ARCHtest

ARCHtest, short for Autoregressive Conditional Heteroskedasticity test, is a statistical procedure used to detect whether a time series exhibits autoregressive conditional heteroskedasticity, i.e., time-varying volatility that depends on past shocks. The test was introduced by Robert F. Engle in 1982 as a means to identify ARCH effects, which are central to modeling volatility in financial and economic data.

The typical procedure starts with a mean model fit to the series and the extraction of residuals. Let e_t be the residuals and z_t = e_t^2. Regress z_t on a constant and a set of lagged z_t terms up to order p (z_{t-1}, ..., z_{t-p}). The Engle ARCH LM statistic is computed as n times the R-squared value of this regression, where n is the number of observations. Under the null hypothesis of no ARCH effects up to lag p, this statistic is asymptotically chi-square distributed with p degrees of freedom. If the statistic exceeds the critical value, the null is rejected, indicating ARCH effects up to order p.

Choosing the lag length p is important and may be guided by prior knowledge, information criteria, or diagnostic testing; results can depend on p. The ARCH test is commonly used to justify the use of ARCH- or GARCH-type models for volatility.

Limitations include sensitivity to the specified mean model, potential distortions from non-stationarity or structural breaks, and the reliance on large-sample theory for the chi-square approximation. The test is widely available in statistical software, sometimes labeled as ArchTest or arch.test.