Loss-Based Bayesian Sequential Prediction of Value-at-Risk with a Long-Memory and Non-Linear Realized Volatility Model

Abstract A long-memory and non-linear realized volatility model class is proposed for direct Value-at-Risk (VaR) forecasting. This model, referred to as RNN-HAR, extends the heterogeneous autoregressive (HAR) model, a framework known for efficiently capturing long memory in realized measures, by integrating a Recurrent Neural Network (RNN) to handle the non-linear dynamics. Quantile loss-based generalized Bayesian method with Sequential Monte Carlo is employed for model estimation and sequential prediction in RNN-HAR. The empirical analysis is conducted using daily closing prices and realized measures with around 12 years of data till 2022, covering 31 market indices. The proposed model’s one-step-ahead VaR forecasting performance is compared against a basic HAR model and its extensions. The results demonstrate that the proposed RNN-HAR model consistently outperforms all other models considered in the study. The implementation code of the HAR-RNN model is publicly available on GitHub: https://github.com/chaowang-usyd/RNN-HAR.