estimate.BVARs: Bayesian estimation of a Bayesian Hierarchical Vector Autoregressions for cubic data using Gibbs sampler

Description

Estimates the Bayesian Hierarchical VARs for cubic data using the Gibbs sampler proposed by Sanchez-Martinez & Woźniak (2024).

Usage

# S3 method for BVARs
estimate(specification, S, thin = 1L, show_progress = TRUE)

Value

An object of class PosteriorBVARs 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, nu, m, w, s.

last_draw

an object of class BVARs 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.

Arguments

specification: an object of class BVARs generated using the specify_bvars$new() function.
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

Author

Tomasz Woźniak wozniak.tom@pm.me

Details

The Bayesian Hierarchical Vector Autoregressive models described in bpvars 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 $c = 1,\dots,C$
$\nu$: 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 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.

Examples

Run this code

# specify the model
specification = specify_bvars$new(
  ilo_dynamic_panel[1:5], 
  exogenous = ilo_exogenous_variables[1:5]
)
burn_in       = estimate(specification, 10)             # run the burn-in; use say S = 10000
posterior     = estimate(burn_in, 10)                   # estimate the model; use say S = 10000

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