DOI: https://doi.org/2025 

DOI: https://doi.org/10.1007/s12312-025-01507-0 

DOI: http://www.ncbi.nlm.nih.gov/pubmed/40870278 

Abstract: In this paper, we propose examining Heterogeneous Autoregressive (HAR) models using five different estimation techniques and four different estimation horizons to decide which performs better in terms of forecasting accuracy. Several different estimators are used to determine the coefficients of three selected HAR-type models. Furthermore, model lags, calculated using 5 min intraday data from the Standard & Poor's 500 (SPX) index and the Chicago Board Options Exchange Volatility (VIX) index as the sole exogenous variable, enrich the models. For comparison and evaluation of the experimental results, we use three metrics: Quasi-Likelihood (QLIKE), Mean Absolute Error (MAE), and Mean Squared Error (MSE). An empirical study reveals that the Entropy Loss Function consistently achieves the best QLIKE results in all the horizons, especially in the weekly horizon. On the other hand, the performance of the Robust Linear Model implies that it can provide an alternative to the Entropy Loss Function when considering the results of the MAE and MSE metrics. Moreover, research shows that adding more informative lags, such as Realized Quarticity for the Heterogeneous Autoregressive model yielding the Realized Quarticity (HARQ) model, and incorporating the VIX index further improve the general results of the models. The results of the proposed Entropy Loss Function and Robust Linear Model suggest that they successfully achieve significant forecasting accuracy for HAR models across multiple forecasting horizons. In this paper, we propose examining Heterogeneous Autoregressive (HAR) models using five different estimation techniques and four different estimation horizons to decide which performs better in terms of forecasting accuracy. Several different estimators are used to determine the coefficients of three selected HAR-type models. Furthermore, model lags, calculated using 5 min intraday data from the Standard & Poor's 500 (SPX) index and the Chicago Board Options Exchange Volatility (VIX) index as the sole exogenous variable, enrich the models. For comparison and evaluation of the experimental results, we use three metrics: Quasi-Likelihood (QLIKE), Mean Absolute Error (MAE), and Mean Squared Error (MSE). An empirical study reveals that the Entropy Loss Function consistently achieves the best QLIKE results in all the horizons, especially in the weekly horizon. On the other hand, the performance of the Robust Linear Model implies that it can provide an alternative to the Entropy Loss Function when considering the results of the MAE and MSE metrics. Moreover, research shows that adding more informative lags, such as Realized Quarticity for the Heterogeneous Autoregressive model yielding the Realized Quarticity (HARQ) model, and incorporating the VIX index further improve the general results of the models. The results of the proposed Entropy Loss Function and Robust Linear Model suggest that they successfully achieve significant forecasting accuracy for HAR models across multiple forecasting horizons. // In this paper, we propose examining Heterogeneous Autoregressive (HAR) models using five different estimation techniques and four different estimation horizons to decide which performs better in terms of forecasting accuracy. Several different estimators are used to determine the coefficients of three selected HAR-type models. Furthermore, model lags, calculated using 5 min intraday data from the Standard & Poor's 500 (SPX) index and the Chicago Board Options Exchange Volatility (VIX) index as the sole exogenous variable, enrich the models. For comparison and evaluation of the experimental results, we use three metrics: Quasi-Likelihood (QLIKE), Mean Absolute Error (MAE), and Mean Squared Error (MSE). An empirical study reveals that the Entropy Loss Function consistently achieves the best QLIKE results in all the horizons, especially in the weekly horizon. On the other hand, the performance of the Robust Linear Model implies that it can provide an alternative to the Entropy Loss Function when considering the results of the MAE and MSE metrics. Moreover, research shows that adding more informative lags, such as Realized Quarticity for the Heterogeneous Autoregressive model yielding the Realized Quarticity (HARQ) model, and incorporating the VIX index further improve the general results of the models. The results of the proposed Entropy Loss Function and Robust Linear Model suggest that they successfully achieve significant forecasting accuracy for HAR models across multiple forecasting horizons.

DOI: https://doi.org/10.3390/e27080806 

Abstract: In this paper, we propose examining Heterogeneous Autoregressive (HAR) models using five different estimation techniques and four different estimation horizons to decide which performs better in terms of forecasting accuracy. Several different estimators are used to determine the coefficients of three selected HAR-type models. Furthermore, model lags, calculated using 5 min intraday data from the Standard & Poor's 500 (SPX) index and the Chicago Board Options Exchange Volatility (VIX) index as the sole exogenous variable, enrich the models. For comparison and evaluation of the experimental results, we use three metrics: Quasi-Likelihood (QLIKE), Mean Absolute Error (MAE), and Mean Squared Error (MSE). An empirical study reveals that the Entropy Loss Function consistently achieves the best QLIKE results in all the horizons, especially in the weekly horizon. On the other hand, the performance of the Robust Linear Model implies that it can provide an alternative to the Entropy Loss Function when considering the results of the MAE and MSE metrics. Moreover, research shows that adding more informative lags, such as Realized Quarticity for the Heterogeneous Autoregressive model yielding the Realized Quarticity (HARQ) model, and incorporating the VIX index further improve the general results of the models. The results of the proposed Entropy Loss Function and Robust Linear Model suggest that they successfully achieve significant forecasting accuracy for HAR models across multiple forecasting horizons. In this paper, we propose examining Heterogeneous Autoregressive (HAR) models using five different estimation techniques and four different estimation horizons to decide which performs better in terms of forecasting accuracy. Several different estimators are used to determine the coefficients of three selected HAR-type models. Furthermore, model lags, calculated using 5 min intraday data from the Standard & Poor's 500 (SPX) index and the Chicago Board Options Exchange Volatility (VIX) index as the sole exogenous variable, enrich the models. For comparison and evaluation of the experimental results, we use three metrics: Quasi-Likelihood (QLIKE), Mean Absolute Error (MAE), and Mean Squared Error (MSE). An empirical study reveals that the Entropy Loss Function consistently achieves the best QLIKE results in all the horizons, especially in the weekly horizon. On the other hand, the performance of the Robust Linear Model implies that it can provide an alternative to the Entropy Loss Function when considering the results of the MAE and MSE metrics. Moreover, research shows that adding more informative lags, such as Realized Quarticity for the Heterogeneous Autoregressive model yielding the Realized Quarticity (HARQ) model, and incorporating the VIX index further improve the general results of the models. The results of the proposed Entropy Loss Function and Robust Linear Model suggest that they successfully achieve significant forecasting accuracy for HAR models across multiple forecasting horizons. // In this paper, we propose examining Heterogeneous Autoregressive (HAR) models using five different estimation techniques and four different estimation horizons to decide which performs better in terms of forecasting accuracy. Several different estimators are used to determine the coefficients of three selected HAR-type models. Furthermore, model lags, calculated using 5 min intraday data from the Standard & Poor's 500 (SPX) index and the Chicago Board Options Exchange Volatility (VIX) index as the sole exogenous variable, enrich the models. For comparison and evaluation of the experimental results, we use three metrics: Quasi-Likelihood (QLIKE), Mean Absolute Error (MAE), and Mean Squared Error (MSE). An empirical study reveals that the Entropy Loss Function consistently achieves the best QLIKE results in all the horizons, especially in the weekly horizon. On the other hand, the performance of the Robust Linear Model implies that it can provide an alternative to the Entropy Loss Function when considering the results of the MAE and MSE metrics. Moreover, research shows that adding more informative lags, such as Realized Quarticity for the Heterogeneous Autoregressive model yielding the Realized Quarticity (HARQ) model, and incorporating the VIX index further improve the general results of the models. The results of the proposed Entropy Loss Function and Robust Linear Model suggest that they successfully achieve significant forecasting accuracy for HAR models across multiple forecasting horizons.

DOI: https://doi.org/10.36253/form-17185