Iterative Covariance Estimation
Core algorithm: GARCH-weighted covariance with eigenvalue decomposition.
Iterative Covariance Estimation
The covariance matrix is estimated using an iterative algorithm that combines linear time-decay weights with GARCH-based volatility adjustments. This gives more weight to recent observations and volatile periods, producing a more responsive covariance estimate.
Key Steps:- Initialize with linear time-decay weights
- Iterate until weight convergence
- Scale risk model factors by volatility
- Perform eigenvalue decomposition
- Fit GARCH to first principal component
- Update weights based on GARCH volatility
Empirical Covariance:
$$
\hat{\Sigma}_{\text{emp}} = \sum_{t=1}^{T} w_t \, r_t r_t^\top
$$
Weighted Covariance:
$$
Q^{(k+1)}_{\text{emp}} = \sum_{t=1}^{T} w_t^{(k)} \, r_t r_t^\top
$$
Garch Weight Update:
$$
w_t^{(k+1)} \propto \left(\frac{t}{T}\right)^{1-\gamma} \cdot \left(\frac{1}{\hat{\sigma}_t^{(k)}}\right)^{\gamma}
$$
Eigendecomposition:
$$
Q_{\text{rm}} = V \Lambda V^\top
$$