set_ssvs: Stochastic Search Variable Selection (SSVS) Hyperparameter...

In bvhar: Bayesian Vector Heterogeneous Autoregressive Modeling

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

Description

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

Usage

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

is.ssvsinput(x)

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

Arguments

coef_spike_grid	Griddy gibbs grid size for scaling factor (between 0 and 1) of spike sd which is Spike sd = c * slab sd
coef_slab_shape	Inverse gamma shape for slab sd
coef_slab_scl	Inverse gamma scale for slab sd
coef_s1	First shape of coefficients prior beta distribution
coef_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).
chol_spike_grid	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
chol_slab_shape	Inverse gamma shape for slab sd in the cholesky factor
chol_slab_scl	Inverse gamma scale for slab sd in the cholesky factor
chol_s1	First shape of cholesky factor prior beta distribution
chol_s2	Second shape of cholesky factor prior beta distribution
x	Any object
digits	digit option to print
...	not used

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.

Value

ssvsinput object

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.

bvhar documentation built on April 4, 2025, 5:22 a.m.