Dynamic conditional score (DCS) models with time-varying shape parameters provide a exible method for volatility measurement. The new models are estimated by using the maximum likelihood (ML) method, conditions of consistency and asymptotic normality of ML are presented, and Monte Carlo simulation experiments are used to study the precision of ML. Daily data from the Standard & Poor's 500 (S&P 500) for the period of 1950 to 2017 are used. The performances of DCS models with constant and dynamic shape parameters are compared. In-sample statistical performance metrics and out-of-sample value-at-risk backtesting support the use of DCS models with dynamic shape.Astrid Ayala and Szabolcs Blazsek acknowledge funding from
the School of Business of Universidad Francisco Marroquín. Alvaro Escribano acknowledges funding
from the Spanish Ministry of Economy, Industry and Competitiveness (ECO2015-68715-R, ECO2016-
00105-001), Consolidation Grant (#2006/04046/002), and Maria de Maeztu Grant (MDM 2014-0431)
