Information criteria for GARCH model selection

Abstract

In this paper, a set of appropriately modified information criteria for selection of models from the AR-GARCH class is derived. It is argued that unmodified or naively modified traditional information criteria cannot be used for order determination in the context of conditionally heteroscedastic models. The models selected using the modified criteria are then used to forecast both the conditional mean and the conditional variance of two high frequency exchange rate series. The analysis indicates that although the use of such model selection methods does lead to significantly improved forecasting accuracies for the conditional variance in some instances, these improvements are by no means universal. The use of these criteria to jointly select conditional mean and conditional variance model orders leads to performance degradation for the conditional mean forecasts compared to models which do not allow for the heteroscedasticity.

Keywords:

• Akaike information criterion
• Schwarz information criterion
• GARCH
• high frequency financial data
• exchange rate prediction
• volatility forecasting

ACKNOWLEDGEMENTS

We are grateful to Chris Adcock, the Editor of this journal, to an Associate Editor, and to two anonymous referees for comments on a previous version of this paper. The work for this paper was carried out on the University of Reading's Supercomputer. We are grateful to the University's High Performance Computing Centre (HPCC) for allowing us to use this facility. Naturally, all errors are ours alone.

Notes

¹ r = 0 if m = 0 to disallow the degenerate cases where there is no stochastic driving force in the conditional variance equation.

²The argument is that the information matrix is block diagonal between the estimated coefficients of the conditional mean and the conditional variance equations.

³It makes no difference, of course, whether we include the constant term coefficients in the conditional mean and the conditional variances in the total number of estimated parameters g or not since this will add the same number of additional parameters to every model, so that the optimal model would be the same regardless.

⁴Modified information criteria can also be constructed for the Hannan–Quinn (1979) or other information criteria in a similar fashion.

⁵The sample size used (600 observations) is somewhat arbitrary, and results from the desire to use all the information contained in the dataset (over 12 000 observations). But the procedure of estimating 186 different GARCH models for samples larger than this is computationally infeasible, even with the supercomputer that we used for the estimations.

⁶Assuming no model for the conditional mean, the squared returns are equal to the squared errors.

⁷There are six orders possible for each of r and m (integers (0,5)) less the GARCH(r,0) models for r > 0 which have no stochastic driving element in the conditional variance equation.

⁸Note that the old criteria are not used in attempting to determine the conditional variance orders, since as Brooks and Burke (1997) show, these criteria will always select the lowest model order permitted. In this case, it would be a zero order for both r and m, identical to a fixed variance (homoscedastic) model.

Fig. 1 Forecasts for the conditional mean generated by model orders selected by HSIC, for USD_DEM together with actual returns

⁹This may not be so for more complicated models, in particular the ARCH in mean model.

¹⁰This is the origin of the substitution of T log(σ˜²) in Equation 2 by an expression that recognizes the heteroscedasticity of the disturbances of the model.

¹¹It has been assumed that p and m are not smaller than their true values.

¹²A ∼ over a parameter or variable indicates its maximum likelihood estimate.

¹³That is

is an asymptotically pivotal quantity for θ₀.