An Improved Garch-Type Model with Combined Weighted Volatility Measure and Weighted Volatility Indicator: Evidence from German DAX

This paper proposes an unbiased combined weighted (CW) volatility measure and weighted volatility indicator (WVI) that integrates the return- and range-based volatility measures to model the dynamics volatility of stock returns. The main feature of the CW measure is that it is formulated based on the weighted inter- and intra-price information to quantify the volatility directly, while the WVI effectively identifies signals on the shift of inclining volatility. Empirical analysis using the Deutscher Aktienindex demonstrates that the CW measure, utilising squared returns in combination with either Garman-Klass or Rogers-Satchell volatility measure, exhibits the lowest losses based on root mean squared error and quasi-likelihood when compared to weekly realised volatility as a proxy for true volatility. Furthermore, we investigate the feasibility of incorporating the CW measure and WVI as the exogenous variable(s) in the generalised autoregressive conditional heteroscedasticity (GARCH)-type models to enhance the forecasting performance. The findings indicate that the GARCH-CW-WVI and GJRGARCHCW-WVI models exhibit superior in-sample model fit based on the Akaike information criterion and other criteria. Morevever, these two GARCH-type models also offer the best out-of-sample forecasts, tested based on the mean squared error loss using Hansen’s model confidence set. Different risk levels of value-at-risk and expected shortfall forecasts are conducted and validated with various backtests to assess the unbiasedness of value-at-risk and expected shortfall forecasts.