Robustness of copula-correction models in causal analysis: Exploiting between-regressor correlation

Abstract

Causal analysis in management and marketing often faces the challenge of endogeneity, which can result in biased estimates when methods that assume independence between regressors and errors are applied. The joint copula modeling approach proposed by Park and Gupta (Marketing Science, 2012, 31(4), 567–586) provides a practical solution to this issue by modeling the joint distribution of endogenous regressors and errors. This paper proposes a generalisation of their approach with an endogeneity correction that involves the exogenous variables. We first show that the estimator by Park and Gupta requires the strong assumption of independence between the endogenous and the exogenous regressors, and suffers from an omitted variables bias when this assumption is violated. We also quantify this bias. The distinguishing characteristic of the proposed approach is that we use a first-stage auxiliary regression to generate copula correction functions by exploiting the informational content of the exogenous variables in a similar spirit as instrumental-based identification. As this first-stage regression does not generate an additional identification problem, is not more restrictive than the Park and Gupta model. The approach is least-squares-based and thus neither requires numerical optimisation nor the search for starting values. Monte Carlo simulations reveal that the proposed approach performs well in finite samples. We demonstrate the practical applicability by reassessing the empirical example in Park and Gupta using the proposed approach.