estimate_residual_cov_poet_local: Estimate Local Covariance
In TVMVP: Time-Varying Minimum Variance Portfolio

estimate_residual_cov_poet_local

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: `loadings`: a list where each element is a `p × m` matrix of factor loadings. `f_hat`: a `T × m` matrix of estimated factors. `weights`: a list of kernel weight vectors.
`returns`	A numeric matrix of asset returns with dimensions `T × p`.
`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 `\rho` used in `adaptive_poet_rho()`. Default is `seq(0.005, 2, length.out = 30)`.
`floor_value`	A small positive number specifying the lower bound for eigenvalues in the final positive semidefinite repair. Default is `1e-12`.
`epsilon2`	A small positive tuning parameter for the adaptive thresholding. Default is `1e-6`.

Details

The function follows these steps:

**Local Residuals:** Extract the local loadings \Lambda_t from the last element of localPCA_results\$loadings and factors \hat{F} from localPCA_results\$f_hat. Let w_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() on U_{\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.