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.

Since returns of financial assets generally exhibit skewness and kurtosis, modelling returns using a distribution with the ability to capture both of these statistical aspects will increase the accuracy of risk forecasts based on these distributions. The authors propose the use of the multivariate skew normal (MVSN) mixture model to fit asset returns in order to increase the accuracy of Value-at-Risk (VaR) estimates. This paper presents a novel application of the MVSN mixture model to estimate VaR. There is generally no explicit analytical solution for the parameters of the MVSN mixture model via maximum likelihood estimation (MLE), therefore the use of the Expectation Maximization (EM) Algorithm is proposed in order to find the parameter estimates of the model. The example provided in this paper consists of a portfolio of monthly returns of six shares listed on the Australian Securities Exchange (ASX). The shares are BHP Billiton Limited (BHP), Commonwealth Bank of Australia (CBA), Cochlear Limited (COH), News Corporation (NWS), Origin Energy (ORG), and Wesfarmers Limited (WES). Hence, the dimensionality, p, of this portfolio is six. The period of analysis for the data is 01/01/1998-01/04/2011. This paper models the MVSN mixture model with a number of mixtures ranging from one to four. A mixture of multivariate normal densities is modelled for comparison to the MVSN mixture model. We find that for one to three mixtures, the MVSN mixture model provides an improved fit. The improved fit of the MVSN mixture model is translated to the performance of the VaR models, where the results show that for one to three numbers of mixtures, the VaR model using the MVSN mixture model assumption indicates improved risk forecasts when compared to the mixture of multivariate normal densities. Furthermore, for the example examined, we find that the model which incorporates the skewness parameter (MVSN mixture model) requires a fewer number of mixtures when compared to a mixture of normal densities. This is an interesting result as reduced model complexity requires less computational ability, computation time, and will results in decreased computational anomalies.