Print a copreg validity check

Prints the full report built by validity.copreg: the non-normality (or Assumption 3) step, the uncorrelatedness assumption where relevant, exogenous regressors as identifying variation where relevant, the error-term diagnostics, and the ICON standard-error inflation.

Usage

# S3 method for class 'copreg.validity'
print(x, digits = 4, ...)

Arguments

x

an object of class "copreg.validity", as returned by validity.copreg.

digits

number of significant digits to print.

...

currently unused.

Value

x, invisibly. Called for the side effect of printing.

References

Becker, J.-M., D. Proksch, and C. M. Ringle (2022). Revisiting Gaussian copulas to handle endogenous regressors. Journal of the Academy of Marketing Science 50, 46-66.

Yang, F., Y. Qian, and H. Xie (2025). Addressing endogeneity using a two-stage copula generated regressor approach. Journal of Marketing Research 62(4), 601-623.

Qian, Y., A. Koschmann, and H. Xie (2025). A practical guide to endogeneity correction using copulas. Journal of Marketing.

Examples

set.seed(1)
n  <- 150
w  <- rnorm(n)
p  <- 0.4 * w + rt(n, df = 3)
xi <- 0.5 * p + rnorm(n)
y  <- 1 + 2 * p + 1.5 * w + xi
dat <- data.frame(y = y, p = p, w = w)

fit <- endogCopula:::.copreg_fit(
  formula = y ~ p | w, data = dat,
  ctor = endogCopula:::.ctor_twostage(TRUE),
  method = "2sCOPE", cdf = "rank.n", ties = "max",
  nboots = 25, verbose = FALSE)

print(validity(fit))
#> 
#> Validity check for 2sCOPE
#> n = 150, intercept: yes, target power 80%
#> Sources: Becker, Proksch & Ringle (2022); Yang, Qian & Xie (2025);
#>          Qian, Koschmann & Xie (2025)
#> 
#> [1] Nonnormality of the endogenous regressors
#>   skewness ex.kurtosis    AD   CvM  KS p Yang ok Becker ok
#> p    0.962       5.978 1.604 0.261 0.343   FALSE     FALSE
#>     Becker et al. at n = 150: |skewness| >= not attainable, or AD > 18.964, or CvM > 3.488
#>     Yang et al.: KS p < .05
#> 
#> [2] Assumption: correlation of the copula transformation term
#>     with the exogenous regressors
#>   corr(W, CTT) p (Holm)
#> w      -0.1761   0.0309
#>     Joint test: R2 = 0.03102, F = 4.737, p = 0.0311
#>     => violated. Park & Gupta is inconsistent here (Haschka, 2025).
#> 
#> [3] Exogenous regressors as identifying variation
#>     (continuous, KS p < .001, first-stage F > 10)
#>   continuous  KS p  F: p qualifies
#> w       TRUE 0.847 4.797     FALSE
#>     => none qualifies. The conditions are conservative and not
#>        necessary; Yang et al. (2025) propose a bootstrap procedure to gauge
#>        the finite-sample bias in this situation.
#> 
#> [4] Error term: structural residuals xi
#>     skewness = -0.02121, excess kurtosis = -0.3975, AD = 0.2139 (p = 0.849)
#>     Becker et al. (2022) find that with a nonnormal error the approach
#>     is no longer consistent in models with an intercept. Yang et al.
#>     (2025) and Qian et al. (2025) do permit a nonnormal error, but only
#>     under the decomposition xi = U + V into a normally distributed
#>     endogenous part U, which carries the entire dependence with the
#>     regressors, and an independent nonnormal V. The residuals show xi,
#>     not U, so their skewness neither establishes nor rules out a
#>     violation; whether that decomposition holds has to be argued from
#>     the suspected sources of endogeneity.
#> 
#> [5] ICON: standard error inflation relative to uncorrected OLS
#>             SE (corrected) SE (uncorrected)  ICON
#> (Intercept)        0.09116          0.08483 1.075
#> p                  0.22755          0.05394 4.219
#> w                  0.11580          0.08547 1.355
#>     => largest ICON = 4.219, below the threshold of 6.
#>