I am currently researching the following statement:
The ARMA-GARCH model better captures the skewness and leptokurtosis of financial time series than a GARCH model would.
What I know so far:
For a GARCH(p,q) process defined by:
$\epsilon_t = \sigma_t \eta_t$
$\sigma^2_t = \omega + \sum_{i=1}^{q}\alpha_i \epsilon^2_{t-i} + \sum_{j=1}^p \beta_j \sigma^2_{t-j} $
The kurtosis is given by:
$K^{(\epsilon)} = \frac{\mathbb{E}(\eta_t^4)}{\mathbb{E}(\eta_t^4)-[\mathbb{E}(\eta_t^4)-1]\sum_{j=0}^\infty \Psi^2_j}$, where $\Psi_j$ are the constants of the infinite ARCH representation.
For an ARMA(P,Q)-GARCH(p,q) process defined by:
$y_t = \mu + \sum_{i=1}^P \alpha_i (y_{t-i}-\mu) + \epsilon_t + \sum_{j=1}^Q \beta_j \epsilon_{t-j}$
$\epsilon_t = \sigma_t \eta_t$
$\sigma^2_t = \omega + \sum_{i=1}^q \phi_i \epsilon_{t-i}^2 +\sum_{j=1}^p \theta_j \sigma^2_{t-j} $
The kurtosis is given by:
$K^{(y)} = \frac{K^{(\epsilon)}[\sum\limits^\infty_{j=0}\Psi^4_j]=6 \sum\limits_{i<j}^\infty \Psi^2_i \Psi^2_j}{(\sum\limits^\infty_{j=0} \Psi^2_j)^2}$
Where $K^{(\epsilon)}$ is given by:
$K^{(\epsilon)} = \frac{\mathbb{E}(\eta_t^4)}{\mathbb{E}(\eta_t^4)-[\mathbb{E}(\eta_t^4)-1]\sum_{j=0}^\infty \Psi^2_j}$
In both equations the $\Psi$ are the constants of the infinite Moving Average representation.
My questions:
The relationship between the ARMA-GARCH kurtosis and GARCH kurtosis is clear. How can I quantify the difference though?
Why does an ARMA-GARCH process capture the kurtosis of financial time series better and how is this clear from these equations?
How do I find the equations for the skewness of a ARMA-GARCH/GARCH process?