Sampling properties of criteria for evaluating GARCH volatility forecasts

There is considerable evidence that GARCH models do not forecast financial volatility well out of sample when evaluated by the R2 from the Mincer and Zarnowitz (1969) regression. Andersen and Bollerslev (1998) argued that although the R2s tend to be small, they are consistent with the population value of the criterion for a correctly specified GARCH model. We extend the Andersen and Bollerslev result and derive the population moments of the mean squared error, the mean absolute error and a heteroscedasticity adjusted mean square error for the GARCH volatility forecasts. We state existence conditions for the moments. The criteria and their population values are illustrated with empirical examples. Using Monte Carlo simulation, we analyse the sampling properties of these criteria. When volatility is highly persistent, we find that the sampling distribution of the R2 is highly skewed to the right, which indicates that the majority of the realized R2s lie below the population R2. Among the accuracy criteria, we find the heteroscedasticity adjusted mean-squared error is preferable because it has the weakest existence condition and its sampling distribution is reflective of the population value. 'A Good Volatility Model Forecasts Volatility' Engle and Patton (2001)