The three most popular univariate conditional volatility models are the generalized autoregressive conditional … models are important in estimating and forecasting volatility, as well as in capturing asymmetry, which is the different … effects on conditional volatility of positive and negative effects of equal magnitude, and purportedly in capturing leverage …

One of the most popular univariate asymmetric conditional volatility models is the exponential GARCH (or EGARCH …) specification. In addition to asymmetry, which captures the different effects on conditional volatility of positive and negative … subsequent shocks to volatility. However, the statistical properties of the (quasi-) maximum likelihood estimator of the EGARCH …

An early development in testing for causality (technically, Granger non-causality) in the conditional variance (or … volatility) associated with financial returns was the portmanteau statistic for non-causality in the variance of Cheng and Ng … (1996). A subsequent development was the Lagrange Multiplier (LM) test of non-causality in the conditional variance by …