Modeling time‐varying higher‐order conditional moments: A survey

Since the introduction of the Autoregressive Conditional Heteroscedasticity (ARCH) model, the literature on modeling the time‐varying second‐order conditional moment has become increasingly popular in the last four decades. Its popularity is partly due to its success in capturing volatility in financial time series, which is useful for modeling and predicting risk for financial assets. A natural extension of this is to model time variation in higher‐order conditional moments, such as the third and fourth moments, which are related to skewness and kurtosis (tail risk). This leads to an emerging literature on time‐varying higher‐order conditional moments in the last two decades. This paper outlines recent developments in modeling time‐varying higher‐order conditional moments in the economics and finance literature. Using the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) framework as a foundation, this paper provides an overview of the two most common approaches for modeling time‐varying higher‐order conditional moments: autoregressive conditional density (ARCD) and autoregressive conditional moment (ARCM). The discussion covers both the theoretical and empirical aspects of the literature. This includes the identification of the associated skewness–kurtosis domain by using the solutions to the classical moment problems, the structural and statistical properties of the models used to model the higher‐order conditional moments and the computational challenges in estimating these models. We also advocate the use of a maximum entropy density (MED) as an alternative method, which circumvents some of the issues prevalent in these common approaches.