Autoregressive Conditional Heteroskedasticity models (ARCH)

When regression model has (conditional) heteroskedasticity i.e. variance of the error in a particular time-series model in one period depends on the variance of the error in previous periods, standard errors of the regression coefficients in AR, MA or ARMA models will be incorrect, and hypothesis tests would be invalid.


ARCH model: 
ARCH model must be used to test the existence of conditional heteroskedasticity. An ARCH (1) time series is the one in which the variance of the error in one period depends on  size of the squared error in the previous period i.e. if a large error occurs in one period, the variance of the error in the next period will be even larger.

To test whether time series is ARCH (1), the squared residuals from a previously estimated time-series model are regressed on the constant and first lag of the squared residuals



Decision Rule: If the estimate ofα1 is statistically significantly different from zero, the time series is ARCH (1). 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.


Consequences of ARCH: 
• Standard errors for the regression parameters will not be correct.
• When ARCH exists, we can predict the variance of the error terms.



Generalized least squares or other methods that correct for heteroskedasticity must be used to estimate the correct standard error of the parameters in the time-series model.

Autoregressive model versus ARCH model:

• Using AR (1) model implies that model is correctly specified.
• Using ARCH (1) implies that model can not be correctly specified due to existence of conditional heteroskedasticity in the residuals; therefore, ARCH (1) model is used to forecast variance/volatility of residuals.