lm_semi_Bayes_PCV: Semiparametric Bayesian Shrinkage Estimation Method for...

In VARshrink: Shrinkage Estimation Methods for Vector Autoregressive Models

Semiparametric Bayesian Shrinkage Estimation Method for Multivariate Regression

Description

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.

Usage

lm_semi_Bayes_PCV(
  Y,
  X,
  dof = Inf,
  lambda = NULL,
  lambda_var = NULL,
  prior_type = c("NCJ", "CJ"),
  num_folds = 5,
  m0 = ncol(Y)
)

Arguments

Y	An N x K matrix of dependent variables.
X	An N x M matrix of regressors.
dof	Degrees-of-freedom, \nu, 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.
lambda	If NULL or a vector of length >=2, it is selected by PCV.
lambda_var	If NULL, it is selected by a Stein-type shrinkage method.
prior_type	"NCJ" for non-conjugate prior and "CJ" for conjugate prior for scale matrix Sigma.
num_folds	Number of folds for PCV.
m0	A hyperparameter for inverse Wishart distribution for Sigma

Details

Consider the multivariate regression:

\mathbf{Y} = \mathbf{X} \mathbf{\Psi} + \mathbf{e}, \quad \mathbf{e} \sim MVT(0, \nu, \mathbf{\Sigma}).

\mathbf{\Psi} is a (M \times K) matrix of regression coefficients and \mathbf{\Sigma} is a (K \times K) scale matrix for multivariate t-distribution for noise.

Sampling distribution for noise \mathbf{e} is the multivariate t-distribution with the degrees-of-freedom \nu and scale matrix \mathbf{\Sigma}: \mathbf{e} \sim MVT(0, \nu, \mathbf{\Sigma}). The priors are informative priors: 1) a shrinkage prior for regression coefficients \mathbf{Psi}, and 2) inverse Wishart prior for scale matrix \mathbf{\Sigma}, which can be either non-conjugate ("NCJ") or conjugate ("CJ") to the shrinkage prior for coefficients \mathbf{\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.

References

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

VARshrink documentation built on Jan. 10, 2026, 1:06 a.m.