Instrumental variable approaches address the endogeneity dilemma by decomposing the observed variation in a predictor into an exogenous component and an endogenous component using an external instrument. Depending on the source of endogeneity, researchers may rely on two-stage least squares, control functions, Heckman treatment correction, or Heckman selection correction. All of these approaches share a common logic: identification hinges on using an IV to isolate variation in the predictor that is plausibly unrelated to unobserved determinants of the outcome. Implementing any IV approach requires at least one instrument that is strong, valid, and granular.
Because suitable instrumental variables are often difficult to identify, instrument-free approaches have attracted increasing attention. Copula correction follows the control-function logic that the observed variation in a predictor can be decomposed into exogenous and endogenous components, but it does so without requiring an external instrument. The approach constructs a copula-based control function from the conditional distribution of the predictor, typically using its cumulative distribution function or a model-free estimate, and includes this term alongside the predictor in the outcome equation. By capturing dependence between the predictor and the error term, the copula term removes regressor–error correlation and enables consistent estimation, while its coefficient provides an explicit diagnostic for endogeneity. However, copula correction relies on untestable assumptions about unobservables, most notably that the structural error or its endogenous component is approximately normally distributed and that the dependence between the endogenous predictor and the error, conditional on controls, is adequately represented by a Gaussian copula.