A Practical Guide to Endogeneity Correction Using Copulas

Abstract

Causal inference is of central interest in many empirical applications, yet often challenging because of the presence of endogenous regressors. The classical approach to the problem requires using instrumental variables that must satisfy the stringent condition of exclusion restriction. In recent research, instrument-free copula methods have been increasingly used to handle endogenous regressors. This article aims to provide a practical guide for how to handle endogeneity using copulas. The authors give an overview of copula endogeneity correction, outlining its theoretical rationales, advantages, and limitations for empirical research. They also discuss recent advances that enhance the understanding, applicability, and robustness of copula correction, and address implementation aspects of copula correction such as constructing proper and robust copula control functions, handling higher-order terms of endogenous regressors and noncontinuous endogenous regressors, choosing between control function and likelihood-based joint estimation methods, and extending the approach to panel data and nonlinear models. To facilitate the appropriate usage of copula correction in order to realize its full potential, the authors detail a process of checking data requirements and identification assumptions to determine when and how to use copula correction methods, and illustrate its usage using empirical examples.

Many research questions in marketing, management, economics, and health sciences concern causality rather than mere association. Such questions are often addressed by estimating structural regression models that represent causal relationships. A pervasive issue in these analyses is regressor endogeneity, which arises when regressors representing the causes (e.g., an economic program to be evaluated, marketing-mix variables) are not randomly assigned and thus correlate with unobservables (e.g., unobserved product characteristics or common market shocks) in the structural error term (Villas-Boas and Winer 1999). Ignoring the regressor–error dependence can lead to severely biased parameter estimates.

Given the ubiquity of endogenous regressors and the need to address endogeneity bias, extensive research is devoted to developing suitable correction methods. The instrumental variable (IV) method is the classical econometric solution. It depends on valid and strong IVs that satisfy the stringent requirement of exclusion restriction (ER), making IVs difficult to identify and justify in practice (Ebbes et al. 2005). Concerns over IV availability and quality have spurred growing interest in IV-free endogeneity correction methods (Ebbes, Wedel, and Böckenholt 2009; Papies, Ebbes, and Feit 2023; Papies, Ebbes, and Van Heerde 2017; Rutz and Watson 2019). These methods exploit higher moments (HM; Lewbel 1997), identification via heteroskedastic error structures (IH; Rigobon 2003), latent IVs (LIV; Ebbes et al. 2005), semiparametric odds ratio endogeneity correction (SORE; Qian and Xie 2024), and copulas, ¹ starting from the seminal work of Park and Gupta (2012). ²

Copula correction methods provide substantial advantages for addressing the prevalent and thorny issue of endogenous regressors. These methods directly address the regressor–error dependence using copulas, a widely used multivariate dependence model applicable in many practical applications (Danaher and Smith 2011). Unlike the IV approach and other IV-free methods, copula correction does not require the endogenous regressor to contain an exogenous component (either observed or latent) that must satisfy the stringent ER condition. Thus, copula correction is feasible in many situations when appropriate conditions are met. Moreover, it can be implemented by incorporating copula-based control functions—derived from existing regressors—into the structural model as additional control variables to address endogeneity. Thus, copula correction using control functions is computationally tractable and straightforward to apply in a wide array of settings, including both linear and nonlinear models (e.g., discrete choice models), multiple endogenous regressors, panel data, endogenous interaction and higher-order terms, and the slope endogeneity problem.

Largely due to these advantages, copula correction has gained popularity in empirical research. Beyond marketing, it is increasingly adopted in other fields such as economics, management, and information systems (e.g., Ananthakrishnan et al. 2025; Becerra and Markarian 2021; Christopoulos, McAdam, and Tzavalis 2021). The pie chart in Figure 1, Panel A, breaks down by discipline book chapters and journal publications (n = 615) using copula endogeneity correction, according to Google Scholar. Each slice in the pie chart matches journals and journal fields as defined by the Australian Business Dean's Council. Outside marketing, strategy and information systems are the two most common business disciplines adopting copula correction. Within marketing, the bar chart in Figure 1, Panel B, displays the distribution of articles using copula correction (n = 100) by substantive area in leading marketing journals ³ from 2013 to 2025 (see Web Appendix A for the full list of publications).

Figure 1.

Publications Using Copula Correction.

Like other causal methods for nonexperimental data, copula correction relies on specific underlying assumptions and data requirements. Earlier studies (Becker, Proksch, and Ringle 2022; Eckert and Hohberger 2023; Papies, Ebbes, and Van Heerde 2017) have reviewed and evaluated the assumptions and limitations of the original method by Park and Gupta (2012). Since then, methodological advances have significantly relaxed these constraints, enabling copula correction to operate under less strict conditions than previously believed. Hu, Qian, and Xie (2025) show that copula correction using control functions does not require the error to be normally distributed or follow a specific copula structure jointly with endogenous regressors (see Web Appendix B), making the approach more robust and widely applicable than previously thought. Although copula correction originally required endogenous regressors to be uncorrelated with exogenous regressors and have sufficient nonnormality, limiting its applicability, the recent two-stage copula endogeneity correction (2sCOPE) approach by Yang, Qian, and Xie (2025) simultaneously relaxes these restrictions and provides a general framework for further development (see the subsequent “Methodological Background” section). Haschka (2022) and Yang, Qian, and Xie (2024) generalize copula correction to panel data. Hu, Qian, and Xie (2025) introduce nonparametric copula control functions that generalize and unify existing copula correction methods. Qian and Xie (2024) and Hu, Qian, and Xie (2025) develop IV-free methods for handling noncontinuous endogenous regressors (e.g., binary treatment) that current copula control function methods cannot accommodate.

Given the substantial advances since Park and Gupta (2024), updated guidelines are needed to clarify the scope and boundaries of these new methods and to guide the use of the expanded copula correction toolbox. Without randomization or ERs, the trade-off in copula correction lies in the need to explicitly model the regressor–error dependence. While recent advances enhance the copula approach to address endogeneity, it is not a panacea for every instance. We discuss its boundary conditions and limitations to help researchers make a conscious choice of when to use the copula approach.

Focusing on assisting potential users of copula correction, the objectives of this article are to (1) raise awareness of the importance of addressing endogenous regressors in empirical studies and demystify theoretical rationales of copula correction; (2) synthesize recent advances that enhance the understanding, applicability, and robustness of copula correction; (3) provide updated guidance and delineate a process of checking data requirements and identification assumptions to aid proper usage of copula correction; and (4) demonstrate the use of copula correction in practical applications.

The rest of the article proceeds as follows. First, we provide an overview of the theoretical rationale for endogeneity correction using copulas. This addresses how, when, and why copulas work, including identification assumptions, data requirements, and boundary conditions. Second, we present the methodological background: how copula correction assumptions might be relaxed, its usage for panel data, how it is constructed, optimal estimation for moderators and nonlinear effects, and obtaining standard errors. Third, we provide guidance for practical usage, including a flowchart “cookbook” to check data requirements and assumptions at key steps. Fourth, we present two examples that follow the flowchart, using real-world sales data. Finally, we conclude with discussion and future research directions.

Theoretical Rationale for Endogeneity Correction Using Copulas

As an entry point, we first provide an overview of how copulas address endogeneity correction: Why and when should they be used? How do they work? We examine what assumptions and data requirements are actually needed for model identification and discuss boundary conditions to guide appropriate use. This leads to the impact of copula correction.

Why and When Should Copula Correction Be Used?

Empirical examples of endogenous regressors abound. For concreteness, consider a running example of estimating the following linear structural model using nonexperimental data:

$${\mathrm{Y}}_{\mathrm{i}}=\mu +\alpha {\mathrm{P}}_{\mathrm{i}}+{\beta }^{\mathrm{\prime }}{\mathrm{W}}_{\mathrm{i}}+{\mathrm{E}}_{\mathrm{i}},$$

(1)

where i = 1, …, n indexes cross-sectional units or markets across regions or time; Y_i is a scalar response variable (e.g., log sales volume of ice cream in market i); P_i is the endogenous regressor (e.g., log price), and W_i is a vector of exogenous control variables affecting both P_i and Y_i (as shown by the two arrows from W to P and Y in Figure 2, Panel A). The model parameters are (μ, α, β), among which α captures the causal or independent effect of P_i and is of primary interest. The exogenous control variables in W are determined outside the system (e.g., weather) or under control by researchers such that no dependence between W_i and E_i exists (i.e., no arrow between W and E in Figure 2, Panel A), and thus cov(W_i, E_i) = 0. In contrast, P_i may depend on unobserved factors in E_i (e.g., market shocks or product attributes), creating endogeneity through the E → P link in Figure 2, Panel A.

Figure 2.

Directed Acyclic Graphs for Endogeneity.

Copula endogeneity correction's advantages contributing to its wide usage include its broad applicability and high feasibility, as compared with alternative methods (Table 1). The directed acyclic graphs (DAGs) in Figure 2 explicitly include the unobserved error term E and highlights the important role of P–E dependence. Conceptually, copula correction addresses the general cases represented by the DAGs in Panels A, D, and E of Figure 2, while relaxing key assumptions and restrictions required by alternative methods shown in Panels B and C. Next, we highlight some common use cases for copula correction.

Table 1.

An Overview of Copula Correction with Alternative Approaches.

Approach	Estimator	Description	Data Requirements	Main Assumptions	When to Use
Experiment (lab or field)	Between-group or within-group before–after comparisons for treatment effect.	Random group assignment avoids spurious association.	Random assignment of treatment. Categorical focal variable (not continuous).	No treatment–error dependence. Randomization balances all confounders with good compliance to assigned treatment.	When treatment is categorical and it is feasible to manipulate focal variables without concerns of ethics and external validity.
Natural experiment	Regression discontinuity, interrupted time series, difference-in-difference.	Leverage random event/threshold to determine causal effects.	Availability of random and exogenous event/threshold.	No concomitant occurrence of confounding events around the focal event/threshold.	When natural event is available and special design and data requirements are fulfilled, and all confounders are accounted for.
Rich data	Regression adjustment, matching, or weighting.a	Use a rich set of control variables or panel data to control for observed and unobserved effects.	All potential confounders are accurately measured or proxied for by the control variables or by panel data.	No regressor–error dependence given observed control variables and unobserved panel fixed effects; all control variables are exogenous.	When researchers are confident that the set of control variables or panel data modeling captures all potential confounders such that no regressor–error dependence exists.
Instrumental variables (IVs)	Two-stage least squares (2SLS), generalized method of moments, control function.b	Use observed IVs to address unobserved confounders.	Few control variables. Requires IVs satisfying exclusion restriction (ER) and relevance.	No direct effects of IVs on the outcome (ER); IVs affect the endogenous regressor; all control variables are exogenous.	When the endogeneity concern exists in the data at hand, and strong and valid IVs are identified and supported by institutional knowledge.
Latent IV (LIV)c	Likelihood-based estimation.	Use latent discrete IVs to address unobserved confounders.	Few control variables. No observed IVs required. Endogenous regressors are required to be continuous and nonnormal with a normal error distribution.	Latent IVs are discrete and satisfy ER and relevance conditions; all control variables are exogenous.	When the endogeneity concern exists in the data at hand, the latent discrete IVs can be justified by institutional knowledge, and endogenous regressors are continuous.
SORE	Likelihood-based estimation.	Address regressor–error dependence via distribution-free odds ratio multivariate models that nest copulas as special cases.	Few control variables. No requirement of ER. Endogenous regressors are required to be nonnormal with a normal error distribution.	Distribution-free odds ratio multivariate models adequately capture regressor–error dependence; all control variables are exogenous.	When the endogeneity concern exists in the data at hand, and valid or strong IVs are not available for every endogenous regressor. Can handle both continuous and noncontinuous endogenous regressors.
Copula	Control function. Likelihood-based estimation. See the flowchart in Figure 5.	Address the regressor–error dependence via copulas. Copula control functions permit both copula and noncopula regressor–error dependence.	Few control variables. No requirement of ER. Either the model error or the endogenous part of error is symmetrically distributed. See the flowchart in Figure 5 for specific data requirements.	Gaussian copula (GC) adequately captures the dependence between regressor and the error (or the endogenous part of the error) (i.e., double robustness). A GC regressor–error dependence is sufficient but not necessary. All control variables are exogenous.	When the endogeneity concern exists in the data at hand, and valid or strong IVs are not available for every endogenous regressor. Can handle normally distributed endogenous regressors. See Table 2 for data requirements.

a

Regression adjustment includes methods such as ordinary least squares, random effects, and fixed effects for panel data with unobserved effects. Matching (via propensity score, Mahalanobis matching, synthetic control, and other methods) and weighting (via inverse probability weighting) control for confounding effects by balancing the distributions of a rich set of control variables.

b

See Petrin and Train (2010) for control function using IVs.

c

Like LIV, the HM and IH methods are also IV-free but require additional higher-order moment or heteroskedastic error conditions (see note 4). Methods in the table can be combined in a multimethod approach to improve the applicability, robustness, and quality of causal inference.

Case 1: leveraging experiments or acquiring perfect observational data is infeasible

Ensuring independence between P and E—such as through random assignment of P in experiments or by measuring and controlling for all confounders (the rich data approach)—are often impractical (Germann, Ebbes, and Grewal 2015). Randomized experiments are widely regarded as the gold standard for causal inference due to their high internal validity. However, they can be costly, ethically constrained, and often limited to discrete treatment levels (Table 1). Even when feasible, experiments may face challenges such as limited external validity, treatment noncompliance, and failure to balance all confounders. For example, a firm's randomized pricing experiment may not account for competitors’ strategic responses (Rutz and Watson 2019), and natural experiments may rely on events or thresholds that coincide with other events (Table 1). Similarly, rich observational data can be costly or infeasible to obtain, and may fail to fully or accurately capture all relevant variables.

In such cases, the distribution of the endogenous regressor P, via its dependence on E, carries useful information about model parameters. Standard methods that ignore this dependence—such as ordinary least squares (OLS), matching or weighting based on observables, or fixed effects models—assume exogeneity (Figure 2, Panel B) and can suffer from endogeneity bias due to omitted variables, measurement error, or simultaneity. By contrast, copula correction addresses this bias (Eckert and Hohberger 2023; Rutz and Watson 2019) by relaxing the exogeneity assumption and modeling the more general DAG in Figure 2, Panel A, which nests Figure 2, Panel B, as a special case. It does so without requiring experiments or exhaustive measurement of all confounders (Table 1).

Case 2: suitable IVs are unavailable

As the classical approach to addressing endogeneity bias, the IV method is based on the DAG shown in Figure 2, Panel C—a special case of Figure 2, Panel A. Like control variables W, the instrument Z must be relevant (affecting the endogenous regressor P) and exogenous (uncorrelated with the error term E). The crucial distinction is the ER: Z must not directly affect the outcome Y (i.e., β = 0 in Figure 2, Panel C), meaning Z is excluded from the outcome model in Equation 1. This ER condition, which is untestable in practice, is what differentiates valid instruments from merely exogenous variables and makes good IVs hard to find and justify. For instance, in the example of ice cream sales, weather is exogenous but would likely violate the ER condition, as it directly influences demand—people tend to buy more ice cream on hot days, even if prices remain constant. Moreover, the relevance and ER conditions often conflict: Variables that strongly predict the endogenous regressor may also directly influence the outcome. Thus, despite the theoretical appeal of the IV approach, identifying valid instruments in practice remains a major challenge, highlighting the need for more flexible methods to address regressor endogeneity.

Other IV-free methods (LIV, IH, HM) do not require an observed instrument Z but assume that P can be linearly decomposed as P = Z + v, where Z is unobserved and meets the ER condition (Park and Gupta 2012). Identification relies on distributional assumptions: The instrument Z is discrete in LIV, skewed in HM, and heteroskedastic in IH. ⁴ Although these methods circumvent the need to find instruments, researchers must still justify the ER condition for an unobserved instrument Z, whose nature and interpretation are ambiguous.

Unlike IV and these other IV-free methods, copula correction requires no instrument—observed or latent—and thus avoids justifying an instrument Z that satisfies the ER condition and causally affects P. Exogenous control variables in W can enhance the precision and identification of copula correction. A good starting place to find such W is the existing exogenous control variables in OLS or IV models. ⁵ Unlike instruments, these control variables (e.g., exogenous demand shocks) need not satisfy the strict ER condition. That is, these Ws do not have to be excluded from the structural model and can affect the outcome directly. Such Ws are much more readily available than IVs, and because empirical association between the candidate W and P is sufficient (Figure 2, Panel E), researchers using copula correction do not need to argue for the causal pathways between W and P as they do in the case of IVs. Thus, copula correction substantially enhances the feasibility of addressing endogeneity.

Case 3: multimethod causal inference is conducted as a robustness check

Examples of this case are when IVs exist but are imperfect with questionable validity or weak relevance, or when control variables used in rich data methods have questionable comprehensiveness, accuracy, or validity of exogeneity. In such situations, copula correction can be used alongside these approaches to compare results and cross-validate findings (Germann, Ebbes, and Grewal 2015; Papies, Ebbes, and Van Heerde 2017; Qian and Xie 2024).

Case 4: a combination of multiple methods is required to address endogeneity

In this case, for instance, an IV may be available for the treatment variable, while potential moderators are endogenous and lack valid instruments. In such cases, copula correction can be used in conjunction with the IV to address multiple endogenous regressors. Similarly, copula methods can be combined with methods like regression adjustment or SORE to address residual endogeneity and mixed (continuous or discrete) endogenous variables, respectively.

Summary and trade-offs of copula correction

As discussed previously, copula correction can serve as either a primary or complementary method to address regressor endogeneity. It requires no experiments, exhaustive measurement of all confounders, or IVs satisfying the ER condition, and is feasible and tractable to use in a wide variety of settings. However, these advantages come with trade-offs. In exchange for randomization, exhaustive measurement, or ER, copula correction requires adequately capturing regressor–error dependence with copulas—a process outlined subsequently.

How Does Copula Correction Work? A Primer on Copula Correction

Copula correction, first proposed by Park and Gupta (2012), is based on the idea that adequately capturing regressor–error dependence can yield unbiased causal estimates. To address the endogeneity of P in Equation 1, Park and Gupta (2012) (henceforth PG) propose two estimation methods based on a GC dependence model for the joint distribution of (P_i, E_i) and a normal structural error. The first maximizes the likelihood derived from this joint distribution. The second uses a simpler control function approach that rewrites the maximum likelihood estimation (MLE) as a regression augmented with a copula-generated term ${\mathrm{P}}_{\mathrm{i}}*={\mathrm{\Phi }}^{-1}(\hat{\mathrm{F}}({\mathrm{P}}_{\mathrm{i}}))$ , where Φ⁻¹ is the inverse standard normal cumulative distribution function (CDF) and $\hat{\mathrm{F}}({\mathrm{P}}_{\mathrm{i}})$ is the empirical CDF of P. Both estimators were derived under fully specified likelihood, leading to the belief that copula correction is likelihood-based (Table 2). Consequently, estimation and inference may be sensitive to likelihood misspecifications and depend on a set of strict data requirements and five traditional assumptions (TAs; see Table 2) for identification.

Table 2.

An Overview of Identification Assumptions, Data Requirements, and Methodological Aspects of Copula Correction.

Points to Consider	Traditional Assumptions	New Understandings and Recent Advances
Identification assumptions and data requirements	TA1. The structural error distribution is normal. a	The structural error distribution can be nonnormal and left unspecified. (See Assumption 1 in Table 4, Web Appendix B, and Feature 3 in Table 5.)
	TA2. The GC model describes the joint distribution of regressors and the error term. a	Permit both GC and non-GC regressor–error joint distributions. (See Assumption 2 in Table 4, Web Appendix B, and Features 4 and 5 in Table 5.)
	TA3. Endogenous regressors and exogenous regressors are independent. The copula term is Cp = P*=Φ−1(F(P))a, where F(P) is the marginal CDF of the endogenous regressor P.	Endogenous regressors and exogenous regressors can be correlated. With correlated endogenous and exogenous regressors, the copula correction term Cp|w = P* – δW*, which is the residual from a first-stage model that regresses P* on W* and removes the effects of exogenous regressors from P*. (See Features 6 and 9 in Table 5.)b
	TA4. Endogenous regressors must be nonnormal. a	Endogenous regressors can be normal. Relevant exogenous regressors can be leveraged to handle normally distributed endogenous regressors. (See Feature 7 in Table 5.)
	TA5. Endogenous regressors have sufficient support. a	Noncontinuous endogenous regressors (binary, discrete regressors with few levels, or semicontinuous regressors) can be handled via the SORE method. (See Feature 8 in Table 5.)
Methodological Aspects
Panel data	Copula correction for panel data can be handled in the same way as cross-sectional data.	Copula correction for panel data requires special handling, such as fixed-effect transformation of panel outcome models (Haschka 2022) or proper calculation of copula correction terms based on time-demeaned regressors (Table 6).
Estimation approach	Estimation is likelihood-based. Even if control functions are used, one needs to specify the joint distribution of the error term and regressors (Eckert and Hohberger 2023).	Estimation can be likelihood-based, or alternatively performed via control functions without specifying the error and regressor distributions, the joint distribution of the structural error term and regressors, or any likelihood function. (See Features 1 and 2 in Table 5.) Thus, copula correction can be rendered less sensitive to distributional assumptions, computationally simpler, and applicable to broader applications.
Models with intercept	The sample size needs to be very large to avoid estimation bias if the structural model includes the intercept term ( Becker, Proksch, and Ringle 2022 ).	Judicious handling of copula transformation of regressors removes the bias described in Becker, Proksch, and Ringle (2022) . (See the “Methodological Background” section and Equation 9 therein.)
Endogenous interaction terms	Adding copula correction terms for endogenous interaction terms is thought to help control their endogeneity, despite it being sufficient to add copula correction terms for first-order endogenous terms only (Papies, Ebbes, and Van Heerde 2017).	Theoretical proof and empirical evaluation described in the “Methodological Background” section (1) extends the results of Papies, Ebbes, and Van Heerde (2017) to more general copula correction methods and (2) demonstrates a stronger result that adding the unnecessary high-order copula correction terms is suboptimal and has significant adverse effects.

Notes: TA: traditional assumption.

^aSee Park and Gupta (2012, 2024), Papies, Ebbes, and Van Heerde (2017), Becker, Proksch, and Ringle (2022), Haschka (2022), Eckert and Hohberger (2023), Papies, Ebbes, and Feit (2023), Qian and Xie (2024), Breitung et al. (2024), and Liengaard et al. (2024) for description of these traditional assumptions.

^bThe copula term can be generally expressed as Φ^-1(F(P|W)), where F(P|W) is the conditional CDF of P given the exogenous regressors in W and can be replaced with model-free nonparametric estimates (Hu, Qian, and Xie 2025).

Contrary to common belief, we show that copula control function methods require neither a normal error distribution nor GC regressor–error dependence, and can work under substantially less strict conditions (Table 2). To highlight how copula correction works under these weaker conditions (and derive general copula control functions), consider the DAG in Figure 2, Panel D, which decomposes the structural error as E_i = U_i + ξ_i. Here, U_i is the error's endogenous part as the combined effect of unobserved confounders, ξ_i is an exogenous disturbance satisfying $\mathbb{E}({\xi }_{\mathrm{i}}|{\mathrm{P}}_{\mathrm{i}},{\mathrm{W}}_{\mathrm{i}})=0$ , and $\mathbb{E}$ is the expectation operator. We rewrite Equation 1 as

$${\mathrm{Y}}_{\mathrm{i}}=\mathbb{E}({\mathrm{Y}}_{\mathrm{i}}|{\mathrm{P}}_{\mathrm{i}},{\mathrm{W}}_{\mathrm{i}})+{ϵ}_{\mathrm{i}}=\mu +\alpha {\mathrm{P}}_{\mathrm{i}}+{\beta }^{\mathrm{\prime }}{\mathrm{W}}_{\mathrm{i}}+\stackrel{\mathrm{Structural}\mathrm{error}\mathrm{E}}{\overbrace{\underset{\mathrm{Endogenous}}{\underbrace{\mathbb{E}({\mathrm{U}}_{\mathrm{i}}|{\mathrm{P}}_{\mathrm{i}},{\mathrm{W}}_{\mathrm{i}})}}+\underset{\mathrm{Exogenous}}{\underbrace{{ϵ}_{\mathrm{i}}}}}}.$$

Equation 2 decomposes the error E_i into two parts: (1) $\mathbb{E}({\mathrm{U}}_{\mathrm{i}}|{\mathrm{P}}_{\mathrm{i}},{\mathrm{W}}_{\mathrm{i}})$ , the expected omitted effect U_i given regressors (P_i, W_i), which is the error's endogenous part correlated with regressors, and (2) the exogenous part ${ϵ}_{\mathrm{i}}$ uncorrelated with all regressors and $\mathbb{E}({\mathrm{U}}_{\mathrm{i}}|{\mathrm{P}}_{\mathrm{i}},{\mathrm{W}}_{\mathrm{i}})$ . ⁶ Important from Equation 2 is that one does not need to know unobserved U_i to control for endogeneity: $\mathbb{E}({\mathrm{U}}_{\mathrm{i}}|{\mathrm{P}}_{\mathrm{i}},{\mathrm{W}}_{\mathrm{i}})$ sufficiently captures the error's endogenous part and completely controls the confounding effects of omitted variables. If $\mathbb{E}({\mathrm{U}}_{\mathrm{i}}|{\mathrm{P}}_{\mathrm{i}},{\mathrm{W}}_{\mathrm{i}})$ is known and added to the outcome model as an offset term, the resulting OLS regression yields consistent model estimates, as the new error term ${ϵ}_{\mathrm{i}}$ is uncorrelated with all regressors $({\mathrm{P}}_{\mathrm{i}},{\mathrm{W}}_{\mathrm{i}},\mathbb{E}({\mathrm{U}}_{\mathrm{i}}|{\mathrm{P}}_{\mathrm{i}},{\mathrm{W}}_{\mathrm{i}}))$ . This approach clearly hinges on whether the control function $\mathbb{E}({\mathrm{U}}_{\mathrm{i}}|{\mathrm{P}}_{\mathrm{i}},{\mathrm{W}}_{\mathrm{i}})$ can be recovered.

Copula correction based on Equation 2 proceeds by noting that the dependence between the endogenous regressor P_i and the omitted term U_i, unexplained by the control variables in W_i, can be captured by copula models. This copula dependence structure and economic theory ⁷ enable the derivation and recovery of control functions that break the dependence between endogenous regressors and the structural error. As discussed in the next subsection, the copula dependence model is chosen for a number of reasons, including its flexibility, its wide applicability, and the ability to faithfully maintain regressor distributional features and multivariate dependence crucial for model identification.

Yang, Qian, and Xie (2025) introduce the 2sCOPE correction procedure that simultaneously recovers the control function and structural parameters. Under a joint GC model for all regressors and the error, they show that the residual from the first-stage model for the endogenous regressor P_i breaks the regressor–error dependence, serving as the control function up to a constant. As a result, Equation 2 becomes

$$\begin{array}{r}{\mathrm{Y}}_i=\mu +\alpha {\mathrm{P}}_{\mathrm{i}}+{\beta }^{\mathrm{\prime }}{\mathrm{W}}_{\mathrm{i}}+\gamma {\mathrm{C}}_{\mathrm{i},\mathrm{p}|\mathrm{w}}+{ϵ}_{\mathrm{i}},\\ \mathrm{where}{\mathrm{C}}_{\mathrm{i},\mathrm{p}|\mathrm{w}}={\mathrm{P}}_{\mathrm{i}}*-{\hat{\delta }}^{\mathrm{\prime }}{\mathrm{W}}_{\mathrm{i}}*;\end{array}$$

(3)

γC_i,p|w is the control function $\mathbb{E}({\mathrm{U}}_{\mathrm{i}}|{\mathrm{P}}_{\mathrm{i}},{\mathrm{W}}_{\mathrm{i}})$ capturing the error's endogenous part, and C_i,p|w is the copula term; ${\mathrm{P}}_{\mathrm{i}}^{*}={\mathrm{\Phi }}^{-1}({\hat{\mathrm{F}}}_{\mathrm{P}}({\mathrm{P}}_{\mathrm{i}}))$ and ${\mathrm{W}}_{\mathrm{i},l}*={\mathrm{\Phi }}^{-1}({\hat{\mathrm{F}}}_{{\mathrm{W}}_l}({\mathrm{W}}_{\mathrm{i},l}))$ for the l^th (l = 1, …, L) variable in W are copula transformed regressors. Recall that ${\mathrm{P}}_{\mathrm{i}}^{*}$ is the copula term from the PG method. The 2sCOPE approach (Equation 3) removes the part of ${\mathrm{P}}_{\mathrm{i}}^{*}$ that is correlated with exogenous regressors and uses the remaining cleaned part to control for endogeneity. Conditioning on this cleaned part in P_i makes the new error ${ϵ}_{\mathrm{i}}$ independent of all regressors (P_i, W_i, C_i,p|w). As a result, adding C_i,p|w as a control variable allows standard methods like OLS to yield consistent estimates, with $\hat{\gamma }{\mathrm{C}}_{\mathrm{i},\mathrm{p}|\mathrm{w}}$ acting as the estimated control function.

Operationally, 2sCOPE proceeds in two steps: (1) regresses ${\mathrm{P}}_{\mathrm{i}}^{*}$ on ${\mathrm{W}}_{\mathrm{i}}^{*}$ , and (2) adds the first-stage residuals ${\mathrm{C}}_{\mathrm{i},\mathrm{p}|\mathrm{w}}={\mathrm{P}}_{\mathrm{i}}^{*}-{\hat{\delta }}^{\mathrm{\prime }}{\mathrm{W}}_{\mathrm{i}}^{*}$ to control for endogeneity, where $\hat{\delta }$ denotes first-stage coefficient estimates. ⁸ If no endogeneity exists, the true coefficient γ on C_i,p|w is zero. Thus, with the copula control function approach, one can test the presence of endogeneity by statistically testing whether the coefficient γ for the copula term C_i,p|w is zero or not.

The two-stage residual approach has been adapted to various settings (see the subsequent “Methodological Background” section). Hu, Qian, and Xie (2025) develop a two-stage nonparametric copula control function (2sCOPE-np) that generalizes and unifies these methods, expressing the copula term in Equation 3 generally as ${\mathrm{C}}_{\mathrm{i},\mathrm{p}|\mathrm{w}}={\mathrm{\Phi }}^{-1}(\hat{\mathrm{F}}({\mathrm{P}}_{\mathrm{i}}|{\mathrm{W}}_{\mathrm{i}}))$ . Here, the conditional CDF of P given W, F(P_i | W_i), can be consistently estimated using a nonparametric conditional CDF estimator, $\hat{\mathrm{F}}({\mathrm{P}}_{\mathrm{i}}|{\mathrm{W}}_{\mathrm{i}})$ . Thus, one can estimate the control function nonparametrically without specifying first-stage auxiliary models for regressors.

The copula control function offers an alternative to the control function of Petrin and Train (2010). Unlike their approach, copula correction requires no IVs that must satisfy the strict ER condition, a stronger requirement than exogeneity. No arguments for the nature, direction, or forms of relationships between W and P are needed: Empirical association between P and W is sufficient, and 2sCOPE-np employs first-stage model-free control functions. Thus, copula correction greatly increases the feasibility of endogeneity correction.

Table 3 summarizes the 2sCOPE estimation algorithm for the general case of multiple endogenous regressors. For K continuous endogenous regressors (P₁, …, P_k), the copula control function approach estimates the following augmented regression model:

$${\mathrm{Y}}_{\mathrm{i}}=\mu +\sum _{\mathrm{k}=1}^{\mathrm{K}}{\alpha }_{\mathrm{k}}{\mathrm{P}}_{\mathrm{i},\mathrm{k}}+{\beta }^{\mathrm{\prime }}{\mathrm{W}}_i+\sum _{\mathrm{k}=1}^{\mathrm{K}}{\gamma }_{\mathrm{k}}{\mathrm{C}}_{\mathrm{i},{\mathrm{p}}_{\mathrm{k}}|\mathrm{w}}+{ϵ}_{\mathrm{i}},$$

(4)

where the copula term ${\mathrm{C}}_{\mathrm{i},{\mathrm{p}}_{\mathrm{k}}|\mathrm{w}}={\mathrm{P}}_{\mathrm{i},\mathrm{k}}^{*}-{\hat{\delta }}_{\mathrm{k}}^{\mathrm{\prime }}{\mathrm{W}}_{\mathrm{i}}*$ . The algorithm in the subsequent Equation 9 is used for the copula transformation of these regressors, including both continuous and noncontinuous exogenous regressors in W (Web Appendices E.2 and E.3 in Yang, Qian, and Xie 2025). ⁹ The 2sCOPE-np replaces Stage 1 with the following: For the k^th endogenous regressor P_k, perform nonparametric kernel conditional CDF estimation for P_k given W (Web Appendix D, Equation W35 or Equation W37) and obtain the copula term ${\mathrm{C}}_{\mathrm{i},{\mathrm{p}}_{\mathrm{k}}|\mathrm{w}}={\mathrm{\Phi }}^{-1}(\hat{\mathrm{F}}({\mathrm{P}}_{\mathrm{i},\mathrm{k}}|{\mathrm{W}}_{\mathrm{i}}))$ , which involves no copula transformations of individual regressors. In Equation 4, $\sum _{\mathrm{k}=1}^{\mathrm{K}}{\gamma }_{\mathrm{k}}{\mathrm{C}}_{\mathrm{i},{\mathrm{p}}_{\mathrm{k}}|\mathrm{w}}$ is the linear combination of the K copula terms $\{{\mathrm{C}}_{\mathrm{i},{\mathrm{p}}_{\mathrm{k}}|\mathrm{w}}\}$ used to control for endogenous regressors and is the copula control function with multiple endogenous regressors.

Table 3.

Summary of the 2sCOPE Estimation Procedure.

Stage 1 Obtain empirical CDFs for each regressor in Pi and Wi, ${\hat{\mathrm{F}}}_{{\mathrm{P}}_{\mathrm{k}}}(\cdot )$ and ${\hat{\mathrm{F}}}_{{\mathrm{W}}_{\mathrm{l}}}(\cdot )$ . Compute ${\mathrm{P}}_{\mathrm{i},\mathrm{k}}*={\mathrm{\Phi }}^{-1}({\hat{\mathrm{F}}}_{{\mathrm{P}}_{\mathrm{k}}}({\mathrm{P}}_{\mathrm{i},\mathrm{k}}))$ and ${\mathrm{W}}_{\mathrm{i},\mathrm{l}}^{*}={\mathrm{\Phi }}^{-1}({\hat{\mathrm{F}}}_{{\mathrm{W}}_{\mathrm{l}}}({\mathrm{W}}_{\mathrm{i},\mathrm{l}}))$ using the copula transformation algorithm defined in Equation 9. Regress ${\mathrm{P}}_{\mathrm{i},\mathrm{k}}^{*}$ on ${\mathrm{W}}_{\mathrm{i}}^{*}$ and obtain residual ${\mathrm{C}}_{\mathrm{i},{\mathrm{p}}_{\mathrm{k}}|\mathrm{w}}={\mathrm{P}}_{\mathrm{i},\mathrm{k}}^{*}-{{\delta }^{\mathrm{\prime }}}_{\mathrm{k}}{\mathrm{W}}_{\mathrm{i}}^{*}$ , which removes the component related to exogenous regressors.

Stage 2 Add ${\mathrm{C}}_{\mathrm{i},{\mathrm{p}}_{\mathrm{k}}|\mathrm{w}}$ to the outcome structural regression model as a control variable to correct for endogeneity of Pk. The augmented regression model takes the form of Equation 4 (or Equation 12 when the model contains higher-order or interaction terms of regressors).

Notes: W can contain a mixture of continuous and noncontinuous variables (also see note 9 for alternative 2sCOPE implementations that bypass copula transformations of noncontinuous variables in W).

Identification Assumptions and Data Requirements for Copula Correction

As shown previously, copula correction can be achieved using control functions via the method of moments ¹⁰ estimation of the augmented regression in Equation 4, without needing to specify distributions for the error and individual regressors, the regressor–error joint distribution, or any likelihood function (Table 2). This increases the robustness of copula correction and enables us to focus on the most essential assumptions and data requirements for copula correction. We next contrast the assumptions (Table 4) used to derive the preceding general 2sCOPE procedures with the TAs listed in Table 2.

Table 4.

Assumptions Used to Derive 2sCOPE/2sCOPE-np.

Assumptions

Assumption 1. Either the error Ei or its endogenous part Ui is normally distributed (Double Robustness 1).

Assumption 2. Either (P, E) | W or (P, U) | W follows a Gaussian copula (Double Robustness 2).a

Assumption 3. Full rank of all regressors and cov(W, E) = 0.b

Assumption 4. P is continuous and the copula term Cp|w is linearly independent of (1, P, W).

Assess Assumptions Assumptions 1 and 2 are unverifiable from data alone. Though used in method derivation, Assumptions 1 and 2 are not strictly required: Copula correction is robust to symmetric nonnormal distributions of U and to a range of departures from GC dependence (Web Appendix B). Their plausibility should be evaluated based on identified sources of endogeneity and theoretical reasoning, along with using residual inspection and Boundary Condition 1 to detect signs of possible violations. If needed, revise the model (e.g., transform outcomes/regressors if U is suspected to be highly skewed) or copula correction strategies (Web Appendix C). To satisfy Assumption 3, researchers should take care to properly specify the structural model, such as avoiding mistakenly including redundant regressors. To ensure cov(W, E) = 0, include only exogenous control variables in W based on institutional knowledge. Assumption 4 for 2sCOPE requires that the continuous P or one correlated and continuous regressor in W is nonnormal, which can be verified by Boundary Condition 2.c

Boundary Conditions Dependence model misspecification: Inspect the inflation of standard errors of copula-corrected estimates relative to those of uncorrected estimates; inflation > 6 flags GC dependence model misspecification issues when other assumptions hold. Regressor distribution and relevance requirement: The continuous P is nonnormal (normality test p < .05), or for nearly normal P, one exogenous W is continuous, sufficiently nonnormal (normality test p < .001), and sufficiently relevant (first-stage F-statistic > 10). A minimum sample size of 300 is recommended for satisfactory performance of 2sCOPE-np.d

^a

Assumption 2 means that the variation in P unexplained by W follows a GC model jointly with E (or its endogenous part U). Further, 2sCOPE assumes a joint GC model for all regressors (P, W), an assumption not required by 2sCOPE-np.

^b

Full rank means rank(X X) = Q, where Q is the number of columns in X = (1, P, W).

^c

Assumption 4 allows normally distributed P to be identified via exogenous control regressors. Also, 2sCOPE-np can leverage exogenous regressors to handle noncontinuous P.

^d

See Web Appendix D, Figure W4.

Nonnormal error distribution and noncopula regressor–error dependence

Assumptions 1 and 2 (Table 4) mean that the copula control function methods require neither a normal error distribution nor GC regressor–error dependence (TAs 1 and 2 in Table 2) and can work under substantially less rigid conditions than previously believed. In fact, the same 2sCOPE/2sCOPE-np procedure can be derived under both Figure 2, Panel A, and Figure 2, Panel D, and hence possesses a desirable property of double robustness (Web Appendix B): When a GC model adequately captures either the regressor–error dependence or regressor–confounder dependence unexplained by exogenous regressors, the copula corrects endogeneity bias. The reason is intuitive: The exogenous part of E_i, ξ_i, simply adds noise but does not affect endogeneity correction. Because ξ_i does not need to follow a normal distribution (or any GC assumption) for the augmented regression to correct bias, copula control functions do not require the error E_i to be normally distributed or follow a specific copula structure jointly with P_i. Consequently, the normal error distribution and the GC regressor–error dependence are only sufficient but not necessary conditions for copula control functions to work.

In many settings, it is plausible to assume that E_i is normally distributed (Ebbes et al. 2005; Yang, Chen, and Allenby 2003) or U_i is normally distributed as a sum of many confounders’ effects (Breitung, Mayer, and Wied 2024; Qian and Xie 2024; Yang, Qian, and Xie 2025), ¹¹ satisfying Assumption 1. Furthermore, in many settings, the GC model can adequately capture the dependence between U_i (or E_i) and P_i unexplained by exogenous regressors, satisfying Assumption 2. The GC model has desirable properties, making it widely used and applicable in empirical research to robustly capture multivariate dependence that traditional models, such as linear additive dependence models, often fail to capture (Danaher and Smith 2011; Qian and Xie 2024). GC permits the full (−1,1) range of correlation coefficients and is readily extensible and computationally scalable to more than two variables.

Moreover, GC separates modeling dependence from modeling individual variables’ distributions. Thus, distribution-free GC models can capture regressor–error dependence irrespective of (potentially complex) regressors’ distributional features, while nonparametrically preserving these distributional features essential for model identification. Copula correction also demonstrates robustness to a range of departures from the GC assumption. Consequently, copula correction has broad applicability and has become a valuable resource in the toolkit for handling regressor endogeneity in various fields (Figure 1). In many applications, including those in marketing (Web Appendix A), copula correction yields credible findings that are consistent with theoretical predictions, attesting to its effectiveness and applicability.

Correlated endogenous and exogenous regressors

The 2sCOPE method extends the PG method to account for correlated exogenous and endogenous regressors. When P and W are independent, the first-stage coefficient δ = 0 and the 2sCOPE copula term C_i,p|w in Equation 3 reduces to ${\mathrm{P}}_{\mathrm{i}}*={\mathrm{C}}_{\mathrm{i},\mathrm{p}}={\mathrm{\Phi }}^{-1}(\hat{\mathrm{F}}({\mathrm{P}}_{\mathrm{i}}))$ used in the PG method (TA 3 in Table 2). ¹² But when P and W are dependent, using P* as the copula correction term will confound the control function with effects of exogenous regressors, biasing both the control function and model parameter estimates.

The 2sCOPE control function removes the exogenous regressors’ influences on the entire distribution of P rather than just on some aspects such as its mean. In contrast, some methods use a mean regression model for P given W, ¹³ implicitly assuming that W only affects the mean of P, which is known to be violated for bounded, truncated, or discrete endogenous regressors ¹⁴ and can be questionable for unbounded continuous regressors (Chen 2007; Danaher and Smith 2011), leading to biased control function and model estimates. The 2sCOPE/2sCOPE-np procedures are more flexible, permitting W to affect not only the mean but also higher moments of P (Danaher and Smith 2011; Hu, Qian, and Xie 2025; Yang, Qian, and Xie 2025) and providing a model-free nonparametric adjustment. While correlated exogenous regressors complicate copula correction and need to be carefully dealt with, they also provide opportunities to relax key identification constraints, as seen subsequently.

Normally distributed endogenous regressors

Under the PG method, the source of identification comes from distributional features of the endogenous regressor P (nonnormally distributed). If P is normally distributed, then the PG method fails by violating the full rank condition of the regressor matrix (TA 4 in Table 2). Although distributional shapes of endogenous regressors are observed and regressor normality can be tested, this requirement limits the applicability of copula correction because many important endogenous regressors have close to normal distributions (Eckert and Hohberger 2023).

The 2sCOPE procedure relaxes this restriction by leveraging relevant exogenous regressors to identify the effects of endogenous regressors with insufficient nonnormality. With 2sCOPE, the source of identification comes from either the nonnormal distributional features of the variations in P unexplained by W (a nonnormal conditional distribution for F(P | W)) or nonlinear relationships between P and W, ensuring full rank of the copula-augmented model (Assumption 4 in Table 4). Even if P is normally distributed, Assumption 4 is satisfied and the copula model is identified when a continuous exogenous regressor in W is nonnormally distributed and nonlinearly related to $\mathrm{P}$ (Yang, Qian, and Xie 2025).

Noncontinuous endogenous regressors

Noncontinuous endogenous regressors—such as binary variables, counts with small means, or semicontinuous variables—have many ties (observations with the same value) and limited support, creating identification issues in copula correction (TA 5 in Table 2). These issues arise from plateaus in discrete CDFs and the nonuniqueness of their inverses, which can bias copula-based control functions. To address this, Qian and Xie (2024) proposed a likelihood-based SORE approach to bypass inverse mapping and accommodate such regressors. ¹⁵

Limitations and Boundary Conditions of Copula Correction

The preceding considerations show that copula control function methods work under substantially less strict conditions than previously believed, increasing robustness and applicability. With the underlying assumptions and data requirements being met, copula correction can be a powerful tool for addressing endogeneity bias using nonexperimental data. However, some boundary conditions warrant checking to minimize potential pitfalls and inappropriate use of copula correction (Table 4).

Limitation 1: multicollinearity

Like weak instruments (Staiger and Stock 1997), copula correction can be ill-behaved when copula terms exhibit severe collinearity with existing regressors. The copula terms capture the error's endogenous part and thus are expected to correlate with endogenous regressors. However, copula-augmented models become weakly identified or nonidentified when copula terms are severely or perfectly collinear with existing regressors (i.e., failure of Assumption 4 in Table 4). This can occur when regressors and the error jointly follow a multivariate normal distribution (Web Appendix C, Table W6). Even in identified models, strong collinearity inflates standard errors, reduces statistical power of hypothesis testing, and inflicts finite-sample bias. While 2sCOPE identifies nearly normal endogenous regressors, it requires sufficiently relevant and nonnormal exogenous control regressors to prevent severe collinearity, as described in the following guideline (Yang, Qian, and Xie 2025):

Guideline 1: Check Boundary Condition 2 (requirements on regressor distributions and relevance) in Table 4 on the threshold values of the p-value (for nonnormality) of endogenous regressors; for nearly normal endogenous regressors, check the first-stage F-statistic (for relevance) and the p-value (for nonnormality) of candidate exogenous control regressors. ¹⁶

Limitation 2: violations of Assumptions 1 and 2

Assumption 1 is not strictly required, as copula correction is robust to symmetric nonnormality in U and skewed E (Web Appendix B, Table W5). Yet, strong skewness in both U and E can cause bias. Since neither is observed, this assumption should be assessed using theory (see note 11), aided by inspecting residuals. Because copula control functions permit asymmetric error E (Web Appendix B, Tables W4 and W5), skewed residuals alone do not indicate failure. However, if U is also suspected to be highly skewed, revise model specifications (e.g., transform variables or add controls) to avoid misspecification bias.

Assumption 2 is also not strictly necessary, as copula correction is robust to a range of non-GC dependence structures (Web Appendix B, Table W5; Haschka 2022; Park and Gupta 2012; Yang, Qian, and Xie 2025). Nonetheless, gross violations can cause bias (Eckert and Hohberger 2023). Since neither E nor U is observed, its plausibility should be evaluated based on the sources of endogeneity in the application—such as theoretical knowledge about the nature of omitted variables and their links to endogenous regressors, institutional context, and diagnostics (Boundary Condition 1 in Table 4). Broadly speaking, copula models flexibly capture this dependence via unbounded copula-transformed variables without assuming any specific parametric forms for regressors’ marginal distributions, making copula models widely applicable. By contrast, in linear additive endogeneity models an endogenous regressor is itself linearly related to the error; this often creates logical inconsistencies and fails to capture multivariate dependence, limiting its practical applicability (see Danaher and Smith 2011; Web Appendix D.7 in Qian and Xie 2024). Hence, theoretical considerations can help guide the selection of appropriate dependence models, indicating that copula dependence models are suitable for capturing multivariate regressor–error dependence in a much broader range of applications. Meanwhile, we emphasize that the GC assumption warrants attention from users of copula methods as its misspecification may introduce bias.

Misspecifying regressor–error dependence can weaken model identification and inflate standard errors (Haschka 2022; Park and Gupta 2012; Qian and Xie 2024). When the true dependence follows a linear model, copula correction produces biased estimates with huge standard errors even if other assumptions hold (Haschka 2022; Qian and Xie 2024). Theoretical and empirical evidence from these studies as summarized in Web Appendix C indicates that significantly inflated standard errors serve as warning signs of dependence misspecification. We introduce a diagnostic statistic, ICON (Web Appendix C), as the ratios of the standard errors of copula-corrected estimates to those of uncorrected estimates to detect GC misspecifications. Prior studies (Haschka 2022; Qian and Xie 2024; Yang, Qian, and Xie 2025) and our own evaluation (Web Appendix C) have shown that in misspecified models, standard errors of copula-corrected estimates are typically more than eight to ten times those of uncorrected estimates. To be conservative, we suggest ICON > 6 as a threshold value for detecting GC misspecification, following this guideline:

Guideline 2: In addition to theoretical considerations, check Boundary Condition 1 (dependence model misspecification) in Table 4 to detect warning signs of potential regressor–error dependence misspecification and mitigate the risk of misspecification bias.

When GC misspecification is suspected, researchers can add relevant control variables, refine copula correction (Web Appendix C, Table W8), or use other correction methods. Assumption 2 only requires that residual dependence between P and U given W follows GC, which may hold after adding suitable controls even if unconditionally P and U do not.

Limitation 3: exogeneity of control variables in W

Assumption 3 is shared by methods such as OLS and IV regression. While these methods and copula correction do not require W, relevant control variables are often included to justify exogeneity of all regressors in OLS, satisfy ER for IV, or enhance precision and identification in copula correction. The selection of suitable control variables should be guided by institutional knowledge and study goals (see Yang, Qian, and Xie [2025] for examples of suitable and unsuitable Ws). These control variables need to be exogenous to ensure the consistency of these methods and copula correction, following this guideline:

Guideline 3: Use only necessary exogenous control variables for causal estimation. Control variables suspected to be endogenous (e.g., potential mediators and colliders) should be modeled as endogenous or excluded to avoid overcontrol bias.

Impacts of Copula Correction

In many cases, copula correction provides a feasible approach to controlling for the thorny regressor endogeneity issue and offers opportunities for optimal managerial decision-making, as illustrated in the following running example of price sensitivity estimation.

Managers and policymakers are often interested in understanding price sensitivity for category demand. This example (Example 1) estimates price sensitivity in the diaper category using the IRI Academic store scanner purchase data for a focal store in the Buffalo, New York, market from 2002 to 2006 (261 weeks). Price may be endogenous due to unobserved variables (e.g., product characteristics, retailer pricing decisions, number of shelf facings) that, when omitted from a model, become part of the structural error. These unobserved characteristics should induce positive correlation between price and the error term, thereby causing the OLS estimate of price sensitivity to bias toward zero (i.e., be less negative). We show subsequently that the OLS price elasticity estimate here is −1.367, significantly less than the copula-corrected price elasticity estimate of −2.205, a 61% difference reflecting a large impact of a “wrong” estimate. The manager underestimates consumer price sensitivity using OLS, and mistakenly sets the price too high, resulting in lost revenue and profit. The subsequent analysis shows that the OLS price estimate will yield 30% less profit compared with using the copula-corrected price sensitivity estimate (Figure 3).

Figure 3.

Example 1: Impact of Copula Correction on Price Sensitivity Estimation.

Meta-analyses of studies that compare estimates after endogeneity correction to uncorrected estimates also find similar differences. Bijmolt, Van Heerde, and Pieters (2005) find price elasticity of −2.47 without endogeneity correction, but −3.74 with correction. Sethuraman, Tellis, and Briesch (2011) find that “advertising elasticity is lower when endogeneity in advertising is not incorporated in the model” (p. 470). ¹⁷ With personal selling (i.e., sales force), models that account for endogeneity have lower elasticity (.282) than models without endogeneity correction (.373), a significant difference of .091 that importantly represents an overestimation of 32% (Albers, Mantrala, and Sridhar 2010). The importance of endogeneity correction is apparent: Without correction, managers and academics likely experience underestimated effects of pricing and advertising and overestimated effects of sales force.

Methodological Background

In this section, we discuss methodological aspects of implementing copula correction. We first acquaint readers with recent advances in copula correction, then speak to copula correction in panel data, show proper construction of the copula, address inconsistencies in copula correction for higher-order endogenous terms, and explain how to obtain standard errors.

Methods to Relax Assumptions of Copula Correction

Recent methodological advances relax key assumptions and data requirements of the PG method, broadening the applicability of copula correction. These methods differ in their features (Table 5) and fall into two broad classes: moment-based two-stage control function methods and likelihood-based methods. We contribute to this growing literature by systematically comparing these approaches to highlight their strengths and limitations and to guide practitioners in selecting suitable methods for their empirical context.

Table 5.

Copula Correction Methods with Relaxed Assumptions and Reduced Data Requirements.

Features	Methods
1. Control function approach without specifying model likelihood	2sCOPE proceduresa
2. Likelihood-based joint estimation	Haschka (2022); SORE
3. Permit nonnormal structural error	2sCOPE proceduresa
4. Permit both GC and non-GC regressor–error dependence	SORE; 2sCOPE-np; Yang, Qian, and Xie (2024, 2025); Liengaard et al. (2025);
5. Permit both GC and non-GC regressor–confounder dependence	SORE; 2sCOPE-np
6. First-stage model-free control function	2sCOPE-np
7. Handle normal endogenous regressors	2sCOPE-np; Yang, Qian, and Xie (2024, 2025); Liengaard et al. (2025)
8. Handle discrete or semicontinuous endogenous regressors	SOREb
9. Handle noncontinuous and continuous exogenous regressors correlated with endogenous regressors and handle nonlinear terms such as interactions.	Haschka (2022); SORE; 2sCOPE proceduresa

^a

For succinctness, 2sCOPE procedures are defined broadly here to include those of Yang, Qian, and Xie (2024, 2025), Liengaard et al. (2025), Breitung, Mayer, and Wied (2024), Mayer and Wied (2026), and Hu, Qian, and Xie (2025). Hu, Qian, and Xie (2025) (2sCOPE-np) unifies the 2sCOPE procedures by employing first-stage model-free nonparametric copula control functions.

^b

When relevant exogenous regressors are available, the 2sCOPE-np control function can leverage their variations to handle noncontinuous endogenous regressors.

Notes: SORE = semiparametric odds ratio endogeneity correction (Qian and Xie 2024).

Two-stage control function methods

The 2sCOPE method introduces a two-stage copula control function approach. Assumptions 1 and 2 in Table 4 mean that 2sCOPE has the double robustness property: The error term does not need to be normally distributed, and regressor–error dependence does not need to follow a GC relationship as long as GC adequately captures the dependence between regressors and U_i (Features 3 and 4 in Table 5). However, the pairwise dependence between the endogenous regressor P_i and U_i unconditioned on W is restricted to a GC relationship. In this aspect, 2sCOPE-np is more general and permits both GC and non-GC pairwise dependence between P_i and U_i (Feature 5 in Table 5). Assumption 4 means 2sCOPE can handle endogenous regressors that are normally distributed or correlated with W (Features 7 and 9 in Table 5). Even if the endogenous regressor is normally distributed, 2sCOPE can identify the model as long as one correlated W is continuous ¹⁸ and nonnormally distributed, which is feasible in many empirical applications. The 2sCOPE originally assumes that the regressor–error GC correlation structure is constant and does not vary in the population. Recent studies (Liengaard et al. 2025; Yang, Qian, and Xie 2024) extend 2sCOPE through robustness checks that allow the GC dependence structure and 2sCOPE copula terms to vary across the levels of combinations of discrete exogenous regressors. Web Appendix G, Table W19, summarizes the general-location heterogeneous GC approach of Yang, Qian, and Xie (2024), which offers greater generality and flexibility than that of Liengaard et al. (2025). These robustness checking methods require a sufficient sample size and boundary conditions (shown subsequently in a flowchart) being met within each level of combinations of discrete exogenous regressors to prevent inflated estimation variance or bias.

Breitung, Mayer, and Wied (2024) and Mayer and Wied (2026) propose using a first-stage mean regression model for endogenous regressors to account for correlated exogenous regressors. Like 2sCOPE, their approach is relatively easy to apply and allows nonnormal error. Interestingly, although it originates from copula correction and permits non-GC regressor–error dependence, their approach does not permit GC regressor–error dependence in general (Table 5) and can yield biased estimates when regressor error follows GC dependence (Hu, Qian, and Xie 2025). Implicitly, their approach assumes (1) a degenerated GC dependence ¹⁹ between U_i and the unobserved error parts in the first-stage models for endogenous regressors and (2) W affects only the mean but not higher moments of the conditional distribution for P | W. Copula procedures using more flexible multivariate dependence models (i.e., methods with Feature 4 in Table 5) can better account for correlated regressors and also permit both GC and non-GC types of regressor–error dependence. In particular, the nonparametric 2sCOPE-np procedure fully nests their approach as a special case.

As noted previously, the 2sCOPE-np unifies existing copula correction methods. It employs nonparametric copula control functions that generalize and make robust the existing copula correction methods using model-based first-stage residuals. It is model-free in the first stage, accommodates both GC and non-GC regressor–error (or regressor–confounder) dependence, and handles normal or discrete endogenous regressors (Features 3–9 in Table 5). The nonparametric feature of 2sCOPE-np does require larger samples (≥300; Table 4) and greater computation cost for kernel conditional CDF estimation (Hu, Qian, and Xie 2025).

Likelihood-based copula correction procedures

Haschka (2022) and Qian and Xie (2024) develop likelihood-based methods that generalize the PG method (Table 5). Here, we first describe the Qian and Xie (2024) method, and then we describe the Haschka (2022) method, which was developed in the context of panel data.

Qian and Xie (2024) propose a bias correction procedure that accounts for regressor–error dependence using a flexible SORE model. The semiparametric model is often used in marketing and other fields as a flexible multivariate model to measure dependence (Chen 2007), handle missing data and selective sampling (Qian and Xie 2011, 2022), and combine sensitive data (Feit and Bradlow 2022; Qian and Xie 2014, 2015). SORE encompasses a number of existing dependence models (including commonly used copulas), capable of capturing GC, non-Gaussian copula, and noncopula dependence structures, thereby enhancing the robustness of copula correction. SORE requires a special estimation algorithm that eliminates potentially high-dimensional nuisance parameters in the nonparametric baseline distribution function, and maximizes the profile likelihood concentrating on the parameter of interest. Likelihood-based model selection measures (such as the Akaike information criterion and the Bayesian information criterion) guide the choice of appropriate odds ratio dependence functions and identification strategies.

Unlike other IV-free methods except 2sCOPE-np, ²⁰ SORE can handle noncontinuous endogenous regressors (Feature 8 in Table 5). It avoids the inverse CDF mapping of discrete endogenous regressors required by copula control functions, enabling identification for noncontinuous endogenous regressors without any help from exogenous regressors. SORE encompasses likelihood-based GC models and expands identification strategies for noncontinuous endogenous regressors. Furthermore, implementing SORE is straightforward, as it conditions on exogenous regressors and uses a simple likelihood involving no integrals with respect to latent copula data. Thus SORE is applicable to many applications involving noncontinuous endogenous regressors. By contrast, more methods can handle noncontinuous exogenous control regressors (Feature 9 in Table 5 and note 9).

Control function versus likelihood-based correction methods

Generally speaking, SORE employs one-step estimation, potentially offering greater efficiency (smaller standard errors) and allows well-established likelihood-based tests and model comparisons. In contrast, the moment-based 2sCOPE requires fewer assumptions and is computationally simpler. While SORE nests copula dependence models and can capture both GC and noncopula dependence structures, the 2sCOPE also permits departures from GC regressor–error dependence by modeling regressor–confounder dependence. Thus, SORE and 2sCOPE are complementary to each other and nonhierarchical (one not nesting the other).

Copula Correction in Panel/Clustered Data

Copula correction can also address various sources of bias in panel data (Haschka 2022; Park and Gupta 2012; Yang, Qian, and Xie 2024, 2025). Haschka (2022) generalizes copula endogeneity correction to the following fixed-effects panel data model:

$${\mathrm{y}}_{\mathrm{it}}={\mu }_{\mathrm{i}}+{{\mathrm{P}}^{\mathrm{\prime }}}_{\mathrm{it}}\alpha +{{\mathrm{W}}^{\mathrm{\prime }}}_{\mathrm{it}}\beta +{\mathrm{e}}_{\mathrm{it}},$$

(5)

where y_it denotes the dependent variable (e.g., store sales) for cross-sectional unit i = 1, …, N at occasions t = 1, …, T; the fixed-effect parameter μ_i captures the effects of time-constant (unobserved) variables (e.g., store size and market characteristics that do not change over time); P_it denotes endogenous regressors (e.g., price) such that cov(P_it, e_it) ≠ 0 due to time-varying unobservables (e.g., unmeasured consumer tastes or brand attributes varying over time), where the error ${\mathrm{e}}_{\mathrm{it}}\sim \mathrm{N}(0,{\sigma }_{\mathrm{e}}^2)$ ; W_it denotes exogenous control variables (e.g., prearranged promotions, quarter time periods). The parameters α and β capture the effect of P_it and W_it, respectively. Given fixed-effects μ_i, all regressors in (P_it, W_it) must be time-varying. Since fixed-effect parameters μ_i can be correlated with the regressors P_it and W_it, the fixed-effects transformation (Wooldridge 2010, pp. 302–03) is often used to eliminate these incidental intercept parameters. Because fixed-effects transformation changes the panel error structure to be nonspherical (nondiagonal covariance matrix), the generalized least squares transformation is applied to handle nonspherical errors and collapses panel data to pooled observations with spherical errors ${\tilde{\xi }}_{\mathrm{it}}\stackrel{\mathrm{i}.\mathrm{i}.\mathrm{d}.}{\sim }\mathrm{N}(0,{\sigma }_{\xi }^2)$ . After eliminating the nonspherical error problem, Haschka (2022) develops an efficient MLE procedure that maximizes the likelihood of a GC model for the error and all explanatory variables to address regressor endogeneity.

Panel studies often face slope heterogeneity. As shown in extant marketing studies, consumers’ heterogeneous responses to marketing-mix variables (e.g., price slope coefficients) are ubiquitous and substantial bias can arise when ignoring such slope heterogeneity. Thus, it is important to allow for individual-specific slope coefficients, by using random coefficients or mixed effects (i.e., both fixed-effects and random coefficients). Extending the copula MLE method to these more general models with endogenous regressors can be challenging, as the model likelihood contains new intractable integrals of complex functions that involve products of copula density functions (Yang, Qian, and Xie 2025). Copula correction for these general panel data models remains to be developed.

For greater generality and computational tractability, Yang, Qian, and Xie (2024, 2025) propose copula control function approaches for the following more general panel data model:

$${\mathrm{y}}_{\mathrm{it}}={\mu }_{\mathrm{i}}+{\mathrm{P}}_{\mathrm{it}}^{\mathrm{\prime }}{\alpha }_{\mathrm{i}}+{\mathrm{W}}_{\mathrm{it}}^{\mathrm{\prime }}{\beta }_{\mathrm{i}}+{\mathrm{e}}_{\mathrm{it}},$$

(6)

where individual-specific parameters (μ_i, α_i, β_i) can be treated as fixed, random, or mixed effects. The model includes the fixed-effects panel model in Equation 5 as a special case. Their copula control functions involve no numerical integrals and can be implemented straightforwardly using standard software programs, assuming all regressors are exogenous.

In principle, the general copula term can extend to the panel data setting as Φ⁻¹(F(P_it,k | W_it, D_i), where D_i is the dummy variable for unit i to account for panel data structure. This may involve a high-dimensional conditional CDF estimation, which can be computationally intensive. To balance robustness and computational ease, we propose using the following general location GC model (Yang, Qian, and Xie 2024) to calculate proper copula terms in multilevel data ²¹ when cov(P_it, e_it) ≠ 0, cov(P_it, μ_i) ≠ 0, and/or cov(P_it, W_it) ≠ 0:

$${\mathrm{P}}_{\mathrm{it}}={\alpha }_{\mathrm{i},\mathrm{p}}+{\mathrm{e}}_{\mathrm{it},\mathrm{p}},\mathrm{and}{\mathrm{W}}_{\mathrm{it}}={\alpha }_{\mathrm{i},\mathrm{w}}+{\mathrm{e}}_{\mathrm{it},\mathrm{w}},$$

(7)

where P_it and W_it are allowed to depend on unit-specific mean levels α_i,p and α_i,w. The fixed-effects α_i,p and α_i,w capture the dependence between μ_i and regressors (cov(P_it, μ_i) ≠ 0 and cov(W_it, μ_i) ≠ 0) and remove the effects of time-constant confounders. The error terms in Equations 6 and 7 then follow the GC model, capturing the endogeneity of P_it (cov(P_it, e_it) ≠ 0) and the dependence among endogenous and exogenous regressors (cov(P_it, W_it) ≠ 0). A two-stage copula control function approach estimates the following augmented panel regression model:

$${\mathrm{y}}_{\mathrm{it}}={\mu }_{\mathrm{i}}+{\mathrm{P}}_{\mathrm{it}}^{\mathrm{\prime }}{\alpha }_{\mathrm{i}}+{\mathrm{W}}_{\mathrm{it}}^{\mathrm{\prime }}{\beta }_{\mathrm{i}}+\sum _{\mathrm{k}=1}^{\mathrm{K}}{\gamma }_{\mathrm{k}}{\mathrm{C}}_{\mathrm{it},\mathrm{k}}+{\omega }_{\mathrm{it}},$$

(8)

where C_it,k is the copula term in Table 6; ${\tilde{\mathrm{P}}}_{\mathrm{it},\mathrm{k}}$ and ${\tilde{\mathrm{W}}}_{\mathrm{it}}$ in the copula term are the time-demeaned value of P_it,k and W_it (i.e., fixed-effect transformation). Thus, the procedure is to apply the 2sCOPE in Table 3 to the time-demeaned regressors. The new error term ω_it is shown to be uncorrelated with all regressors and the fixed-effects μ_i in the augmented panel regression model in Equation 8, eliminating the regressor–error dependence. Standard panel regression estimators assuming all regressors are exogenous can be applied to Equation 8 and yield consistent estimates. Yang, Qian, and Xie (2024) formally demonstrate the consistency, asymptotic normality, and standard error formula for the model parameter estimates. Copula correction is found to be robust to heterogeneous endogeneity across panel units (Haschka 2022; Yang, Qian, and Xie 2024). When the panel is sufficiently long, Yang, Qian, and Xie (2024) explicitly permit the copula dependence to vary across panel units and recover estimates of panel-specific endogeneity; they further employ the mean group estimator to estimate the model in Equation 8 with slope endogeneity (cov(P_it, α_i) ≠ 0) and extend their approach to dynamic panel models.

Table 6.

Copula Control Function Estimation Procedures for Panel/Clustered Data.

Stage 1 Do time demeaning of Pit,k and Wit within each panel unit, and obtain the demeaned regressors $({\tilde{\mathrm{P}}}_{\mathrm{it},\mathrm{k}},{\tilde{\mathrm{W}}}_{\mathrm{it}})$ . Apply Stage 1 of 2sCOPE (Table 3) to the demeaned regressors $({\tilde{\mathrm{P}}}_{\mathrm{it},\mathrm{k}},{\tilde{\mathrm{W}}}_{\mathrm{it}})$ and obtain ${\mathrm{C}}_{\mathrm{it},\mathrm{k}}=({\tilde{\mathrm{P}}}_{\mathrm{it},\mathrm{k}})*-{\hat{\delta }}_{\mathrm{k}}^{\mathrm{\prime }}({\tilde{\mathrm{W}}}_{\mathrm{it}})*$ .

Stage 2 Add copula term Cit,k as the generated regressor to control for endogeneity of Pit,k. The augmented panel regression model takes the form of Equation 8.

Notes: One can also apply 2sCOPE-np to the time-demeaned regressors and obtain ${\mathrm{C}}_{\mathrm{it},\mathrm{k}}={\mathrm{\Phi }}^{-1}(\hat{\mathrm{F}}({\tilde{\mathrm{P}}}_{\mathrm{it},\mathrm{k}}|{\tilde{\mathrm{W}}}_{\mathrm{it}}))$ . Time demeaning subtracts each unit's averages over time of P_it,k and W_it from the original values of P_it,k and W_it. Thus, time demeaning removes the effects of all time-constant confounders correlated with μ_i and is needed for handling endogenous regressors that vary over both i and t. Endogenous regressors that vary only over t or only over i do not need time demeaning. The algorithm can also be applied to grouped/clustered data formed by discrete exogenous regressors, in which i indexes the ith group/cluster.

Copula Correction in Discrete Choice Models

Copula correction can also be applied to address regressor endogeneity in random coefficients logit (RCL) models for panel discrete choice outcomes (Park and Gupta 2012; Yang, Qian, and Xie 2025). In RCL models, the endogeneity of price is modeled as the dependence between product price and unobserved time-varying product characteristics. One can then map an RCL model specified at the consumer level to an aggregate linear model for the product utility averaged across all consumers, for which copula correction for linear models can be directly applied to address regressor endogeneity.

Proper Construction of Nonparametric Rank-Based Copula Transformation

Most applications of copula-based endogeneity correction use rank-based copula transformations based on empirical marginal distributions of regressors. While convenient and robust to misspecification, this approach requires care when mapping ranks to latent copula data. Becker, Proksch, and Ringle (2022) report that the PG method can yield biased estimates in models with intercepts, especially in small to moderate samples, and provide a decision flowchart to guide its use based on the finding. We examine this issue and assess an alternative copula transformation with strong theoretical grounding that avoids such bias.

The empirical rank-based copula transformation involves two steps: assigning percentile ranks to observations, then applying the inverse normal CDF. However, the inverse normal of the 100th percentile—assigned to the maximum rank—is undefined (see the example in Web Appendix D, Table W9). To prevent this, one can adjust the copula transformation of the maximum value of the regressor (e.g., P) for a sample size n as follows (Yang et al. 2025):

$${\mathrm{P}}_{\mathrm{i}}*={\mathrm{\Phi }}^{-1}({\mathrm{F}}_{\mathrm{P}}({\mathrm{P}}_{\mathrm{i}}))=\{\begin{array}{cc}{\mathrm{\Phi }}^{-1}(\mathrm{Rank}({\mathrm{P}}_{\mathrm{i}})/\mathrm{n})& \mathrm{if}{\mathrm{P}}_{\mathrm{i}}<\max (\mathrm{P})\\ {\mathrm{\Phi }}^{-1}(\mathrm{n}/(\mathrm{n}+1))& \mathrm{if}{\mathrm{P}}_{\mathrm{i}}=\max (\mathrm{P}).\end{array}$$

(9)

The preceding percentile adjustment ensures a theoretically valid maximum for the copula-transformed data. This is justified by approximating the expected maximum of a standard normal sample of size n by ${\mathrm{\Phi }}^{-1}(\frac{\mathrm{n}-\alpha }{\mathrm{n}+1-2\alpha })$ , with α = .375 recommended by Royston (1982). Using ${\mathrm{\Phi }}^{-1}(\frac{\mathrm{n}}{\mathrm{n}+1})$ simplifies the formula (α = 0) and yields a result nearly identical to that of setting α = .375 for typical sample sizes (i.e., N >> α).

To demonstrate the importance of the empirical copula transformation, consider an alternative approach used in Becker, Proksch, and Ringle (2022), which sets the percentile of the highest-ranked observation to a fixed value of .9999999:

$${\mathrm{P}}_{\mathrm{i},\mathrm{Fix}}*={\mathrm{\Phi }}^{-1}({\mathrm{F}}_{\mathrm{P}}({\mathrm{P}}_{\mathrm{i}}))=\{\begin{array}{cc}{\mathrm{\Phi }}^{-1}(\mathrm{Rank}({\mathrm{P}}_{\mathrm{i}})/\mathrm{n})& \mathrm{if}{\mathrm{P}}_{\mathrm{i}}<\max (\mathrm{P})\\ {\mathrm{\Phi }}^{-1}(.9999999)=5.1999& \mathrm{if}{\mathrm{P}}_{\mathrm{i}}=\max (\mathrm{P}),\end{array}$$

(10)

where ${\mathrm{P}}_{\mathrm{Fix}}^{*}$ refers to using a fixed percentile (.9999999) for the highest rank to preserve rank order unless the sample size exceeds one million. However, in smaller samples, this can lead to substantial deviations from the theoretical maximum, creating a high-leverage outlier far from the centroid of covariate distributions. Such an outlier can distort coefficient estimates and weaken the performance of copula correction.

To assess the impact of empirical copula construction on the performance of copula correction, we compare the algorithms in Equations 9 and 10 using simulation studies ²² in which the true parameter values are known. The simulation studies consider situations both without correlated exogenous regressors, as described in Becker, Proksch, and Ringle (2022), and with correlated exogenous regressors, as described in Web Appendix D.

Results in Web Appendix D reveal that judicious copula transformation is crucial for effective copula correction. Notably, including an intercept poses no issue if the highest-ranked value is properly adjusted using the recommended copula transformation algorithm. A key finding is that the substantial bias in models with intercepts, reported by Becker, Proksch, and Ringle (2022), is largely resolved by applying the adjustment in Equation 9.

Together, our analysis offers theoretical and empirical rationales for optimal copula transformation. The new insights help demystify misinterpretations about copula correction and promote optimal copula transformation for effective copula correction. We recommend avoiding fixed percentiles for the highest rank, suggesting instead to use the algorithm in Equation 9 for valid transformation regardless of sample sizes.

Optimal Copula Estimation of Endogenous Moderating and Nonlinear Effects

Higher-order terms of endogenous regressors (e.g., interactions with moderators) are common in empirical studies aimed at understanding causal mechanisms or informing optimal policy. While copula methods can address such nonlinear terms (Table 5), practices vary widely (Web Appendix A, Table W3). Some studies omit copula-generated terms without explanation, while others include them to account for endogeneity. The following example (Example 2) of a moderator of price sensitivity illustrates the impact of these differences.

Price may work together with a retail store's feature advertising to achieve synergistic effects on sales. This can be tested by estimating the interaction term between price and feature advertisement in a sales model, with feature advertisement as a potential moderator of price. Blattberg and Neslin (1990) note that feature advertising “may interact with price discounts. If the consumer is not informed that a price discount is offered, the price elasticity is likely to be small” (p. 347). This suggests a negative sign for the interaction term between price and feature advertisement.

Figure 4 plots mean price sensitivity estimates by feature advertising intensity quartile in the peanut butter category, predicted from a sales model with an interaction term between price and feature advertising, using the IRI Academic data for a New York City store. The black (gray) bars are price sensitivity estimates estimated with (without) a copula term for the interaction term. Including the copula term for the interaction yields similar price sensitivity estimates across different feature advertising intensity (i.e., lack of interactive effect), while excluding the copula term yields a statistically significant and negative interaction effect. In this section, we examine the best approach to handling these higher-order endogenous terms via both theoretical proof and empirical evaluations. As shown next, adding the copula term for the interaction term can induce bias and increase parameter estimate variability.

Figure 4.

Mean Price Sensitivity Estimates by Quartile of Feature Advertising Intensity.

Consider the following general model containing higher-order terms of regressors:

$${\mathrm{Y}}_{\mathrm{i}}=\mu +{\alpha }_1^{\mathrm{\prime }}{\mathrm{P}}_{\mathrm{i}}+{\alpha }_2^{\mathrm{\prime }}{\mathrm{f}}_1({\mathrm{P}}_{\mathrm{i}})+{\alpha }_3^{\mathrm{\prime }}{\mathrm{f}}_2({\mathrm{P}}_{\mathrm{i}},{\mathrm{W}}_{\mathrm{i}})+{\beta }^{\mathrm{\prime }}{\mathrm{W}}_{\mathrm{i}}+\eta {\mathrm{f}}_3({\mathrm{W}}_{\mathrm{i}})+{\mathrm{E}}_{\mathrm{i}},$$

(11)

where P_i is a vector of K continuous and endogenous regressors, and W_i is a vector of exogenous regressors. The structural model in Equation 11 expands the model in Equation 1 to include higher-order endogenous terms, namely f₁(P_i) and f₂(P_i, W_i), and higher-order exogenous terms, f₃(W_i). The following are examples of these higher-order terms:

• Polynomial functions of a scalar P_i: ${\alpha }_2^{\mathrm{\prime }}{\mathrm{f}}_1({\mathrm{P}}_{\mathrm{i}})={\alpha }_2{\mathrm{P}}_{\mathrm{i}}^2$

• Interaction of two endogenous regressors P_i = (P_1i, P_2i): ${\alpha }_2^{\mathrm{\prime }}{\mathrm{f}}_1({\mathrm{P}}_{\mathrm{i}})={\alpha }_2{\mathrm{P}}_{1\mathrm{i}}{\mathrm{P}}_{2\mathrm{i}}$

• Interaction of endogenous and exogenous regressors: ${\alpha }_3^{\mathrm{\prime }}{\mathrm{f}}_2({\mathrm{P}}_{\mathrm{i}},{\mathrm{W}}_{\mathrm{i}})={\alpha }_3{\mathrm{P}}_{\mathrm{i}}{\mathrm{W}}_{\mathrm{i}}$ .

Because higher-order terms of endogenous regressors, f₁(P_i) and f₂(P_i, W_i), are also endogenous, it is tempting to control their endogeneity by adding separate copula correction terms for them. However, the point of not needing these copula correction terms for these higher-order terms is clearly shown in the following copula-augmented regression, including only copula correction terms for the first-order endogenous terms (i.e., main effects):

$${\mathrm{Y}}_{\mathrm{i}}=\mu +{\alpha }_1^{\mathrm{\prime }}{\mathrm{P}}_{\mathrm{i}}+{\alpha }_2^{\mathrm{\prime }}{\mathrm{f}}_1({\mathrm{P}}_{\mathrm{i}})+{\alpha }_3^{\mathrm{\prime }}{\mathrm{f}}_2({\mathrm{P}}_{\mathrm{i}},{\mathrm{W}}_{\mathrm{i}})+{\beta }^{\mathrm{\prime }}{\mathrm{W}}_{\mathrm{i}}+\eta {\mathrm{f}}_3({\mathrm{W}}_{\mathrm{i}})+{\gamma }^{\mathrm{\prime }}{\mathrm{C}}_{\mathrm{i},\mathrm{main}}+{ϵ}_{\mathrm{i}},$$

(12)

where C_i,main = (C_i,1, …, C_i,K) contains copula correction terms for main terms P_i only (Table 3). Because the new error term $ϵ$ is independent of P and W, it is also independent of their functions, such as f₁(P), f₂(P, W), and f₃(W). Thus, once the copula correction terms for main effects C_main are included as control variables in Equation 12, no additional correction for these higher-order terms is needed. This simplicity of handling higher-order endogenous regressors is a merit of copula correction. ²³

Although it is unnecessary to add the copula correction terms for higher-order terms, ²⁴ a further question is what might happen if the additional copula-generated regressors for the higher-order terms are included. Will doing this lead to better or worse performance of copula correction? The issue with adding unnecessary regressors ${\mathrm{C}}_{{\mathrm{f}}_1({\mathrm{P}}_{\mathrm{i}})}$ and ${\mathrm{C}}_{{\mathrm{f}}_2({\mathrm{P}}_{\mathrm{i}},{\mathrm{W}}_{\mathrm{i}})}$ is the significant collinearity between these higher-order copula terms and their covarying constituents (P, f₁(P), f₂(P, W), and C_main). This substantially decreases the precision of coefficient estimates, making copula correction methods perform worse than otherwise, shown formally by Theorem 1 in Web Appendix E. Consistent with the theoretical results, simulation studies (Web Appendix F) demonstrate substantial harmful effects if copula terms for higher-order regressors are added to control for their endogeneity. These effects include large finite sample bias and inflated variability of structural model parameter estimates.

Obtaining Standard Errors

For methods performing joint estimation in one step (Qian and Xie 2024), standard errors can be directly obtained by inverting the Hessian matrix as a byproduct of the estimation process. For two-step copula methods, bootstrapping is applied to obtain proper standard errors in order to account for additional uncertainty from obtaining generated regressors in the first step. Specifically, the data are resampled with replacement to form bootstrap samples, on which copula-corrected estimates are recomputed repeatedly. The standard deviation of these estimates provides the standard error. For panel data, cluster bootstrapping should be used to resample independent cross-sectional units rather than individual observations (Haschka 2022). That is, only the cross-sectional units (clusters) are resampled, while all the observations within the sampled clusters are retained and unchanged. This ensures the bootstrap samples retain dependence structures among panel observations existing in the original data, and simulation studies have shown that bootstrapping produces reliable standard error estimates (Haschka 2022; Park and Gupta 2012; Yang, Qian, and Xie 2025).

Guidance for Practical Use

Based on recent advances, this section describes a procedure guiding practical usage of copula correction methods. Figure 5 presents a step-by-step flowchart ²⁵ of key steps and checkpoints. Before using it, clearly define the causal structure—specifying the outcome, main explanatory variables, and relevant control covariates. Ensure the model is theoretically sound, with pertinent control variables included in W and the regressor matrix being full rank. To ensure exogeneity of W, include only necessary exogenous control variables. Control variables suspected to be endogenous should be modeled as such or excluded from the model.

Figure 5.

Flowchart for Copula Procedure.

When the need to use copula correction is affirmed at the start of the flowchart (Step 0a of Figure 5), assess the plausibility of the underlying assumptions in the focal application (Step 0b of Figure 5) according to Item 1 under “Assess Assumptions” in Table 4. The double robustness property of copula correction using control functions means that copula correction can be used with departures from GC regressor–error dependence, as long as GC adequately captures unexplained dependence between endogenous regressors and U (the combined effects of all unobserved confounders) given exogenous regressors. Copula correction also works with a nonnormal error distribution. However, out of an abundance of caution and for optimal robustness, consider revising model specifications (e.g., transform variables or add more control variables) if the error distribution is suspected to be highly skewed.

If copula correction is chosen, follow the rest of the flowchart to determine appropriate copula correction methods. As shown previously, copula correction only needs to include copula control functions corresponding to the first-order terms of endogenous regressors, P_main, even when the structural model contains higher-order terms of endogenous regressors. Thus, the flowchart only needs to consider P_main. Furthermore, when the structural model includes an intercept, the copula transformation should use the algorithm in Equation 9 to avoid estimation bias. For panel data, copula terms are computed using time-demeaned regressors. When conditions are met, the PG method can be followed, but recent research relaxes these conditions and presents the path to perform copula correction even when these conditions are not met.

Step 1 checks whether the endogenous regressor P_main has sufficient support. If P_main is noncontinuous (binary, discrete with only a few levels, or semicontinuous), use likelihood-based SORE. Otherwise, continue to Step 2.

Step 2 checks whether P_main is normally distributed or not. If P_main is normally distributed, the PG method cannot be used because the model is unidentified. However, a normally distributed P_main can still be a candidate for copula correction through 2sCOPE. Yet, this route follows a different path, as seen in Figure 5 and discussed more in Step 3b. The literature notes that more powerful tests for normality, such as the Shapiro–Wilk test or the Anderson–Darling test, might not fully rule out nonidentification, because these tests can detect small departures from normality that are insufficient for copula correction (Becker, Proksch, and Ringle 2022; Eckert and Hohberger 2023). Yet, the Kolmogorov–Smirnov (KS) test is relatively conservative among the most commonly used normality tests; a p-value less than .05 from the KS normality test has been shown to perform well for ruling out finite sample bias due to insufficient regressor nonnormality (Yang, Qian, and Xie 2025).

Step 3 marks one of the biggest shifts in copula usage since PG, consisting of two disjoint steps (3a and 3b), depending on the outcome of Step 2. The data requirements in this step are established using comprehensive factorial design simulation experiments to ensure satisfactory performance of copula correction, across a wide range of conditions in finite samples (Web Appendix E.8 in Yang, Qian, and Xie 2025).

Step 3a proceeds as follows: If the endogenous regressor P_main has sufficient nonnormality (KS p-value < .05) in Step 2, Step 3a will check an additional condition of no correlation between P_main and all exogenous regressors (using Fisher's Z test for correlation) to determine if the PG method can be used. ²⁶ When this condition is met and sample size is small, the PG method may be preferred because a simpler and valid model is more efficient than a more general method. As sample size increases, 2sCOPE has negligible efficiency loss relative to the PG method and is the preferred method. If P_main is correlated with any exogenous regressors, one should use 2sCOPE to handle correlated exogenous regressors. Alternatively, an MLE copula procedure (either the one-step SORE or the two-step procedure of Haschka 2022) can be used.

Step 3b proceeds as follows: If the endogenous regressor P_main is found to have insufficient nonnormality (KS p-value > .05) in Step 2, then one cannot use the PG method, but can use 2sCOPE to leverage correlated exogenous regressors to achieve model identification. To compensate for the lack of nonnormality of endogenous regressor P in 2sCOPE, at least one exogenous and continuous regressor W needs to satisfy the following two conditions: (1) sufficient nonnormality, and (2) sufficient association with the endogenous regressor P. A conservative rule of thumb for such a W is the p-value from the KS test on W being less than .001 and a strong association with P (F statistic for the effect of W* on ${\mathrm{P}}_{\mathrm{main}}*>10$ in the first-stage regression). When these conditions are met, 2sCOPE is expected to yield estimates with negligible bias even if P_main is normally distributed. When these conditions are not met, Yang, Qian, and Xie (2025) suggest gauging potential bias of 2sCOPE for data at hand via a bootstrap procedure described there, and using 2sCOPE only if the potential bias is small.

As seen previously, only one of Steps 3a or 3b is used. Importantly, if P already has sufficient nonnormality that leads to Step 3a, there is no need to do Step 3b to check if any continuous W has sufficient nonnormality and is associated with P. These conditions are only checked to find a useful W to compensate for the lack of nonnormality of P. In Step 3b, 2sCOPE uses W to tease out an exogenous part of the endogenous regressor for model identification.

Step 4 applies the appropriate copula procedures using either control functions or likelihood-based joint estimation. To choose between likelihood-based methods and moment-based control function methods when both can be used, see the previous section “Control function versus likelihood-based correction methods.” Among copula control function methods, 2sCOPE is relatively easy to apply and reasonably robust (e.g., assumes no particular error distribution and no specific copula structure for regressor–error dependence). This balance of simplicity and robustness makes it well suited for a first-line method. The more general 2sCOPE-np is model-free in the first stage and allows for broader types of regressor–confounder dependence, making it preferred when more robustness is desired. The nonparametric kernel control function estimation employed in 2sCOPE-np does require larger sample sizes (Boundary Condition 3 in Table 4) and greater computation power, compared with the simpler 2sCOPE. For control function methods, if the generated regressor is not statistically significant, this suggests the endogenous regressor P_main is not sufficiently correlated with the error term, and endogeneity is unlikely. Thus, nonsignificant generated regressors should be dropped and the model reestimated. Marketing studies have dropped copula correction terms at the p > .10 level (e.g., Datta et al. 2022), suggesting that even marginally significant copula correction terms are still worth retaining. If no generated regressor is significant, the model can be estimated in a more traditional manner (i.e., OLS).

Finally, Step 5 checks the inflation of standard errors of copula-corrected estimates relative to those of uncorrected estimates using the ICON statistics. An inflation of more than six times flags potential model misspecification issues or lack of model identification.

Copula Implementation Examples

This section illustrates use of the flowchart to guide copula implementation via two examples using weekly store sales data from the IRI Academic dataset (Bronnenberg, Kruger, and Mela 2008). To correct for price endogeneity, the first example examines the main effect of price, while the second example examines higher-order moderating effects captured by the interaction between price and store feature advertising (i.e., weekly store flyer promoting products).

Example 1: main effects application of copula correction

Returning to our running Example 1, the outcome of interest is the weekly sale volume in the diaper category for one focal store in the Buffalo, New York, market in the years 2002–2006, where volume is measured in diaper counts. Price is defined on an equitable volume across UPCs, since pack sizes vary in diapers per pack. IRI additionally collected information on whether UPCs were featured in the store's weekly flyer that week. Category price and feature advertising are evaluated as market-share weighted averages of UPC-level price and feature advertising, respectively.

Knowledge of category price elasticity is critical for retailers or category managers to set optimal pricing and increase category demand, which is the first source of profitable growth, and for policymakers to design interventions (e.g., gasoline tax). Price is commonly considered endogenous in category demand models (Li, Linn, and Muehlegger 2014; Nijs et al. 2001; Park and Gupta 2012). In this example, price was treated as endogenous because of unobserved variables (e.g., retailer pricing decisions, number of shelf facings) that, when omitted from a model, become part of the structural error. For brevity, we use “Price” and “Volume” hereafter to refer to the log-transformed category price and sales volume, respectively. The impacts of price and feature advertising appear in the following model:

$${\mathrm{Volume}}_{\mathrm{t}}=\mu +\alpha {\mathrm{P}}_{\mathrm{t}}+{\beta }^{\mathrm{\prime }}{\mathrm{W}}_{\mathrm{t}}+{\mathrm{E}}_{\mathrm{t}}.$$

(13)

In the model, P_t is the endogenous regressor as log-transformed price. W_t is a vector of control variables including feature advertising, week, and binary variables for Quarters 2, 3, and 4. We treat feature advertising as exogenous because decisions to promote items in the store flyer are made well in advance of implementation, and are likely uncorrelated with weekly unobservables (Chintagunta 2002; Sriram, Balachander, and Kalwani 2007). The week variable is included as a control variable to account for a small but significant trend in price increases over time.

One solution to price endogeneity is to use IVs, where the diaper price of another store in the same market was used as an IV. Prices are correlated for both stores, with the belief that wholesale prices are similar for products sold by the two stores (relevance), but uncaptured product characteristics (including retailer decisions like shelf facings and shelf location) are unlikely related to wholesale prices (ER). However, the ER assumption is untestable, and the IV may be not strong enough. This is one of the use cases for copula correction as shown in Figure 5: Use multiple methods (both IV estimation and copula correction here) to cross-validate results and increase robustness of causal inference.

We next assess the plausibility of the underlying assumptions in the application (Step 0b in Figure 5), following Item 1 under “Assess Assumptions” in Table 4. Since the model includes feature advertising and quarters to control for planned promotion activities affecting sales, the omitted variables mainly involve unmeasured product attributes tied to retailer decisions (like shelf facings and locations). Their joint effect (U) can be expected to follow a normal distribution. ²⁷ We then use both theoretical reasoning and diagnostic tools to assess regressor–error dependence. The endogenous regressor, Price, is likely bounded within a feasible range due to bounded price-setting (Kocherlakota 2021). As discussed previously, the GC dependence is empirically plausible because it can flexibly capture regressor–error dependence irrespective of the bounded nature of endogenous regressors. We will further apply ICON (Boundary Condition 1 in Table 4) to inspect standard errors from copula correction to confirm empirical identification and check for signs of regressor–error dependence misspecifications. Before we present the results, we first walk through the steps of the Figure 5 flowchart.

Step 1: Is P_main continuous? The endogenous regressor, Price, is a continuous measure, ranging from $.140 to $.262 per diaper, with a mean of $.221, median of $.224, and standard deviation of $.018.

Step 2: Is P_main normally distributed? Figure 6 shows some skewness to the left for the price variable. However, the skewness is not strong enough to reject the KS test for normality (D = .08, p > .05) at the .05 level of significance. This means that the endogenous regressor may not have sufficient nonnormality. One solution is to leverage related exogenous regressors with sufficient nonnormality via 2sCOPE as described next.

Step 3b: Is at least one W sufficiently nonnormal and correlated with P_main? The first-stage regression shows that only one exogenous regressor is sufficiently correlated with price (F-statistic > 10): feature advertising (F = 16.8). The regressor, feature advertising, is highly skewed (Figure 6) and nonnormally distributed based on the KS test (D = .14, p < .0001).

Step 4: Perform 2sCOPE estimation. The preceding steps show that conditions have been verified such that 2sCOPE can be used to handle the price endogeneity. ²⁸ The standard errors are obtained using 500 bootstrap samples.

Step 5: Check the inflation of standard errors using the ICON statistics. All ICON statistics are far less than 6 (Table 7), showing no signs of weak identification.

Figure 6.

Distributions of Price and Feature Advertising in Example 1.

Table 7.

Estimation Results for Example 1.

Parameters	OLS		2SLS		2sCOPE
	Est. (SE)	p	Est. (SE)	p	Est. (SE)	p	ICON
Intercept	6.005 (.205)	.000	4.371 (.978)	.000	4.763 (.668)	.000	3.3
Price	−1.367 (.137)	.000	−2.470 (.661)	.000	−2.205 (.446)	.000	3.3
Feature advertising	.298 (.095)	.002	059 (.178)	.738	.124 (.124)	.317	1.3
Week	−.002 (.000)	.000	−.002 (.000)	.000	−.002 (.000)	.000	1.1
Q2	−.019 (.031)	.550	−.014 (.035)	.693	−.018 (.036)	.617	1.2
Q3	−.018 (.032)	.567	−.034 (.036)	.349	−.029 (.035)	.407	1.1
Q4	−.018 (.032)	.576	−.061 (.041)	.140	−.044 (.035)	.209	1.1
Cprice					.077 (.037)	.037	—
ρ					.366 (.160)	.022	—

Notes: The table presents estimates, bootstrapped standard errors in parentheses, and the p-values. ICON is the ratio of standard errors of 2sCOPE estimates to those of the OLS estimates.

Table 7 compares 2sCOPE to OLS and 2SLS using the IV. The 2sCOPE estimation results show that the copula correction term C_price (i.e., the first-stage residual) is significant (Est. = .077, SD = .037, p < .05), indicating the presence of price endogeneity, so we retain the copula term in the model to control for price endogeneity.

The results show that while price has the smallest absolute effect in the OLS model (Est. = −1.367, SE = .137, p < .01), the effect is greatest in the 2SLS model (Est. = −2.470, SE = .661, p < .01); the 2sCOPE price estimate falls in between and is much closer to the 2SLS price estimate (Est. = −2.205, SE = .446, p < .01). Compared with 2SLS using IV, the 2sCOPE results are not unlike that of 2SLS, within one standard deviation of the 2SLS price estimates. The 2SLS price estimate differs somewhat from the 2sCOPE price estimate by 12.0%. Although the correlation in prices between the two stores is significant and passes the weak instruments test (F = 13.89, p < .01), the correlation is not especially strong (r = .218). Thus, the difference between 2sCOPE and 2SLS seen here could be because the other store's price as an IV is not particularly strong, and a strong IV is not always readily available. In such cases, cross-validating results from different methods (IV correction and IV-free copula correction) can increase the robustness of causal estimation. The 2sCOPE result shows that price is positively correlated with the error term (Est. = .366, SE = .160, p < .05), indicating the presence of price endogeneity. This finding is consistent with the result of the Wu–Hausman test (H = 3.56, p < .07) from 2SLS, which also suggests endogeneity was likely present. Overall, the comparison with 2sCOPE shows that without endogeneity correction, managers would severely underestimate price elasticity based on the OLS findings for this store, by 38.0%.

Example 2: copula estimation of endogenous interactions

Example 2 illustrates how copula correction is applied with endogenous interaction terms and examines the adverse effects (estimation bias and inflated estimation variability) of including higher-order copula terms. This empirical application extends the sales response model in Equation 13 to include an interaction term between price and feature advertising. See Web Appendix G.2 for detailed analysis and results of Example 2.

Managerial and Academic Implications

The two examples highlight how copulas can correct for endogeneity to remove bias in estimation, as well as how copulas should be correctly specified in models with interactions. Example 1 showed that without the copula, the OLS estimate for price elasticity was severely underestimated (Est. = −1.367) compared with both 2SLS (Est. = −2.470) and 2sCOPE (Est. = −2.205). The result showed price elasticity that was 38% lower in size in OLS than in 2sCOPE. We also noted that the instrument was significant but not particularly strong, attributing this finding to the difference between 2SLS and 2sCOPE estimates.

Controlling for endogeneity in price elasticity estimates can have important managerial implications. Price elasticity estimates are often a crucial piece of information for managers to set the optimal pricing that maximizes profit. Let the profit function p(Price) = V × (Price − Cost), where V is sales volume and Cost is the marginal cost. The maximum profit is then the value of Price that satisfies the condition $\frac{\mathrm{\partial }\ln \mathrm{p}(\mathrm{Price})}{\mathrm{\partial }\mathrm{Price}}=0$ . Following the Amoroso–Robinson relation, the profit-maximizing price is ${\mathrm{Price}}_{\mathrm{optim}}=\frac{\alpha }{1+\alpha }\mathrm{Cost}$ , where α is the price elasticity. In Example 1, we find that the optimal pricing is ${\mathrm{Price}}_{\mathrm{ols}}=\frac{-1.367}{-1.367+1}\mathrm{Cost}=3.72\times \mathrm{Cost}$ if the OLS price elasticity estimate is used, and ${\mathrm{Price}}_{\mathrm{cop}}=\frac{-2.205}{-2.205+1}\mathrm{Cost}=1.83\times \mathrm{Cost}$ if the 2sCOPE estimate is used. Because of the price endogeneity associated with the scanner panel data, the biased OLS estimate underestimates the size of price elasticity, meaning that OLS considers consumers less price sensitive than they actually are. Thus, the manager will set the price more aggressively; in Example 1, using the OLS price elasticity estimate means the manager will set the price at 103% higher than the actual optimal price.

This considerable difference in optimal pricing based on the OLS and 2sCOPE price elasticity estimates results in a substantial profit difference as well. It can be shown that the profits achieved at the different prices have the following relationship: $\ln \frac{{\mathrm{p}}_{\mathrm{cop}}}{{\mathrm{p}}_{\mathrm{ols}}}=\alpha \ln [{\mathrm{Price}}_{\mathrm{cop}}/{\mathrm{Price}}_{\mathrm{ols}}]+\ln [({\mathrm{Price}}_{\mathrm{cop}}-\mathrm{Cost})/({\mathrm{Price}}_{\mathrm{ols}}$ $-\mathrm{Cost})]$ , where p_cop and p_ols refer to the profit achieved when using the 2sCOPE and OLS price elasticity estimates, respectively. For Example 1, $\frac{{\mathrm{p}}_{\mathrm{cop}}}{{\mathrm{p}}_{\mathrm{ols}}}=1.46$ , which corresponds to a loss of 31% in profit when using the incorrect OLS price elasticity estimate, compared with using the correct 2sCOPE estimate (Figure 3).

Conclusion

Endogeneity correction is a key concern for academics and practitioners, and the instrument-free copula correction has been increasingly used to address endogeneity bias. Copula correction has practical advantages and feasible implementation. Yet, like other causal estimation procedures designed for use with nonexperimental data, the validity of copula correction requires correct implementation of the method, needing boundary conditions and data requirements to be met in its empirical applications.

This study contributes to the field in three areas. First, we advance the discussion regarding the theoretical rationales of copula correction and provide a review of how copula correction has been used in marketing and other fields to correct for endogeneity, across substantive areas, and how it has been applied (and misapplied). Second, we elucidate the identification assumptions and data requirements of copula correction and build on recent advances to provide an updated best-practices “cookbook” for both managers and academics to follow in applying and implementing the copula procedures (Tables 1–6; Figure 5). The cookbook also explains how to modify analysis when certain conditions are not met. Third, we evaluate implementation variations (such as optimal copula transformations and higher-order effects of moderation) and demystify misconceptions of copula correction, showing theoretically and with real-world data best practices for copula correction usage.

We demonstrate that existing variations in the implementation of copula correction have substantial impacts on its performance. Our discussion on the methodological aspects of the copula method informs optimal and theoretically sound implementation for copula correction. We present a theoretically grounded way of constructing copula transformation that avoids the potential finite sample bias problem and substantially improves the performance of copula correction. We show that excluding the copula terms for higher-order endogenous regressors (i.e., interactions) is optimal and considerably outperforms the method that includes these copula terms. To our knowledge, these are the first theoretical results justifying the optimal implementation of these aspects affecting the performance of copula correction.

We also discuss the latest extensions that expand the applicability, flexibility, and robustness of copula correction, highlighting endogeneity correction when the conditions and data requirements of earlier copula correction approaches are not met (Table 2); for cases where the endogenous regressors have insufficient nonnormality or correlate with exogenous regressors (and the traditional PG method fails to work), we describe how a two-stage copula correction (2sCOPE) and its extensions, as well as other copula correction procedures, can still work by leveraging relevant exogenous regressors.

We synthesize the preceding discussion into a flowchart with easy-to-follow checkpoints and data requirements. This guide is practical for researchers—in both academia and industry—to employ copula correction methods. In addition to making the copula code available, we illustrate its usage in two empirical examples for two different product categories.

Future avenues of research are teeming, for example, extending current copula correction frameworks for more generality and for handling discrete endogenous regressors such as endogenous treatment selection (Hu, Qian, and Xie 2025; Qian and Xie 2024). Future research directions also include adapting copula correction to Bayesian inference (Haschka 2025), exploring methods to further reduce the dependence on the GC assumption (Qian and Xie 2024), and improving computational efficiency especially for computationally intensive procedures (e.g., the MLE procedures), to name a few. While copula correction has made advances, and a great variety of quantitative models have utilized copulas, new models are regularly emerging. Thus, new opportunities to adapt copula correction to new types of data, models, or applications abound.

Supplemental Material

sj-pdf-1-jmx-10.1177_00222429251410844 - Supplemental material for A Practical Guide to Endogeneity Correction Using Copulas

Supplemental material, sj-pdf-1-jmx-10.1177_00222429251410844 for A Practical Guide to Endogeneity Correction Using Copulas by Yi Qian, Anthony Koschmann and Hui Xie in Journal of Marketing