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Specify prior on PHI

Source: R/utility_functions.R
specify_prior_phi.Rd

Configures prior on PHI, the matrix of reduced-form VAR coefficients.

Usage

specify_prior_phi(
  data = NULL,
  M = ncol(data),
  lags = 1L,
  prior = "HS",
  priormean = 0,
  PHI_tol = 1e-18,
  DL_a = "1/K",
  DL_tol = 0,
  R2D2_a = 0.1,
  R2D2_b = 0.5,
  R2D2_tol = 0,
  NG_a = 0.1,
  NG_b = 1,
  NG_c = 1,
  NG_tol = 0,
  SSVS_c0 = 0.01,
  SSVS_c1 = 100,
  SSVS_semiautomatic = TRUE,
  SSVS_p = 0.5,
  HMP_lambda1 = c(0.01, 0.01),
  HMP_lambda2 = c(0.01, 0.01),
  normal_sds = 10,
  global_grouping = "global",
  ...
)

Arguments

data

Optional. Data matrix (can be a time series object). Each of \(M\) columns is assumed to contain a single time-series of length \(T\).

M

positive integer indicating the number of time-series of the VAR.

lags

positive integer indicating the order of the VAR, i.e. the number of lags of the dependent variables included as predictors.

prior

character, one of "HS", "R2D2", "NG", "DL", "SSVS", "HMP" or "normal".

priormean

real numbers indicating the prior means of the VAR coefficients. One single number means that the prior mean of all own-lag coefficients w.r.t. the first lag equals priormean and 0 else. A vector of length M means that the prior mean of the own-lag coefficients w.r.t. the first lag equals priormean and 0 else. If priormean is a matrix of dimension c(lags*M,M), then each of the \(M\) columns is assumed to contain lags*M prior means for the VAR coefficients of the respective VAR equations.

PHI_tol

Minimum number that the absolute value of a VAR coefficient draw can take. Prevents numerical issues that can appear when strong shrinkage is enforced if chosen to be greater than zero.

DL_a

(Single) positive real number. The value is interpreted as the concentration parameter for the local scales. Smaller values enforce heavier shrinkage. If the argument global_grouping specifies e.g. k groups, then DL_a can be a numeric vector of length k and the elements indicate the shrinkage in each group. A matrix of dimension c(s,2) specifies a discrete hyperprior, where the first column contains s support points and the second column contains the associated prior probabilities. DL_a has only to be specified if prior="DL".

DL_tol

Minimum number that a parameter draw of one of the shrinking parameters of the Dirichlet Laplace prior can take. Prevents numerical issues that can appear when strong shrinkage is enforced if chosen to be greater than zero. DL_tol has only to be specified if prior="DL".

R2D2_a

(Single) positive real number. The value is interpreted as the concentration parameter for the local scales. Smaller values enforce heavier shrinkage. If the argument global_grouping specifies e.g. k groups, then R2D2_a can be a numeric vector of length k and the elements indicate the shrinkage in each group. A matrix of dimension c(s,2) specifies a discrete hyperprior, where the first column contains s support points and the second column contains the associated prior probabilities. R2D2_a has only to be specified if prior="R2D2".

R2D2_b

(Single) positive real number. The value indicates the shape parameter of the inverse gamma prior on the (semi-)global scales. If the argument global_grouping specifies e.g. k groups, then NG_b can be a numeric vector of length k and the elements determine the shape parameter in each group. R2D2_b has only to be specified if prior="R2D2".

R2D2_tol

Minimum number that a parameter draw of one of the shrinking parameters of the R2D2 prior can take. Prevents numerical issues that can appear when strong shrinkage is enforced if chosen to be greater than zero. R2D2_tol has only to be specified if prior="R2D2".

NG_a

(Single) positive real number. The value is interpreted as the concentration parameter for the local scales. Smaller values enforce heavier shrinkage. If the argument global_grouping specifies e.g. k groups, then NG_a can be a numeric vector of length k and the elements indicate the shrinkage in each group. A matrix of dimension c(s,2) specifies a discrete hyperprior, where the first column contains s support points and the second column contains the associated prior probabilities. NG_a has only to be specified if prior="NG".

NG_b

(Single) positive real number. The value indicates the shape parameter of the inverse gamma prior on the (semi-)global scales. If the argument global_grouping specifies e.g. k groups, then NG_b can be a numeric vector of length k and the elements determine the shape parameter in each group. NG_b has only to be specified if prior="NG".

NG_c

(Single) positive real number. The value indicates the scale parameter of the inverse gamma prior on the (semi-)global scales. If the argument global_grouping specifies e.g. k groups, then NG_c can be a numeric vector of length k and the elements determine the scale parameter in each group. Expert option would be to set the scale parameter proportional to NG_a. E.g. in the case where a discrete hyperprior for NG_a is chosen, a desired proportion of let's say 0.2 is achieved by setting NG_c="0.2a" (character input!). NG_c has only to be specified if prior="NG".

NG_tol

Minimum number that a parameter draw of one of the shrinking parameters of the normal-gamma prior can take. Prevents numerical issues that can appear when strong shrinkage is enforced if chosen to be greater than zero. NG_tol has only to be specified if prior="NG".

SSVS_c0

single positive number indicating the (unscaled) standard deviation of the spike component. SSVS_c0 has only to be specified if prior="SSVS". It should be that \(SSVS_{c0} \ll SSVS_{c1}\)! SSVS_c0 has only to be specified if prior="SSVS".

SSVS_c1

single positive number indicating the (unscaled) standard deviation of the slab component. SSVS_c0 has only to be specified if prior="SSVS". It should be that \(SSVS_{c0} \ll SSVS_{c1}\)!

SSVS_semiautomatic

logical. If SSVS_semiautomatic=TRUE both SSVS_c0 and SSVS_c1 will be scaled by the variances of the posterior of PHI under a FLAT conjugate (dependent Normal-Wishart prior). SSVS_semiautomatic has only to be specified if prior="SSVS".

SSVS_p

Either a single positive number in the range (0,1) indicating the (fixed) prior inclusion probability of each coefficient. Or numeric vector of length 2 with positive entries indicating the shape parameters of the Beta distribution. In that case a Beta hyperprior is placed on the prior inclusion probability. SSVS_p has only to be specified if prior="SSVS".

HMP_lambda1

numeric vector of length 2. Both entries must be positive. The first indicates the shape and the second the rate of the Gamma hyperprior on own-lag coefficients. HMP_lambda1 has only to be specified if prior="HMP".

HMP_lambda2

numeric vector of length 2. Both entries must be positive. The first indicates the shape and the second the rate of the Gamma hyperprior on cross-lag coefficients. HMP_lambda2 has only to be specified if prior="HMP".

normal_sds

numeric vector of length \(n\), where \(n = lags M^2\) is the number of all VAR coefficients (excluding the intercept), indicating the prior variances. A single number will be recycled accordingly! Must be positive. normal_sds has only to be specified if prior="normal".

global_grouping

One of "global", "equation-wise", "covariate-wise", "olcl-lagwise" "fol" indicating the sub-groups of the semi-global(-local) modifications to HS, R2D2, NG, DL and SSVS prior. Works also with user-specified indicator matrix of dimension c(lags*M,M). Only relevant if prior="HS", prior="DL", prior="R2D2", prior="NG" or prior="SSVS".

...

Do not use!

Value

A baysianVARs_prior_phi-object.

Details

For details concerning prior-elicitation for VARs please see Gruber & Kastner (2023).

Currently one can choose between six hierarchical shrinkage priors and a normal prior: prior="HS" stands for the Horseshoe-prior, prior="R2D2 for the R\(^2\)-induced-Dirichlet-decompostion-prior, prior="NG" for the normal-gamma-prior, prior="DL" for the Dirichlet-Laplace-prior, prior="SSVS" for the stochastic-search-variable-selection-prior, prior="HMP" for the semi-hierarchical Minnesota prior and prior=normal for the normal-prior.

Semi-global shrinkage, i.e. group-specific shrinkage for pre-specified subgroups of the coefficients, can be achieved through the argument global_grouping.

References

Gruber, L. and Kastner, G. (2023). Forecasting macroeconomic data with Bayesian VARs: Sparse or dense? It depends! arXiv:2206.04902.

Examples

# Access a subset of the usmacro_growth dataset
data <- usmacro_growth[,c("GDPC1", "CPIAUCSL", "FEDFUNDS")]

# Horseshoe prior for a VAR(2)
phi_hs <- specify_prior_phi(data = data, lags = 2L ,prior = "HS")

# Semi-global-local Horseshoe prior for a VAR(2) with semi-global shrinkage parameters for
# cross-lag and own-lag coefficients in each lag
phi_hs_sg <- specify_prior_phi(data = data, lags = 2L, prior = "HS",
global_grouping = "olcl-lagwise")

# Semi-global-local Horseshoe prior for a VAR(2) with equation-wise shrinkage
# construct indicator matrix for equation-wise shrinkage
semi_global_mat <- matrix(1:ncol(data), 2*ncol(data), ncol(data),
byrow = TRUE)
phi_hs_ew <- specify_prior_phi(data = data, lags = 2L, prior = "HS",
global_grouping = semi_global_mat)
# (for equation-wise shrinkage one can also use 'global_grouping = "equation-wise"')

# \donttest{
# Estimate model with your prior configuration of choice
mod <- bvar(data, lags = 2L, prior_phi = phi_hs_sg, quiet = TRUE)
# }