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.