Analytic moments of TGARCH models with polynomially adjusted densities
62 Pages Posted: 30 Nov 2021 Last revised: 3 May 2023
Date Written: November 29, 2021
Abstract
This paper extends He et al. (2008) and Francq and Zakoian (2010) by providing analytical expressions for the moments of the unconditional distribution of the TGARCH(1,1) under alternative specifications for the conditional mean and different skewed distributions for the innovations. We consider polynomially adjusted (PA) densities, such as the PA Logistic, PA hyperbolic secant and the PA Gaussian or Gram-Charlier (GC), along with the skewed Student-t. For the GC density, we find: (i) the skewness of the TGARCH is mainly driven by the innovations' skewness, having the excess kurtosis a smaller effect. However, both skewness and kurtosis of the innovations affect significantly the TGARCH kurtosis; (ii) if the conditional mean is not constant, returns can be asymmetric even if innovations are symmetric; (iii) skewed innovations can generate cross-correlations different from zero, indicating leverage effect, even when the volatility model is symmetric. Finally, we illustrate our results with an empirical application.
Keywords: asymmetry, cross-correlation, leverage effect, unconditional skewness, unconditional kurtosis
JEL Classification: C22, C58, G10
Suggested Citation: Suggested Citation