Set SSVS hyperparameters for VAR or VHAR coefficient matrix and Cholesky factor.
set_ssvs(
coef_spike_grid = 100L,
coef_slab_shape = 0.01,
coef_slab_scl = 0.01,
coef_s1 = c(1, 1),
coef_s2 = c(1, 1),
shape = 0.01,
rate = 0.01,
chol_spike_grid = 100,
chol_slab_shape = 0.01,
chol_slab_scl = 0.01,
chol_s1 = 1,
chol_s2 = 1
)# S3 method for ssvsinput
print(x, digits = max(3L, getOption("digits") - 3L), ...)
is.ssvsinput(x)
# S3 method for ssvsinput
knit_print(x, ...)
ssvsinput object
Griddy gibbs grid size for scaling factor (between 0 and 1) of spike sd which is Spike sd = c * slab sd
Inverse gamma shape for slab sd
Inverse gamma scale for slab sd
First shape of coefficients prior beta distribution
Second shape of coefficients prior beta distribution
Gamma shape parameters for precision matrix (See Details).
Gamma rate parameters for precision matrix (See Details).
Griddy gibbs grid size for scaling factor (between 0 and 1) of spike sd which is Spike sd = c * slab sd in the cholesky factor
Inverse gamma shape for slab sd in the cholesky factor
Inverse gamma scale for slab sd in the cholesky factor
First shape of cholesky factor prior beta distribution
Second shape of cholesky factor prior beta distribution
Any object
digit option to print
not used
Let \(\alpha\) be the vectorized coefficient, \(\alpha = vec(A)\). Spike-slab prior is given using two normal distributions. $$\alpha_j \mid \gamma_j \sim (1 - \gamma_j) N(0, \tau_{0j}^2) + \gamma_j N(0, \tau_{1j}^2)$$ As spike-slab prior itself suggests, set \(\tau_{0j}\) small (point mass at zero: spike distribution) and set \(\tau_{1j}\) large (symmetric by zero: slab distribution).
\(\gamma_j\) is the proportion of the nonzero coefficients and it follows $$\gamma_j \sim Bernoulli(p_j)$$
coef_spike: \(\tau_{0j}\)
coef_slab: \(\tau_{1j}\)
coef_mixture: \(p_j\)
\(j = 1, \ldots, mk\): vectorized format corresponding to coefficient matrix
If one value is provided, model function will read it by replicated value.
coef_non: vectorized constant term is given prior Normal distribution with variance \(cI\). Here, coef_non is \(\sqrt{c}\).
Next for precision matrix \(\Sigma_e^{-1}\), SSVS applies Cholesky decomposition. $$\Sigma_e^{-1} = \Psi \Psi^T$$ where \(\Psi = \{\psi_{ij}\}\) is upper triangular.
Diagonal components follow the gamma distribution. $$\psi_{jj}^2 \sim Gamma(shape = a_j, rate = b_j)$$ For each row of off-diagonal (upper-triangular) components, we apply spike-slab prior again. $$\psi_{ij} \mid w_{ij} \sim (1 - w_{ij}) N(0, \kappa_{0,ij}^2) + w_{ij} N(0, \kappa_{1,ij}^2)$$ $$w_{ij} \sim Bernoulli(q_{ij})$$
shape: \(a_j\)
rate: \(b_j\)
chol_spike: \(\kappa_{0,ij}\)
chol_slab: \(\kappa_{1,ij}\)
chol_mixture: \(q_{ij}\)
\(j = 1, \ldots, mk\): vectorized format corresponding to coefficient matrix
\(i = 1, \ldots, j - 1\) and \(j = 2, \ldots, m\): \(\eta = (\psi_{12}, \psi_{13}, \psi_{23}, \psi_{14}, \ldots, \psi_{34}, \ldots, \psi_{1m}, \ldots, \psi_{m - 1, m})^T\)
chol_ arguments can be one value for replication, vector, or upper triangular matrix.
George, E. I., & McCulloch, R. E. (1993). Variable Selection via Gibbs Sampling. Journal of the American Statistical Association, 88(423), 881-889.
George, E. I., Sun, D., & Ni, S. (2008). Bayesian stochastic search for VAR model restrictions. Journal of Econometrics, 142(1), 553-580.
Ishwaran, H., & Rao, J. S. (2005). Spike and slab variable selection: Frequentist and Bayesian strategies. The Annals of Statistics, 33(2).
Koop, G., & Korobilis, D. (2009). Bayesian Multivariate Time Series Methods for Empirical Macroeconomics. Foundations and Trends® in Econometrics, 3(4), 267-358.