We compare the predictive ability of several volatility models for a long series of weekly log-returns of the Dow Jones Industrial Average Index from 1902 to 2016. Our focus is particularly on predicting one and multi-step ahead conditional and aggregated conditional densities. Our set of competing models includes: Well-known GARCH specifications, Markov switching GARCH, sempiparametric GARCH, Generalised Autoregressive Score (GAS), the plain stochastic volatility (SV) as well as its more flexible extensions such as SV with leverage, in-mean effects and Student-t distributed errors. We find that: (i) SV models generally outperform the GARCH specifications, (ii): The SV model with leverage effect provides very strong out-of-sample performance in terms of one and multi-steps ahead density prediction, (iii) Differences in terms of Value-at-Risk (VaR) predictions accuracy are less evident. Thus, our results have an important implication: the best performing model depends on the evaluation criterion
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