Forecasting Expected Shortfall and Value-at-Risk With Cross-Sectional Aggregation

The combination of the conditional autoregressive value-at-risk (CAViaR) process with the Fissler–Ziegel (FZ) loss function generates a recently emerging framework (CAViaR-FZ) for forecasting value-at-risk (VaR) and expected shortfall (ES). However, existing CAViaR-FZ models typically overlook the presence of long-range dependence, a stylized fact of financial time series. This paper proposes a long-memory CAViaR-FZ model using the cross-sectional aggregation (CSA) method. The CSA method is well-recognized for its ability to generate a long-memory process by aggregating an infinite number of short-memory processes cross-sectionally. The proposed CSA-CAViaR-FZ model flexibly captures long-memory dynamics in both VaR and ES and includes the original short-memory CAViaR-FZ model as a special case. Simulation and empirical results demonstrate that the proposed model outperforms various competing models.