In volatility forecasting literature, Markov switching multifractal (MSM) models are well known for capturing many important stylized facts such as long memory and fat tails. MSM delivers stronger performance both in- and out-of-sample than GARCH-type models in long-term forecasts. However, the literature shows that MSM forecasts only slightly improve on GARCH(1,1) at short-term intervals. This indicates that there may exist certain patterns to be discovered in the innovation part ${\epsilon }_t$. To enhance MSM's prediction accuracy at the short-term level with higher frequency data, a hybrid model of the MSM model and support vector regression (SVR) is proposed, in which a particle swarm optimization (PSO) algorithm is applied to optimize hyperparameters of the support vector regression in the scope of constraint permission. The method is referred to as MSM-PSO-SVR. Further, we introduce the Fourier kernel MSM-PSO-SVR and evaluate the performance of various MSM-PSO-SVR models in terms of mean absolute error (MAE) and the mean squared error (MSE) with one-minute data of the exchange traded fund (ETF) SPDR S&P 500 Trust ETF (ticker symbol: SPY). The experimental results show that the proposed approach outperforms the other competing peer models and in particular, the selection of SVR kernel might yield significant boosts in forecasting ability. Results of Hansen's Superior Predictive Ability test further validate the conclusion.

在波动率预测文献中，马尔可夫切换多重分形 (MSM) 模型以捕捉许多重要的程式化事实而闻名，例如长记忆和肥尾。在长期预测中，MSM 在样本内和样本外都提供了比 GARCH 类型模型更强的性能。然而，文献表明，MSM 对 GARCH(1,1) 的预测在短期内仅略有改善。这表明创新部分可能存在某些模式有待发现${\epsilon }_{吨}$. 为提高MSM对高频数据的短期预测精度，提出了一种MSM模型和支持向量回归（SVR）的混合模型，其中应用粒子群优化（PSO）算法来优化模型的超参数。约束许可范围内的支持向量回归。该方法称为 MSM-PSO-SVR。此外，我们介绍了傅里叶核 MSM-PSO-SVR，并根据交易所交易的一分钟数据的平均绝对误差 (MAE) 和均方误差 (MSE) 评估了各种 MSM-PSO-SVR 模型的性能基金 (ETF) SPDR S&P 500 Trust ETF（股票代码：SPY）。实验结果表明，所提出的方法优于其他竞争对等模型，特别是，SVR 核的选择可能会显着提高预测能力。汉森超强预测能力测试的结果进一步验证了这一结论。