Package: VARshrink 0.5.0

Introduction | Backgrounds | Main Purpose | Models | Shrinkage Estimation Methods | Multivariate Ridge Regression | Nonparametric Shrinkage (NS) | Full Bayesian Shrinkage | Then, conditional posteriors can be expressed as:\begin{equation}\label{expr_fbayes_psihat}\begin{split}( \boldsymbol\psi | \mathbf{Q}, \mathbf{\Sigma}, w, \delta ; \mathbf{Y} )& \sim \text{N}{J} \left(\widehat{\boldsymbol\psi}^\text{F}\text{NCJ}(\mathbf{\Sigma}, \mathbf{Q}, \delta),\left( \mathbf{\Sigma}^{-1} \otimes \mathbf{X}^\top \mathbf{Q} \mathbf{X} + \delta^{-1} \mathbf{I} \right)^{-1}\right) ,\\pi( \mathbf{\Sigma} | \boldsymbol\psi, \mathbf{Q}, w, \delta ; \mathbf{Y} )& \propto\frac{\exp\left( - 1 / 2 , \text{trace} (\mathbf{\Sigma}^{-1}\mathbf{E}(\mathbf{\Psi})^\top \mathbf{Q} \mathbf{E}(\mathbf{\Psi})) \right)}{|\mathbf{\Sigma}|^{N/2+1} \prod_{1\leq i < j \leq d} (\lambda_i - \lambda_j)},\( q_t | \boldsymbol\psi, \mathbf{\Sigma}, w, \delta ; \mathbf{Y} )& \sim \text{Gamma}\left(w + \frac{K}{2}, w + \frac{[\mathbf{E}(\mathbf{\Psi}) \mathbf{\Sigma}^{-1} \mathbf{E}(\mathbf{\Psi})^\top ]{tt}}{2} \right) ,\\pi( w | \boldsymbol\psi, \mathbf{Q}, \mathbf{\Sigma}, \delta ; \mathbf{Y} )& \propto \frac{w^{Nw + a_0 - 1}}{\Gamma(w)^N} \left( \prod{t=p+1}^T q_t \right)^w \exp\left( - \left( b_0 + \sum_{t=p+1}^T q_t \right) w \right),\( \delta | \boldsymbol\psi, \mathbf{Q}, \mathbf{\Sigma}, w ; \mathbf{Y} )& \sim \text{InvGamma} \left( \frac{J}{2} - 1, \frac{1}{2} \text{trace} (\mathbf{\Psi}^\top \mathbf{\Psi}) \right) ,\end{split}\end{equation}where$$\widehat{\boldsymbol\Psi}^\text{F}_\text{NCJ} (\mathbf{\Sigma}, \mathbf{Q}, \delta) | Semiparametric Bayesian Shrinkage | K-fold Cross Validation for Semiparametric Shrinkage | Numerical Experiments | Benchmark Data | Conclusions | References