estimate.BSVARSIGN {bsvarSIGNs}

Bayesian estimation of a Structural Vector Autoregression with traditional and narrative sign restrictions via Gibbs sampler

Description

Estimates Bayesian Structural Vector Autoregression model using the Gibbs sampler proposed by Waggoner & Zha (2003) with traditional sign restrictions following Rubio-Ramírez, Waggoner & Zha (2010) and narrative sign restrictions following Antolín-Díaz & Rubio-Ramírez (2018). 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.

Given sign restrictions, in each Gibbs sampler iteration, the sampler draws rotation matrix Q uniformly from the space of NxN orthogonal matrices and checks if the sign restrictions are satisfied. If a valid Q is found within max_tries (defined in specify_bsvarSIGN), the sampler saves the current A and B draw and proceeds to the next iteration. Otherwise, the sampler then proceeds to next iteration without saving the current A and B draw. If a narrative sign restriction is given, the posterior draws are resampled with algorithm 1 in Antolín-Díaz & Rubio-Ramírez (2018).

See section Details for the model equations.

Usage

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

Arguments

specification

an object of class BSVARSIGN generated using the specify_bsvarSIGN$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

Details

The Structural VAR 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. More specifically,

B = Q'P

where Q is an NxN rotation matrix and P is an NxN lower triangular matrix.

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

Value

An object of class PosteriorBSVARSIGN 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

skipped

an integer of the total skipped iterations, the Gibbs sampler performs a total of S+skipped iterations, when the sampler does not find a valid rotation matrix Q within max_tries, the current iteration is skipped (i.e. the current draw of A,B is not saved). A message is shown when skipped/(skipped+S/thin) > 0.05, where S/thin is the total number of draws returned.

last_draw an object of class BSVARSIGN 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], Xiaolei Wang [e-mail]

References

Sampling from the generalised-normal full conditional posterior distribution of matrix B is implemented using the Gibbs sampler by:

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.

Sampling from the multivariate normal full conditional posterior distribution of each of the A matrix row is implemented using the sampler by:

Chan, J.C.C., Koop, G, and Yu, X. (2021) Large Order-Invariant Bayesian VARs with Stochastic Volatility.

Narrative sign restrictions are implemented using algorithm 1 in:

Antolín-Díaz, J., & Rubio-Ramírez, J. F. (2018). Narrative sign restrictions for SVARs.

Examples

# investigate the effects of the optimism shock
data(optimism)

# specify identifying restrictions:
# + no effect on productivity (zero restriction)
# + positive effect on stock prices (positive sign restriction) 
sign_irf       = matrix(c(0, 1, rep(NA, 23)), 5, 5)

# specify the model and set seed
set.seed(123)
specification  = specify_bsvarSIGN$new(optimism * 100,
                                       p        = 12,
                                       sign_irf = sign_irf)
                                       
# estimate the model
posterior      = estimate(specification, S = 10)