3.1. Conditional higher-order moments of returns

GARCH-based modelling is an important tool to analyze financial data while accounting for volatility clustering. Standard GARCH models estimate the time-variation in the second moment of the return distribution, while assuming that the third and fourth moments are constant. In this section, we estimate and extract the time-varying conditional volatility, skewness, and kurtosis using the GARCHSK model proposed by León et al. [21] and later employed by Kräussl et al. [30]. This model allows for modelling the excess of volatility shocks when the proxy for actual volatility is not fully modelled by standard GARCH processes. As such, it captures asymmetry and excess kurtosis, which is very suitable to the context of the extremely highly volatile Bitcoin market and volatile stock market. Specifically, time-varying and conditional higher-order moments are obtained based on the GARCHSK process with the assumption that error terms follow a distribution of Gram-Charlier series expansion of the normality [21]. Accordingly, it relaxes the strict and somewhat assumption that third and fourth moments, which is realistic given the frequent occurrence of extreme events and shocks and the strong deviation of returns from the normal distribution. The GARCHSK model has been applied in numerous studies covering energy and stock markets [31,32]:

where r_t is the log-returns of the index under study (i.e., Bitcoin or S&P500), and h_t is the conditional variance that follows a GARCH (1,1) structure. Notably, the results remain unchanged when we used the asymmetric GARCH (1,1) model. Furthermore, s_t and k_t denote the skewness and kurtosis corresponding to the conditional distribution of the standardized residuals ηt(ηt=εtht−1/2). Using the Gram–Charlier series expansion, we estimate the model and truncate at the fourth moment. For more details, refer to [21,30].