Conditional Autoregressive Expected Shortfall (CAESar)

• CAESar is a distribution‐free framework for jointly modeling Value-at-Risk (VaR) and Expected Shortfall (ES) via autoregressive recursions and consistent scoring rules.
• It employs a robust three-stage estimation process—quantile regression, residual ES modeling, and joint optimization—to enhance forecasting accuracy and model stability.
• The model effectively captures heteroskedasticity, autocorrelation, and tail asymmetry in financial returns, demonstrating superior empirical performance and regulatory backtesting results.

Conditional Autoregressive Expected Shortfall (CAESar) is a distribution-free, dynamic framework for jointly modeling and forecasting Value at Risk (VaR) and Expected Shortfall (ES), two cornerstone risk metrics in quantitative finance. CAESar achieves this via time series recursions inspired by the CAViaR approach, direct joint estimation using strictly consistent scoring functions, and robust regularization methodologies that account for realistic financial data characteristics including heteroskedasticity, autocorrelation, and tail asymmetry. CAESar resolves the issue of the non-elicitability of ES by harnessing its joint elicitability with VaR, enabling the minimization of joint loss functions for consistent estimation and backtesting (Gatta et al., 9 Jul 2024, Wang et al., 2019, Wang et al., 2018, Patton et al., 2017).

1. Model Structure and Mathematical Formulation

The canonical CAESar model expresses the dynamics of VaR and ES using conditional autoregressive recursions with no parametric assumption on the underlying return distribution. Let $y_t$ denote the portfolio return at time $t$, and $\alpha\in(0,1)$ the risk level.

(a) VaR Process (CAViaR-style Asymmetric Slope, 1,1):

$$Q_t(\alpha) = \beta_0 + \beta_1(y_{t-1})^+ + \beta_2(y_{t-1})^- + \beta_3 Q_{t-1}(\alpha)$$

where $(y_{t-1})^+ = \max\{0, y_{t-1}\}$, $(y_{t-1})^- = \max\{0, -y_{t-1}\}$. The parameter vector $\beta = (\beta_0, \beta_1, \beta_2, \beta_3)$ governs intercept, return slopes, and autoregressive weight.

(b) ES Process (Autoregressive Asymmetric Slope, 1,1):

$$ES_t(\alpha) = \gamma_0 + \gamma_1(y_{t-1})^+ + \gamma_2(y_{t-1})^- + \gamma_3 Q_{t-1}(\alpha) + \gamma_4 ES_{t-1}(\alpha)$$

with $\gamma = (\gamma_0, ..., \gamma_4)$. This recursion introduces both return-dependent and lagged VaR/ES heteroskedasticity.

(c) Joint Loss and Constraints:

VaR and ES are jointly estimated by minimizing the Patton–Pflug–Nolde consistent scoring rule: $$\mathcal{L}_P^\alpha(\beta, \gamma) = \frac{1}{T} \sum_{t=1}^T \left\{ \frac{\hat Q_t}{\hat E_t} - \frac{\hat Q_t - y_t}{\alpha \hat E_t} \mathbf{1}_{\{y_t \le \hat Q_t\}} + \log(-\hat E_t) \right\}$$ Monotonicity and negativity constraints ($\hat E_t \le \hat Q_t \le 0$) are enforced via soft penalty terms: $$\mathcal{L}_{q,e}^\alpha(\beta, \gamma) = \mathcal{L}_P^\alpha(\beta, \gamma) + \lambda_e \sum_{t=1}^T (\hat E_t - \hat Q_t)^+ + \lambda_q \sum_{t=1}^T (\hat Q_t)^+$$ where $(x)^+ = \max\{0, x\}$ and $\lambda_e, \lambda_q$ are penalty weights (Gatta et al., 9 Jul 2024).

2. Estimation Procedure

CAESar estimation follows a robust three-stage process:

1. Stage 1 (VaR): Estimate $\beta$ by minimizing the quantile loss (pinball loss):

$$\mathcal{L}_q^\alpha(\beta) = \frac{1}{T} \sum_{t=1}^T (y_t - Q_t(\alpha)) \left(\alpha - \mathbf{1}_{\{y_t < Q_t(\alpha)\}} \right)$$

using standard quantile regression algorithms.

2. Stage 2 (Preliminary ES Residual): Define the residual series $\hat r_t = \hat E_t - \hat Q_t$ and fit an analogous AR(1,1) model to $r_t$. Minimize the Barrera–Cosso–Genton–Nolde squared-penalty loss:

$$\mathcal{L}_B^\alpha(\tilde\gamma) = \frac{1}{T} \sum_{t=1}^T \left(\tilde r_t + \frac{1}{\alpha} (\hat Q_t - y_t)^+ \right)^2 + \lambda_r \sum_{t=1}^T (\tilde r_t)^+$$

$\tilde\gamma$ is mapped to the joint model parameters.

3. Stage 3 (Joint VaR–ES): Initialize $(\beta, \gamma)$ at the solutions from Stages 1/2. Minimize the full penalized joint loss $\mathcal{L}_{q,e}^\alpha(\beta, \gamma)$ using a constrained nonlinear optimizer (e.g., SLSQP with random restarts) (Gatta et al., 9 Jul 2024).

Convergence sensitivity to initial values is mitigated by this multi-step coarse-to-fine fitting approach.

3. Properties, Assumptions, and Theoretical Foundations

• No Conditional Distributional Assumption: The model is fully distribution-free, applying solely to time series of returns and their lags.
• Joint Elicitability: ES is not separately elicitable, but joint scoring rules (e.g., Fissler–Ziegel, Patton–Pflug–Nolde) cover both VaR and ES, allowing proper joint estimation (Patton et al., 2017).
• Heteroskedastic Effects: Recursion on lagged VaR and ES, as well as asymmetric past returns, flexibly captures volatility clustering.
• Monotonicity and Negativity: Imposed via soft penalties to rule out nonphysical forecasts (e.g., ES exceeding VaR, or positive VaR forecasts under loss scenarios).
• Parameter Complexity: With the (1,1) specification, the model balances parsimony and flexibility.

4. Backtesting and Validation Techniques

CAESar incorporates an array of industry-standard and advanced statistical backtests:

• VaR:
  • Unconditional Coverage (Kupiec): Verifies that empirical violation rate matches nominal level $\alpha$.
  • Dynamic Quantile (Christoffersen–Engle): Tests for independence in violation sequences.
• ES:
  • Direct Approximation:
  • McNeil–Frey bootstrap: Test $$\mathbb{E}[y_t - \hat E_t \mid y_t \le \hat Q_t] \ge 0$$.
  • Acerbi–Székely $Z_1, Z_2$ statistics: Evaluate $$\mathbb{E}[y_t / \hat E_t \mid y_t \le \hat Q_t] = 1$$ and $$\mathbb{E}[y_t / (\alpha \hat E_t) \mathbf{1}_{y_t \le \hat Q_t}] = 1$$.
  • Model-Comparison (Scoring function based):
  • Diebold–Mariano: Statistical loss-difference testing under stationarity.
  • Corrected Resampled Student’s $t$ (Nadeau–Bengio).
  • Loss-Difference Bootstrap and Encompassing Forecasts (Gatta et al., 9 Jul 2024).

A comprehensive portfolio of backtests ensures the model's adequacy both in likelihood-fit and out-of-sample predictive performance.

5. Empirical Performance and Comparative Evaluation

Simulation experiments (GARCH-Gaussian and GARCH-$t$ data-generating processes) demonstrate that CAESar and the K-CAViaR ensemble approach yield the lowest mean absolute error (MAE), root mean squared error (RMSE), and joint loss ($\mathcal{L}_B, \mathcal{L}_P$) among a wide array of competing models including neural network approaches (BCGNS, QRNN) and GAS-based models, which are often less stable or outperformed.

Empirical studies on daily financial data (10 international indices, 24 rolling 7-year windows) indicate:

• CAESar ranks among the top models in both joint and ES losses for multiple risk levels ($\alpha \in \{0.05, 0.025, 0.01\}$).
• ES direct-approximation tests yield the lowest rejection rates for CAESar, especially at deep tails ($\alpha=0.01$).
• Diebold–Mariano and related model-comparison tests overwhelmingly favor CAESar forecasts.
• Joint estimation slightly improves VaR performance over separate quantile regression.
• Further application to US banking stocks corroborates these findings (Gatta et al., 9 Jul 2024).

6. Implementation Guidelines and Practical Advice

• Software: Reference Python code is available at [github.com/fgt996/CAESar].
• Specification: Use AS–(1,1) (Asymmetric Slope with one lag) for $(y^+, y^-)$ to balance estimation efficiency and model fidelity.
• Penalty Weights: Choose $\lambda_r, \lambda_e, \lambda_q \in [10, 10^3]$, optimally tuned via cross-validation.
• Optimization: Employ SLSQP with 3–5 random starts to mitigate local minima; parallelize initializations for computational efficiency.
• Computational Tips: Most computations are CPU-bound except for neural network variants where GPU acceleration is beneficial.
• Convergence and Stability: The three-stage fit is crucial for convergence, particularly in the presence of nonconvexity.
• Limitations: Extremely low $\alpha$ levels (e.g., 0.01) present challenges under data scarcity; models involving higher lag orders risk overfitting and increased estimation error (Gatta et al., 9 Jul 2024).

7. Variations and Related Models

CAESar belongs to a class of joint VaR–ES models that include:

• GAS-type CAESar Models: Directly specify dynamic updating via Generalized Autoregressive Score methods, drawing on the Fissler–Ziegel scoring rule for consistent joint estimation (Patton et al., 2017).
• Measurement-Equation/Expectile Extensions: Incorporate contemporaneous volatility measurements (e.g., realized variance/range) and link expectiles to ES, often using Bayesian MCMC for parameter inference (Wang et al., 2019, Wang et al., 2018).
• Empirical Optimization and Realized Measures: Sub-sampled realized variance/range are found to enhance ES forecast sharpness and robustness (Wang et al., 2019, Wang et al., 2018).

A comparative implication is that CAESar’s distribution-free, multistage estimation approach consistently delivers state-of-the-art tail risk prediction and passes rigorous probabilistic and backtesting diagnostics across broad benchmarks. Competing approaches relying on parametric innovations or less robust loss functions may be systematically outperformed, especially in stressed-market or heavy-tailed conditions.

Summary Table: Typical CAESar Specification

Component	Functional Form	Penalty/Constraint
VaR ($Q_t(\alpha)$)	$$\beta_0 + \beta_1(y_{t-1})^+ + \beta_2(y_{t-1})^- + \beta_3Q_{t-1}$$	$\hat Q_t \le 0$ (soft penalty)
ES ($ES_t(\alpha)$)	$$\gamma_0 + \gamma_1(y_{t-1})^+ + \gamma_2(y_{t-1})^- + \gamma_3Q_{t-1} + \gamma_4 ES_{t-1}$$	$\hat E_t \le \hat Q_t$ (soft penalty)
Joint Loss	Patton–Pflug–Nolde scoring function $\mathcal{L}_P^\alpha$	Soft penalty additions
Optimization	Three stage (quantile, residual ES, joint)	SLSQP with random restarts

CAESar defines a modern, empirically validated, and theoretically rigorous standard for dynamic tail risk modeling without restrictive distributional assumptions, supporting both regulatory and tactical financial risk management (Gatta et al., 9 Jul 2024, Patton et al., 2017, Wang et al., 2019, Wang et al., 2018).