Estimate regression coefficients and scale matrix for noise by using a parameterized cross validation (PCV). The function assumes 1) multivariate t-distribution for noise as a sampling distribution, and 2) informative priors for regression coefficients and scale matrix for noise.
lm_semi_Bayes_PCV(Y, X, dof = Inf, lambda = NULL, lambda_var = NULL,
prior_type = c("NCJ", "CJ"), num_folds = 5, m0 = ncol(Y))An N x K matrix of dependent variables.
An N x M matrix of regressors.
Degree of freedom for multivariate t-distribution. If dof = Inf (default), then multivariate normal distribution is applied and weight vector q is not estimated. If dof = NULL or a numeric vector, then dof is selected by K-fold CV automatically and q is estimated.
If NULL or a vector of length >=2, it is selected by PCV.
If NULL, it is selected by a Stein-type shrinkage method.
"NCJ" for non-conjugate prior and "CJ" for conjugate prior for scale matrix Sigma.
Number of folds for PCV.
A hyperparameter for inverse Wishart distribution for Sigma
Consider the multivariate regression: $$Y = X Psi + e, \quad e ~ mvt(0, dof, Sigma).$$ Psi is a M-by-K matrix of regression coefficients and Sigma is a K-by-K scale matrix for multivariate t-distribution for noise.
Sampling distribution for noise e is the multivariate t-distribution with degree of freedom dof and scale matrix Sigma: e ~ mvt(0, dof, Sigma). The priors are informative priors: 1) a shrinkage prior for regression coefficients Psi, and 2) inverse Wishart prior for scale matrix Sigma, which can be either non-conjugate ("NCJ") or conjugate ("CJ") to the shrinkage prior for coefficients Psi.
The function implements parameterized cross validation (PCV) for selecting a shrinkage parameter lambda for estimating regression coefficients (0 < lambda <= 1). In addition, the function uses a Stein-type shrinkage method for selecting a shrinkage parameter lambda_var for estimating variances of time series variables.
N. Lee, H. Choi, and S.-H. Kim (2016). Bayes shrinkage estimation for high-dimensional VAR models with scale mixture of normal distributions for noise. Computational Statistics & Data Analysis 101, 250-276. doi: 10.1016/j.csda.2016.03.007