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Symmetric α-stable distribution GARCH-type models Indirect inference Maximum likelihood Leverage effects Student's t distribution a b s t r a c t Financial returns exhibit conditional heteroscedasticity, asymmetric responses of their volatility to negative and positive returns (leverage effects) and fat tails. The α-stable distribution is a natural candidate for capturing the tail-thickness of the conditional distribution of financial returns, while the GARCH-type models are very popular in depicting the conditional heteroscedasticity and leverage effects. However, practical implementation of α-stable distribution in finance applications has been limited by its estimation difficulties.The performance of the indirect inference approach using GARCH models with Student's t distributed errors as auxiliary models is compared to the maximum likelihood approach for estimating GARCH-type models with symmetric α-stable innovations. It is shown that the expected efficiency gains of the maximum likelihood approach come at high computational costs compared to the indirect inference method.

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In this article, we introduce a new method of forecasting large-dimensional covariance matrices by exploiting the theoretical and empirical potential of mixing forecasts derived from different information sets. The main theoretical contribution of the article is to find the conditions under which a mixed approach (MA) provides a smaller mean squared forecast error than a standard one. The conditions are general and do not rely on distributional assumptions of the forecasting errors or on any particular model specification. The empirical contribution of the article regards a comprehensive comparative exercise of the new approach against standard ones when forecasting the covariance matrix of a portfolio of thirty stocks. The implemented MA uses volatility forecasts computed from high-frequency-based models and correlation forecasts using realized-volatility-adjusted dynamic conditional correlation models. The MA always outperforms the standard methods computed from daily returns and performs equally well to the ones using high-frequency-based specifications, however at a lower computational cost.

This article proposes a simple approach to estimate quantiles of daily financial returns directly from highfrequency data. We denote the resulting estimator as realized quantile (RQ) and use it to forecast tail risk measures, such as Value at Risk (VaR) and Expected Shortfall (ES). The RQ estimator is built on the assumption that financial logarithm prices are subordinated self-similar processes in intrinsic time. The intrinsic time dimension stochastically transforms the clock time in order to capture the real "heartbeat" of financial markets in accordance with their trading activity and/or riskiness. The self-similarity assumption allows to compute daily quantiles by simply scaling up their intraday counterparts, while the subordination technique can easily accommodate numerous empirical features of financial returns, such as volatility persistence and fat-tailedness. Our method, which is built on a flexible assumption, is simple to implement and exploits the rich information content of high-frequency data from another time perspective than the classical clock time. In a comprehensive empirical exercise, we show that our forecasts of VaR and ES are more accurate than the ones from a large set of up-to-date comparative models, for both, stocks and foreign exchange rates.