The ARCH(p) model is simply an AR(p) model applied to the variance of the residuals of a time series model that makes this variance at time t conditional on lagged observations of the variance. More specifically, the error terms, ε_t, are residuals of a linear model, such as ARIMA, on the original time series and are split into a time-dependent standard deviation, σ_t, and a disturbance, z_t, as follows:

An ARCH(p) model can be estimated using OLS. Engle proposed a method to identify the appropriate ARCH order using the Lagrange multiplier test that corresponds to the F-test of the hypothesis that all coefficients in linear regression are zero (see Chapter 7, Linear Models).

One strength of the model is that it produces volatility, estimates positive excess kurtosis—that is, fat tails relative to the normal distribution—which in turn is in line with empirical observations about returns. Weaknesses include that the model assumes the same effect for positive and negative volatility shocks because it depends on the square of the previous shocks, whereas asset prices are known to respond differently to positive and negative shocks. The ARCH model also does not offer new insight into the source of variations of a financial time series because it just mechanically describes the conditional variance. Finally, ARCH models are likely to overpredict the volatility because they respond slowly to large, isolated shocks to the return series.

For a properly-specified ARCH model, the standardized residuals (divided by the model estimate for the period of standard deviation) should resemble white noise and can be subjected to a Ljung-Box Q test.