Specify prior on Sigma

Configures prior on the variance-covariance of the VAR.

Usage

specify_prior_sigma(
  data = NULL,
  M = ncol(data),
  type = c("factor", "cholesky"),
  factor_factors = 1L,
  factor_restrict = c("none", "upper"),
  factor_priorfacloadtype = c("rowwiseng", "colwiseng", "normal"),
  factor_priorfacload = 0.1,
  factor_facloadtol = 1e-18,
  factor_priorng = c(1, 1),
  factor_priormu = c(0, 10),
  factor_priorphiidi = c(10, 3),
  factor_priorphifac = c(10, 3),
  factor_priorsigmaidi = 1,
  factor_priorsigmafac = 1,
  factor_priorh0idi = "stationary",
  factor_priorh0fac = "stationary",
  factor_heteroskedastic = TRUE,
  factor_priorhomoskedastic = NA,
  factor_interweaving = 4,
  cholesky_U_prior = c("HS", "DL", "R2D2", "NG", "SSVS", "normal", "HMP"),
  cholesky_U_tol = 1e-18,
  cholesky_heteroscedastic = TRUE,
  cholesky_priormu = c(0, 100),
  cholesky_priorphi = c(20, 1.5),
  cholesky_priorsigma2 = c(0.5, 0.5),
  cholesky_priorh0 = "stationary",
  cholesky_priorhomoscedastic = as.numeric(NA),
  cholesky_DL_a = "1/n",
  cholesky_DL_tol = 0,
  cholesky_R2D2_a = 0.4,
  cholesky_R2D2_b = 0.5,
  cholesky_R2D2_tol = 0,
  cholesky_NG_a = 0.5,
  cholesky_NG_b = 0.5,
  cholesky_NG_c = 0.5,
  cholesky_NG_tol = 0,
  cholesky_SSVS_c0 = 0.001,
  cholesky_SSVS_c1 = 1,
  cholesky_SSVS_p = 0.5,
  cholesky_HMP_lambda3 = c(0.01, 0.01),
  cholesky_normal_sds = 10,
  expert_sv_offset = 0,
  quiet = FALSE,
  ...
)

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.

type

character, one of "factor" (the default) or "cholesky", indicating which decomposition to be applied to the covariance-matrix.

factor_factors

Number of latent factors to be estimated. Only required if type="factor".

factor_restrict

Either "upper" or "none", indicating whether the factor loadings matrix should be restricted to have zeros above the diagonal ("upper") or whether all elements should be estimated from the data ("none"). Setting restrict to "upper" often stabilizes MCMC estimation and can be important for identifying the factor loadings matrix, however, it generally is a strong prior assumption. Setting restrict to "none" is usually the preferred option if identification of the factor loadings matrix is of less concern but covariance estimation or prediction is the goal. Only required if type="factor".

factor_priorfacloadtype

Can be "normal", "rowwiseng", "colwiseng". Only required if type="factor".

"normal":

Normal prior. The value of priorfacload is interpreted as the standard deviations of the Gaussian prior distributions for the factor loadings.

"rowwiseng":

Row-wise Normal-Gamma prior. The value of priorfacload is interpreted as the shrinkage parameter a.

"colwiseng":

Column-wise Normal-Gamma prior. The value of priorfacload is interpreted as the shrinkage parameter a.

For details please see Kastner (2019).

factor_priorfacload

Either a matrix of dimensions M times factor_factors with positive elements or a single number (which will be recycled accordingly). Only required if type="factor". The meaning of factor_priorfacload depends on the setting of factor_priorfacloadtype and is explained there.

factor_facloadtol

Minimum number that the absolute value of a factor loadings draw can take. Prevents numerical issues that can appear when strong shrinkage is enforced if chosen to be greater than zero. Only required if type="factor".

factor_priorng

Two-element vector with positive entries indicating the Normal-Gamma prior's hyperhyperparameters c and d (cf. Kastner (2019)). Only required if type="factor".

factor_priormu

Vector of length 2 denoting prior mean and standard deviation for unconditional levels of the idiosyncratic log variance processes. Only required if type="factor".

factor_priorphiidi

Vector of length 2, indicating the shape parameters for the Beta prior distributions of the transformed parameters (phi+1)/2, where phi denotes the persistence of the idiosyncratic log variances. Only required if type="factor".

factor_priorphifac

Vector of length 2, indicating the shape parameters for the Beta prior distributions of the transformed parameters (phi+1)/2, where phi denotes the persistence of the factor log variances. Only required if type="factor".

factor_priorsigmaidi

Vector of length M containing the prior volatilities of log variances. If factor_priorsigmaidi has exactly one element, it will be recycled for all idiosyncratic log variances. Only required if type="factor".

factor_priorsigmafac

Vector of length factor_factors containing the prior volatilities of log variances. If factor_priorsigmafac has exactly one element, it will be recycled for all factor log variances. Only required if type="factor".

factor_priorh0idi

Vector of length 1 or M, containing information about the Gaussian prior for the initial idiosyncratic log variances. Only required if type="factor". If an element of factor_priorh0idi is a nonnegative number, the conditional prior of the corresponding initial log variance h0 is assumed to be Gaussian with mean 0 and standard deviation factor_priorh0idi times \(sigma\). If an element of factor_priorh0idi is the string 'stationary', the prior of the corresponding initial log volatility is taken to be from the stationary distribution, i.e. h0 is assumed to be Gaussian with mean 0 and variance \(sigma^2/(1-phi^2)\).

factor_priorh0fac

Vector of length 1 or factor_factors, containing information about the Gaussian prior for the initial factor log variances. Only required if type="factor". If an element of factor_priorh0fac is a nonnegative number, the conditional prior of the corresponding initial log variance h0 is assumed to be Gaussian with mean 0 and standard deviation factor_priorh0fac times \(sigma\). If an element of factor_priorh0fac is the string 'stationary', the prior of the corresponding initial log volatility is taken to be from the stationary distribution, i.e. h0 is assumed to be Gaussian with mean 0 and variance \(sigma^2/(1-phi^2)\).

factor_heteroskedastic

Vector of length 1, 2, or M + factor_factors, containing logical values indicating whether time-varying (factor_heteroskedastic = TRUE) or constant (factor_heteroskedastic = FALSE) variance should be estimated. If factor_heteroskedastic is of length 2 it will be recycled accordingly, whereby the first element is used for all idiosyncratic variances and the second element is used for all factor variances. Only required if type="factor".

factor_priorhomoskedastic

Only used if at least one element of factor_heteroskedastic is set to FALSE. In that case, factor_priorhomoskedastic must be a matrix with positive entries and dimension c(M, 2). Values in column 1 will be interpreted as the shape and values in column 2 will be interpreted as the rate parameter of the corresponding inverse gamma prior distribution of the idiosyncratic variances. Only required if type="factor".

factor_interweaving

The following values for interweaving the factor loadings are accepted (Only required if type="factor"):

0:

No interweaving.

1:

Shallow interweaving through the diagonal entries.

2:

Deep interweaving through the diagonal entries.

3:

Shallow interweaving through the largest absolute entries in each column.

4:

Deep interweaving through the largest absolute entries in each column.

For details please see Kastner et al. (2017). A value of 4 is the highly recommended default.

cholesky_U_prior

character, one of "HS", "R2D2", "NG", "DL", "SSVS", "HMP" or "normal". Only required if type="cholesky".

cholesky_U_tol

Minimum number that the absolute value of an free off-diagonal element of an \(U\)-draw can take. Prevents numerical issues that can appear when strong shrinkage is enforced if chosen to be greater than zero. Only required if type="cholesky".

cholesky_heteroscedastic

single logical indicating whether time-varying (cholesky_heteroscedastic = TRUE) or constant (cholesky_heteroscedastic = FALSE) variance should be estimated. Only required if type="cholesky".

cholesky_priormu

Vector of length 2 denoting prior mean and standard deviation for unconditional levels of the log variance processes. Only required if type="cholesky".

cholesky_priorphi

Vector of length 2, indicating the shape parameters for the Beta prior distributions of the transformed parameters (phi+1)/2, where phi denotes the persistence of the log variances. Only required if type="cholesky".

cholesky_priorsigma2

Vector of length 2, indicating the shape and the rate for the Gamma prior distributions on the variance of the log variance processes. (Currently only one global setting for all \(M\) processes is supported). Only required if type="cholesky".

cholesky_priorh0

Vector of length 1 or M, containing information about the Gaussian prior for the initial idiosyncratic log variances. Only required if type="cholesky". If an element of cholesky_priorh0 is a nonnegative number, the conditional prior of the corresponding initial log variance h0 is assumed to be Gaussian with mean 0 and standard deviation cholesky_priorh0 times \(sigma\). If an element of cholesky_priorh0 is the string 'stationary', the prior of the corresponding initial log volatility is taken to be from the stationary distribution, i.e. h0 is assumed to be Gaussian with mean 0 and variance \(sigma^2/(1-phi^2)\).

cholesky_priorhomoscedastic

Only used if cholesky_heteroscedastic=FALSE. In that case, cholesky_priorhomoscedastic must be a matrix with positive entries and dimension c(M, 2). Values in column 1 will be interpreted as the shape and values in column 2 will be interpreted as the scale parameter of the corresponding inverse gamma prior distribution of the variances. Only required if type="cholesky".

cholesky_DL_a

(Single) positive real number. The value is interpreted as the concentration parameter for the local scales. Smaller values enforce heavier shrinkage. 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. cholesky_DL_a has only to be specified if cholesky_U_prior="DL".

cholesky_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 cholesky_U_prior="DL".

cholesky_R2D2_a

(Single) positive real number. The value is interpreted as the concentration parameter for the local scales. Smaller values enforce heavier shrinkage. 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. cholesky_R2D2_a has only to be specified if cholesky_U_prior="R2D2".

cholesky_R2D2_b

single positive number, where greater values indicate heavier regularization. cholesky_R2D2_b has only to be specified if cholesky_U_prior="R2D2".

cholesky_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. cholesky_R2D2_tol has only to be specified if cholesky_U_prior="R2D2".

cholesky_NG_a

(Single) positive real number. The value is interpreted as the concentration parameter for the local scales. Smaller values enforce heavier shrinkage. 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. cholesky_NG_a has only to be specified if cholesky_U_prior="NG".

cholesky_NG_b

(Single) positive real number. The value indicates the shape parameter of the inverse gamma prior on the global scales. cholesky_NG_b has only to be specified if cholesky_U_prior="NG".

cholesky_NG_c

(Single) positive real number. The value indicates the scale parameter of the inverse gamma prior on the global scales. 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!). cholesky_NG_c has only to be specified if cholesky_U_prior="NG".

cholesky_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. cholesky_NG_tol has only to be specified if cholesky_U_prior="NG".

cholesky_SSVS_c0

single positive number indicating the (unscaled) standard deviation of the spike component. cholesky_SSVS_c0 has only to be specified if choleksy_U_prior="SSVS". It should be that \(SSVS_{c0} \ll SSVS_{c1}\)!

cholesky_SSVS_c1

single positive number indicating the (unscaled) standard deviation of the slab component. cholesky_SSVS_c1 has only to be specified if choleksy_U_prior="SSVS". It should be that \(SSVS_{c0} \ll SSVS_{c1}\)!

cholesky_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. cholesky_SSVS_p has only to be specified if choleksy_U_prior="SSVS".

cholesky_HMP_lambda3

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 the contemporaneous coefficients. cholesky_HMP_lambda3 has only to be specified if choleksy_U_prior="HMP".

cholesky_normal_sds

numeric vector of length \(\frac{M^2-M}{2}\), indicating the prior variances for the free off-diagonal elements in \(U\). A single number will be recycled accordingly! Must be positive. cholesky_normal_sds has only to be specified if choleksy_U_prior="normal".

expert_sv_offset

... Do not use!

quiet

logical indicating whether informative output should be omitted.

...

Do not use!

Value

Object of class bayesianVARs_prior_sigma.

Details

bvar offers two different specifications for the errors: The user can choose between a factor stochastic volatility structure or a cholesky stochastic volatility structure. In both cases the disturbances \(\boldsymbol{\epsilon}_t\) are assumed to follow a \(M\)-dimensional multivariate normal distribution with zero mean and variance-covariance matrix \(\boldsymbol{\Sigma}_t\). In case of the cholesky specification \(\boldsymbol{\Sigma}_t = \boldsymbol{U}^{\prime -1} \boldsymbol{D}_t \boldsymbol{U}^{-1}\), where \(\boldsymbol{U}^{-1}\) is upper unitriangular (with ones on the diagonal). The diagonal matrix \(\boldsymbol{D}_t\) depends upon latent log-variances, i.e. \(\boldsymbol{D}_t=diag(exp(h_{1t}),\dots, exp(h_{Mt})\). The log-variances follow a priori independent autoregressive processes \(h_{it}\sim N(\mu_i + \phi_i(h_{i,t-1}-\mu_i),\sigma_i^2)\) for \(i=1,\dots,M\). In case of the factor structure, \(\boldsymbol{\Sigma}_t = \boldsymbol{\Lambda} \boldsymbol{V}_t \boldsymbol{\Lambda}^\prime + \boldsymbol{G}_t\). The diagonal matrices \(\boldsymbol{V}_t\) and \(\boldsymbol{G}_t\) depend upon latent log-variances, i.e. \(\boldsymbol{G}_t=diag(exp(h_{1t}),\dots, exp(h_{Mt})\) and \(\boldsymbol{V}_t=diag(exp(h_{M+1,t}),\dots, exp(h_{M+r,t})\). The log-variances follow a priori independent autoregressive processes \(h_{it}\sim N(\mu_i + \phi_i(h_{i,t-1}-\mu_i),\sigma_i^2)\) for \(i=1,\dots,M\) and \(h_{M+j,t}\sim N(\phi_ih_{M+j,t-1},\sigma_{M+j}^2)\) for \(j=1,\dots,r\).

References

Kastner, G. (2019). Sparse Bayesian Time-Varying Covariance Estimation in Many Dimensions Journal of Econometrics, 210(1), 98–115, doi:10.1016/j.jeconom.2018.11.007

Kastner, G., Frühwirth-Schnatter, S., and Lopes, H.F. (2017). Efficient Bayesian Inference for Multivariate Factor Stochastic Volatility Models. Journal of Computational and Graphical Statistics, 26(4), 905–917, doi:10.1080/10618600.2017.1322091 .

See also

specify_prior_phi().

Examples

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

# examples with stochastic volatility (heteroscedasticity) -----------------
# factor-decomposition with 2 factors and colwise normal-gamma prior on the loadings
sigma_factor_cng_sv <- specify_prior_sigma(data = data, type = "factor",
factor_factors = 2L, factor_priorfacloadtype = "colwiseng", factor_heteroskedastic = TRUE)
#> 
#> Since argument 'type' is specified with 'factor', all arguments starting with 'cholesky_' are being ignored.

# cholesky-decomposition with Dirichlet-Laplace prior on U
sigma_cholesky_dl_sv <- specify_prior_sigma(data = data, type = "cholesky",
cholesky_U_prior = "DL", cholesky_DL_a = 0.5, cholesky_heteroscedastic = TRUE)
#> 
#> Since argument 'type' is specified with 'cholesky', all arguments starting with 'factor_' are being ignored.

# examples without stochastic volatility (homoscedasticity) ----------------
# factor-decomposition with 2 factors and colwise normal-gamma prior on the loadings
sigma_factor_cng <- specify_prior_sigma(data = data, type = "factor",
factor_factors = 2L, factor_priorfacloadtype = "colwiseng",
factor_heteroskedastic = FALSE, factor_priorhomoskedastic = matrix(c(0.5,0.5),
ncol(data), 2))
#> 
#> Since argument 'type' is specified with 'factor', all arguments starting with 'cholesky_' are being ignored.
#> 
#> Cannot do deep factor_interweaving if (some) factor_factors are homoskedastic. Setting 'factor_interweaving' to 3.

# cholesky-decomposition with Horseshoe prior on U
sigma_cholesky_dl <- specify_prior_sigma(data = data, type = "cholesky",
cholesky_U_prior = "HS", cholesky_heteroscedastic = FALSE)
#> 
#> Since argument 'type' is specified with 'cholesky', all arguments starting with 'factor_' are being ignored.
#> 
#> Argument 'cholesky_priorhomoscedastic' not specified. Setting both shape and rate of inverse gamma prior equal to 0.01.

# \donttest{
# Estimate model with your prior configuration of choice
mod <- bvar(data, prior_sigma = sigma_factor_cng_sv, quiet = TRUE)
# }