Samples random draws from a generalised normal distribution using the Gibbs sampler by Waggoner & Zha (2003a)

Samples random numbers from a generalised normal distribution for a restricted $N\times N$ full rank matrix $X$. The matrix is drawn row-by-row from their full conditional distributions using the Gibbs sampler by Waggoner & Zha (2003a). The density is proportional to $$det(X)^{\nu - N}\exp(-\frac{1}{2}\sum_{n=1}^{N}x_nV_nS_nV_n'x_n')$$ specified by the $N\times N$ scale matrices $S_n$, a positive scalar-valued shape parameter $\nu$, and $r_n\times N$ selection matrices $V_n$ defining the restrictions. Zero restrictions may be imposed on th $X$ matrix row-by-row as $$X_{n.} = x_nV_n$$ where $1\times r_n$ vectors $x_n$ collect the unrestricted elements of $X_{n.}$ and the matrices $V_n$ place them in appropriate spots of the $1\times N$ row $X_{n.}$. The output may be normalised using the method by Waggoner & Zha (2003b) ensuring positive diagonal elements of the output matrices.

This method is useful for sampling the structural matrix of the structural vector autoregressive models identified by exclusion restrictions, sign and exclusion restrictions, heteroskedasticity, and instrumental variables.

Usage

rgennorm(n, X, S, nu, V, normalise = TRUE)

Arguments

n: a positive integer with the number of draws to be sampled. C++: a int object.
X: an $N\times N$ structural matrix with the starting values to initiate the Gibbs sampler. C++: an arma::mat matrix object.
S: an $N\times N\times N$ array with $N\times N$ row-specific precision matrices $S_n$. C++: an arma::cube object.
nu: a positive integer number with the shape parameter $nu$. C++: an arma::int object.
V: a list with $N$ elements each including $r_n\times N$ matrices $V_n$. C++: an arma::field<arma::mat> object.
normalise: a logical value of whether the output should be normalised following the method by Waggoner & Zha (2003b) ensuring positive signs of the diagonal elements of the sampled matrices. C++: an bool object.

Value

an $N\times N\times n$-array with random draws from the generalised normal distribution. C++: an arma::cube object.

Details

This function is based on C++ code from the R package bsvars by Woźniak (2024,2025) and is using objects and commands from the armadillo library by Sanderson & Curtin (2025) thanks to the RcppArmadillo package by Eddelbuettel, Francois, Bates, Ni, & Sanderson (2025).

References

Eddelbuettel D., Francois R., Bates D., Ni B., Sanderson C. (2025). RcppArmadillo: 'Rcpp' Integration for the 'Armadillo' Templated Linear Algebra Library. R package version 15.0.2-2. <doi:10.32614/CRAN.package.RcppArmadillo>

Sanderson C., Curtin R. (2025). Armadillo: An Efficient Framework for Numerical Linear Algebra. International Conference on Computer and Automation Engineering, 303-307, <doi:10.1109/ICCAE64891.2025.10980539>

Waggoner D.F., Zha T., (2003a). A Gibbs Sampler for Structural Vector Autoregressions. Journal of Economic Dynamics and Control, 28(2), 349-366, <doi:10.1016/S0165-1889(02)00168-9>.

Waggoner, D.F., Zha, T. (2003b). Likelihood Preserving Normalization in Multiple Equation Models. Journal of Econometrics, 114(2), 329-347. <doi:10.1016/S0304-4076(03)00087-3>

Woźniak T. (2024). bsvars: Bayesian Estimation of Structural Vector Autoregressive Models, R package version 3.2, <doi:10.32614/CRAN.package.bsvars>.

Woźniak T. (2025). Fast and Efficient Bayesian Analysis of Structural Vector Autoregressions Using the R Package bsvars. University of Melbourne Working Paper, 1–25. <doi:10.48550/arXiv.2410.15090>.

Author

Tomasz Woźniak wozniak.tom@pm.me

Examples

rgennorm(1, diag(2), array(diag(2), c(2,2,2)), 3, list(diag(2), matrix(1,1,2)))
#> , , 1
#> 
#>           [,1]       [,2]
#> [1,] 1.1593962 -3.2777784
#> [2,] 0.9654515  0.9654515
#>