Garch Models of Dynamic Volatility and Correlation

Economic and financial time series typically exhibit time varying conditional (given the past) standard deviations and correlations. The conditional standard deviation is also called the volatility. Higher volatilities increase the risk of assets, and higher conditional correlations cause an increased risk in portfolios. Therefore, models of time varying volatilities and correlations are essential for risk management. GARCH (Generalized AutoRegressive Conditional Heteroscedastic) processes are dynamic models of conditional standard deviations and correlations. This tutorial begins with univariate GARCH models of conditional variance, including univariate APARCH (Asymmetric Power ARCH) models that feature the leverage effect often seen in asset returns. The leverage effect is the tendency of negative returns to increase the conditional variance more than do positive returns of the same magnitude. Multivariate GARCH models potentially suffer from the curse of dimensionality, because there are d(d + 1)/2 variances and covariances for a d-dimensional process, but most multivariate GARCH models reduce the dimensionality in some way. A number of multivariate GARCH models are reviewed: the EWMA model, the diagonal VEC model, the dynamic conditional correlations model, orthogonal GARCH, and the dynamic orthogonal components model. In a case study, the dynamic orthogonal components model was found to provide the best fit.