Score function scaling for QAR plus Beta-t-EGARCH: an empirical application to the S&P 500

In the literature on score-driven models, alternative choices of the scaling parameters of the conditional score terms are used, but the optimal choice of those parameters is an open question. Although there are score-driven models for which the choice of the scaling parameters is irrelevant, there are important score-driven models for which score-driven scale filters appear in the information matrix, and the choice of the scaling parameters is relevant. We focus on the quasi-autoregressive (QAR) plus Beta-$t$t-EGARCH (exponential generalized autoregressive conditional heteroscedasticity) model, by using data on the Vanguard Standard & Poor’s 500 (S&P 500) exchange-traded fund (VOO) and all available S&P 500 stocks for the period of 2013–2023. For QAR plus Beta-$t$t-EGARCH, each updating term is the product of a scaling parameter and a conditional score, and we use specific alternative scaling parameters from the literature. For different scaling parameters in the scale filter (volatility), alternative location and scale filters coincide. For different scaling parameters in the location filter (expected return), alternative location and scale filters differ significantly. For the statistical and volatility forecasting performances of VOO and most of the S&P 500 stocks, the best-performing scaling parameter for the score-driven location is the conditional inverse information matrix.