verify_autoregression.PosteriorBSVARMSH function

Verifies hypotheses involving autoregressive parameters

Computes the logarithm of Bayes factor for the joint hypothesis, $H_0$, possibly for many autoregressive parameters represented by argument hypothesis via Savage-Dickey Density Ration (SDDR). 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]$log p(H_0 | data) - log p(H_0) [REMOVE_ME_2]$

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

## S3 method for class 'PosteriorBSVARMSH'
verify_autoregression(posterior, hypothesis)

Arguments

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

• hypothesis: an NxK matrix of the same dimension as the autoregressive matrix $A$ with numeric values for the parameters to be verified, in which case the values represent the joint hypothesis, and missing value NA

  for these parameters that are not tested

Returns

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

logSDDR a scalar with values of the logarithm of the Bayes factors for the autoregressive 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
• log_denominator_s: an NxS matrix of the log-full conditional posterior density ordinates computed to estimate the denominator
• se_components: a 30-vector containing the log-Bayes factors on the basis of which the standard errors are computed

Description

Computes the logarithm of Bayes factor for the joint hypothesis, $H_0$, possibly for many autoregressive parameters represented by argument hypothesis via Savage-Dickey Density Ration (SDDR). 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 hypothesis. 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 autoregression
H0             = matrix(NA, ncol(us_fiscal_lsuw), ncol(us_fiscal_lsuw) + 1)
H0[1,3]        = 0        # a hypothesis of no Granger causality from gdp to ttr
sddr           = verify_autoregression(posterior, H0)

# workflow with the pipe |>
############################################################
set.seed(123)
us_fiscal_lsuw |>
  specify_bsvar_msh$new(p = 1, M = 2) |>
  estimate(S = 10) |> 
  verify_autoregression(hypothesis = H0) -> sddr

References

Woźniak, T., and Droumaguet, M., (2024) Bayesian Assessment of Identifying Restrictions for Heteroskedastic Structural VARs

Author(s)

Tomasz Woźniak wozniak.tom@pm.me