Jointly Forecasting Value-at-Risk and Expected Shortfall with Score-Driven Dynamic Relationships

For the lack of elicitability of expected shortfall (ES), many prediction models based on the joint loss function of ES and value-at-risk (VaR) have been proposed. However, these models typically assume that the ratio of ES to VaR is constant, which implies a static relationship between ES and VaR. To capture the dynamic relationship between ES and VaR, this paper proposes two scoredriven forms: dynamic additive and dynamic multiplicative. The score-driven updating equation is derived from the joint loss function of ES and VaR. Empirical tests with five international stock indices and four probability levels indicate that the joint model with a score-driven dynamic relationship can achieve better out-of-sample performance than various benchmark models