Volatility forecasting plays an integral role in risk management, investments and security valuation for all assets with uncertain future payoffs. We enrich the literature by presenting computationally intensive variations of the heterogeneous autoregressive (HAR) volatility model: the complete subset linear/quantile regression HAR models, HAR-CSLR and HAR-CSQR. Predictions of 1- to 22-day-ahead volatility of four major market indices (NIKKEI 225, S&P 500, SSEC and STOXX 50) show that both models tend to outperform several benchmark HAR models. Forecasting accuracy improvements tend to stabilize for longer forecasting horizons: e.g., five-day-ahead improvements range from 6.57% (SSEC) to 35.62% (NIKKEI 225) and from 3.99% (STOXX) to 9.54% for mean square error (MSE) and QLIKE loss functions. In terms of MSE, the HAR-CSQR model outperforms several standard benchmark HAR models across all market indices and forecast horizons.

对于未来收益不确定的所有资产，波动率预测在风险管理、投资和证券估值方面发挥着不可或缺的作用。我们通过呈现异构自回归 (HAR) 波动率模型的计算密集型变化来丰富文献：完整的子集线性/分位数回归 HAR 模型、HAR-CSLR 和 HAR-CSQR。四个主要市场指数（NIKKEI 225、S&P 500、SSEC 和 STOXX 50）的 1 至 22 天前波动率的预测表明，这两种模型的表现往往优于几个基准 HAR 模型。对于更长的预测范围，预测准确性的改进趋于稳定：例如，五天前的改进范围从 6.57% (SSEC) 到 35.62% (NIKKEI 225) 和从 3.99% (STOXX) 到 9.54% 的均方误差 (MSE)和 QLIKE 损失函数。在 MSE 方面，