The Hurst exponent is a key parameter for characterizing the long memory of high-frequency time series. However, traditional estimators often exhibit systematic biases due to the influence of high-frequency endogenous noise and low-frequency trend shifts. Theoretical derivations show that endogenous noise contemporaneously correlated with the latent signal possesses a spectral density in the first-differenced series that is asymptotically equivalent to a squared sine functional form. Accordingly, the proposed estimator incorporates a corresponding spectral density component to fit the high-frequency error. Simultaneously, the model introduces a SCAD penalty term to control the low-frequency spectral divergence caused by structural breaks, thereby mitigating spurious long memory in parameter estimation. Monte Carlo simulations demonstrate that the Penalized Mixture Whittle estimator yields smaller finite-sample biases and root mean square errors in scenarios involving both trend disturbances and endogenous noise. Empirical analysis shows that the estimates obtained using this method are robust to changes in sampling frequency. In further volatility forecasting experiments on commodity futures, the linear forecasting model constructed based on the parameter set achieves higher prediction accuracy than benchmark models such as HAR, as confirmed by the Diebold-Mariano test. This paper provides an effective econometric tool for high-frequency data inference in the presence of composite statistical disturbances.

Widely used volatility forecasting methods are usually based on low frequency time series models. Although some of them employ high frequency observations, these intraday data are often summarized into a point low frequency statistic, e.g., a daily realized measure, before being incorporated into a forecasting model. This paper contributes to the volatility forecasting literature by instead predicting the next-period intraday volatility curve via a functional time series forecasting approach. In contrast with non-functional methods, the proposed functional approach fully exploits the rich intraday information and hence leads to more accurate volatility forecasts. This is further confrmed by extensive comparisons between the proposed functional method and those widely used non-functional methods in out-of-sample volatility forecasting for a number of stocks and equity indices from the Chinese market.

This paper proposes the two-component realized EGARCH model with dynamic jump intensity (hereafter REGARCH-C-DJI model) to model and forecast stock market volatility. The key feature of our REGARCH-C-DJI model is its ability to exploit the high-frequency information as well as to capture the long memory volatility and jump dynamics. An empirical application to Shanghai Stock Exchange Composite (SSEC) index data shows the presence of high persistence of volatility and dynamic jumps in China’s stock market. More importantly, the REGARCH-C-DJI model dominates the GARCH, EGARCH, REGARCH and REGARCH-C models in terms of out-of-sample forecast performance. Our findings highlight the importance of accommodating the realized volatility, volatility components and jump dynamics in forecasting stock market volatility.