Improved gradient scaling for score-driven filters with an application to stock market volatility

Score-driven filters are updated by the scaled gradient of the log-likelihood (LL). The gradient is with respect to a dynamic parameter and the scaling parameter is 1, or the information quantity or its square root in the literature. The information quantity is minus the expected value of the Hessian of the LL with respect to a dynamic parameter, i.e. the Hessianis smoothed using a probability-weighted average for each period. We suggest an alternative approach and scale the gradients using novel Hessian-driven filters, i.e. Hessian smoothing is performed over time. The method can be used for score-driven models in general. We illustrate it for Beta-t-EGARCH (exponential generalized autoregressive conditional heteroscedasticity). Weuse Standard & Poor's 500 (S&P 500) data. We show empirical results for in-sample statistical performance from 2015 to 2025, and out-of-sample forecasting performance from 2021 to 2025. We find for the S&P 500 that the Hessian-driven scaling is superior to the existing scaling methods for Beta-t-EGARCH. We find similar results for a Monte Carlo simulation experimentwhere misspecified Beta-t-EGARCH models with constant and Hessian-driven gradient scaling are estimated for returns generated by a Markov-switching (MS) Beta-t-EGARCH. Hessianbased gradient scaling captures regime-switching dynamics better than constant gradient scaling.