estimate_residual_cov_poet_local {TVMVP} | R Documentation |
Estimate Local Covariance
Description
This internal function computes a time-varying covariance matrix estimate for a given
window of asset returns by combining factor-based and sparse residual covariance estimation.
It uses results from a local PCA to form residuals and then applies an adaptive thresholding
procedure (via adaptive_poet_rho()
) to shrink the residual covariance.
Usage
estimate_residual_cov_poet_local(
localPCA_results,
returns,
M0 = 10,
rho_grid = seq(0.005, 2, length.out = 30),
floor_value = 1e-12,
epsilon2 = 1e-06
)
Arguments
localPCA_results |
A list containing the results from local PCA, with components:
|
returns |
A numeric matrix of asset returns with dimensions |
M0 |
Integer. The number of observations to leave out between the two sub-samples in the adaptive thresholding procedure. Default is 10. |
rho_grid |
A numeric vector of candidate shrinkage parameters |
floor_value |
A small positive number specifying the lower bound for eigenvalues in the final positive semidefinite repair. Default is |
epsilon2 |
A small positive tuning parameter for the adaptive thresholding. Default is |
Details
The function follows these steps:
**Local Residuals:** Extract the local loadings
\Lambda_t
from the last element oflocalPCA_results\$loadings
and factors\hat{F}
fromlocalPCA_results\$f_hat
. Letw_t
denote the corresponding kernel weights. The local residuals are computed as:U_{\text{local}} = R - F \Lambda_t,
where
R
is the returns matrix.**Adaptive Thresholding:** The function calls
adaptive_poet_rho()
onU_{\text{local}}
to select an optimal shrinkage parameter\hat{\rho}_t
.**Residual Covariance Estimation:** The raw residual covariance is computed as:
S_{u,\text{raw}} = \frac{1}{T} U_{\text{local}}^\top U_{\text{local}},
and a threshold is set as:
\text{threshold} = \hat{\rho}_t × \text{mean}(|S_{u,\text{raw}}|),
where the mean is taken over the off-diagonal elements. Soft-thresholding is then applied to obtain the shrunk residual covariance matrix
\hat{S}_u
.**Total Covariance Estimation:** The final covariance matrix is constructed by combining the factor component with the shrunk residual covariance:
\Sigma_R(t) = \Lambda_t \left(\frac{F^\top F}{T}\right) \Lambda_t^\top + \hat{S}_u.
**PSD Repair:** A final positive semidefinite repair is performed by flooring eigenvalues at
floor_value
and symmetrizing the matrix.
Value
A list containing:
-
best_rho
: The selected shrinkage parameter\hat{\rho}_t
for the local residual covariance. -
residual_cov
: The shrunk residual covariance matrix\hat{\Sigma}_e(T)
. -
total_cov
: The final estimated time-varying covariance matrix\Sigma_R(t)
. -
loadings
: The local factor loadings\Lambda_t
from the local PCA. -
naive_resid_cov
: The raw (unshrunk) residual covariance matrix.