Conditional Volatility, Skewness and Kurtosis: Existence and Persistence

Abstract

Recent portfolio choice, asset pricing, and option valuation models highlight the importance of skewness and kurtosis. Since skewness and kurtosis are related to extreme variations, they are also important for Value-at-Risk measurements. Our framework builds on a GARCH model with a conditional generalized-t distribution for residuals. We compute the skewness and kurtosis for this model and compare the range of these moments with the maximal theoretical moments. Our model, thus allows for time-varying conditional skewness and kurtosis. We implement the model as a constrained optimization with possibly several thousand restrictions on the dynamics. A sequential quadratic programming algorithm successfully estimates all the models, on a PC, within at most 50 seconds. Estimators, obtained with logistically-constrained dynamics, have different properties. We apply this model to daily and weekly foreign exchange returns, stock returns, and interest-rate changes. This finding is consistent with findings from extreme value theory. Kurtosis exists on fewer dates and for fewer series. There is little evidence, at the weekly frequency, of time-variability of conditional higher moments. Transition matrices document that agitated stares come as a surprise and that there is a certain persistence in moments beyond volatility. For exchange-rate and stock-market data, cross-sectionally and at daily frequency, we also document co-variability of moments beyond volatility.

Keywords: Garch, stock indices, exchange rates, interest rates, SNOPT, VaR

JEL Classification: C22, C51, G12