Computes the logarithm of Bayes factor for the normality hypothesis for each of the structural shocks via Savage-Dickey Density Ration (SDDR). The hypothesis of normality, \(H_0\), is represented by restriction that the equation-specific degrees of freedom parameter is equal to infinity, \(\nu_n\rightarrow\infty\). 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 normality of the structural shock. The estimation of th first element relies on kernel density estimation of the marginal posterior density, whereas the second element is equal to the log of value 1.
Value
An object of class SDDRnormality that is a list of three components:
logSDDR an N-vector with values of the logarithm of the Bayes factors for
the normality 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.
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
############################################################
# specify the model
specification = specify_bsvar_sv$new(us_fiscal_lsuw, distribution = "t")
#> The identification is set to the default option of lower-triangular structural matrix.
# estimate the model
posterior = estimate(specification, 10)
#> **************************************************|
#> bsvars: Bayesian Structural Vector Autoregressions|
#> **************************************************|
#> Gibbs sampler for the SVAR-SV model |
#> Non-centred SV model is estimated |
#> **************************************************|
#> Progress of the MCMC simulation for 10 draws
#> Every draw is saved via MCMC thinning
#> Press Esc to interrupt the computations
#> **************************************************|
# verify heteroskedasticity
sddr = verify_normality(posterior)
# workflow with the pipe |>
############################################################
us_fiscal_lsuw |>
specify_bsvar_sv$new(distribution = "t") |>
estimate(S = 10) |>
verify_normality() -> sddr
#> The identification is set to the default option of lower-triangular structural matrix.
#> **************************************************|
#> bsvars: Bayesian Structural Vector Autoregressions|
#> **************************************************|
#> Gibbs sampler for the SVAR-SV model |
#> Non-centred SV model is estimated |
#> **************************************************|
#> Progress of the MCMC simulation for 10 draws
#> Every draw is saved via MCMC thinning
#> Press Esc to interrupt the computations
#> **************************************************|