Identification and convergence checks for a copregbayes fit — validity.copregbayes • endogCopulaBayes

validity() for a CopRegBAYES fit reports two things at once. First, the identification requirement of the model: the copula correction is identified off the nonnormality of the endogenous regressors, since under normality \(E(e|z)\) is linear and the copula cannot separate regressor variation from error variation; the posterior of the endogeneity correlations (the correlation between an endogenous regressor's normal score and that of the structural error) is reported alongside, since the Bayesian version is expected to degrade gracefully – concentrating near zero rather than producing confident nonsense – when identification is weak. Second, convergence of the MCMC chain itself: Geweke's statistic, the effective sample size and the lag-one autocorrelation, all computed from the chain that is already there, plus – on request, since it needs several chains from dispersed starting values – the Gelman-Rubin statistic, which costs one further run of the sampler per additional chain. The autocorrelation sum behind the effective sample size is truncated at the first negative lag, so the reported value never exceeds the number of retained draws, and is NA for a chain that does not move.

print() on the result formats all of that.

Usage

# S3 method for class 'copregbayes'
validity(object, chains = FALSE, power = 0.8, verbose = interactive(), ...)

# S3 method for class 'copregbayes.validity'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

Arguments

object: An object of class "copregbayes", as returned by CopRegBAYES.
chains: Whether to also compute the Gelman-Rubin statistic from dispersed starting values. FALSE (the default) skips it. TRUE runs 4 additional chains; a number runs that many additional chains instead (at least 2). Each additional chain re-runs the sampler with the same call but a fresh, dispersed starting point drawn from \(N\)(coefficients, posterior sd) for the regression coefficients, an LKJ(1) draw for the copula correlation matrix, and a fresh Dir(1) draw per regressor for the probability masses. The extra chains re-evaluate the original call in the frame validity() was called from, so a fit made inside a function still finds its data.
power: Statistical power used for the sample-size-dependent nonnormality thresholds (Becker, Proksch & Ringle 2022). Defaults to 0.8.
verbose: If TRUE, print a message before each additional chain required by chains. Defaults to interactive().
...: Not used; present for S3 method consistency.
x: An object of class "copregbayes.validity", as returned by validity.copregbayes.
digits: Number of significant digits to print.

Value

For validity.copregbayes, an object of class "copregbayes.validity", a list with nonnormality (a table of skewness, excess kurtosis, Anderson-Darling and Cramer-von Mises statistics and a Kolmogorov-Smirnov p-value for each endogenous regressor, with pass/fail flags against two thresholds), thresholds (the sample-size-and-power-dependent thresholds used), endogeneity (posterior mean, 95% interval and \(P(\rho > 0)\) for each endogeneity correlation), convergence (a data frame of Geweke, effective sample size and lag-one autocorrelation for every retained parameter), gelman.rubin (a named vector of Gelman-Rubin statistics, or NULL if chains = FALSE), acceptance, ndraws and endo.names.

For print.copregbayes.validity, x is returned invisibly; called for its side effect of printing.

References

Haschka, R. E. (2025). Bayesian inference for joint estimation models using copulas to handle endogenous regressors. Oxford Bulletin of Economics and Statistics. doi:10.1111/obes.70023

Examples

set.seed(1)
n <- 60
x <- rnorm(n); z <- x + rnorm(n); y <- 1 + z + x + rnorm(n)
fit <- CopRegBAYES(y ~ z | x, data = data.frame(y, z, x),
                    iterations = 200, burnin = 50, thin = 5, verbose = FALSE)
validity(fit)
#> 
#> Checks for the Bayesian copula correction
#> 
#> [1] Nonnormality of the endogenous regressors
#>     Identification comes from the nonlinearity of E(e|z), which is
#>     present only when z is nonnormal; under normality the copula cannot
#>     tell regressor variation from error variation.
#>   skewness ex.kurtosis    AD  CvM  KS p Yang ok Becker ok
#> z   -0.691       0.741 0.451 0.06 0.812   FALSE     FALSE
#> 
#> [2] Posterior of the endogeneity correlations
#>              P. Mean 2.5%    97.5%   P(rho > 0)
#> rho(z*, xi*) -0.4564 -0.7113  0.1041  0.1000   
#>     An interval covering zero says the data carry no evidence of
#>     endogeneity, which is the honest reading; it is not a test decision.
#> 
#> [3] Convergence of the chain
#>     Geweke compares the first tenth with the last half of the draws and
#>     is a z statistic, so |Geweke| > 2 is a warning sign. ESS is the
#>     effective number of independent draws behind 30 retained ones.
#>                Geweke    ESS    AC(1)
#> (Intercept)    4.5229 18.240  0.29615
#> z            -11.1126  5.897  0.76881
#> x             13.0992  5.391  0.66270
#> sigma2        -7.5163  5.991  0.58845
#> rho(z*, x*)   -0.9601 30.000 -0.06026
#> rho(z*, xi*)   7.1434  6.304  0.76326
#>     Acceptance rates: coefficients 0.835, sigma2 0.865
#> 
#> [4] Gelman-Rubin was not computed
#>     It needs several chains from dispersed starts, so it costs one
#>     further run of the sampler per chain. Pass chains = TRUE, or a
#>     number, to compute it.
#> 
#> Source: Haschka (2025), Oxford Bulletin of Economics and Statistics