^{2}\]See: