Combining Value-at-Risk and Expected Shortfall forecasts via the Model Confidence Set

To comply with increasingly stringent international standards in risk management and regulation, several approaches have been developed in the literature for forecasting tail-risk measures such as Value-at-Risk (VaR) and Expected Shortfall (ES). However, the accuracy of these measures can be significantly affected by multiple sources of uncertainty, including model misspecification, data limitations and estimation procedures. To address these challenges and enhance the predictive performance of individual models, this study introduces novel forecast combination strategies based on the Model Confidence Set (MCS) methodology. Specifically, a strictly consistent joint VaR-ES loss function is employed to identify the best-performing models, which constitute the Set of Superior Models (SSM). Subsequently, the VaR and ES forecasts of the models included in the SSM are combined using various weighting schemes. An empirical analysis based on nine stock market indices at the 2.5% and 1% risk levels provides evidence that the proposed combined predictors are a robust alternative for forecasting tail-risk measures, successfully passing standard backtests and consistently entering the SSM of the MCS.

💡 Research Summary

This paper addresses the growing need for accurate tail‑risk forecasts under stringent regulatory frameworks by proposing a novel forecast‑combination methodology that leverages the Model Confidence Set (MCS) procedure. The authors first adopt the strictly consistent joint VaR‑ES loss function introduced by Fissler and Ziegel (2016) – both in an unweighted form and in an exponentially weighted version that places greater emphasis on recent observations. Using these loss functions, a “training” MCS is performed over a rolling window to identify a Set of Superior Models (SSM) at each point in time. The SSM contains only those models that cannot be statistically rejected as inferior according to the chosen loss.

Once the SSM is established, the VaR and ES forecasts of its constituent models are combined using three weighting schemes: (i) a simple arithmetic average, (ii) the Relative Score (RS‑Comb) approach, where each model’s weight is a non‑linear function of its loss, and (iii) the Minimum Score (MS‑Comb) approach, which directly minimizes the joint VaR‑ES loss across the selected models. By pairing each weighting scheme with either the unweighted or weighted loss used in the training MCS, the authors generate six distinct MCS‑based combined predictors.

The empirical study evaluates these predictors on nine major equity indices (S&P 500, Shanghai Composite, Euro Stoxx 50, NASDAQ, Nikkei, Hang Seng, MXX, BSESN, Bovespa) at the 2.5 % and 1 % risk levels. A total of 32 competing models—including parametric GARCH‑type specifications, semi‑parametric quantile regressions (CAViaR), filtered historical simulation, and various realized‑volatility‑based approaches—provide the raw VaR and ES forecasts. The authors compare the six MCS‑based combinations with four benchmark aggregators that use the full model universe: simple mean, median, global RS‑Comb, and global MS‑Comb.

Back‑testing is conducted using the Kupiec Unconditional Coverage test, the Christoffersen Conditional Coverage test, the Engle‑Manganelli Dynamic Quantile test, and the Bayer‑Dimitriadis regression‑based ES test. Forecast accuracy is also assessed via the FZ loss itself. Results show that the MCS‑based combinations consistently pass all back‑tests, especially at the 2.5 % level, and are frequently retained in the evaluation MCS throughout the sample. In contrast, individual models rarely dominate across all indices, and the global combination methods sometimes suffer from the inclusion of poorly performing forecasts and higher computational burden. The weighted‑loss training MCS (with λ = 0.06) slightly improves responsiveness to recent market shifts, while the unweighted version offers more stable performance.

The paper’s contributions are threefold: (1) it integrates the MCS framework with a strictly consistent joint VaR‑ES loss to dynamically filter out inferior models, (2) it demonstrates that combining only the superior models yields more robust and accurate tail‑risk forecasts than combining the entire model set, and (3) it provides a practical, computationally efficient alternative to existing joint VaR‑ES combination methods. Limitations include the need for manual selection of the smoothing parameter λ and potential scalability issues when the model pool becomes very large. Future research directions suggested are automated hyper‑parameter tuning, extensions to multivariate tail‑risk measures, and the application of the methodology to other domains such as climate risk or sports analytics.