Bayesian estimation of a Bayesian Hierarchical Panel Vector Autoregression using Gibbs sampler

Estimates the Bayesian Hierarchical Panel VAR using the Gibbs sampler proposed by Sanchez-Martinez & Woźniak (2024).

Usage

# S3 method for class 'PosteriorBVARPANEL'
estimate(specification, S, thin = 1, show_progress = TRUE)

Arguments

specification: an object of class PosteriorBVARPANEL generated using the estimate.BVARPANEL() function. This setup facilitates the continuation of the MCMC sampling starting from the last draw of the previous run.
S: a positive integer, the number of posterior draws to be generated
thin: a positive integer, specifying the frequency of MCMC output thinning
show_progress: a logical value, if TRUE the estimation progress bar is visible

Value

An object of class PosteriorBVARPANEL containing the Bayesian estimation output and containing two elements:

posterior: a list with a collection of S draws from the posterior distribution generated via Gibbs sampler. Elements of the list correspond to the parameters of the model listed in section Details and are named respectively: A_c, Sigma_c, A, Sigma, V, nu, m, w, s.
last_draw: an object of class BVARPANEL with the last draw of the current MCMC run as the starting value to be passed to the continuation of the MCMC estimation using the estimate() method.

Details

The Bayesian Hierarchical Panel Vector Autoregressive model described in bvarPANELs is estimated using the Gibbs sampler. In this estimation procedure all the parameters of the model are estimated jointly. The list of parameters of the model includes:

\(\mathbf{A}_c\): a KxN country-specific autoregressive parameters matrix for each of the countries \(c = 1,\dots,C\)
\(\mathbf{\Sigma}_c\): an NxN country-specific covariance matrix for each of the countries \(c = 1,\dots,C\)
\(\mathbf{A}\): a KxN global autoregressive parameters matrix
\(\mathbf{\Sigma}\): an NxN global covariance matrix
\(\mathbf{V}\): a KxK covariance matrix of prior for global autoregressive parameters
\(\nu\): prior degrees of freedom parameter
\(m\): prior average global persistence parameter
\(w\): prior scaling parameter
\(s\): prior scaling parameter

Gibbs sampler is an algorithm to sample random draws from the posterior distribution of the parameters of the model given the data. The algorithm is briefly explained on an example of a two-parameter model with parameters \(\theta_1\) and \(\theta_2\). In order to sample from the joint posterior distribution \(p(\theta_1,\theta_2|\mathbf{Y})\) the Gibbs sampler proceeds by sampling from full-conditional posterior distributions of each parameter given data and all the other parameters, denoted by \(p(\theta_1|\theta_2,\mathbf{Y})\) and \(p(\theta_2|\theta_1,\mathbf{Y})\). These distributions are available from derivations and should be in a form of distributions that are easy to sample random numbers from.

To obtain S draws from the posterior distribution:

Set the initial values of the parameters \(\theta_2^{(0)}\)
At each of the s iterations:
1. Sample \(\theta_1^{(s)}\) from \(p(\theta_1|\theta_2^{(s-1)},\mathbf{Y})\)
2. Sample \(\theta_2^{(s)}\) from \(p(\theta_2|\theta_1^{(s)},\mathbf{Y})\)
Repeat step 2. S times. Return \(\{\theta_1^{(s)},\theta_2^{(s)}\}_{s=1}^{S}\) as a sample drawn from the posterior distribution \(p(\theta_1,\theta_2|\mathbf{Y})\).

The estimate() function returns the draws from the posterior distribution of the parameters of the hierarchical panel VAR model listed above.

Thinning. Thinning is a procedure to reduce the dependence in the returned sample from the posterior distribution. It is obtained by returning every thin draw in the final sample. This procedure reduces the number of draws returned by the estimate() function.

Author

Tomasz Woźniak wozniak.tom@pm.me

Examples

data(ilo_dynamic_panel)                                 # load the data
data(ilo_exogenous_variables)                           # load the exogenous variables
set.seed(123)
# specify the model
specification = specify_bvarPANEL$new(ilo_dynamic_panel, exogenous = ilo_exogenous_variables)
burn_in       = estimate(specification, 10)             # run the burn-in; use say S = 10000
#> **************************************************|
#> bvarPANELs: Forecasting with Bayesian Hierarchical|
#>             Panel Vector Autoregressions          |
#> **************************************************|
#>  Progress of the MCMC simulation for 10 draws
#>     Every draw is saved via MCMC thinning
#>  Press Esc to interrupt the computations
#> **************************************************|
posterior     = estimate(burn_in, 10)                   # estimate the model; use say S = 10000
#> **************************************************|
#> bvarPANELs: Forecasting with Bayesian Hierarchical|
#>             Panel Vector Autoregressions          |
#> **************************************************|
#>  Progress of the MCMC simulation for 10 draws
#>     Every draw is saved via MCMC thinning
#>  Press Esc to interrupt the computations
#> **************************************************|