Stochastic Search Variable Selection (SSVS) Hyperparameter for Coefficients Matrix and Cholesky Factor

Set SSVS hyperparameters for VAR or VHAR coefficient matrix and Cholesky factor.

Usage

set_ssvs(
  spike_grid = 100L,
  slab_shape = 0.01,
  slab_scl = 0.01,
  s1 = c(1, 1),
  s2 = c(1, 1),
  shape = 0.01,
  rate = 0.01
)

# S3 method for class 'ssvsinput'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

is.ssvsinput(x)

# S3 method for class 'ssvsinput'
knit_print(x, ...)

Arguments

spike_grid: Griddy gibbs grid size for scaling factor (between 0 and 1) of spike sd which is Spike sd = c * slab sd
slab_shape: Inverse gamma shape for slab sd
slab_scl: Inverse gamma scale for slab sd
s1: First shape of coefficients prior beta distribution
s2: Second shape of coefficients prior beta distribution
shape: Gamma shape parameters for precision matrix (See Details).
rate: Gamma rate parameters for precision matrix (See Details).
x: Any object
digits: digit option to print
...: not used

Value

ssvsinput object

Details

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.

References

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.