Analytic Moments of TGARCH(1,1) Models with Polynomially Adjusted Densities

Abstract This article extends He, Silvennoinen, and Teräsvirta (2008, J Finan Econ, 6, 208–230) and Francq and Zakoïan (2010, GARCH Models) 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, along with the skewed Student-t. Our results show that (i) the main driver of the skewness of the TGARCH(1,1) is the skewness of the innovations, while the excess kurtosis has a comparatively lesser impact. However, both skewness and kurtosis of the innovations significantly affect the TGARCH(1,1) 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 theoretical results with an empirical application to stock indices.