Abstract: Considering the multi-fractal, time-varying and heteroscedasticity of daytime volatility, this paper constructs HAR-HV, HAR-FV, HAR-BSMFV and HAR-TSMFV models and evaluates and compares the goodness of fit and prediction accuracy of these four HAR models. The empirical results and the MCS test confirm that the HAR-SMFV models with Markov-switching multi-fractal volatility have significantly improved predictive abilities and the results are robust. Among them, the HAR-TSMFV model not only describes the time-varying and heteroscedasticity of daytime volatility, but also captures the conversion between the three multi-fractal volatility forms, showing the highest prediction accuracy.