Modeling Autoregressive Conditional Skewness and Kurtosis with Multi-Quantile CAViaR

This chapter extends Engle and Manganelli's (2004) univariate CAViaR model to a multi-quantile version, MQ-CAViaR. This allows for both a general vector autoregressive structure in the conditional quantiles and the presence of exogenous variables. The MQ-CAViaR model is then used to specify conditional versions of the more robust skewness and kurtosis measures discussed in Kim and White (2004). The chapter is organized as follows. Section 2 develops the MQ-CAViaR data generating process (DGP). Section 3 proposes a quasi-maximum likelihood estimator for the MQ-CAViaR process, and proves its consistency and asymptotic normality. Section 4 shows how to consistently estimate the asymptotic variance-covariance matrix of the MQCAViaR estimator. Section 5 specifies conditional quantile-based measures of skewness and kurtosis based on MQ-CAViaR estimates. Section 6 contains an empirical application of our methods to the S&P 500 index. The chapter also reports results of a simulation experiment designed to examine the finite sample behavior of our estimator. Section 7 contains a summary and concluding remarks.