verify_volatility.PosteriorBSVARMSH function

Verifies heteroskedasticity of structural shocks equation by equation

This function will be deprecated starting from version 4.0. It is replaced by verify_identification function.

Computes the logarithm of Bayes factor for the homoskedasticity hypothesis for each of the structural shocks via Savage-Dickey Density Ration (SDDR). The hypothesis of homoskedasticity is represented by restriction: [REMOVE_ME]$H_0: \sigma^2_{n.1} = ... = \sigma^2_{n.M} = 1 [REMOVE_ME_2]$

The logarithm of Bayes factor for this hypothesis can be computed using the SDDR as the difference of logarithms of the marginal posterior distribution ordinate at the restriction less the marginal prior distribution ordinate at the same point: [REMOVE_ME]$log p(\omega_n = 0 | data) - log p(\omega_n = 0) [REMOVE_ME_2]$

Therefore, a negative value of the difference is the evidence against homoskedasticity of the structural shock. The estimation of both elements of the difference requires numerical integration.

## S3 method for class 'PosteriorBSVARMSH'
verify_volatility(posterior)

Arguments

• posterior: the posterior element of the list from the estimation outcome

Returns

An object of class SDDRvolatility that is a list of three components:

logSDDR an N-vector with values of the logarithm of the Bayes factors for the homoskedasticity hypothesis for each of the shocks

log_SDDR_se an N-vector with estimation standard errors of the logarithm of the Bayes factors reported in output element logSDDR that are computed based on 30 random sub-samples of the log-ordinates of the marginal posterior and prior distributions.

components a list of three components for the computation of the Bayes factor

• log_denominator: an N-vector with values of the logarithm of the Bayes factor denominators
• log_numerator: an N-vector with values of the logarithm of the Bayes factor numerators
• log_numerator_s: an NxS matrix of the log-full conditional posterior density ordinates computed to estimate the numerator
• se_components: an Nx30 matrix containing the log-Bayes factors on the basis of which the standard errors are computed

Description

This function will be deprecated starting from version 4.0. It is replaced by verify_identification function.

Computes the logarithm of Bayes factor for the homoskedasticity hypothesis for each of the structural shocks via Savage-Dickey Density Ration (SDDR). The hypothesis of homoskedasticity is represented by restriction:

The logarithm of Bayes factor for this hypothesis can be computed using the SDDR as the difference of logarithms of the marginal posterior distribution ordinate at the restriction less the marginal prior distribution ordinate at the same point:

Therefore, a negative value of the difference is the evidence against homoskedasticity of the structural shock. The estimation of both elements of the difference requires numerical integration.

Examples

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

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

# estimate the model
posterior      = estimate(specification, 10)

# verify heteroskedasticity
sddr           = verify_volatility(posterior)

# workflow with the pipe |>
############################################################
set.seed(123)
us_fiscal_lsuw |>
  specify_bsvar_msh$new(p = 1, M = 2) |>
  estimate(S = 10) |> 
  verify_volatility() -> sddr

References

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, tools:::Rd_expr_doi("10.1016/j.jedc.2020.103862") .

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, tools:::Rd_expr_doi("10.48550/arXiv.2404.11057") .

See Also

specify_bsvar_msh, estimate

Author(s)

Tomasz Woźniak wozniak.tom@pm.me