estimate.PosteriorBSVARMIX {bsvars}

Bayesian estimation of a Structural Vector Autoregression with shocks following a finite mixture of normal components via Gibbs sampler

Description

Estimates the SVAR with non-normal residuals following a finite M mixture of normal distributions proposed by Woźniak & Droumaguet (2022). Implements the Gibbs sampler proposed by Waggoner & Zha (2003) for the structural matrix B and the equation-by-equation sampler by Chan, Koop, & Yu (2021) 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 thanks to a hierarchical prior distribution. The finite mixture of normals model is estimated using the prior distributions and algorithms proposed by Woźniak & Droumaguet (2022). See section Details for the model equations.

Usage

## S3 method for class 'PosteriorBSVARMIX'
estimate(specification, S, thin = 10, show_progress = TRUE)

Arguments

specification

an object of class PosteriorBSVARMIX 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

Details

The heteroskedastic 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 finite-mixture of normals distributed with zero mean. The conditional variance of the nth shock at time t is given by:

Var_{t-1}[u_{n.t}] = s^2_{n.s_t}

where s_t is a the regime indicator of the regime-specific conditional variances of structural shocks s^2_{n.s_t}. In this model, the variances of each of the structural shocks sum to M.

The regime indicator s_t is either such that:

• the regime probabilities are non-zero which requires all regimes to have a positive number occurrences over the sample period, or

• sparse with potentially many regimes with zero occurrences over the sample period and in which the number of regimes is estimated.

These model selection also with this respect is made using function specify_bsvar_mix.

Value

An object of class PosteriorBSVARMIX 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

sigma2

an NxMxS array with the posterior draws for the structural shocks conditional variances

PR_TR

an MxMxS array with the posterior draws for the transition matrix.

xi

an MxTxS array with the posterior draws for the regime allocation matrix.

pi_0

an MxS matrix with the posterior draws for the ergodic probabilities

sigma

an NxTxS array with the posterior draws for the structural shocks conditional standard deviations' series over the sample period

last_draw an object of class BSVARMIX 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

The model, prior distributions, and estimation algorithms were proposed by

Woźniak, T., and Droumaguet, M., (2022) Bayesian Assessment of Identifying Restrictions for Heteroskedastic Structural VARs

Some more analysis on heteroskedastic SVAR models was proposed by:

Lütkepohl, H., and Woźniak, T., (2020) Bayesian Inference for Structural Vector Autoregressions Identified by Markov-Switching Heteroskedasticity. Journal of Economic Dynamics and Control 113, 103862, doi:10.1016/j.jedc.2020.103862.

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.

The estimation of the mixture of normals heteroskedasticity closely follows procedures described by:

Song, Y., and Woźniak, T., (2021) Markov Switching. Oxford Research Encyclopedia of Economics and Finance, Oxford University Press, doi:10.1093/acrefore/9780190625979.013.174.

and

Frühwirth-Schnatter, S., (2006) Finite Mixture and Markov Switching Models. Springer Series in Statistics. New York: Springer, doi:10.1007/978-0-387-35768-3.

The sparse model is inspired by:

Malsiner-Walli, G., Frühwirth-Schnatter, S., and Grün, B. (2016) Model-based clustering based on sparse finite Gaussian mixtures. Statistics and Computing, 26(1–2), 303–324, doi:10.1007/s11222-014-9500-2.

The forward-filtering backward-sampling is implemented following the proposal by:

Chib, S. (1996) Calculating posterior distributions and modal estimates in Markov mixture models. Journal of Econometrics, 75(1), 79–97, doi:10.1016/0304-4076(95)01770-4.

See Also

specify_bsvar_mix, specify_posterior_bsvar_mix, normalise_posterior

Examples

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

# specify the model and set seed
specification  = specify_bsvar_mix$new(us_fiscal_lsuw, p = 1, M = 2)
set.seed(123)

# run the burn-in
burn_in        = estimate(specification, 10)

# estimate the model
posterior      = estimate(burn_in, 50)

# workflow with the pipe |>
############################################################
set.seed(123)
us_fiscal_lsuw |>
  specify_bsvar_mix$new(p = 1, M = 2) |>
  estimate(S = 10) |> 
  estimate(S = 50) |> 
  compute_impulse_responses(horizon = 4) -> irf