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