Verifies identification through heteroskedasticity or non-normality of of structural shocks

Computes the logarithm of Bayes factor for the hypothesis of normality for each of the structural shocks via Savage-Dickey Density Ration (SDDR). The hypothesis of normality in this mixture of normals model is represented by restriction: $$H_0: \sigma^2_{n.1} = ... = \sigma^2_{n.M} = 1$$ 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: $$log p(H_0 | data) - log p(H_0)$$ 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.

Usage

# S3 method for class 'PosteriorBSVARMIX'
verify_identification(posterior)

Arguments

posterior: the estimation outcome obtained using estimate function

Value

An object of class SDDRid* that is a list with components:

logSDDR a vector with values of the logarithm of the Bayes factors

log_SDDR_se a vector with numerical 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.

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, 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, doi:10.48550/arXiv.2404.11057 .

Author

Tomasz Woźniak wozniak.tom@pm.me

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)
#> The identification is set to the default option of lower-triangular structural matrix.
set.seed(123)

# estimate the model
posterior      = estimate(specification, 10)
#> **************************************************|
#> bsvars: Bayesian Structural Vector Autoregressions|
#> **************************************************|
#>  Gibbs sampler for the SVAR-finiteMIX model             |
#> **************************************************|
#>  Progress of the MCMC simulation for 10 draws
#>     Every draw is saved via MCMC thinning
#>  Press Esc to interrupt the computations
#> **************************************************|

# verify heteroskedasticity
sddr           = verify_identification(posterior)

# workflow with the pipe |>
############################################################
set.seed(123)
us_fiscal_lsuw |>
  specify_bsvar_mix$new(p = 1, M = 2) |>
  estimate(S = 10) |> 
  verify_identification() -> sddr
#> The identification is set to the default option of lower-triangular structural matrix.
#> **************************************************|
#> bsvars: Bayesian Structural Vector Autoregressions|
#> **************************************************|
#>  Gibbs sampler for the SVAR-finiteMIX model             |
#> **************************************************|
#>  Progress of the MCMC simulation for 10 draws
#>     Every draw is saved via MCMC thinning
#>  Press Esc to interrupt the computations
#> **************************************************|