estimate.PosteriorBSVART {bsvars}

Bayesian estimation of a homoskedastic Structural Vector Autoregression with t-distributed structural shocks via Gibbs sampler

Description

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 'PosteriorBSVART'
estimate(specification, S, thin = 1, show_progress = TRUE)

Arguments

specification

an object of class PosteriorBSVART generated using the estimate.BSVART() 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

Details

The homoskedastic SVAR model with t-distributed structural shocks 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 Student-t distributed with zero mean, unit variances, and an estimated degrees-of-freedom parameter.

Value

An object of class PosteriorBSVART 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

df

an S vector with the posterior draws for the degrees-of-freedom parameter of the Student-t distribution

lambda

a TxS matrix with the posterior draws for the latent variable

last_draw an object of class BSVART 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().

Author(s)

Tomasz Woźniak [e-mail]

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.

See Also

specify_bsvar_t, specify_posterior_bsvar_t, normalise_posterior

Examples

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

# specify the model and set seed
specification  = specify_bsvar_t$new(us_fiscal_lsuw, p = 1)
set.seed(123)

# run the burn-in
burn_in        = estimate(specification, 5)

# estimate the model
posterior      = estimate(burn_in, 10, thin = 2)

# workflow with the pipe |>
############################################################
set.seed(123)
us_fiscal_lsuw |>
  specify_bsvar_t$new(p = 1) |>
  estimate(S = 5) |> 
  estimate(S = 10, thin = 2) -> posterior