Heteroskedasticity is the dependence of the variance of the error term on the independent variable. Homoskedasticity is the independence of the variance of the error term on the independent variable. It is assumed that time series follows the homoskedasticity assumption.

At times, however, this assumption is violated. The variance of the error term is not constant.

Autoregressive Conditional Heteroskedasticity (ARCH) exists if the variance of the residuals in one period is dependent on the variance of the residuals in a previous period. An ARCH(1) model is an AR(1) model with conditional heteroskedasticity.

Heteroskedasticity in the autoregressive model makes the standard errors of the regression coefficients of the model invalid. Our hypothesis tests would be invalid.

If a time-series model has ARCH(1) errors, then the variance of the errors in period t + 1 can be predicted in period t using the formula:

If a time series model has been determined to contain ARCH errors, regression procedures that correct for heteroskedasticity such as Generalized Least Squares (GLS) must be used in order to develop a predictive model. Otherwise, the standard error of the model's coefficients will be incorrect, leading to invalid conclusions.

Engle and other researchers have suggested many generalizations of the ARCH (1) model which include ARCH (p) and generalized autoregressive conditional heteroskedasticity (GARCH) models. GARCH models are similar to ARMA models of the error variance in a time series. Just like ARMA models, GARCH models can be finicky and unstable.