Bayesian estimation of a homoskedastic Structural Vector Autoregression via Gibbs sampler

Estimates the homoskedastic SVAR using the Gibbs sampler proposed by Waggoner & Zha (2003) for the structural matrix $B$ and the equation-by-equation sampler by Chan, Koop, & Yu (2024) for the autoregressive slope parameters $A$. Additionally, the parameter matrices $A$ and $B$ follow a Minnesota prior and generalised-normal prior distributions respectively with the matrix-specific overall shrinkage parameters estimated using a hierarchical prior distribution as in Lütkepohl, Shang, Uzeda, and Woźniak (2024). See section Details for the model equations.

Usage

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

Arguments

specification: an object of class PosteriorBSVAR generated using the estimate.BSVAR() 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 PosteriorBSVAR 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 containing:

A: an NxKxS array with the posterior draws for matrix $A$
B: an NxNxS array with the posterior draws for matrix $B$
hyper: a 5xS matrix with the posterior draws for the hyper-parameters of the hierarchical prior distribution

last_draw an object of class BSVAR with the last draw of the current MCMC run as the starting value to be passed to the continuation of the MCMC estimation using estimate().

Details

The homoskedastic SVAR model is given by the reduced form equation: $$Y = AX + E$$ where $Y$ is an NxT matrix of dependent variables, $X$ is a KxT matrix of explanatory variables, $E$ is an NxT matrix of reduced form error terms, and $A$ is an NxK matrix of autoregressive slope coefficients and parameters on deterministic terms in $X$.

The structural equation is given by $$BE = U$$ where $U$ is an NxT matrix of structural form error terms, and $B$ is an NxN matrix of contemporaneous relationships.

Finally, the structural shocks, U, are temporally and contemporaneously independent and jointly normally distributed with zero mean and unit variances.

References

Chan, J.C.C., Koop, G, and Yu, X. (2024) Large Order-Invariant Bayesian VARs with Stochastic Volatility. Journal of Business & Economic Statistics, 42, doi:10.1080/07350015.2023.2252039 .

Lütkepohl, H., Shang, F., Uzeda, L., and Woźniak, T. (2024) Partial Identification of Heteroskedastic Structural VARs: Theory and Bayesian Inference. University of Melbourne Working Paper, 1–57, doi:10.48550/arXiv.2404.11057 .

Waggoner, D.F., and Zha, T., (2003) A Gibbs sampler for structural vector autoregressions. Journal of Economic Dynamics and Control, 28, 349–366, doi:10.1016/S0165-1889(02)00168-9 .

Author

Tomasz Woźniak wozniak.tom@pm.me

Examples

# simple workflow
############################################################
# upload data
data(us_fiscal_lsuw)

# specify the model and set seed
specification  = specify_bsvar$new(us_fiscal_lsuw, p = 1)
#> The identification is set to the default option of lower-triangular structural matrix.
set.seed(123)

# run the burn-in
burn_in        = estimate(specification, 5)
#> **************************************************|
#> bsvars: Bayesian Structural Vector Autoregressions|
#> **************************************************|
#>  Gibbs sampler for the SVAR model                 |
#> **************************************************|
#>  Progress of the MCMC simulation for 5 draws
#>     Every draw is saved via MCMC thinning
#>  Press Esc to interrupt the computations
#> **************************************************|

# estimate the model
posterior      = estimate(burn_in, 10, thin = 2)
#> **************************************************|
#> bsvars: Bayesian Structural Vector Autoregressions|
#> **************************************************|
#>  Gibbs sampler for the SVAR model                 |
#> **************************************************|
#>  Progress of the MCMC simulation for 10 draws
#>     Every 2nd draw is saved via MCMC thinning
#>  Press Esc to interrupt the computations
#> **************************************************|

# workflow with the pipe |>
############################################################
set.seed(123)
us_fiscal_lsuw |>
  specify_bsvar$new(p = 1) |>
  estimate(S = 5) |> 
  estimate(S = 10, thin = 2) |> 
  compute_impulse_responses(horizon = 4) -> irf
#> The identification is set to the default option of lower-triangular structural matrix.
#> **************************************************|
#> bsvars: Bayesian Structural Vector Autoregressions|
#> **************************************************|
#>  Gibbs sampler for the SVAR model                 |
#> **************************************************|
#>  Progress of the MCMC simulation for 5 draws
#>     Every draw is saved via MCMC thinning
#>  Press Esc to interrupt the computations
#> **************************************************|
#> **************************************************|
#> bsvars: Bayesian Structural Vector Autoregressions|
#> **************************************************|
#>  Gibbs sampler for the SVAR model                 |
#> **************************************************|
#>  Progress of the MCMC simulation for 10 draws
#>     Every 2nd draw is saved via MCMC thinning
#>  Press Esc to interrupt the computations
#> **************************************************|