Theoretical Framework

The HAR-CAESar model combines the robust tail risk forecasting of CAESar (Conditional Autoregressive Expected Shortfall) with the long-memory capabilities of the HAR (Heterogeneous Autoregressive) volatility model.

The HAR-CAESar Model

The core innovation is the modification of the autoregressive dynamics to condition on returns aggregated over different time horizons, inspired by the Heterogeneous Market Hypothesis (Muller et al., 1997).

Specification

The joint dynamics for Value at Risk ($Q_t$) and Expected Shortfall ($ES_t$) at probability level $theta$ use an Asymmetric Slope (AS) structure at all horizons to capture leverage effects:

VaR Equation:

\[Q_t = \beta_0 + \beta_1^{(d)} (r_{t-1}^{(d)})^+ + \beta_2^{(d)} (r_{t-1}^{(d)})^- + \beta_1^{(w)} (r_{t-1}^{(w)})^+ + \beta_2^{(w)} (r_{t-1}^{(w)})^- + \beta_1^{(m)} (r_{t-1}^{(m)})^+ + \beta_2^{(m)} (r_{t-1}^{(m)})^- + \beta_3 Q_{t-1} + \beta_4 ES_{t-1}\]

ES Equation:

\[ES_t = \gamma_0 + \gamma_1^{(d)} (r_{t-1}^{(d)})^+ + \gamma_2^{(d)} (r_{t-1}^{(d)})^- + \gamma_1^{(w)} (r_{t-1}^{(w)})^+ + \gamma_2^{(w)} (r_{t-1}^{(w)})^- + \gamma_1^{(m)} (r_{t-1}^{(m)})^+ + \gamma_2^{(m)} (r_{t-1}^{(m)})^- + \gamma_3 Q_{t-1} + \gamma_4 ES_{t-1}\]

Where:

  • Daily: $r_{t-1}^{(d)} = r_{t-1}$

  • Weekly: $r_{t-1}^{(w)} = frac{1}{5} sum_{j=1}^{5} r_{t-j}$

  • Monthly: $r_{t-1}^{(m)} = frac{1}{22} sum_{j=1}^{22} r_{t-j}$

  • Positive/Negative: $(x)^+ = max(0, x)$ and $(x)^- = max(0, -x)$

This specification has 9 parameters per equation (intercept + 6 return coefficients + 2 autoregressive terms), allowing positive and negative returns to have different impacts at each horizon.

Estimation Strategy

The model is estimated using a three-stage approach to ensure stability and consistency:

Stage 1: VaR Initialization

Estimate VaR parameters using CAViaR with the Tick (quantile) Loss:

\[L_{tick}(r_t, Q_t) = (r_t - Q_t)(\theta - \mathbf{1}_{\{r_t < Q_t\}})\]

Stage 2: ES Residual Estimation

Rather than estimating ES directly, Stage 2 estimates the ES residual $r_t = ES_t - Q_t$ using the Barrera loss function. This reparametrisation ensures monotonicity ($ES_t le Q_t < 0$) since $r_t < 0$.

Stage 3: Joint Refinement

Re-estimate all parameters simultaneously by minimizing the joint Fissler-Ziegel Loss function:

\[L_{FZ}(r_t, Q_t, ES_t) = \frac{1}{T} \sum_{t=1}^T \left[ \frac{1}{\theta ES_t} (r_t - Q_t) \mathbf{1}_{\{r_t \le Q_t\}} + \frac{Q_t}{ES_t} + \log(-ES_t) - 1 \right]\]

Implementation Details:

  • Penalty weights: $lambda_q = lambda_e = 10$ enforce the monotonicity constraint $ES_t le Q_t$

  • Multiple random initializations ensure global convergence

  • Convergence verified using gradient tolerance and loss function stability

Backtesting

The implementation provides comprehensive backtesting tools:

VaR Backtests:

  • Kupiec (1995): Unconditional coverage test for correct violation rate

  • Christoffersen (1998): Conditional coverage test combining correct rate and independence

ES Backtests:

  • McNeil-Frey (2000): Bootstrap test for ES calibration using exceedance residuals

  • Acerbi-Szekely (2014): Z1 and Z2 tests for ES specification

Forecast Comparison:

  • Diebold-Mariano: HAC-robust test for predictive accuracy comparison

  • Bootstrap loss differential: One-sided test for forecast encompassing