Score-driven time series models with dynamic shape : an application to the Standard & Poor's 500 index

Abstract

We introduce new dynamic conditional score (DCS) volatility models with dynamic scale and shape parameters for the effective measurement of volatility. In the new models, we use the EGB2 (exponential generalized beta of the second kind), NIG (normal-inverse Gaussian) and Skew-Gen-t (skewed generalized-t) probability distributions. Those distributions involve several shape parameters that control the dynamic skewness, tail shape and peakedness of financial returns. We use daily return data from the Standard & Poor's 500 (S&P 500) index for the period of January 4, 1950 to December 30, 2017. We estimate all models by using the maximum likelihood (ML) method, and we present the conditions of consistency and asymptotic normality of the ML estimates. We study those conditions for the S&P 500 and we also perform diagnostic tests for the residuals. The statistical performances of several DCS specifications with dynamic shape are superior to the statistical performance of the DCS specification with constant shape. Outliers in the shape parameters are associated with important announcements that affected the United States (US) stock market. Our results motivate the application of the new DCS models to volatility measurement, pricing financial derivatives, or estimation of the value-at-risk (VaR) and expected shortfall (ES) metrics.