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

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

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().

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.

References

Antolín-Díaz & Rubio-Ramírez (2018) Narrative Sign Restrictions for SVARs, American Economic Review, 108(10), 2802-29, <doi:10.1257/aer.20161852>.

Arias, Rubio-Ramírez, & Waggoner (2018), Inference Based on Structural Vector Autoregressions Identified With Sign and Zero Restrictions: Theory and Applications, Econometrica, 86(2), 685-720, <doi:10.3982/ECTA14468>.

Giannone, Lenza, Primiceri (2015) Prior Selection for Vector Autoregressions, Review of Economics and Statistics, 97(2), 436-451 <doi:10.1162/REST_a_00483>.

Rubio-Ramírez, Waggoner & Zha (2010) Structural Vector Autoregressions: Theory of Identification and Algorithms for Inference, The Review of Economic Studies, 77(2), 665-696, <doi:10.1111/j.1467-937X.2009.00578.x>.

Author

Tomasz Woźniak wozniak.tom@pm.me, Xiaolei Wang adamwang15@gmail.com

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)
#> **************************************************|
#>  bsvarSIGNs: Bayesian Structural VAR with sign,   |
#>              zero and narrative restrictions      |
#> **************************************************|
#>  Progress of simulation for 10 independent draws
#>  Press Esc to interrupt the computations
#> **************************************************|