Chapter Hands-on Code

Regularized regression with the Lasso penalty, the role of thresholding, and the coordinate-descent algorithm — motivated through simple models and real-data examples.

Matrix factorizations of covariance matrices — spectral, Cholesky, and variance–correlation separation — alongside copulas as counterparts of correlation for nonlinear dependence.

General linear models with unstructured covariance, with shrinkage of the least-squares estimator and regression-based methods such as CAPME for sparse high-dimensional estimation.

Progress in covariance regression from linear covariance models toward GLMs for covariance matrices, with parsimony-driven parameterizations and generalized estimating equations (GEE).

Classical PCA and factor analysis alongside modern approximate factor models for econometric panels, with the POET estimator handling sparse idiosyncratic covariance in high dimensions.

Ledoit–Wolf shrinkage that pulls the sample covariance toward a simple target, and thresholding that zeros out small entries — affecting only eigenvalues versus both eigenvalues and eigenvectors.

Sparse Gaussian graphical models via penalized likelihood and the GLasso algorithm, with extensions including thresholded GLasso, latent variable GLasso, and covariance graphical models.

Bayesian networks and DAGs where structural equation models determine sparse Cholesky factors of covariance matrices, with constraint-based, score-based, and functional structure-learning approaches.

Stationary multivariate time series with cross-sectional and temporal dependence — ARMA via Cholesky orthogonalization, time-series GLasso, sparse high-dimensional VAR, and Granger causality.