estimate.PosteriorBSVARSV {bsvars}

Bayesian estimation of a Structural Vector Autoregression with Stochastic Volatility heteroskedasticity via Gibbs sampler

Description

Estimates the SVAR with Stochastic Volatility (SV) heteroskedasticity proposed by Lütkepohl, Shang, Uzeda, and Woźniak (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 SV model is estimated using a range of techniques including: simulation smoother, auxiliary mixture, ancillarity-sufficiency interweaving strategy, and generalised inverse Gaussian distribution summarised by Kastner & Frühwirth-Schnatter (2014). See section Details for the model equations.

Usage

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

Arguments

specification

an object of class PosteriorBSVARSV 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 jointly normally distributed with zero mean.

Two alternative specifications of the conditional variance of the nth shock at time t can be estimated: non-centred Stochastic Volatility by Lütkepohl, Shang, Uzeda, and Woźniak (2022) or centred Stochastic Volatility by Chan, Koop, & Yu (2021).

The non-centred Stochastic Volatility by Lütkepohl, Shang, Uzeda, and Woźniak (2022) is selected by setting argument centred_sv of function specify_bsvar_sv$new() to value FALSE. It has the conditional variances given by:

Var_{t-1}[u_{n.t}] = exp(w_n h_{n.t})

where w_n is the estimated conditional standard deviation of the log-conditional variance and the log-volatility process h_{n.t} follows an autoregressive process:

h_{n.t} = g_n h_{n.t-1} + v_{n.t}

where h_{n.0}=0, g_n is an autoregressive parameter and v_{n.t} is a standard normal error term.

The centred Stochastic Volatility by Chan, Koop, & Yu (2021) is selected by setting argument centred_sv of function specify_bsvar_sv$new() to value TRUE. Its conditional variances are given by:

Var_{t-1}[u_{n.t}] = exp(h_{n.t})

where the log-conditional variances h_{n.t} follow an autoregressive process:

h_{n.t} = g_n h_{n.t-1} + v_{n.t}

where h_{n.0}=0, g_n is an autoregressive parameter and v_{n.t} is a zero-mean normal error term with variance s_{v.n}^2.

Value

An object of class PosteriorBSVARSV 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

h

an NxTxS array with the posterior draws of the log-volatility processes

rho

an NxS matrix with the posterior draws of SV autoregressive parameters

omega

an NxS matrix with the posterior draws of SV process conditional standard deviations

S

an NxTxS array with the posterior draws of the auxiliary mixture component indicators

sigma2_omega

an NxS matrix with the posterior draws of the variances of the zero-mean normal prior for omega

s_

an S-vector with the posterior draws of the scale of the gamma prior of the hierarchical prior for sigma2_omega

last_draw an object of class BSVARSV 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

Lütkepohl, H., Shang, F., Uzeda, L., and Woźniak, T. (2022) Partial Identification of Heteroskedastic Structural VARs: Theory and Bayesian Inference.

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.

Many of the techniques employed for the estimation of the Stochastic Volatility model are summarised by:

Kastner, G. and Frühwirth-Schnatter, S. (2014) Ancillarity-Sufficiency Interweaving Strategy (ASIS) for Boosting MCMC Estimation of Stochastic Volatility Models. Computational Statistics & Data Analysis, 76, 408–423, doi:10.1016/j.csda.2013.01.002.

See Also

specify_bsvar_sv, specify_posterior_bsvar_sv, normalise_posterior

Examples

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

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

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

# estimate the model
posterior      = estimate(burn_in, 20)

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