set_ssvs {bvhar}

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 = 0.1,
  coef_slab = 5,
  coef_mixture = 0.5,
  coef_s1 = 1,
  coef_s2 = 1,
  mean_non = 0,
  sd_non = 0.1,
  shape = 0.01,
  rate = 0.01,
  chol_spike = 0.1,
  chol_slab = 5,
  chol_mixture = 0.5,
  chol_s1 = 1,
  chol_s2 = 1
)

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

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

Arguments

coef_spike

Standard deviance for Spike normal distribution (See Details).

coef_slab

Standard deviance for Slab normal distribution (See Details).

coef_mixture

Bernoulli parameter for sparsity proportion (See Details).

coef_s1

First shape of coefficients prior beta distribution

coef_s2

Second shape of coefficients prior beta distribution

mean_non

Prior mean of unrestricted coefficients

sd_non

Standard deviance for unrestricted coefficients

shape

Gamma shape parameters for precision matrix (See Details).

rate

Gamma rate parameters for precision matrix (See Details).

chol_spike

Standard deviance for Spike normal distribution, in the cholesky factor (See Details).

chol_slab

Standard deviance for Slab normal distribution, in the cholesky factor (See Details).

chol_mixture

Bernoulli parameter for sparsity proportion, in the cholesky factor (See Details).

chol_s1

First shape of cholesky factor prior beta distribution

chol_s2

Second shape of cholesky factor prior beta distribution

x

ssvsinput

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.

Koop, G., & Korobilis, D. (2009). Bayesian Multivariate Time Series Methods for Empirical Macroeconomics. Foundations and Trends® in Econometrics, 3(4), 267–358.