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Revisiting Gaussian copulas to handle endogenous regressors

Revisiting Gaussian copulas to handle endogenous regressors Marketing researchers are increasingly taking advantage of the instrumental variable (IV)-free Gaussian copula approach. They use this method to identify and correct endogeneity when estimating regression models with non-experimental data. The Gaussian copula approach’s original presentation and performance demonstration via a series of simulation studies focused primarily on regression models without intercept. However, marketing and other disciplines’ researchers mainly use regression models with intercept. This research expands our knowledge of the Gaussian copula approach to regression models with intercept and to multilevel models. The results of our simulation studies reveal a fundamental bias and concerns about statistical power at smaller sample sizes and when the approach’s primary assumptions are not fully met. This key finding opposes the method’s potential advantages and raises concerns about its appropriate use in prior studies. As a remedy, we derive boundary conditions and guidelines that contribute to the Gaussian copula approach’s proper use. Thereby, this research contributes to ensuring the validity of results and conclusions of empirical research applying the Gaussian copula approach. Keywords Endogeneity · Gaussian copula · Intercept · Linear regression · Multilevel models · Sample size · Simulation Introduction correlate with the error term violating a fundamental causal modeling assumption of regression analysis (Wooldridge, Endogeneity is a key concern when using regression mod- 2020). Potential reasons for error term correlations are meas- els in marketing studies with non-experimental data (Rutz urement errors, simultaneous causality, and omitted vari- & Watson, 2019; Sande & Ghosh, 2018). In a regression ables that correlate with one or more independent variable(s) model, endogeneity occurs when one or more regressors and with the dependent variable(s) in the regression model (e.g., Papies et al., 2017; Rutz & Watson, 2019). Endoge- neity problems lead to biased and inconsistent coefficients, This manuscript was handled by Editor John Hulland. which become causally uninterpretable. The best approach to overcome endogeneity is to specify * Christian M. Ringle the model correctly according to the underlying (causal) c.ringle@tuhh.de data-generating mechanism. In practice, however, the Jan‑Michael Becker required data is usually not available, unless researchers jan-michael.becker@bi.no use experimental designs (Ebbes et al., 2016). Marketing Dorian Proksch literature has extensively discussed methods of dealing with d.e.proksch@utwente.nl endogeneity (e.g., Rutz & Watson, 2019; Sande & Ghosh, Department of Marketing, BI Norwegian Business School, 2018; Zaefarian et al., 2017). Of these methods, the use Nydalsveien 37, 0484 Oslo, Norway of instrumental variables (IVs) is particularly well-known Netherlands Institute for Knowledge-Intensive for addressing endogeneity problems (Wooldridge, 2010). Entrepreneurship (NIKOS), University of Twente, Despite its frequent application, the IV approach has sev- Drienerlolaan 5, 7522 NB Enschede, Netherlands eral drawbacks, as it requires identifying strong and valid Department of Management Science and Technology, instruments (i.e., they correlate strongly with the endog- Institute of Human Resource Management enous regressor but do not correlate with the error term). and Organizations, Hamburg University of Technology, Am Schwarzenberg‑Campus 4, 21073 Hamburg, Germany In applications, however, researchers often fail to revert to suitable variables whose appropriateness as instruments they Waikato Management School, University of Waikato, Walikato, New Zealand Vol.:(0123456789) 1 3 Journal of the Academy of Marketing Science can sufficiently justify theoretically (Rossi, 2014). Methodo- 89.9%) appeared in 2018, 2019, and 2020 and 58.0% (40 of logically, only the instrument’s strength can be empirically 69) of these journal articles appeared in marketing journals. tested and not its validity (Wooldridge, 2010). Thus, the Besides Marketing Science, which initially published the results of the IV approach do not allow for assessing whether method, premier marketing journals such as International the endogeneity problem has improved or worsened (Papies Journal of Research in Marketing (11), Journal of the et al., 2017). Academy of Marketing Science (7), Journal of Marketing To remedy these concerns, Park and Gupta (2012)—fur- Research (6), Journal of Retailing (6), and Journal of Mar- ther referred to as P&G—introduced the Gaussian copula keting (5) predominantly published Gaussian copula appli- approach to cope with endogeneity in regression models. cations. In addition, the Gaussian copula approach has been This IV-free method has the advantage of not requiring addi- disseminated across disciplines, for example, in management tional variables, because it directly models the correlation with its subfields like human resources, information systems, between the potentially endogenous regressor and the error and tourism (23 of 69 articles; 33.3%), and in economics (3 term using a Gaussian copula. Thereby, the approach pro- articles; 4.3%). vides a relatively simple way of identifying and correcting One of the observations from our literature review is that endogeneity biases in regression models (Rutz & Watson, researchers not only use the Gaussian copula approach to 2019). improve the precision of the endogenous regressor’s parame- There are two variants of the Gaussian copula approach. ter estimate, but also to identify whether endogeneity poses a The original approach by P&G suggests the regression problem for a regression model. They do so by assessing the model’s estimation by using an adapted maximum likeli- copula term’s statistical significance to determine whether hood function that accounts for the correlation between endogeneity is at a critical level in their empirical study or the regressor and the error term using the Gaussian copula not (e.g., Bornemann et al., 2020; Keller et al., 2018). This (P&G, Eq. 8). The disadvantage of the maximum likelihood approach seems plausible, as the copula term’s parameter approach is that it can only account for one endogenous estimate is a scaled version of the unobservable correla- regressor in the model. In practice, almost all applications tion between the endogenous regressor and the error term therefore use the second variant, which adds a “copula term” (P&G, p. 572) and can be compared to the Hausman test in to the regression equation (P&G, Eq. 10)—like the control a normal IV estimation (Papies et al., 2017). If this correla- function approach for IV model estimation. The Gaussian tion is significant, the Gaussian copula approach indicates copula control function approach can also account for mul- a potential endogeneity problem and the inclusion of the tiple endogenous regressors, which require the simultaneous Gaussian copula term in the regression model should cor- inclusion of multiple copula terms, one for each regressor rect the endogeneity problem. Otherwise, if the correlation (P&G, Eq. 12). In this variant, the copula term is a non- is not significant, researchers assume that endogeneity does linear transformation of the endogenous regressor, using not substantially affect the regression model ’s results and, −1 the inverse normal cumulative distribution function Φ therefore, they often do not include the respective copula and the empirical cumulative distribution function H as fol- terms in their final model (e.g., Campo et al., 2021; van ∗ −1 ∗ lows (P&G, p. 572): P =Φ H(P )  where P is the addi- Ewijk et al., 2021; Wlömert & Papies, 2019). t t tional copula term added to the model. This copula term’s Our literature review shows that using this procedure has parameter estimate is the estimated correlation between a surprising result. Endogeneity only seems to be an issue in the regressor and the error term scaled by the error’s vari- studies with larger sample sizes (Fig. 1). More specifically, ance (P&G, Eq. 10). On the basis of bootstrapped standard 15 of the 18 studies (83.3%) with a samples size of 5,000 and errors, a statistical test of this parameter estimate allows for larger report a significant copula term at a p -value smaller assessing whether this correlation is statistically significant than 5%. In contrast, of the 24 studies with samples sizes and endogeneity problems are therefore present (Hult et al., of fewer than 500 observations, only eight (33.3%) identify 2018; Papies et al., 2017). a significant copula term. Based on these findings, some Marketing researchers seem to increasingly adopt the of the studies conclude that endogeneity does not seem to IV-free Gaussian copula approach to address endogeneity be an issue. This conclusion may be questionable given the problems in their empirical studies. A literature search of overall pattern that our literature review reveals in Fig. 1. the use of the Gaussian copula approach to address endoge- The larger the sample size, the larger the share of signic fi ant neity problems in regression models, and a citation analy- copula terms in applications. Consequently, there seems to sis of P&G’s article in literature databases, such as ABI/ be a sample-size-related problem with the Gaussian copula’s Informs, EBSCO, Web of Science, Scopus, and Google statistical power to identify endogeneity issues. P&G’s ini- Scholar, reveal 69 publications by the end of 2020 (see the tial simulation results do not suggest that there are sample complete list of these publications and Fig. WA.1.1 in the size restrictions. They find that the approach performs very Web Appendix 1). Most of the publications (62 of the 69, well on sample sizes as low as 200 observations, which calls 1 3 Journal of the Academy of Marketing Science copula’s performance in a broader application with differ - ent endogeneity levels, alternative levels of explained vari- ance, and different endogenous regressor distributions, also taking different sample sizes into account. We find that, in addition to the sample size, the endogenous regressor’s nonnormality and its assessment are another important area of concern regarding the method’s performance. Finally, in Study 5, we reevaluate the Gaussian copula approach’s robustness against misspecification of the error term distri- bution and the endogenous variable’s correlation structure with the error term, when estimated on models with inter- cept. We find that the model with intercept is much less robust against misspecification than P&G’ s original model without intercept. Fig. 1 Significant copulas (p < 0.05) per sample size. Note: This anal- Overall, we find that when estimating models with inter - ysis is based on 58 of the 69 reviewed journal articles, excluding 11 cept (or fully centered, or standardized data), which, accord- additional studies that do not report the Gaussian copula term’s sig- nificance ing to our literature review, is most common in marketing applications, researchers need to be far more careful when applying the Gaussian copula approach. In models with for a more detailed analysis of sample size’s role in combi- intercept, the approach is much more sensitive to violations nation with other relevant factors for estimating Gaussian of its fundamental assumptions. A careless application of copula models. the Gaussian copula approach—that is, without adhering Another important observation from our literature review sufficiently to its identification conditions (i.e., assessing the addresses the use of a regression intercept. Our literature endogenous regressor’s sufficient nonnormality, the error review reveals that 53 of the 69 applications (76.8%) use term distribution’s normality, and the Gaussian correla- regression models with intercept. In the remaining 16 stud- tion structure)—poses a potential threat to the validity of ies, 13 (18.8%) use fully centered or standardized data, and this approach’s findings. However, our simulation results only three studies (4.3%) do not include an intercept in an also allow us to determine boundary conditions that serve unstandardized model (Web Appendix 1, Fig. WA.1.2). This as guidelines for the Gaussian copula approach’s appro- observation appears meaningful, because regression models priate use and allow to obtain reliable results (e.g., with without intercept are a restricted version of the more gen- the expected error rate) when identifying and correcting eral model with intercept. They require strong assumptions endogeneity problems in regression models with intercept. to yield meaningful and unbiased estimates for the regres- While the correlation structure is inherently unobservable, sion parameters. In most cases, researchers do not have suf- researchers should ensure the following prerequisites: First, ficient support for these assumptions and therefore estimate they should assess the normality of the error term by test- a model with intercept (or fully center or standardize their ing the regression residual’s normality. Second, researchers data). However, the predominant use of regression models should confirm sufficient (and not only significant) non- with intercept requires special attention, because P&G did normality of the endogenous regressor. Third, they should not consider models with intercept in their initial simulation consider far larger sample sizes than originally expected. studies. For each of these steps, we derive clear guidelines on how This research presents five simulation studies to sub- researchers can verify that they have met each requirement. stantially extend and deepen our knowledge of the Gauss- The results of this research extend our knowledge of the ian copula approach to address endogeneity issues. Study 1 Gaussian copula approach considerably and call for it to be replicates P&G’s original simulation with a larger sample far more judiciously applied by also taking additional key size variation and, additionally, estimates the results of a factors, which had been previously ignored, into account. model with intercept. The study indicates that the Gaussian We provide recommendations and a flowchart illustration copula approach for models with intercept has a consider- regarding when and how the method should be appropri- able performance issue, and that the method requires much ately applied to support decision making. Researchers could larger sample sizes than originally expected. In Study 2, we therefore take advantage of the Gaussian copula approach’s investigate the performance of models with intercept further benefits to identify and correct endogeneity issues in order by varying the true underlying intercept. Study 3 allows us to ensure that their marketing studies’ results are valid, while to confirm these findings in respect of multilevel models also carefully considering its limitations. with a random intercept. In Study 4, we test the Gaussian 1 3 Journal of the Academy of Marketing Science with and without intercept) for each dataset. We apply the Simulation study 1: Intercept extension Gaussian copula approach’s two different versions to each of P&G’s case 1 of the two models. These versions are the control function approach, which adds a copula term to the regression, and In our first simulation study, we use the basic design of the maximum likelihood approach. P&G’s simulations to investigate the intercepts’ influence on the Gaussian copula’s performance. This allows us to Evaluation criteria replicate and extend the originally presented simulations, allowing the results to be compared. To evaluate the Gaussian copula approach’s performance, we examine three performance criteria: mean bias, statis- Design tical power, and relative bias. For a parameter θ (e.g., a regression coefficient), the bias is defined as   –  , which In this study, we replicate the data generating process (DGP) is the difference between its estimate  and its true value θ in P&G’s Case 1 (i.e., their “Linear Regression Model”), but in the DGP. The bias denotes the accuracy of a parameter without the additional IV: estimate. The closer the bias is to zero, the closer the esti- 0 1 0.50 mate is to the true value, and the more accurate the estimate = N , (1) P 0 0.50 1 is. Positive values indicate overestimation, while negative values indicate an underestimation of the parameter. In our −1 ∗ simulation studies, we determine the mean bias of the focal =Φ (Φ( )) (2) (endogenous) regressor’s parameter estimate over the 1,000 simulation datasets per factor-level combination. P = Φ(P ) (3) In addition, we also investigate the relative bias, which depicts a parameter’s bias in the copula model divided by Y =−1P + that in the model without copula (i.e., with untreated endo- (4) t t t geneity problem). The relative bias allows for assessing whereby Y represents the dependent regressor, P the endog- t t how much of the original endogeneity bias remains after enous regressor, and ξ the error term in the regression the copula approach has corrected it. This evaluation crite- model. The DGP specifies a linear model without intercept, rion is particularly useful for comparing models with dif- with a uniform distribution of the endogenous regressor P , ferent endogeneity bias in the untreated model. In Study 4, and a correlation of 0.50 between the endogenous regressor some design factors, such as the error term correlation and and the model’s error term. P&G generated 1,000 datasets the error term variance, affect the original endogeneity bias for sample sizes of T = 200 and T = 400, and estimated a and, therefore, the amount of bias that the copula approach linear regression without intercept. needs to correct. A remaining bias of -0.19 might therefore We pursue three main objectives with this simulation be differently evaluated, depending on whether the original study. First, we estimate regression models without inter- endogeneity bias was -0.20 or -0.90, resulting in a relative cept. Our simulations thereby replicate and confirm P&G’ s bias of 95% or 21%. results. Second, our literature review indicates that most Statistical power is the probability of a hypothesis test researchers (76.8%) include an intercept when estimating rejecting the null hypothesis when a specific alternative their models. While the DGP does not need an intercept for hypothesis is true. In other words, it is the likelihood of the reliable model estimation, because the true intercept is obtaining a significant parameter estimate at a given α-level zero, researchers do not usually know this a-priori. They (i.e., type I error level), when the true parameter is different therefore usually estimate a model with intercept. Third, we from zero. A high statistical power implies a low chance of consider a much wider range of sample sizes from 100 up making a type II error (i.e., failing to reject the null hypoth- to 60,000 observations (i.e., 100; 200; 400; 600; 800; 1,000; esis when the alternative is true). In our simulations, we 2,000; 4,000; 6,000; 8,000; 10,000; 20,000; 40,000; 60,000), estimate the statistical power by the number of significant because our literature review indicates that the sample size parameter estimates for the endogenous regressor or the might play a far more important role than initially assumed. copula term at a given α-level (e.g., p < 0.05) divided by We use the DGP to obtain 1,000 datasets for every fac- the number of sampled datasets (i.e., 1,000 per factor-level tor-level combination and estimate the two models (i.e., combination). We make the R-Code for the simulation and the results datasets available in the paper’s online repository at https:// t1p. de/ euqn. 1 3 Journal of the Academy of Marketing Science Fig. 2 Bias of the endogenous regressor choice of p-value is consistent with the level commonly used Results in studies to assess the Gaussian copula’s significance. In line with P&G’s results, the Gaussian copula models with- The results replicate and confirm P&G’ s simulation study for out intercept perform exceptionally well. Both the copula models without intercept. We also find that the Gaussian cop- term and the endogenous regressor’s power levels are close ula approach accounts for the endogeneity problem and esti- to 100% across all the sample sizes (Fig.  3, Panel A and mates the endogenous regressor’s coeci ffi ent without notice - B), regardless of estimation method used. When extending able bias, regardless of the sample size and the estimation the original study in terms of Gaussian copula models with method used (Fig. 2 shows the results of the more popular intercept, the statistical power of small to medium-sized control function approach; Web Appendix 2, Table WA.2.1, samples is less satisfactory and depends on the type of esti- provides the outcomes of the maximum likelihood approach). mation method used. The results of the control function The situation changes fundamentally when we extend approach require more than 800 observations for the copula P&G's simulation study by estimating a regression model term’s parameter estimate to achieve power levels of 80% with intercept. The results show that, in many situations, the and higher, and more than 2,000 observations for the endog- endogeneity problem has not been resolved. A substantial enous regressor to reach these power levels. The power of bias remains in the copula model for smaller to medium identifying a significant copula parameter is slightly higher samples. The endogenous regressor’s parameter bias only in small sample sizes when using the maximum likelihood reaches a negligible level for sample sizes of 4,000 and approach. This method needs only 600 observations for the more. At sample sizes of about 40,000 observations and copula term to achieve power levels of 80% and higher. The more, Gaussian copula models with intercept achieve a per- endogenous regressors’ power does not improve much, as formance level comparable to those without intercept. This it still requires about 2,000 observations to achieve power finding holds for both estimation methods (i.e., the control levels of 80%. function and the maximum likelihood approach yield almost Mean-centering could be a naïve strategy for coping with the the same parameter estimates for the endogenous regressor). intercept model’s problems. A fully mean-centered (or standard- To determine the copula term’s and the endogenous ized) model would not need an intercept for the reliable estima- regressor’s statistical power, we use bootstrapping with 500 tion of the regression parameters. We therefore also estimated resamples (for further details see P&G; they used 50 and a model without intercept, in which we mean-centered all the 100 resamples). Based on the bootstrap standard errors, we variables before entering the estimation, which 13 (18.8%) of the consider parameters significant if their p -value is smaller published studies also do. We find that the results of both methods than the 5% level. As shown in our literature review, this 1 3 Journal of the Academy of Marketing Science Fig. 3 Statistical power of the copula term and the endogenous regressor are equivalent to the model with intercept (see Web Appendix 2, endogeneity bias. Nevertheless, researchers should be aware Tables WA.2.1 to WA.2.3). that the method is less effective at reducing endogeneity bias in models with intercept when the sample sizes are small, Discussion and that they need to interpret such results more carefully. Another important issue is the method’s ability to iden- Our discussion addresses two important aspects of our tify endogeneity problems. Some recommendations suggest study’s results. First, we discuss the impact of including testing for the presence of endogeneity in regression models intercepts in regression models regarding the Gaussian cop- by using the copula term’s significance (Hult et al., 2018; ula approach’s performance. Thereafter, we address poten- Papies et al., 2017). The copula approach’s low power to tial reasons for why an intercept weakens the performance. identify a significant parameter for the copula term in mod- els with intercept makes this practice highly problematic Consequences of intercept inclusion if the sample sizes are not large enough. Researchers need to be far more careful when using the copula term’s sig- Our analysis reveals two important findings. First, the nificance to decide whether endogeneity poses a problem Gaussian copula approach cannot always correct the endo- and whether or not to include the copula term. This is par- geneity problem when estimating the regression model with ticularly important, since our literature review revealed that intercept. Smaller sample sizes are still subject to substan- researchers currently also use this approach with relatively tial bias. Second, when estimating the model with intercept, small sample sizes and, to some extent, probably mistak- the Gaussian copula approach has low power to identify enly conclude that endogeneity is not a problematic issue significant error term correlations in smaller sample sizes. in their model. There are two main reasons for the copula This finding not only holds when using the maximum like- term’s low statistical power. First, small sample sizes lead lihood approach but is also somewhat more pronounced to a substantial underestimation of the copula term’s param- when using the control function approach. Our literature eter, which is a scaled version of the correlation between the review reveals that 64 of the 69 studies (92.8%) use the endogenous regressor and the error term (Web Appendix 2, latter approach when applying the copula approach empiri- Table WA.2.2). Second, in the control function approach, cally, which underlines the importance of these findings. the estimated parameter also comprises the error term’s While the remaining bias can be substantial, some research- variance (P&G, Eq. 10). The parameter estimate contains ers could argue that correcting some bias is still better than additional noise that inflates the standard errors and makes not correcting any. The Gaussian copula approach might the estimation unreliable, especially in smaller sample sizes. therefore still be valuable although it cannot fully correct the Consequently, this approach has a slightly weaker power compared to the original maximum likelihood approach, which allows for estimating the parameter without the error Three studies do not report how they include the copula, while only variance’s scaling. two report the use of the original maximum likelihood approach. 1 3 Journal of the Academy of Marketing Science Overall, the simulation results show the importance of a also provide a solution to this problem in their Appendix I, sufficient sample size if the Gaussian copula approach is to where they show that even models with endogenous regres- perform well in regression models with intercept in terms of sors that are normally distributed can be identified if (1) identifying and correcting endogeneity problems. This is a the normal variable has a non-zero mean, and (2) the esti- novel n fi ding that has not yet been reported. While this n fi d - mated model does not include an intercept. Nonnormality ing imposes limitations on the method in finite samples, our is therefore only required in models that are estimated with simulations also show that when increasing the sample size intercept. The endogenous regressor’s availability of a non- toward infinity, the method’ s bias is reduced to zero (i.e., it zero mean (i.e., the uniform distribution has a mean of 0.50) is a consistent estimator) and has sufficient power to iden- and the absence of an intercept can therefore compensate tify endogeneity issues. Furthermore, we show that using for smaller sample sizes’ lack of sufficient information from mean-centered (or standardized) data is not a valid strategy nonnormality. However, in models with intercept (or when for coping with this issue, which is also in line with previ- mean-centering the data), this mechanism is not at play and ous research on mean-centering (Echambadi & Hess, 2007). the lack of information from sufficiently strong nonnormal- ity makes it harder to separately identify the copula term Potential reasons and the regressor’s parameter, which results in the pattern of bias that we observe. Consequently, in models with inter- The pronounced differences between the models estimated cept (or mean-centered data), the regressor’s nonnormality with and without intercept raise the question: why is there needs to be much stronger than in models without intercept. is such a big difference in their performance? Identifica- tion problems could be a potential reason for the weaker performance in models with intercept. P&G highlight two Simulation study 2: Different intercept levels important pre-requisites for identifying the Gaussian copula model. The rs fi t is the endogenous regressor’s nonnormality: In Study 1, we replicate the original simulation results by “If P follows a normal distribution, P is a linear transfor- P&G and show that including a regression intercept in the ∗ −1 mation of P since P =Φ H(P ) . Hence, we cannot sepa- estimation reduces the copula approach’s performance (both t t rately identify α and  ⋅  in (10). As the true distribution of in terms of bias correction and statistical power). In this P approaches a normal distribution, the correlation between study, we extend these findings by varying the level of the P and P increases, causing a multicollinearity problem” intercept. (P&G, p. 572). The second is the error term’s normality: While P&G’s original DGP does not include an intercept “We assume that the marginal distribution of the structural (i.e., the intercept is zero), it is unlikely that the true intercept error term is normal” (P&G, p. 570). Both assumptions are will be zero in practice. Estimating a regression model with- fulfilled in our simulations’ DGPs. The error term follows a out intercept requires strong assumptions that are untestable normal distribution with N(0,1), and the endogenous regres- a-priori. In their applications, researchers usually estimate sor a uniform distribution with U(0,1). However, there is regression models with intercept. In addition, similar to an still substantial correlation between the regressor P and the ignored endogeneity problem, ignoring an intercept when copula term P (e.g., on average we observe a correlation of it is necessary is also likely to induce strong bias. Conse- 0.973 in the model with intercept). Consequently, smaller quently, it is usually not recommended to simply estimate sample sizes seem to have not enough information on the a regression without intercept, and it is unclear whether the difference between the nonnormal distribution of the regres- copula approach can compensate for this type of bias. sor P and the normal distribution of the copula term P to allow a robust estimation of the parameters. If the differ - Design ences are too small, it is difficult to distinguish the variation that is a result of endogenous regressor from the variation We use the same DGP as in Study 1, but instead of using that stems from the error term. In our study, we observed Eq. 4, which does not include an intercept, we add the inter- that the copula term’s parameter is underestimated propor- cept i to Eq. 5 constituting Y : tional to the parameter overestimation of the endogenous Y = i − 1P + (5) t t t regressor. When the sample size increases, this makes more information available about the differences between the two In this simulation study, we vary i ϵ {−10, −3, −0.50, predictors, and their bias shrinks toward zero. −0.10, 0, 0.10, 0.50, 1, 3, 10}. But why can the model without intercept be more eas- ily identified than the model with intercept? P&G might 1 3 Journal of the Academy of Marketing Science Fig. 4 Bias of the endogenous regressor with varying intercepts in the copula regression copula’s performance is independent of the intercept’s size, Results and all of Study 1’s findings apply here as well. The results show that the intercept variations do not affect the Gaussian copula approach’s bias (Fig. 4, Panel A) and Simulation study 3: Multilevel model power when we estimate the model with intercept. We find the same performance as in Study 1, with smaller sample To extend the simple linear model in Studies 1 and 2, this sizes showing relatively high bias and low power, both of simulation study utilizes a multilevel model to assess the which improve with sufficiently large sample sizes. sample size’s effect in more depth. In particular, we investi- In contrast, we find that the intercept’ s variation affects gate the effect of different sample sizes within-cluster (level the Gaussian copula approach when estimated by means of a 1) and between-clusters (level 2) on the Gaussian copula model without intercept (Fig. 4, Panel B). When the difference model’s performance. For this purpose, we use a two-level, between the true intercept and zero increases, the model’s bias random-intercept model (often referred to as a panel data also increases as expected. Similar to Study 1, the performance model in economics). The endogeneity problem occurs at is not dependent on the sample size. However, the bias from the within-cluster level as a result of a correlation between the omitted intercept can be larger than the endogeneity bias the within-cluster (level 1) predictor and the within-cluster depending on the intercept’s size. The regression model is, (level 1) structural error. Although other endogeneity prob- of course, misspecified when estimated without intercept on lems could arise in multilevel models (e.g., correlations the basis of a DGP that includes an intercept. Constraining a between level 1 predictors and level 2 error terms, etc.), parameter (in this case to zero) without suc ffi ient prior assump - in our literature review, the abovementioned endogeneity tions will cause this bias in the estimation. Nevertheless, it problem seems to be marketers’ most common concern, as is interesting that the Gaussian copula in a model without they introduce copulas to level 1 (within-cluster) predic- intercept cannot correct the bias of an omitted intercept. If tors to avoid correlation with the level 1 structural error. researchers simply omit the intercept, they will trade one bias Moreover, other instrument-free methods, such as the gen- for another. eralized method of moments approach by Kim and Frees (2007), might address the correlations between regressors Discussion and higher-level error terms. If the DGP includes a non-zero intercept, estimating the Design model without intercept is not an option, because the results are biased, even if the model contains a Gaussian copula We use a similar DGP as in Study 1, but extend it to the term. In contrast, when the estimated model includes an two-level, random-intercept model. Instead of Eq. 4, which intercept, the Gaussian copula approach can correct the does neither include an intercept nor does it consider the endogeneity bias if the sample size is large enough. The 1 3 Journal of the Academy of Marketing Science clustering of level 1 (within-cluster) observations, we use on non-parametric bootstrapping, we considered two differ - the following Eq. 6: ent alternatives of sampling the cases in the bootstrapping. We subsequently report the results of sampling the cases at Y = u + u − 1P +  , jt 0 j jt jt (6) the cluster level (level 2), which is advised when estimating multilevel data models (Goldstein, 2011). However, most of where the outcome Y and regressor P are observed at jt jt the studies in our literature review do not reveal the kind of the within-cluster level (level 1; e.g., time) with t = 1…T bootstrapping strategy they use. Consequently, we also use observations in each cluster j (level 2; e.g., brands). The a different bootstrapping strategy in which we sample the random intercept u ∼ N 0,  denotes an error component level 1 observations directly (i.e., ignoring the hierarchical that is specific to the cluster and captures all unobserved data structure) and find very similar results. Finally, we do level 2 specific effects (e.g., all effects that are specific for a not estimate models without intercept in this study, as the brand, but do not vary over time). Both the error component original DGP includes a random intercept and ignoring this at level 2 (i.e., the random intercept u ) and the structural random-intercept structure could itself induce bias and inef- level 1 error component  need to be uncorrelated with the jt 3 ficiency (similar to Study 2). level 1 regressor P for efficient and consistent estimation. jt However, similar to Study 1, we assume an error correla- Results tion between P and  of 0.50 in this DGP, so that Eqs. 1–3 jt jt become: The results of the endogenous regressor’s bias in Fig.  5 (Panel  A) show that there are basically no differences 0 1 0.50 jt = N , (7) ∗ between the copula models in this study and the simple lin- P 0 0.50 1 jt ear model in Study 1 when using the total sample size (i.e., the combination of within-cluster and between-cluster obser- −1 ∗ vations) as a reference. Both the random-intercept multilevel =Φ Φ( ) (8) jt jt model and the fixed-effects panel model follow the same pattern as the simple linear model with copula and intercept P = Φ(P ) (9) in Study 1, with a bias that only reaches a negligible level jt jt for sample sizes of 4,000 and more. In addition, we hardly We systematically vary both the level 1 and level 2 sam- observe any variations in the bias in different combinations ple sizes T and J ϵ{5, 10, 20, 40, 60, 80, 100, 200, 400, of level 1 (within) and level 2 (between) sample sizes, which 600, 800}, excluding total sample sizes lower than 100 result in the same total sample size. and larger than 40,000 for reasons of efficient estimation. The results of the copula term’s power and the endog- In addition, we set the non-random intercept u to zero and enous regressor’s power also follow very similar patters as the variance of the random intercept to one (i.e.,  = 1). their counterparts in the simple linear model in Study 1. Consequently, u is uncorrelated with both P and  . We j jt jt Figure 5 (Panel B) illustrates this pattern in respect of the estimate the model with a random-intercept, multilevel copula term’s power in the random-intercept multilevel model using maximum likelihood estimation. Moreover, model and fixed-effect panel model compared to the cop- we also consider a fixed-effects panel estimator. We esti- ula term’s power in the model with intercept in Study 1. In mate both models with and without the control function contrast to the bias, we observe a slightly larger variation approach by adding an additional copula term. Because the in power for different combinations of level 1 (within) and estimation of the copula model’s standard errors is based level 2 (between) sample sizes, which result in the same total sample size. More specifically, the power seems to be 3 slightly larger if the number of level 1 observations (i.e., If the within-cluster level 1 regressor P correlates only with the jt within cluster observations, e.g., the time series) is larger random intercept u , but not with the structural error   , the fixed- j jt effects panel model estimator is consistent, but not efficient. and the number of level 2 observations (i.e., the number of Although J is usually large in typical panel data models and T cross-sectional units, e.g., brands) is smaller. Table WA.3.1 small, the opposite is true of multilevel models employed in the social (Web Appendix 3) illustrates this effect for exemplary total sciences where researchers, for example, investigate many students sample sizes ranging from 100 to 4,000 observations. How- clustered within a few schools. In our literature review, most of the ever, most of the variation in the copula term’s statistical studies that use Gaussian copulas to address endogeneity in multi- level data have larger J and smaller T, although we also found studies power comes from the total sample size. Overall, the gen- with large T and small J. Consequently, we systematically vary both eral finding is the same as in Study 1: a sufficient copula components. term power of 80% is only reached with 800 and more total To focus our analyses, we do not consider the alternative maximum observations. likelihood method for copula estimation in this study, because it is rarely used in empirical application and complex to implement. 1 3 Journal of the Academy of Marketing Science Fig. 5 Simulation results for the multilevel model Second, the approach requires nonnormality of the endog- Discussion enous regressor, and Study 1 has highlighted that even the uniform distribution might not be sufficiently nonnormal to This study shows that both the bias and the statistical pow- identify the model in smaller sample sizes. We therefore er’s pattern of results are very similar to Study 1 when the vary the endogenous regressor’s distribution. Third, we total sample size is considered. Different level 1 and level systematically vary the ratio of explained to unexplained 2 sample sizes resulting in the same total sample size only variance (i.e., the R ) in the regression model. This is poten- marginally affect the bias and statistical power. We can tially important because the endogenous regressor’s different therefore conclude that the total sample size is the important distributions imply different variance for this variable. Com- criterion to consider when evaluating the appropriateness bined with a fixed error term variance, this would lead to dif- of the Gaussian copula approach. Further, the findings from ferent ratio of explained to unexplained variance, potentially the simple cross-sectional model are generalizable to the confounding the effect of the distribution with R levels. multilevel model’s total sample size. We therefore continue In addition, the ratio of explained to unexplained variance to explore this much simpler model and extend it in other influences the uncertainty in the parameter estimates (i.e., important ways. the parameters’ standard errors), potentially influencing the approach’s statistical power. Since most researchers use an intercept to estimate their regression models in practice, we Simulation study 4: Extension by additional will only focus on the performance of models estimated with factors intercept in this study. Finally, we again estimate our mod- els with the control function and the maximum likelihood Our previous simulation models investigated the role of the approach. The simulation’s detailed design, which is very intercept when estimating the Gaussian copula approach. similar to that of the previous studies, can be found in Web While these focused studies help us understand the role of Appendix WA.4. the intercept and sample size, they only use a single non- Since the endogenous regressor’s nonnormality is a pre- normal distribution of the endogenous regressor (i.e., the requisite to apply the Gaussian copula approach, in practice, uniform distribution) and a fixed error term correlation of researchers usually test whether the encountered distribution 0.50. In this study, we broaden our scope and investigate is significantly different from a normal distribution. How - three additional factors that are potentially important for ever, it is currently unknown when the endogenous regres- the performance of the Gaussian copula approach. First, the sor’s distribution is sufficiently nonnormal to allow the level of the error correlation with the endogenous regres- application of the Gaussian copula approach. We therefore sor defines the endogeneity problem’ s severity, potentially also assess different nonnormality tests and simple moment affecting both the bias and the power of the Gaussian copula measures, like skewness and kurtosis, to identify situations approach (for detailed expectations regarding the different which support the reliable usage of the Gaussian copula assessment criteria, see Web Appendix 4, Table WA.4.1). 1 3 Journal of the Academy of Marketing Science approach. Our literature review reveals that 34 of the 69 results. The endogenous regressor’s power again depends (49.3%) studies use the Shapiro–Wilk test, 4 (5.8%) the Kol- on the sample size, but also, as expected, on the R level: mogorov–Smirnov test, 4 (5.8%) the Kolmogorov–Smirnov the power increases with increasing R . It should be noted test with Lilliefors correction, 2 (2.9%) the Anderson–Dar- that, in this study, the endogenous regressor’s average power ling test, and 2 (2.9%) Mardia’s coefficient. Moreover, only is higher than in the previous studies, because we consider two studies (2.9%) analyze the skewness. The remaining higher R levels than in the original replication model 21 studies (30.4%) do not test or do not report how they (where we have an R of only about 10%). In contrast, the tested nonnormality. To assess which nonnormality test best endogenous regressor’s power only depends marginally on captures the degree of nonnormality needed to identify the the error term’s correlation level Gaussian copula approach, we include these and additional With regard to the endogenous regressor’s bias, we find tests (i.e., Cramer-von Mises, Shapiro-Francia, Jarque–Bera, that it again depends strongly on the sample size. The bias D’Agostino, and Bonett-Seier) that the literature suggests decreases with increasing sample sizes. The bias is on aver- (e.g., Mbah & Paothong, 2015; Yap & Sim, 2011) in our age lower in this simulation than in the previous simulations, simulation study. because we consider different endogeneity and R levels. However, the relative bias (i.e., the copula model’s bias Results divided by the endogenous model’s bias) follows the same pattern as our other simulation studies, reaching about 50% The results presentation begins with the main effects of the for sample sizes of 100 observations, which is similar to potentially relevant factors, namely the sample size, R , and Studies 1 to 3 when we include the intercept in the estima- endogeneity (error correlations), as well as their different tion. Moreover, the endogenous regressor’s bias decreases levels, on the Gaussian copula’s performance (i.e., power with higher R levels, but the relative bias does not depend and bias). Thereafter, we assess the effect of the endogenous on R (i.e., the copula bias decreases with the same magni- regressor’s distribution (nonnormality) on power and bias. tude relative to the original regression’s bias). Finally, the Next, we present the results of skewness and kurtosis as well bias also depends on the endogeneity level, with increasing as different nonnormality tests’ suitability to reliably iden- bias with increasing error correlations. However, the endo- tify endogeneity with Gaussian copula models. We focus geneity level again does not affect the relative bias, because our presentation on the control function approach’s results, the bias in the copula model increases proportionally to the which is the most common approach by far. Overall, the bias in the original regression without copula. maximum likelihood approach yields similar results. The We conclude and reconfirm that the Gaussian copula’ s detailed results of the maximum likelihood approach are pre- performance depends strongly on the sample size, with sented in the Web Appendix 4 (Table WA.4.2, Fig. WA.4.1). substantial effects on both power and bias. In contrast, the endogeneity level does not affect the copula model’ s ability Main effects of design factors to correct the endogeneity bias as indicated by the relative bias, but does affect the copula term ’s power. The higher The results in Table 1 show the main effects of the sample the error term correlation (i.e., the more severe the endo- size, R , and endogeneity levels (error correlations) when geneity problem), the greater the power to identify endo- averaged across the other simulation factors with regard to geneity. Finally, we find that the R level is not relevant for the mean and relative bias of the endogenous regressor and the copula performance, as it neither affects the power nor statistical power of the copula term and endogenous regres- relative bias. In the following analyses, we will therefore sor (at the 5% error level). not further consider R variations and only focus on the We start the analysis by focusing on the copula term and interplay between the level of endogeneity, the sample size, the endogenous regressor’s statistical power. With respect and the distributional form. to the copula term’s power, we confirm that it strongly We substantiate these findings by using a (logistic) regres- depends on the sample size and only reaches acceptable lev- sion with the copula and the endogenous regressor’s power, els beyond 2,000 observations. Moreover, the copula term’s as well as the endogenous regressor’s mean and relative power does not depend on the R level, but, as expected, bias, as dependent variables and the design parameters as depends strongly on the endogeneity level (i.e., the error independent variables. The results indicate that the R level correlation): the higher the error term correlation (i.e., the does not have a significant influence on the copula term’ s more severe the endogeneity problem), the higher the copula power, or on the relative bias, while all the other simulation term’s power to identify endogeneity. This picture changes factors have significant effects (see the Web Appendix 4, somewhat when we examine the endogenous regressor’s Table WA.4.3). 1 3 Journal of the Academy of Marketing Science Endogenous regressor’s distribution Table 1 Effects of key simulation factors on the Gaussian copula approach’s performance Next, we analyze the power and relative bias of different Statistical power Bias Relative bias distributional forms (i.e., different levels of nonnormality) Copula term (%) Endogenous Endog- Endogenous when varying the sample sizes and the endogeneity levels. regressor (%) enous regressor (%) The results show that complex interactions between the regressor distribution, sample size, and endogeneity level influence Sample size a copula term’s power (see Web Appendix 4, Fig. WA.4.2). 100 16 59 0.189 51 For weak endogeneity problems, even heavily nonnormal 200 32 78 0.119 32 distributions, like the log-normal or gamma distribution, 400 49 90 0.064 17 show quite low power unless the sample sizes are very large. 600 59 95 0.042 11 However, for larger error correlations, strongly nonnormal 800 65 97 0.030 8 distributions also have sufficient power if the sample sizes 1,000 70 98 0.023 6 are smaller. 2,000 80 99 0.010 3 In contrast, the endogeneity level does not affect the 4,000 88 100 0.004 1 endogenous regressor’s relative bias. Our analysis indicates 6,000 91 100 0.002 1 that only a combination of sample size and distributional 8,000 93 100 0.001 0 form affects the relative bias and that larger sample sizes 10,000 94 100 0.001 0 and the distributions’ higher nonnormality reduce the endog- enous regressor’s relative bias (Fig. 6). Interestingly, we also 10% 67 84 0.063 11 observe a few situations in which heavily nonnormal distri- 20% 67 88 0.054 12 butions (i.e., some of the gamma, log-normal, and chi ) over- 30% 67 91 0.048 12 compensate the endogeneity bias in smaller sample sizes, 40% 67 93 0.043 12 resulting in a bias in the opposite direction of the original 50% 67 95 0.038 12 endogeneity bias (e.g., underestimating instead of overesti- 60% 67 97 0.034 12 mating the coefficient). 70% 67 98 0.029 12 Endogeneity level (error term correlation) Nonnormality tests 0.10 28 89 0.010 9 0.20 50 90 0.023 11 Since endogeneity is not observable a-priori, researchers can 0.30 62 92 0.033 12 only assess the distribution’s nonnormality and the sample 0.40 70 92 0.043 12 size to decide whether the Gaussian copula approach could 0.50 76 93 0.050 12 be applied. Accordingly, several Gaussian copula applica- 0.60 80 94 0.057 12 tions in our literature review test the endogenous regressor’s 0.70 84 94 0.065 12 nonnormality by using a nonnormality test, mostly the Sha- 0.80 86 95 0.073 12 piro–Wilk test. However, common nonnormality tests’ high sensitivity to small deviations from normality is a problem. In our simulation, for example, the Shapiro–Wilk test reports a significant (at p < 0.05) finding in 96% (94% with p < 0.01) greatly. We therefore also assessed the correlation between of all the cases (Table 2). Only the D’Agostino and Bonett- the copula term’s bootstrap t-statistic and the nonnormality Seier tests have sensitivity rates below 90%. In contrast, the tests’ test statistic. Table 2 shows that the Anderson–Darling copula term is only significant in 67% of the cases. Conse- and Cramer-von Mises tests have the highest correlation with quently, nonnormality test cannot help researchers directly the copula term’s bootstrap t-statistic. In addition, the results decide whether a distribution is sufficiently nonnormal to indicate that kurtosis and skewness alone are not good pre- apply the Gaussian copula approach. Owing to our simu- dictors of the copula term’s t-statistic. Nevertheless, it is lation study, we find that the correspondence between the interesting that skewness seems to be more important than copula and the nonnormality test’s significance is relatively kurtosis. Finally, we also find that the correlation between low (between 61% and 76%), with no test clearly outper- the VIF and copula t-statistic is not very pronounced. forming the other (for the correspondence analysis, see the Web Appendix 4, Table WA.4.4). This outcome is roughly Discussion and boundary conditions analysis equivalent to the copula term’s power (i.e., 67%). The analyzed p-values (i.e., 0.05 and 0.01) represent Based on Study  4’s simulation results, we find that the arbitrary cut-off levels that may reduce the correspondence amount of explained variance has no noticeable influence 1 3 Journal of the Academy of Marketing Science Fig. 6 Relative bias of the endogenous regressor for different dis- (8), blue (14), purple (20); Gamma distribution (α, β): red (1, 0.50), tributions with varying distribution parameters, sample sizes, and green (1, 2), blue (2, 4), purple (4, 2); Log-normal distribution (μ, σ): endogeneity levels. Note: Different colors represent different distri- red (0, 1), green (0, 0.75), blue (0, 0.50), purple (0, 0.25); Student t bution parameters: Beta distribution (p, q): red (0.50, 0.50), green distribution (df): red (3), green (4), blue (5), purple (6) (1, 1), blue (2, 2), purple (4, 4); Chi distribution (df): red (2), green on the Gaussian copula’s power. In contrast, and as expected, failure to identify a significant copula does not necessarily imply the the endogeneity level has a strong effect (i.e., it is harder to absence of endogeneity. It could imply a relatively small endogene- identify a small endogeneity problem). However, even for high ity problem (which might be negligible), but it could also imply an levels of endogeneity the Gaussian copula approach still per- insuc ffi ient sample size or nonnormality. A suc ffi ient sample size forms poorly when sample sizes are small. We also confirm and the careful assessment of nonnormality are therefore particu- the sample size’s strong effect on the Gaussian copula’ s power larly important for the Gaussian copula approach’s application. and bias, and the importance of the endogenous regressor’s Popular nonnormality tests, such as the Shapiro–Wilk nonnormality to identify the Gaussian copula’s parameter test, which, according to our literature review, is the one estimates. Consequently, researchers should use the Gaussian most often used in Gaussian copula applications, do not copula approach cautiously if they suspect the endogeneity identify sufficient nonnormality with common p < 0.05 (or problem is not pronounced (i.e., a small error correlation), the p < 0.01) thresholds. These tests are too sensitive to small sample size is small, or the nonnormality is insufficient. deviations from nonnormality that could lead to insignificant While the sample size is observable and the nonnormality can copula terms, even for substantial endogeneity problems be analyzed, the Gaussian copula approach’s objective is to deter- (i.e., large error correlations). In addition, the nonnormality mine the endogeneity level, which is unknown a-priori. However, a should specifically stem from skewness and not (only) from 1 3 Journal of the Academy of Marketing Science Table 2 Correlation analysis and statistical power of nonnormality and the Gaussian copula’s power level. We reveal, for exam- tests ple, that the lower the number of observations, the higher the skewness levels required to obtain power levels of 80% Correlation p < 0.05 p < 0.01 and higher (Web Appendix, Fig. WA.4.3). Similarly, we find Shapiro–Wilk −0.423 96% 94% that smaller sample sizes require higher levels of the Ander- Kolmogorov–Smirnov −0.029 93% 90% son–Darling and Cramer-von Mises test statistics for a cop- KS-Lilliefors 0.323 93% 89% ula power of at least 80%. These two test statistics’ required Anderson–Darling 0.663 96% 94% levels decrease with a higher number of observations. In Cramer-von Mises 0.653 95% 92% contrast, we observe no clear pattern for the kurtosis, which Shapiro-Francia −0.420 96% 93% is in line with its low correlation with the t-statistic. Jarque–Bera 0.018 99% 99% To turn these findings into more actionable recom- D'Agostino 0.565 73% 71% mendations, we consider all observable characteristics Bonett-Seier 0.477 84% 80% of our models (e.g., sample size, skewness, kurtosis, R , Variance inflation factor (VIF) −0.234 – – and nonnormality test statistics) to derive thresholds that Kurtosis 0.154 – – will ensure that the Gaussian copula approach has a high Skewness 0.313 – – power level. Researchers can use these thresholds as an Absolute skewness* 0.341 – – approximate point of orientation to ensure the method’s effective use in their applications. We do so by employ - The second column shows the correlation between the copula term’s bootstrap t-statistic and the test statistic of the endogenous regressor’s ing decision tree analysis, using the C5.0 algorithm (Kuhn nonnormality test. Columns three and four show the statistical power et al., 2020). Based on our simulation study’s results (i.e., of the nonnormality test (i.e., the percentage of the tests that are sig- Study 4 of regression models with intercept), our goal is to nificant at the given p-level) identify situations where the Gaussian copula approach has *We also assessed the absolute skewness, because it should not mat- a power of at least 80%. Figure WA.4.4 (Web Appendix) ter whether the distribution is skewed to the left or the right side of the mean. Removing the sign might therefore provide a more realistic shows a decision tree result in which we consider sample picture of the actual correlation size, skewness, kurtosis, and R² for predicting the copula’s power (the latter two are not relevant and therefore do not appear in the decision tree). The classification error is 6.4% with 8 false negatives and 6 false positive out of 220 simu- lation design conditions (i.e., 20 distributions times, 11 kurtosis. Our results show that nonnormal distributions with high kurtosis, but small skewness, perform relatively poorly sample sizes). According to the results, the sample size should be larger than 1,000 observations if the skewness regarding identifying the copula term with small to medium sample sizes. Researchers are therefore also advised to is larger than 0.774. If the skewness is equal to or smaller than this level, more than 2,000 observations are required report these more descriptive nonnormality statistics when describing their variables’ nonnormality. Finally, we find to obtain an 80% power level. For smaller sample sizes in the range between 400 to 1,000 observations, a skewness that the Cramer-von Mises tests and the Anderson–Darling test seem to be the most promising candidates for identify- level of 1.932 is required to obtain adequate power. None of our distributions achieves a sufficient power level for ing sufficient nonnormality, because they correlate best with the copula term’s t-statistic. This is not surprising, as both the copula term for sample sizes of 200 observations or smaller. Please note that these findings are derived from the tests build on the empirical cumulative distribution func- tion, which also underlies the Gaussian copula approach. outcomes of the simulation studies, which are constrained by the parameter space of the simulation design. Therefore, The Cramer-von Mises test statistic is the integral of the squared deviation of the endogenous regressor’s empirical these thresholds are an approximate point of reference to guide decision-making. Moreover, researchers must ensure distribution and the theoretical normal distribution. The Anderson–Darling test is an extension of the Cramer-von that their empirical examples meet the other necessary con- ditions for using the Gaussian copula approach that we Mises test that adds a weighting factor to put more weight on the distribution’s tails. investigate in this research (see Fig. 8 for a comprehensive summary). Using our simulation results, we subsequently derive actionable boundary conditions for the required nonnor- We ran similar decision tree analyses that considered the Anderson–Darling and the Cramer-von Mises test statistics mality and sample size, and provide recommendations that could help researchers identify situations with sufficiently (see the Web Appendix 4, Fig. WA.4.5). For example, if the Anderson–Darling (Cramer-von Mises) test statistic has high copula term power in regression models with endoge- neity. In general, we find a complex relationship between a value larger than 18.964 (3.488), the Gaussian copula’s power is 80% and higher. With a sample size of more than the sample size, the endogenous regressor’s nonnormality, 1 3 Journal of the Academy of Marketing Science 1,000 observations, a somewhat lower level of the test sta- Results tistic, but larger than 15.159 (2.628), can achieve this power level. In respect of the error term misspecification, we find the In summary, the endogenous variable’s nonnormality, as same overall pattern of remaining bias and low power at indicated by minimum levels of skewness, and the Ander- smaller sample sizes when the model is estimated with son–Darling or the Cramer-von Mises test statistics, in com- intercept as in our previous studies (for the detailed results, bination with a sufficiently large sample size, may ensure please see Web Appendix 5). However, regarding the bias, that the Gaussian copula approach has adequate power. Our we uncover an additional problem related to the misspecifi- study results suggest that researchers need to ensure that cation of the error term. When the error term is nonnormally there are relatively high nonnormality levels, which should distributed, the Gaussian copula approach is no longer con- stem from the endogenous variable’s skewness, and a rela- sistent (Fig. 7). That is, the remaining bias does not shrink tively large sample size, in order to apply the Gaussian cop- toward zero when the sample size increases. Instead, the ula approach adequately in regression models with intercept. bias approaches an unknown nonzero constant, depending on the error term’s level of nonnormality. In our simulation, this value is positive for negative kurtosis (e.g., beta distri- Simulation study 5: Robustness butions) and negative for positive kurtosis (e.g., student-t to misspecification distributions). In the latter case, the method overcorrects the initially positive endogeneity bias, resulting in a nega- Besides the nonnormality of the endogenous regressor, P&G tive remaining bias. In all our cases, the bias does not occur highlight two additional important criteria to identify the when estimating the model without intercept, reconfirming Gaussian copula approach: 1) the normality of the error P&G’s results on robustness without intercept. The power of term, and 2) the Gaussian copula correlation structure. In the copula term (i.e., the test for the presence of significant their simulations, they show that the method is robust against error correlation) does not seem to be affected beyond the misspecification of the error term and correlation structure. already uncovered issues in our previous simulation studies. However, these simulations are also estimated without inter- Variations in power due to the error term distributions are cept. This study investigates whether including an intercept relatively small and limited to smaller sample sizes. in the estimation retains this robustness or causes additional The results show similar problems in terms of the copula problems. To achieve this objective, we again closely repli- misspecification when estimating the method with inter - cate the simulations from P&G (for detailed design of these cept. For some correlation structures (e.g., Frank and Far- simulations, see Web Appendix 5). For the error term mis- lie-Gumbel-Morgenstern), the method does not correct any specification, we specify several symmetric nonnormal error bias, when estimated with intercept (while we reconfirm its distributions from the Beta and Student-t family, which are robustness when estimated without intercept). Other copula similar to those used in Study 4. We thereby extend the sim- models show a similar pattern as the error term’s misspecifi - ulation by P&G, who only report the uniform distribution’s cation. The bias varies by sample size, decreasing with larger results (i.e., Beta[1,1]). In addition, we evaluate whether the sample sizes but converging to an unknown nonzero con- error term’s nonnormality also manifests in nonnormality of stant, which differs across the analyzed copula models. For the regression residual. If this is the case, researchers could those copulas that correct the bias, the statistical power of evaluate whether their model fulfils this identification crite- the copula term is not affected beyond the already revealed rion. In respect of the copula structure misspecification, we small sample size issues. However, the statistical power of use the same alternative copula models as in P&G’s article those copulas that do not correct the bias (i.e., Frank and (i.e., Ali-Mikhail-Haq distribution with θ = 1, Plackett dis- Farlie-Gumbel-Morgenstern) is low across all sample sizes, tribution with θ = 20, Farlie-Gumbel-Morgenstern distribu- erroneously indicating an absence of endogeneity. tion with θ = 1, Clayton copula with θ = 2, and Frank copula with θ = 2). Comparing the Gaussian copula maximum likelihood approach to the control function approach, we find that the remaining biases (in both directions) are larger in the maximum likelihood approach. This These thresholds become more restrictive for higher power levels. suggest that the Gaussian copula control function approach is slightly For example, to accomplish a 90% power level, the Gaussian copula more robust against error term misspecifications. approach requires more than 600 (2,000) observations at a skew- With respect to the copula model misspecification, both the ness level exceeding 1.974 (0.998). Similarly, the Anderson–Darling approaches (i.e., maximum likelihood and control function) show (Cramer-von Mises) test statistic requires a value of more than 67.875 indistinguishable patterns of bias. This suggests that both approaches (12.246) for sample sizes equal to and smaller than 2,000 and a value are equally affected. of 46.832 (7.994) for sample sizes larger than 2,000 observations. 1 3 Journal of the Academy of Marketing Science Fig. 7 Bias of the endogenous regressor for different error term distributions checked by assessing the regression residual (we find prom- Discussion ising results in this regard, which we report in Web Appen- dix 5), the correlation structure with the unobservable error Researchers estimating models with intercept (which is the term is inherently unobservable, and therefore solely subject standard use case in marketing research) should not only test to assumptions made by the researcher. If these assumptions the endogenous regressor’s nonnormality carefully, but they are violated, the method may experience a strong remaining should also ensure the Gaussian copula approach’s addi- bias, not correct any bias at all, or even overcorrect the initial tional assumptions. While the error term’s normality can be 1 3 Journal of the Academy of Marketing Science bias in the other direction. Hence, the copula model might which may have occurred unintentionally in the past. In not perform better than the original endogenous model. doing so, we contribute to the rigor of regression models’ application and to the accurate presentation and interpreta- tion of marketing research. Summary of key findings In our five studies, we reveal that several factors affect the Gaussian copula approach’s performance. We focus on Researchers in marketing and other disciplines are increas- the interplay between the regression intercept and sample ingly taking advantage of the IV-free Gaussian copula size, as P&G examined regression models without intercept approach to identify and correct endogeneity problems in and, only to a limited extent, the sample size. Our literature regression models. The method’s increasing relevance moti- review reveals that these two factors play an important role vates a closer examination of its adequate performance on when applying the Gaussian copula approach. First, almost the basis of simulation studies. This research replicates and all researchers include an intercept in their model or mean- extends P&G’s initial simulation studies with several new center their data (66 of 69 studies in our literature review and important simulation factors that are highly relevant or 95.7%). Second, our literature review provides indica- in research applications. The results reveal critical issues tions that sample size, and, therefore, the statistical power, and limitations when using the Gaussian copula approach are more important for the results than originally expected. to identify and correct regressions models with intercept. Consequently, our simulation studies shed light on the role The method is not as straightforward and easy to use as pre- of the sample size and the statistical power when using the viously assumed. At the same time, our simulation results Gaussian copula approach to identify and correct endogene- allow us to provide recommendations that are essential to ity problems. ensure that researchers use the Gaussian copula approach In accordance with P&G, our Studies  1 and  2 con- appropriately and obtain valid results on which they can base firm the method’s high performance in regression models their findings and conclusions. Table  3 summarizes our find- without intercept, even in a wider range of sample sizes. ings and provides guidelines to take advantage of the IV-free A very different picture emerges when researchers use Gaussian copula approach while avoiding misapplications, regression models with intercept or mean-centered data, Table 3 Summary of conclusions • Gaussian copula models with intercept are subject to several additional considerations and constraints than those without intercept:   - There is a substantial remaining bias for smaller sample sizes in models with intercept   - There is only low statistical power to identify endogeneity in small samples, especially regarding the control function approach, but to a lesser degree also regarding the maximum likelihood approach   - Beside the sample size, the endogenous regressor’s nonnormality and the error correlation’s size are also important factors that influence performance   - In multilevel (or panel) models, the total sample size imposes the same performance restriction as in cross-sectional models, while only within or between sample sizes are less relevant   - The method is much less robust against misspecifications of the error distribution and the copula structure, resulting in remaining biases that do not vanish when the sample size is increased • Estimating a model without intercept is usually not an option without strong prior assumptions, as this estimation would also induce substantial bias when the true intercept is not zero and the copula approach cannot correct for omitted intercepts • All disclosed limitations concern finite (small) sample sizes, but the method works well within the limits if there is sufficient information to identify the model and the error term and copula are not misspecified • Based on simulation Study 4, we propose the following guidelines as a rough orientation for applying the Gaussian copula approach:   - In general, researchers should consider applying far more conservative nonnormality tests to ensure sufficient (and not only significant) non- normality, especially with sample sizes < 5,000   - The Anderson–Darling and Cramer-van Mieses nonnormality tests are conceptually closest to the copula approach and yield the best cor- respondence   - Sample sizes equal to or less than 200 observations should always be avoided   - For sample sizes below 1,000 observations, only a few very nonnormal distributions (e.g., skewness larger than 2 or Anderson–Darling test statistics above 20) yield sufficient power to identify endogeneity   - All these recommendations are based on continuous distributions. Discrete distributions, such as Poisson or Likert-scale survey data, might require even larger sample sizes, as they contain less information • The Gaussian copula approach is not free of assumptions and researchers need to be very careful when applying the method, especially to smaller sample sizes. Ultimately, researchers will always need to argue that the underlying assumptions have been fulfilled. Some of these assumptions (like the copula correlation structure) are inherently unobservable 1 3 Journal of the Academy of Marketing Science which is common in marketing studies. The Gaussian observations. In contrast, the nonnormality is still impor- copula approach has far less power and higher remaining tant for sample sizes above 1,000 observations, but to a endogeneity bias in these regression models, especially lesser degree. These boundary conditions of factors of key when using the estimation method that most researchers relevance for the Gaussian copula approach’s valid use in prefer: the control function approach (i.e., adding addi- regression models with intercept (i.e., the required sam- tional copula terms as new variables to the regression ple size, the endogenous regressor’s nonnormality, and model). In such models, the Gaussian copula approach’s its identification with suitable nonnormality tests) allow identification and correction of endogeneity problems researchers to effectively identify and correct endogeneity requires a much larger sample size. If this requirement problems. is not met, the approach may not identify an endogene- Using the maximum likelihood approach might be a ity problem even though it is present and has substantial potential solution to remedy some of these concerns, as it endogeneity bias. This finding is of central relevance, has slightly larger power to identify endogeneity (but has because our literature review reveals that most studies the same remaining bias). However, the control function apply the Gaussian copula approach to regression mod- approach has several advantages: (1) It is much faster and els that include an intercept or mean-centered data. It easier to implement in models that go beyond the simple is therefore very likely that studies with smaller sample linear regression model (e.g., panel models, binary choice sizes do not always identify significant copula terms due models, etc.) that might make deriving the appropriate likeli- to their insufficient power, although endogeneity prob- hood function more complex or even impossible, and (2) it lems are present (Fig. 1). Consequently, researchers may allows for including more than one copula term and, there- come to the false conclusion that endogeneity problems fore, for treating several endogenous regressors simultane- do not affect their studies’ results and present invalid ously. For these reasons, the maximum likelihood approach findings and conclusions. might not be a practical solution in many situations, and the In Study 3, we show that the findings of Studies 1 and 2 gains in power are also limited. extend to multilevel models, in which the endogeneity is Finally, Study 5 sheds light on the misspecification of present at the within-cluster level (i.e., the correlation the error term and the copula structure when regression between a within-cluster predictor and the structural error) models are estimated with intercept. This study’s results when the total sample size is taken into account. In our underscore concerns about the method’s estimation accu- literature review, most studies with multilevel data have racy when an intercept is present, and contradict find- rather larger sample sizes, but a few also have total sam- ings about its robustness as presented in P&G's original ple sizes in the range for which we identify the Gaussian study. Researchers should ensure both the presence of an copula approach’s reduced performance. Consequently, in appropriate Gaussian copula correlation structure and a respect of multilevel (or panel) models, the same recom- normally distributed error term. While the analysis of mendations apply regarding a sufficient sample size and the regression residual allows an assessment of the error nonnormality as do for the simpler, cross-sectional regres- term, the correlation structure is inherently unobserv- sion model. able and therefore only subject to untestable theoretical Study 4 aims at helping researchers apply the Gaussian considerations. copula approach appropriately and exploit its advantages We have two recommendations for research that does not effectively. More specifically, in Study 4, we extend the satisfy the boundary conditions identified in this research: simulations to include several additional factors that are first , researchers should carefully assess whether the data relevant for regression analyses, such as the endogenous and model might be prone to empirical identification issues. regressor’s nonnormality, the explained variance (R They can do so by, for example, carefully checking whether level), and the error correlation. Study 4 derives boundary the endogenous regressor has sufficient nonnormality and conditions for these factors to guide the Gaussian copula checking for multicollinearity issues after including the approach’s appropriate use in studies. The findings sub - Gaussian copula, as well as testing the regressions’ residual stantiate that for sample sizes below 1,000 observations, for normality. Second, and more importantly, researchers only a few very nonnormal distributions (e.g., skewness should avoid using the Gaussian copula approach to test above 2 or Anderson–Darling test statistics above 20) lead for endogeneity (i.e., concluding that endogeneity is not to sufficiently high power when using the copula term in a problem due to insignificant copula terms), but should regression models (i.e., larger than 80%). Nevertheless, revert to traditional ways of handling endogeneity problems, none of our considered distributions has sufficiently such as using IVs or other means of identifying the causal large power for sample sizes equal to or less than 200 mechanism. 1 3 Journal of the Academy of Marketing Science application has gained the method the reputation of being an Conclusions and future research easy-to-use add-on in any study that has a potential endoge- neity problem, our results highlight that researchers should The Gaussian copula approach is valuable for identifying use the Gaussian copula approach more cautiously, espe- and correcting endogeneity issues in regression models cially when sample sizes are small and the model includes a when the assumptions are fulfilled. However, when the regression intercept (or mean-centered data). Figure 8 sum- regression models contain an intercept, the method is much marizes our studies’ findings and conclusions in a decision more constrained than initially thought. It is less robust flowchart by illustrating the path of choices that researchers against deviations of the error term’s normality, the Gauss- need to consider when deciding whether to apply the Gauss- ian copula correlation structure between the error and the ian copula approach. Given these new recommendations, regressor, and the regressor’s nonnormality. Even if these researchers might far less often conclude that the Gaussian preconditions are met, the approach requires large sample copula approach is a recommended method for dealing with sizes to perform well in models with intercept. However, endogeneity problems. constraining the intercept to zero is usually not an option, These recommendations represent approximate thresh- because this would also induce substantial bias as high- olds based on the results of our simulation studies that lighted in our Study 2. While the Gaussian copula’s simple Fig. 8 Flowchart for the decision on the application of the Gaussian copula approach 1 3 Journal of the Academy of Marketing Science provide researchers with an indication of whether the copula which the Gaussian copula approach is not applicable. We can be successfully applied. However, they do not replace therefore call for further research on comparing the methods careful theoretical consideration of the nature of endogene- under varying conditions to provide researchers with bet- ity and the fulfillment of the Gaussian copula approach’ s ter guidelines on which method to use when. However, all general assumptions (i.e., the nonnormality of the endog- IV-free approaches demand fulfillment of certain identifica- enous regressor, the normality of the error term, and the tion requirements, which are often untestable. Applying any Gaussian copula correlation structure). Moreover, these rec- of these methods blindly may provide no better results than ommendations are based on models with a single continu- merely ignoring endogeneity problems does. Consequently, ous endogenous regressor variable. It is likely that multiple it is important that researchers are aware of these approaches’ endogenous regressors or discrete variables will increase limitations, because ultimately, they always need to carefully the requirements for identifying the copula and, therefore, argue that the underlying assumptions have been fulfilled. for the method’s successful application. More research is Supplementary Information The online version contains supplemen- needed to extend the recommendations in respect of these tary material available at https://doi. or g/10. 1007/ s11747- 021- 00805-y . areas. Future research should therefore extend our findings by Acknowledgements The authors presented some of the research results adding simulation studies that, for instance, analyze the depicted in this paper at the 2020 AMA Winter Academic Confer- Gaussian copula’s performance with additional endogenous ence, San Diego, CA, February 14-16, 2020, but without publication in the conference proceedings. The authors thank Peter Ebbes, HEC regressor distributions (i.e., additional nonnormality levels) Paris, France, and Dominik Papies, University of Tübingen, Germany, to further substantiate our thresholds. Furthermore, we are for their useful comments on our research presentation. Moreover, the currently not aware of possibilities to test the assumption authors like to thank Edward E. Rigdon, Georgia State University, that the error term and the endogenous variable follow a United States of America, for his helpful comments to improve an ear- lier version of the manuscript. For their simulation studies, the authors Gaussian copula correlation structure. However, a misspeci- used the high-performance computing cluster of Hamburg University fication potentially leads to invalid results as our simula- of Technology, Germany. tion results show. Thus, creating a test for this assumption would greatly enhance confidence in the method’ s accuracy. Funding This research was partly funded by the Deutsche Forschun- gsgemeinschaft (DFG, German Research Foundation)—VO 1555/1-1. In addition, future studies should analyze discrete distribu- tions further (i.e., P&G show that discrete distributions Open Access This article is licensed under a Creative Commons Attri- suffer even more from identification problems and that bution 4.0 International License, which permits use, sharing, adapta- thresholds might therefore be much higher in such cases) tion, distribution and reproduction in any medium or format, as long and revert to more complex regression models with mul- as you give appropriate credit to the original author(s) and the source, tiple endogenous regressors. Additional knowledge about provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are these factors’ relevance will help researchers use the method included in the article's Creative Commons licence, unless indicated adequately to derive valid inferences for marketing decision otherwise in a credit line to the material. If material is not included in making. Moreover, future research should address the core the article's Creative Commons licence and your intended use is not issue that the Gaussian copula approach’s usability is limited permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a regarding finite (small) sample sizes, but works well in the copy of this licence, visit http://cr eativ ecommons. or g/licen ses/ b y/4.0/ . limit when sufficient information is available to identify the model. Methodological research should aim at developing a solution for this limitation. 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Pauwels (Eds.), Advanced methods in jurisdictional claims in published maps and institutional affiliations. modeling markets (pp. 581–627). Springer. Park, S., & Gupta, S. (2012). Handling endogenous regressors by joint estimation using copulas. Marketing Science, 31, 567–586. 1 3 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the Academy of Marketing Science Springer Journals

Revisiting Gaussian copulas to handle endogenous regressors

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0092-0703
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10.1007/s11747-021-00805-y
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Abstract

Marketing researchers are increasingly taking advantage of the instrumental variable (IV)-free Gaussian copula approach. They use this method to identify and correct endogeneity when estimating regression models with non-experimental data. The Gaussian copula approach’s original presentation and performance demonstration via a series of simulation studies focused primarily on regression models without intercept. However, marketing and other disciplines’ researchers mainly use regression models with intercept. This research expands our knowledge of the Gaussian copula approach to regression models with intercept and to multilevel models. The results of our simulation studies reveal a fundamental bias and concerns about statistical power at smaller sample sizes and when the approach’s primary assumptions are not fully met. This key finding opposes the method’s potential advantages and raises concerns about its appropriate use in prior studies. As a remedy, we derive boundary conditions and guidelines that contribute to the Gaussian copula approach’s proper use. Thereby, this research contributes to ensuring the validity of results and conclusions of empirical research applying the Gaussian copula approach. Keywords Endogeneity · Gaussian copula · Intercept · Linear regression · Multilevel models · Sample size · Simulation Introduction correlate with the error term violating a fundamental causal modeling assumption of regression analysis (Wooldridge, Endogeneity is a key concern when using regression mod- 2020). Potential reasons for error term correlations are meas- els in marketing studies with non-experimental data (Rutz urement errors, simultaneous causality, and omitted vari- & Watson, 2019; Sande & Ghosh, 2018). In a regression ables that correlate with one or more independent variable(s) model, endogeneity occurs when one or more regressors and with the dependent variable(s) in the regression model (e.g., Papies et al., 2017; Rutz & Watson, 2019). Endoge- neity problems lead to biased and inconsistent coefficients, This manuscript was handled by Editor John Hulland. which become causally uninterpretable. The best approach to overcome endogeneity is to specify * Christian M. Ringle the model correctly according to the underlying (causal) c.ringle@tuhh.de data-generating mechanism. In practice, however, the Jan‑Michael Becker required data is usually not available, unless researchers jan-michael.becker@bi.no use experimental designs (Ebbes et al., 2016). Marketing Dorian Proksch literature has extensively discussed methods of dealing with d.e.proksch@utwente.nl endogeneity (e.g., Rutz & Watson, 2019; Sande & Ghosh, Department of Marketing, BI Norwegian Business School, 2018; Zaefarian et al., 2017). Of these methods, the use Nydalsveien 37, 0484 Oslo, Norway of instrumental variables (IVs) is particularly well-known Netherlands Institute for Knowledge-Intensive for addressing endogeneity problems (Wooldridge, 2010). Entrepreneurship (NIKOS), University of Twente, Despite its frequent application, the IV approach has sev- Drienerlolaan 5, 7522 NB Enschede, Netherlands eral drawbacks, as it requires identifying strong and valid Department of Management Science and Technology, instruments (i.e., they correlate strongly with the endog- Institute of Human Resource Management enous regressor but do not correlate with the error term). and Organizations, Hamburg University of Technology, Am Schwarzenberg‑Campus 4, 21073 Hamburg, Germany In applications, however, researchers often fail to revert to suitable variables whose appropriateness as instruments they Waikato Management School, University of Waikato, Walikato, New Zealand Vol.:(0123456789) 1 3 Journal of the Academy of Marketing Science can sufficiently justify theoretically (Rossi, 2014). Methodo- 89.9%) appeared in 2018, 2019, and 2020 and 58.0% (40 of logically, only the instrument’s strength can be empirically 69) of these journal articles appeared in marketing journals. tested and not its validity (Wooldridge, 2010). Thus, the Besides Marketing Science, which initially published the results of the IV approach do not allow for assessing whether method, premier marketing journals such as International the endogeneity problem has improved or worsened (Papies Journal of Research in Marketing (11), Journal of the et al., 2017). Academy of Marketing Science (7), Journal of Marketing To remedy these concerns, Park and Gupta (2012)—fur- Research (6), Journal of Retailing (6), and Journal of Mar- ther referred to as P&G—introduced the Gaussian copula keting (5) predominantly published Gaussian copula appli- approach to cope with endogeneity in regression models. cations. In addition, the Gaussian copula approach has been This IV-free method has the advantage of not requiring addi- disseminated across disciplines, for example, in management tional variables, because it directly models the correlation with its subfields like human resources, information systems, between the potentially endogenous regressor and the error and tourism (23 of 69 articles; 33.3%), and in economics (3 term using a Gaussian copula. Thereby, the approach pro- articles; 4.3%). vides a relatively simple way of identifying and correcting One of the observations from our literature review is that endogeneity biases in regression models (Rutz & Watson, researchers not only use the Gaussian copula approach to 2019). improve the precision of the endogenous regressor’s parame- There are two variants of the Gaussian copula approach. ter estimate, but also to identify whether endogeneity poses a The original approach by P&G suggests the regression problem for a regression model. They do so by assessing the model’s estimation by using an adapted maximum likeli- copula term’s statistical significance to determine whether hood function that accounts for the correlation between endogeneity is at a critical level in their empirical study or the regressor and the error term using the Gaussian copula not (e.g., Bornemann et al., 2020; Keller et al., 2018). This (P&G, Eq. 8). The disadvantage of the maximum likelihood approach seems plausible, as the copula term’s parameter approach is that it can only account for one endogenous estimate is a scaled version of the unobservable correla- regressor in the model. In practice, almost all applications tion between the endogenous regressor and the error term therefore use the second variant, which adds a “copula term” (P&G, p. 572) and can be compared to the Hausman test in to the regression equation (P&G, Eq. 10)—like the control a normal IV estimation (Papies et al., 2017). If this correla- function approach for IV model estimation. The Gaussian tion is significant, the Gaussian copula approach indicates copula control function approach can also account for mul- a potential endogeneity problem and the inclusion of the tiple endogenous regressors, which require the simultaneous Gaussian copula term in the regression model should cor- inclusion of multiple copula terms, one for each regressor rect the endogeneity problem. Otherwise, if the correlation (P&G, Eq. 12). In this variant, the copula term is a non- is not significant, researchers assume that endogeneity does linear transformation of the endogenous regressor, using not substantially affect the regression model ’s results and, −1 the inverse normal cumulative distribution function Φ therefore, they often do not include the respective copula and the empirical cumulative distribution function H as fol- terms in their final model (e.g., Campo et al., 2021; van ∗ −1 ∗ lows (P&G, p. 572): P =Φ H(P )  where P is the addi- Ewijk et al., 2021; Wlömert & Papies, 2019). t t tional copula term added to the model. This copula term’s Our literature review shows that using this procedure has parameter estimate is the estimated correlation between a surprising result. Endogeneity only seems to be an issue in the regressor and the error term scaled by the error’s vari- studies with larger sample sizes (Fig. 1). More specifically, ance (P&G, Eq. 10). On the basis of bootstrapped standard 15 of the 18 studies (83.3%) with a samples size of 5,000 and errors, a statistical test of this parameter estimate allows for larger report a significant copula term at a p -value smaller assessing whether this correlation is statistically significant than 5%. In contrast, of the 24 studies with samples sizes and endogeneity problems are therefore present (Hult et al., of fewer than 500 observations, only eight (33.3%) identify 2018; Papies et al., 2017). a significant copula term. Based on these findings, some Marketing researchers seem to increasingly adopt the of the studies conclude that endogeneity does not seem to IV-free Gaussian copula approach to address endogeneity be an issue. This conclusion may be questionable given the problems in their empirical studies. A literature search of overall pattern that our literature review reveals in Fig. 1. the use of the Gaussian copula approach to address endoge- The larger the sample size, the larger the share of signic fi ant neity problems in regression models, and a citation analy- copula terms in applications. Consequently, there seems to sis of P&G’s article in literature databases, such as ABI/ be a sample-size-related problem with the Gaussian copula’s Informs, EBSCO, Web of Science, Scopus, and Google statistical power to identify endogeneity issues. P&G’s ini- Scholar, reveal 69 publications by the end of 2020 (see the tial simulation results do not suggest that there are sample complete list of these publications and Fig. WA.1.1 in the size restrictions. They find that the approach performs very Web Appendix 1). Most of the publications (62 of the 69, well on sample sizes as low as 200 observations, which calls 1 3 Journal of the Academy of Marketing Science copula’s performance in a broader application with differ - ent endogeneity levels, alternative levels of explained vari- ance, and different endogenous regressor distributions, also taking different sample sizes into account. We find that, in addition to the sample size, the endogenous regressor’s nonnormality and its assessment are another important area of concern regarding the method’s performance. Finally, in Study 5, we reevaluate the Gaussian copula approach’s robustness against misspecification of the error term distri- bution and the endogenous variable’s correlation structure with the error term, when estimated on models with inter- cept. We find that the model with intercept is much less robust against misspecification than P&G’ s original model without intercept. Fig. 1 Significant copulas (p < 0.05) per sample size. Note: This anal- Overall, we find that when estimating models with inter - ysis is based on 58 of the 69 reviewed journal articles, excluding 11 cept (or fully centered, or standardized data), which, accord- additional studies that do not report the Gaussian copula term’s sig- nificance ing to our literature review, is most common in marketing applications, researchers need to be far more careful when applying the Gaussian copula approach. In models with for a more detailed analysis of sample size’s role in combi- intercept, the approach is much more sensitive to violations nation with other relevant factors for estimating Gaussian of its fundamental assumptions. A careless application of copula models. the Gaussian copula approach—that is, without adhering Another important observation from our literature review sufficiently to its identification conditions (i.e., assessing the addresses the use of a regression intercept. Our literature endogenous regressor’s sufficient nonnormality, the error review reveals that 53 of the 69 applications (76.8%) use term distribution’s normality, and the Gaussian correla- regression models with intercept. In the remaining 16 stud- tion structure)—poses a potential threat to the validity of ies, 13 (18.8%) use fully centered or standardized data, and this approach’s findings. However, our simulation results only three studies (4.3%) do not include an intercept in an also allow us to determine boundary conditions that serve unstandardized model (Web Appendix 1, Fig. WA.1.2). This as guidelines for the Gaussian copula approach’s appro- observation appears meaningful, because regression models priate use and allow to obtain reliable results (e.g., with without intercept are a restricted version of the more gen- the expected error rate) when identifying and correcting eral model with intercept. They require strong assumptions endogeneity problems in regression models with intercept. to yield meaningful and unbiased estimates for the regres- While the correlation structure is inherently unobservable, sion parameters. In most cases, researchers do not have suf- researchers should ensure the following prerequisites: First, ficient support for these assumptions and therefore estimate they should assess the normality of the error term by test- a model with intercept (or fully center or standardize their ing the regression residual’s normality. Second, researchers data). However, the predominant use of regression models should confirm sufficient (and not only significant) non- with intercept requires special attention, because P&G did normality of the endogenous regressor. Third, they should not consider models with intercept in their initial simulation consider far larger sample sizes than originally expected. studies. For each of these steps, we derive clear guidelines on how This research presents five simulation studies to sub- researchers can verify that they have met each requirement. stantially extend and deepen our knowledge of the Gauss- The results of this research extend our knowledge of the ian copula approach to address endogeneity issues. Study 1 Gaussian copula approach considerably and call for it to be replicates P&G’s original simulation with a larger sample far more judiciously applied by also taking additional key size variation and, additionally, estimates the results of a factors, which had been previously ignored, into account. model with intercept. The study indicates that the Gaussian We provide recommendations and a flowchart illustration copula approach for models with intercept has a consider- regarding when and how the method should be appropri- able performance issue, and that the method requires much ately applied to support decision making. Researchers could larger sample sizes than originally expected. In Study 2, we therefore take advantage of the Gaussian copula approach’s investigate the performance of models with intercept further benefits to identify and correct endogeneity issues in order by varying the true underlying intercept. Study 3 allows us to ensure that their marketing studies’ results are valid, while to confirm these findings in respect of multilevel models also carefully considering its limitations. with a random intercept. In Study 4, we test the Gaussian 1 3 Journal of the Academy of Marketing Science with and without intercept) for each dataset. We apply the Simulation study 1: Intercept extension Gaussian copula approach’s two different versions to each of P&G’s case 1 of the two models. These versions are the control function approach, which adds a copula term to the regression, and In our first simulation study, we use the basic design of the maximum likelihood approach. P&G’s simulations to investigate the intercepts’ influence on the Gaussian copula’s performance. This allows us to Evaluation criteria replicate and extend the originally presented simulations, allowing the results to be compared. To evaluate the Gaussian copula approach’s performance, we examine three performance criteria: mean bias, statis- Design tical power, and relative bias. For a parameter θ (e.g., a regression coefficient), the bias is defined as   –  , which In this study, we replicate the data generating process (DGP) is the difference between its estimate  and its true value θ in P&G’s Case 1 (i.e., their “Linear Regression Model”), but in the DGP. The bias denotes the accuracy of a parameter without the additional IV: estimate. The closer the bias is to zero, the closer the esti- 0 1 0.50 mate is to the true value, and the more accurate the estimate = N , (1) P 0 0.50 1 is. Positive values indicate overestimation, while negative values indicate an underestimation of the parameter. In our −1 ∗ simulation studies, we determine the mean bias of the focal =Φ (Φ( )) (2) (endogenous) regressor’s parameter estimate over the 1,000 simulation datasets per factor-level combination. P = Φ(P ) (3) In addition, we also investigate the relative bias, which depicts a parameter’s bias in the copula model divided by Y =−1P + that in the model without copula (i.e., with untreated endo- (4) t t t geneity problem). The relative bias allows for assessing whereby Y represents the dependent regressor, P the endog- t t how much of the original endogeneity bias remains after enous regressor, and ξ the error term in the regression the copula approach has corrected it. This evaluation crite- model. The DGP specifies a linear model without intercept, rion is particularly useful for comparing models with dif- with a uniform distribution of the endogenous regressor P , ferent endogeneity bias in the untreated model. In Study 4, and a correlation of 0.50 between the endogenous regressor some design factors, such as the error term correlation and and the model’s error term. P&G generated 1,000 datasets the error term variance, affect the original endogeneity bias for sample sizes of T = 200 and T = 400, and estimated a and, therefore, the amount of bias that the copula approach linear regression without intercept. needs to correct. A remaining bias of -0.19 might therefore We pursue three main objectives with this simulation be differently evaluated, depending on whether the original study. First, we estimate regression models without inter- endogeneity bias was -0.20 or -0.90, resulting in a relative cept. Our simulations thereby replicate and confirm P&G’ s bias of 95% or 21%. results. Second, our literature review indicates that most Statistical power is the probability of a hypothesis test researchers (76.8%) include an intercept when estimating rejecting the null hypothesis when a specific alternative their models. While the DGP does not need an intercept for hypothesis is true. In other words, it is the likelihood of the reliable model estimation, because the true intercept is obtaining a significant parameter estimate at a given α-level zero, researchers do not usually know this a-priori. They (i.e., type I error level), when the true parameter is different therefore usually estimate a model with intercept. Third, we from zero. A high statistical power implies a low chance of consider a much wider range of sample sizes from 100 up making a type II error (i.e., failing to reject the null hypoth- to 60,000 observations (i.e., 100; 200; 400; 600; 800; 1,000; esis when the alternative is true). In our simulations, we 2,000; 4,000; 6,000; 8,000; 10,000; 20,000; 40,000; 60,000), estimate the statistical power by the number of significant because our literature review indicates that the sample size parameter estimates for the endogenous regressor or the might play a far more important role than initially assumed. copula term at a given α-level (e.g., p < 0.05) divided by We use the DGP to obtain 1,000 datasets for every fac- the number of sampled datasets (i.e., 1,000 per factor-level tor-level combination and estimate the two models (i.e., combination). We make the R-Code for the simulation and the results datasets available in the paper’s online repository at https:// t1p. de/ euqn. 1 3 Journal of the Academy of Marketing Science Fig. 2 Bias of the endogenous regressor choice of p-value is consistent with the level commonly used Results in studies to assess the Gaussian copula’s significance. In line with P&G’s results, the Gaussian copula models with- The results replicate and confirm P&G’ s simulation study for out intercept perform exceptionally well. Both the copula models without intercept. We also find that the Gaussian cop- term and the endogenous regressor’s power levels are close ula approach accounts for the endogeneity problem and esti- to 100% across all the sample sizes (Fig.  3, Panel A and mates the endogenous regressor’s coeci ffi ent without notice - B), regardless of estimation method used. When extending able bias, regardless of the sample size and the estimation the original study in terms of Gaussian copula models with method used (Fig. 2 shows the results of the more popular intercept, the statistical power of small to medium-sized control function approach; Web Appendix 2, Table WA.2.1, samples is less satisfactory and depends on the type of esti- provides the outcomes of the maximum likelihood approach). mation method used. The results of the control function The situation changes fundamentally when we extend approach require more than 800 observations for the copula P&G's simulation study by estimating a regression model term’s parameter estimate to achieve power levels of 80% with intercept. The results show that, in many situations, the and higher, and more than 2,000 observations for the endog- endogeneity problem has not been resolved. A substantial enous regressor to reach these power levels. The power of bias remains in the copula model for smaller to medium identifying a significant copula parameter is slightly higher samples. The endogenous regressor’s parameter bias only in small sample sizes when using the maximum likelihood reaches a negligible level for sample sizes of 4,000 and approach. This method needs only 600 observations for the more. At sample sizes of about 40,000 observations and copula term to achieve power levels of 80% and higher. The more, Gaussian copula models with intercept achieve a per- endogenous regressors’ power does not improve much, as formance level comparable to those without intercept. This it still requires about 2,000 observations to achieve power finding holds for both estimation methods (i.e., the control levels of 80%. function and the maximum likelihood approach yield almost Mean-centering could be a naïve strategy for coping with the the same parameter estimates for the endogenous regressor). intercept model’s problems. A fully mean-centered (or standard- To determine the copula term’s and the endogenous ized) model would not need an intercept for the reliable estima- regressor’s statistical power, we use bootstrapping with 500 tion of the regression parameters. We therefore also estimated resamples (for further details see P&G; they used 50 and a model without intercept, in which we mean-centered all the 100 resamples). Based on the bootstrap standard errors, we variables before entering the estimation, which 13 (18.8%) of the consider parameters significant if their p -value is smaller published studies also do. We find that the results of both methods than the 5% level. As shown in our literature review, this 1 3 Journal of the Academy of Marketing Science Fig. 3 Statistical power of the copula term and the endogenous regressor are equivalent to the model with intercept (see Web Appendix 2, endogeneity bias. Nevertheless, researchers should be aware Tables WA.2.1 to WA.2.3). that the method is less effective at reducing endogeneity bias in models with intercept when the sample sizes are small, Discussion and that they need to interpret such results more carefully. Another important issue is the method’s ability to iden- Our discussion addresses two important aspects of our tify endogeneity problems. Some recommendations suggest study’s results. First, we discuss the impact of including testing for the presence of endogeneity in regression models intercepts in regression models regarding the Gaussian cop- by using the copula term’s significance (Hult et al., 2018; ula approach’s performance. Thereafter, we address poten- Papies et al., 2017). The copula approach’s low power to tial reasons for why an intercept weakens the performance. identify a significant parameter for the copula term in mod- els with intercept makes this practice highly problematic Consequences of intercept inclusion if the sample sizes are not large enough. Researchers need to be far more careful when using the copula term’s sig- Our analysis reveals two important findings. First, the nificance to decide whether endogeneity poses a problem Gaussian copula approach cannot always correct the endo- and whether or not to include the copula term. This is par- geneity problem when estimating the regression model with ticularly important, since our literature review revealed that intercept. Smaller sample sizes are still subject to substan- researchers currently also use this approach with relatively tial bias. Second, when estimating the model with intercept, small sample sizes and, to some extent, probably mistak- the Gaussian copula approach has low power to identify enly conclude that endogeneity is not a problematic issue significant error term correlations in smaller sample sizes. in their model. There are two main reasons for the copula This finding not only holds when using the maximum like- term’s low statistical power. First, small sample sizes lead lihood approach but is also somewhat more pronounced to a substantial underestimation of the copula term’s param- when using the control function approach. Our literature eter, which is a scaled version of the correlation between the review reveals that 64 of the 69 studies (92.8%) use the endogenous regressor and the error term (Web Appendix 2, latter approach when applying the copula approach empiri- Table WA.2.2). Second, in the control function approach, cally, which underlines the importance of these findings. the estimated parameter also comprises the error term’s While the remaining bias can be substantial, some research- variance (P&G, Eq. 10). The parameter estimate contains ers could argue that correcting some bias is still better than additional noise that inflates the standard errors and makes not correcting any. The Gaussian copula approach might the estimation unreliable, especially in smaller sample sizes. therefore still be valuable although it cannot fully correct the Consequently, this approach has a slightly weaker power compared to the original maximum likelihood approach, which allows for estimating the parameter without the error Three studies do not report how they include the copula, while only variance’s scaling. two report the use of the original maximum likelihood approach. 1 3 Journal of the Academy of Marketing Science Overall, the simulation results show the importance of a also provide a solution to this problem in their Appendix I, sufficient sample size if the Gaussian copula approach is to where they show that even models with endogenous regres- perform well in regression models with intercept in terms of sors that are normally distributed can be identified if (1) identifying and correcting endogeneity problems. This is a the normal variable has a non-zero mean, and (2) the esti- novel n fi ding that has not yet been reported. While this n fi d - mated model does not include an intercept. Nonnormality ing imposes limitations on the method in finite samples, our is therefore only required in models that are estimated with simulations also show that when increasing the sample size intercept. The endogenous regressor’s availability of a non- toward infinity, the method’ s bias is reduced to zero (i.e., it zero mean (i.e., the uniform distribution has a mean of 0.50) is a consistent estimator) and has sufficient power to iden- and the absence of an intercept can therefore compensate tify endogeneity issues. Furthermore, we show that using for smaller sample sizes’ lack of sufficient information from mean-centered (or standardized) data is not a valid strategy nonnormality. However, in models with intercept (or when for coping with this issue, which is also in line with previ- mean-centering the data), this mechanism is not at play and ous research on mean-centering (Echambadi & Hess, 2007). the lack of information from sufficiently strong nonnormal- ity makes it harder to separately identify the copula term Potential reasons and the regressor’s parameter, which results in the pattern of bias that we observe. Consequently, in models with inter- The pronounced differences between the models estimated cept (or mean-centered data), the regressor’s nonnormality with and without intercept raise the question: why is there needs to be much stronger than in models without intercept. is such a big difference in their performance? Identifica- tion problems could be a potential reason for the weaker performance in models with intercept. P&G highlight two Simulation study 2: Different intercept levels important pre-requisites for identifying the Gaussian copula model. The rs fi t is the endogenous regressor’s nonnormality: In Study 1, we replicate the original simulation results by “If P follows a normal distribution, P is a linear transfor- P&G and show that including a regression intercept in the ∗ −1 mation of P since P =Φ H(P ) . Hence, we cannot sepa- estimation reduces the copula approach’s performance (both t t rately identify α and  ⋅  in (10). As the true distribution of in terms of bias correction and statistical power). In this P approaches a normal distribution, the correlation between study, we extend these findings by varying the level of the P and P increases, causing a multicollinearity problem” intercept. (P&G, p. 572). The second is the error term’s normality: While P&G’s original DGP does not include an intercept “We assume that the marginal distribution of the structural (i.e., the intercept is zero), it is unlikely that the true intercept error term is normal” (P&G, p. 570). Both assumptions are will be zero in practice. Estimating a regression model with- fulfilled in our simulations’ DGPs. The error term follows a out intercept requires strong assumptions that are untestable normal distribution with N(0,1), and the endogenous regres- a-priori. In their applications, researchers usually estimate sor a uniform distribution with U(0,1). However, there is regression models with intercept. In addition, similar to an still substantial correlation between the regressor P and the ignored endogeneity problem, ignoring an intercept when copula term P (e.g., on average we observe a correlation of it is necessary is also likely to induce strong bias. Conse- 0.973 in the model with intercept). Consequently, smaller quently, it is usually not recommended to simply estimate sample sizes seem to have not enough information on the a regression without intercept, and it is unclear whether the difference between the nonnormal distribution of the regres- copula approach can compensate for this type of bias. sor P and the normal distribution of the copula term P to allow a robust estimation of the parameters. If the differ - Design ences are too small, it is difficult to distinguish the variation that is a result of endogenous regressor from the variation We use the same DGP as in Study 1, but instead of using that stems from the error term. In our study, we observed Eq. 4, which does not include an intercept, we add the inter- that the copula term’s parameter is underestimated propor- cept i to Eq. 5 constituting Y : tional to the parameter overestimation of the endogenous Y = i − 1P + (5) t t t regressor. When the sample size increases, this makes more information available about the differences between the two In this simulation study, we vary i ϵ {−10, −3, −0.50, predictors, and their bias shrinks toward zero. −0.10, 0, 0.10, 0.50, 1, 3, 10}. But why can the model without intercept be more eas- ily identified than the model with intercept? P&G might 1 3 Journal of the Academy of Marketing Science Fig. 4 Bias of the endogenous regressor with varying intercepts in the copula regression copula’s performance is independent of the intercept’s size, Results and all of Study 1’s findings apply here as well. The results show that the intercept variations do not affect the Gaussian copula approach’s bias (Fig. 4, Panel A) and Simulation study 3: Multilevel model power when we estimate the model with intercept. We find the same performance as in Study 1, with smaller sample To extend the simple linear model in Studies 1 and 2, this sizes showing relatively high bias and low power, both of simulation study utilizes a multilevel model to assess the which improve with sufficiently large sample sizes. sample size’s effect in more depth. In particular, we investi- In contrast, we find that the intercept’ s variation affects gate the effect of different sample sizes within-cluster (level the Gaussian copula approach when estimated by means of a 1) and between-clusters (level 2) on the Gaussian copula model without intercept (Fig. 4, Panel B). When the difference model’s performance. For this purpose, we use a two-level, between the true intercept and zero increases, the model’s bias random-intercept model (often referred to as a panel data also increases as expected. Similar to Study 1, the performance model in economics). The endogeneity problem occurs at is not dependent on the sample size. However, the bias from the within-cluster level as a result of a correlation between the omitted intercept can be larger than the endogeneity bias the within-cluster (level 1) predictor and the within-cluster depending on the intercept’s size. The regression model is, (level 1) structural error. Although other endogeneity prob- of course, misspecified when estimated without intercept on lems could arise in multilevel models (e.g., correlations the basis of a DGP that includes an intercept. Constraining a between level 1 predictors and level 2 error terms, etc.), parameter (in this case to zero) without suc ffi ient prior assump - in our literature review, the abovementioned endogeneity tions will cause this bias in the estimation. Nevertheless, it problem seems to be marketers’ most common concern, as is interesting that the Gaussian copula in a model without they introduce copulas to level 1 (within-cluster) predic- intercept cannot correct the bias of an omitted intercept. If tors to avoid correlation with the level 1 structural error. researchers simply omit the intercept, they will trade one bias Moreover, other instrument-free methods, such as the gen- for another. eralized method of moments approach by Kim and Frees (2007), might address the correlations between regressors Discussion and higher-level error terms. If the DGP includes a non-zero intercept, estimating the Design model without intercept is not an option, because the results are biased, even if the model contains a Gaussian copula We use a similar DGP as in Study 1, but extend it to the term. In contrast, when the estimated model includes an two-level, random-intercept model. Instead of Eq. 4, which intercept, the Gaussian copula approach can correct the does neither include an intercept nor does it consider the endogeneity bias if the sample size is large enough. The 1 3 Journal of the Academy of Marketing Science clustering of level 1 (within-cluster) observations, we use on non-parametric bootstrapping, we considered two differ - the following Eq. 6: ent alternatives of sampling the cases in the bootstrapping. We subsequently report the results of sampling the cases at Y = u + u − 1P +  , jt 0 j jt jt (6) the cluster level (level 2), which is advised when estimating multilevel data models (Goldstein, 2011). However, most of where the outcome Y and regressor P are observed at jt jt the studies in our literature review do not reveal the kind of the within-cluster level (level 1; e.g., time) with t = 1…T bootstrapping strategy they use. Consequently, we also use observations in each cluster j (level 2; e.g., brands). The a different bootstrapping strategy in which we sample the random intercept u ∼ N 0,  denotes an error component level 1 observations directly (i.e., ignoring the hierarchical that is specific to the cluster and captures all unobserved data structure) and find very similar results. Finally, we do level 2 specific effects (e.g., all effects that are specific for a not estimate models without intercept in this study, as the brand, but do not vary over time). Both the error component original DGP includes a random intercept and ignoring this at level 2 (i.e., the random intercept u ) and the structural random-intercept structure could itself induce bias and inef- level 1 error component  need to be uncorrelated with the jt 3 ficiency (similar to Study 2). level 1 regressor P for efficient and consistent estimation. jt However, similar to Study 1, we assume an error correla- Results tion between P and  of 0.50 in this DGP, so that Eqs. 1–3 jt jt become: The results of the endogenous regressor’s bias in Fig.  5 (Panel  A) show that there are basically no differences 0 1 0.50 jt = N , (7) ∗ between the copula models in this study and the simple lin- P 0 0.50 1 jt ear model in Study 1 when using the total sample size (i.e., the combination of within-cluster and between-cluster obser- −1 ∗ vations) as a reference. Both the random-intercept multilevel =Φ Φ( ) (8) jt jt model and the fixed-effects panel model follow the same pattern as the simple linear model with copula and intercept P = Φ(P ) (9) in Study 1, with a bias that only reaches a negligible level jt jt for sample sizes of 4,000 and more. In addition, we hardly We systematically vary both the level 1 and level 2 sam- observe any variations in the bias in different combinations ple sizes T and J ϵ{5, 10, 20, 40, 60, 80, 100, 200, 400, of level 1 (within) and level 2 (between) sample sizes, which 600, 800}, excluding total sample sizes lower than 100 result in the same total sample size. and larger than 40,000 for reasons of efficient estimation. The results of the copula term’s power and the endog- In addition, we set the non-random intercept u to zero and enous regressor’s power also follow very similar patters as the variance of the random intercept to one (i.e.,  = 1). their counterparts in the simple linear model in Study 1. Consequently, u is uncorrelated with both P and  . We j jt jt Figure 5 (Panel B) illustrates this pattern in respect of the estimate the model with a random-intercept, multilevel copula term’s power in the random-intercept multilevel model using maximum likelihood estimation. Moreover, model and fixed-effect panel model compared to the cop- we also consider a fixed-effects panel estimator. We esti- ula term’s power in the model with intercept in Study 1. In mate both models with and without the control function contrast to the bias, we observe a slightly larger variation approach by adding an additional copula term. Because the in power for different combinations of level 1 (within) and estimation of the copula model’s standard errors is based level 2 (between) sample sizes, which result in the same total sample size. More specifically, the power seems to be 3 slightly larger if the number of level 1 observations (i.e., If the within-cluster level 1 regressor P correlates only with the jt within cluster observations, e.g., the time series) is larger random intercept u , but not with the structural error   , the fixed- j jt effects panel model estimator is consistent, but not efficient. and the number of level 2 observations (i.e., the number of Although J is usually large in typical panel data models and T cross-sectional units, e.g., brands) is smaller. Table WA.3.1 small, the opposite is true of multilevel models employed in the social (Web Appendix 3) illustrates this effect for exemplary total sciences where researchers, for example, investigate many students sample sizes ranging from 100 to 4,000 observations. How- clustered within a few schools. In our literature review, most of the ever, most of the variation in the copula term’s statistical studies that use Gaussian copulas to address endogeneity in multi- level data have larger J and smaller T, although we also found studies power comes from the total sample size. Overall, the gen- with large T and small J. Consequently, we systematically vary both eral finding is the same as in Study 1: a sufficient copula components. term power of 80% is only reached with 800 and more total To focus our analyses, we do not consider the alternative maximum observations. likelihood method for copula estimation in this study, because it is rarely used in empirical application and complex to implement. 1 3 Journal of the Academy of Marketing Science Fig. 5 Simulation results for the multilevel model Second, the approach requires nonnormality of the endog- Discussion enous regressor, and Study 1 has highlighted that even the uniform distribution might not be sufficiently nonnormal to This study shows that both the bias and the statistical pow- identify the model in smaller sample sizes. We therefore er’s pattern of results are very similar to Study 1 when the vary the endogenous regressor’s distribution. Third, we total sample size is considered. Different level 1 and level systematically vary the ratio of explained to unexplained 2 sample sizes resulting in the same total sample size only variance (i.e., the R ) in the regression model. This is poten- marginally affect the bias and statistical power. We can tially important because the endogenous regressor’s different therefore conclude that the total sample size is the important distributions imply different variance for this variable. Com- criterion to consider when evaluating the appropriateness bined with a fixed error term variance, this would lead to dif- of the Gaussian copula approach. Further, the findings from ferent ratio of explained to unexplained variance, potentially the simple cross-sectional model are generalizable to the confounding the effect of the distribution with R levels. multilevel model’s total sample size. We therefore continue In addition, the ratio of explained to unexplained variance to explore this much simpler model and extend it in other influences the uncertainty in the parameter estimates (i.e., important ways. the parameters’ standard errors), potentially influencing the approach’s statistical power. Since most researchers use an intercept to estimate their regression models in practice, we Simulation study 4: Extension by additional will only focus on the performance of models estimated with factors intercept in this study. Finally, we again estimate our mod- els with the control function and the maximum likelihood Our previous simulation models investigated the role of the approach. The simulation’s detailed design, which is very intercept when estimating the Gaussian copula approach. similar to that of the previous studies, can be found in Web While these focused studies help us understand the role of Appendix WA.4. the intercept and sample size, they only use a single non- Since the endogenous regressor’s nonnormality is a pre- normal distribution of the endogenous regressor (i.e., the requisite to apply the Gaussian copula approach, in practice, uniform distribution) and a fixed error term correlation of researchers usually test whether the encountered distribution 0.50. In this study, we broaden our scope and investigate is significantly different from a normal distribution. How - three additional factors that are potentially important for ever, it is currently unknown when the endogenous regres- the performance of the Gaussian copula approach. First, the sor’s distribution is sufficiently nonnormal to allow the level of the error correlation with the endogenous regres- application of the Gaussian copula approach. We therefore sor defines the endogeneity problem’ s severity, potentially also assess different nonnormality tests and simple moment affecting both the bias and the power of the Gaussian copula measures, like skewness and kurtosis, to identify situations approach (for detailed expectations regarding the different which support the reliable usage of the Gaussian copula assessment criteria, see Web Appendix 4, Table WA.4.1). 1 3 Journal of the Academy of Marketing Science approach. Our literature review reveals that 34 of the 69 results. The endogenous regressor’s power again depends (49.3%) studies use the Shapiro–Wilk test, 4 (5.8%) the Kol- on the sample size, but also, as expected, on the R level: mogorov–Smirnov test, 4 (5.8%) the Kolmogorov–Smirnov the power increases with increasing R . It should be noted test with Lilliefors correction, 2 (2.9%) the Anderson–Dar- that, in this study, the endogenous regressor’s average power ling test, and 2 (2.9%) Mardia’s coefficient. Moreover, only is higher than in the previous studies, because we consider two studies (2.9%) analyze the skewness. The remaining higher R levels than in the original replication model 21 studies (30.4%) do not test or do not report how they (where we have an R of only about 10%). In contrast, the tested nonnormality. To assess which nonnormality test best endogenous regressor’s power only depends marginally on captures the degree of nonnormality needed to identify the the error term’s correlation level Gaussian copula approach, we include these and additional With regard to the endogenous regressor’s bias, we find tests (i.e., Cramer-von Mises, Shapiro-Francia, Jarque–Bera, that it again depends strongly on the sample size. The bias D’Agostino, and Bonett-Seier) that the literature suggests decreases with increasing sample sizes. The bias is on aver- (e.g., Mbah & Paothong, 2015; Yap & Sim, 2011) in our age lower in this simulation than in the previous simulations, simulation study. because we consider different endogeneity and R levels. However, the relative bias (i.e., the copula model’s bias Results divided by the endogenous model’s bias) follows the same pattern as our other simulation studies, reaching about 50% The results presentation begins with the main effects of the for sample sizes of 100 observations, which is similar to potentially relevant factors, namely the sample size, R , and Studies 1 to 3 when we include the intercept in the estima- endogeneity (error correlations), as well as their different tion. Moreover, the endogenous regressor’s bias decreases levels, on the Gaussian copula’s performance (i.e., power with higher R levels, but the relative bias does not depend and bias). Thereafter, we assess the effect of the endogenous on R (i.e., the copula bias decreases with the same magni- regressor’s distribution (nonnormality) on power and bias. tude relative to the original regression’s bias). Finally, the Next, we present the results of skewness and kurtosis as well bias also depends on the endogeneity level, with increasing as different nonnormality tests’ suitability to reliably iden- bias with increasing error correlations. However, the endo- tify endogeneity with Gaussian copula models. We focus geneity level again does not affect the relative bias, because our presentation on the control function approach’s results, the bias in the copula model increases proportionally to the which is the most common approach by far. Overall, the bias in the original regression without copula. maximum likelihood approach yields similar results. The We conclude and reconfirm that the Gaussian copula’ s detailed results of the maximum likelihood approach are pre- performance depends strongly on the sample size, with sented in the Web Appendix 4 (Table WA.4.2, Fig. WA.4.1). substantial effects on both power and bias. In contrast, the endogeneity level does not affect the copula model’ s ability Main effects of design factors to correct the endogeneity bias as indicated by the relative bias, but does affect the copula term ’s power. The higher The results in Table 1 show the main effects of the sample the error term correlation (i.e., the more severe the endo- size, R , and endogeneity levels (error correlations) when geneity problem), the greater the power to identify endo- averaged across the other simulation factors with regard to geneity. Finally, we find that the R level is not relevant for the mean and relative bias of the endogenous regressor and the copula performance, as it neither affects the power nor statistical power of the copula term and endogenous regres- relative bias. In the following analyses, we will therefore sor (at the 5% error level). not further consider R variations and only focus on the We start the analysis by focusing on the copula term and interplay between the level of endogeneity, the sample size, the endogenous regressor’s statistical power. With respect and the distributional form. to the copula term’s power, we confirm that it strongly We substantiate these findings by using a (logistic) regres- depends on the sample size and only reaches acceptable lev- sion with the copula and the endogenous regressor’s power, els beyond 2,000 observations. Moreover, the copula term’s as well as the endogenous regressor’s mean and relative power does not depend on the R level, but, as expected, bias, as dependent variables and the design parameters as depends strongly on the endogeneity level (i.e., the error independent variables. The results indicate that the R level correlation): the higher the error term correlation (i.e., the does not have a significant influence on the copula term’ s more severe the endogeneity problem), the higher the copula power, or on the relative bias, while all the other simulation term’s power to identify endogeneity. This picture changes factors have significant effects (see the Web Appendix 4, somewhat when we examine the endogenous regressor’s Table WA.4.3). 1 3 Journal of the Academy of Marketing Science Endogenous regressor’s distribution Table 1 Effects of key simulation factors on the Gaussian copula approach’s performance Next, we analyze the power and relative bias of different Statistical power Bias Relative bias distributional forms (i.e., different levels of nonnormality) Copula term (%) Endogenous Endog- Endogenous when varying the sample sizes and the endogeneity levels. regressor (%) enous regressor (%) The results show that complex interactions between the regressor distribution, sample size, and endogeneity level influence Sample size a copula term’s power (see Web Appendix 4, Fig. WA.4.2). 100 16 59 0.189 51 For weak endogeneity problems, even heavily nonnormal 200 32 78 0.119 32 distributions, like the log-normal or gamma distribution, 400 49 90 0.064 17 show quite low power unless the sample sizes are very large. 600 59 95 0.042 11 However, for larger error correlations, strongly nonnormal 800 65 97 0.030 8 distributions also have sufficient power if the sample sizes 1,000 70 98 0.023 6 are smaller. 2,000 80 99 0.010 3 In contrast, the endogeneity level does not affect the 4,000 88 100 0.004 1 endogenous regressor’s relative bias. Our analysis indicates 6,000 91 100 0.002 1 that only a combination of sample size and distributional 8,000 93 100 0.001 0 form affects the relative bias and that larger sample sizes 10,000 94 100 0.001 0 and the distributions’ higher nonnormality reduce the endog- enous regressor’s relative bias (Fig. 6). Interestingly, we also 10% 67 84 0.063 11 observe a few situations in which heavily nonnormal distri- 20% 67 88 0.054 12 butions (i.e., some of the gamma, log-normal, and chi ) over- 30% 67 91 0.048 12 compensate the endogeneity bias in smaller sample sizes, 40% 67 93 0.043 12 resulting in a bias in the opposite direction of the original 50% 67 95 0.038 12 endogeneity bias (e.g., underestimating instead of overesti- 60% 67 97 0.034 12 mating the coefficient). 70% 67 98 0.029 12 Endogeneity level (error term correlation) Nonnormality tests 0.10 28 89 0.010 9 0.20 50 90 0.023 11 Since endogeneity is not observable a-priori, researchers can 0.30 62 92 0.033 12 only assess the distribution’s nonnormality and the sample 0.40 70 92 0.043 12 size to decide whether the Gaussian copula approach could 0.50 76 93 0.050 12 be applied. Accordingly, several Gaussian copula applica- 0.60 80 94 0.057 12 tions in our literature review test the endogenous regressor’s 0.70 84 94 0.065 12 nonnormality by using a nonnormality test, mostly the Sha- 0.80 86 95 0.073 12 piro–Wilk test. However, common nonnormality tests’ high sensitivity to small deviations from normality is a problem. In our simulation, for example, the Shapiro–Wilk test reports a significant (at p < 0.05) finding in 96% (94% with p < 0.01) greatly. We therefore also assessed the correlation between of all the cases (Table 2). Only the D’Agostino and Bonett- the copula term’s bootstrap t-statistic and the nonnormality Seier tests have sensitivity rates below 90%. In contrast, the tests’ test statistic. Table 2 shows that the Anderson–Darling copula term is only significant in 67% of the cases. Conse- and Cramer-von Mises tests have the highest correlation with quently, nonnormality test cannot help researchers directly the copula term’s bootstrap t-statistic. In addition, the results decide whether a distribution is sufficiently nonnormal to indicate that kurtosis and skewness alone are not good pre- apply the Gaussian copula approach. Owing to our simu- dictors of the copula term’s t-statistic. Nevertheless, it is lation study, we find that the correspondence between the interesting that skewness seems to be more important than copula and the nonnormality test’s significance is relatively kurtosis. Finally, we also find that the correlation between low (between 61% and 76%), with no test clearly outper- the VIF and copula t-statistic is not very pronounced. forming the other (for the correspondence analysis, see the Web Appendix 4, Table WA.4.4). This outcome is roughly Discussion and boundary conditions analysis equivalent to the copula term’s power (i.e., 67%). The analyzed p-values (i.e., 0.05 and 0.01) represent Based on Study  4’s simulation results, we find that the arbitrary cut-off levels that may reduce the correspondence amount of explained variance has no noticeable influence 1 3 Journal of the Academy of Marketing Science Fig. 6 Relative bias of the endogenous regressor for different dis- (8), blue (14), purple (20); Gamma distribution (α, β): red (1, 0.50), tributions with varying distribution parameters, sample sizes, and green (1, 2), blue (2, 4), purple (4, 2); Log-normal distribution (μ, σ): endogeneity levels. Note: Different colors represent different distri- red (0, 1), green (0, 0.75), blue (0, 0.50), purple (0, 0.25); Student t bution parameters: Beta distribution (p, q): red (0.50, 0.50), green distribution (df): red (3), green (4), blue (5), purple (6) (1, 1), blue (2, 2), purple (4, 4); Chi distribution (df): red (2), green on the Gaussian copula’s power. In contrast, and as expected, failure to identify a significant copula does not necessarily imply the the endogeneity level has a strong effect (i.e., it is harder to absence of endogeneity. It could imply a relatively small endogene- identify a small endogeneity problem). However, even for high ity problem (which might be negligible), but it could also imply an levels of endogeneity the Gaussian copula approach still per- insuc ffi ient sample size or nonnormality. A suc ffi ient sample size forms poorly when sample sizes are small. We also confirm and the careful assessment of nonnormality are therefore particu- the sample size’s strong effect on the Gaussian copula’ s power larly important for the Gaussian copula approach’s application. and bias, and the importance of the endogenous regressor’s Popular nonnormality tests, such as the Shapiro–Wilk nonnormality to identify the Gaussian copula’s parameter test, which, according to our literature review, is the one estimates. Consequently, researchers should use the Gaussian most often used in Gaussian copula applications, do not copula approach cautiously if they suspect the endogeneity identify sufficient nonnormality with common p < 0.05 (or problem is not pronounced (i.e., a small error correlation), the p < 0.01) thresholds. These tests are too sensitive to small sample size is small, or the nonnormality is insufficient. deviations from nonnormality that could lead to insignificant While the sample size is observable and the nonnormality can copula terms, even for substantial endogeneity problems be analyzed, the Gaussian copula approach’s objective is to deter- (i.e., large error correlations). In addition, the nonnormality mine the endogeneity level, which is unknown a-priori. However, a should specifically stem from skewness and not (only) from 1 3 Journal of the Academy of Marketing Science Table 2 Correlation analysis and statistical power of nonnormality and the Gaussian copula’s power level. We reveal, for exam- tests ple, that the lower the number of observations, the higher the skewness levels required to obtain power levels of 80% Correlation p < 0.05 p < 0.01 and higher (Web Appendix, Fig. WA.4.3). Similarly, we find Shapiro–Wilk −0.423 96% 94% that smaller sample sizes require higher levels of the Ander- Kolmogorov–Smirnov −0.029 93% 90% son–Darling and Cramer-von Mises test statistics for a cop- KS-Lilliefors 0.323 93% 89% ula power of at least 80%. These two test statistics’ required Anderson–Darling 0.663 96% 94% levels decrease with a higher number of observations. In Cramer-von Mises 0.653 95% 92% contrast, we observe no clear pattern for the kurtosis, which Shapiro-Francia −0.420 96% 93% is in line with its low correlation with the t-statistic. Jarque–Bera 0.018 99% 99% To turn these findings into more actionable recom- D'Agostino 0.565 73% 71% mendations, we consider all observable characteristics Bonett-Seier 0.477 84% 80% of our models (e.g., sample size, skewness, kurtosis, R , Variance inflation factor (VIF) −0.234 – – and nonnormality test statistics) to derive thresholds that Kurtosis 0.154 – – will ensure that the Gaussian copula approach has a high Skewness 0.313 – – power level. Researchers can use these thresholds as an Absolute skewness* 0.341 – – approximate point of orientation to ensure the method’s effective use in their applications. We do so by employ - The second column shows the correlation between the copula term’s bootstrap t-statistic and the test statistic of the endogenous regressor’s ing decision tree analysis, using the C5.0 algorithm (Kuhn nonnormality test. Columns three and four show the statistical power et al., 2020). Based on our simulation study’s results (i.e., of the nonnormality test (i.e., the percentage of the tests that are sig- Study 4 of regression models with intercept), our goal is to nificant at the given p-level) identify situations where the Gaussian copula approach has *We also assessed the absolute skewness, because it should not mat- a power of at least 80%. Figure WA.4.4 (Web Appendix) ter whether the distribution is skewed to the left or the right side of the mean. Removing the sign might therefore provide a more realistic shows a decision tree result in which we consider sample picture of the actual correlation size, skewness, kurtosis, and R² for predicting the copula’s power (the latter two are not relevant and therefore do not appear in the decision tree). The classification error is 6.4% with 8 false negatives and 6 false positive out of 220 simu- lation design conditions (i.e., 20 distributions times, 11 kurtosis. Our results show that nonnormal distributions with high kurtosis, but small skewness, perform relatively poorly sample sizes). According to the results, the sample size should be larger than 1,000 observations if the skewness regarding identifying the copula term with small to medium sample sizes. Researchers are therefore also advised to is larger than 0.774. If the skewness is equal to or smaller than this level, more than 2,000 observations are required report these more descriptive nonnormality statistics when describing their variables’ nonnormality. Finally, we find to obtain an 80% power level. For smaller sample sizes in the range between 400 to 1,000 observations, a skewness that the Cramer-von Mises tests and the Anderson–Darling test seem to be the most promising candidates for identify- level of 1.932 is required to obtain adequate power. None of our distributions achieves a sufficient power level for ing sufficient nonnormality, because they correlate best with the copula term’s t-statistic. This is not surprising, as both the copula term for sample sizes of 200 observations or smaller. Please note that these findings are derived from the tests build on the empirical cumulative distribution func- tion, which also underlies the Gaussian copula approach. outcomes of the simulation studies, which are constrained by the parameter space of the simulation design. Therefore, The Cramer-von Mises test statistic is the integral of the squared deviation of the endogenous regressor’s empirical these thresholds are an approximate point of reference to guide decision-making. Moreover, researchers must ensure distribution and the theoretical normal distribution. The Anderson–Darling test is an extension of the Cramer-von that their empirical examples meet the other necessary con- ditions for using the Gaussian copula approach that we Mises test that adds a weighting factor to put more weight on the distribution’s tails. investigate in this research (see Fig. 8 for a comprehensive summary). Using our simulation results, we subsequently derive actionable boundary conditions for the required nonnor- We ran similar decision tree analyses that considered the Anderson–Darling and the Cramer-von Mises test statistics mality and sample size, and provide recommendations that could help researchers identify situations with sufficiently (see the Web Appendix 4, Fig. WA.4.5). For example, if the Anderson–Darling (Cramer-von Mises) test statistic has high copula term power in regression models with endoge- neity. In general, we find a complex relationship between a value larger than 18.964 (3.488), the Gaussian copula’s power is 80% and higher. With a sample size of more than the sample size, the endogenous regressor’s nonnormality, 1 3 Journal of the Academy of Marketing Science 1,000 observations, a somewhat lower level of the test sta- Results tistic, but larger than 15.159 (2.628), can achieve this power level. In respect of the error term misspecification, we find the In summary, the endogenous variable’s nonnormality, as same overall pattern of remaining bias and low power at indicated by minimum levels of skewness, and the Ander- smaller sample sizes when the model is estimated with son–Darling or the Cramer-von Mises test statistics, in com- intercept as in our previous studies (for the detailed results, bination with a sufficiently large sample size, may ensure please see Web Appendix 5). However, regarding the bias, that the Gaussian copula approach has adequate power. Our we uncover an additional problem related to the misspecifi- study results suggest that researchers need to ensure that cation of the error term. When the error term is nonnormally there are relatively high nonnormality levels, which should distributed, the Gaussian copula approach is no longer con- stem from the endogenous variable’s skewness, and a rela- sistent (Fig. 7). That is, the remaining bias does not shrink tively large sample size, in order to apply the Gaussian cop- toward zero when the sample size increases. Instead, the ula approach adequately in regression models with intercept. bias approaches an unknown nonzero constant, depending on the error term’s level of nonnormality. In our simulation, this value is positive for negative kurtosis (e.g., beta distri- Simulation study 5: Robustness butions) and negative for positive kurtosis (e.g., student-t to misspecification distributions). In the latter case, the method overcorrects the initially positive endogeneity bias, resulting in a nega- Besides the nonnormality of the endogenous regressor, P&G tive remaining bias. In all our cases, the bias does not occur highlight two additional important criteria to identify the when estimating the model without intercept, reconfirming Gaussian copula approach: 1) the normality of the error P&G’s results on robustness without intercept. The power of term, and 2) the Gaussian copula correlation structure. In the copula term (i.e., the test for the presence of significant their simulations, they show that the method is robust against error correlation) does not seem to be affected beyond the misspecification of the error term and correlation structure. already uncovered issues in our previous simulation studies. However, these simulations are also estimated without inter- Variations in power due to the error term distributions are cept. This study investigates whether including an intercept relatively small and limited to smaller sample sizes. in the estimation retains this robustness or causes additional The results show similar problems in terms of the copula problems. To achieve this objective, we again closely repli- misspecification when estimating the method with inter - cate the simulations from P&G (for detailed design of these cept. For some correlation structures (e.g., Frank and Far- simulations, see Web Appendix 5). For the error term mis- lie-Gumbel-Morgenstern), the method does not correct any specification, we specify several symmetric nonnormal error bias, when estimated with intercept (while we reconfirm its distributions from the Beta and Student-t family, which are robustness when estimated without intercept). Other copula similar to those used in Study 4. We thereby extend the sim- models show a similar pattern as the error term’s misspecifi - ulation by P&G, who only report the uniform distribution’s cation. The bias varies by sample size, decreasing with larger results (i.e., Beta[1,1]). In addition, we evaluate whether the sample sizes but converging to an unknown nonzero con- error term’s nonnormality also manifests in nonnormality of stant, which differs across the analyzed copula models. For the regression residual. If this is the case, researchers could those copulas that correct the bias, the statistical power of evaluate whether their model fulfils this identification crite- the copula term is not affected beyond the already revealed rion. In respect of the copula structure misspecification, we small sample size issues. However, the statistical power of use the same alternative copula models as in P&G’s article those copulas that do not correct the bias (i.e., Frank and (i.e., Ali-Mikhail-Haq distribution with θ = 1, Plackett dis- Farlie-Gumbel-Morgenstern) is low across all sample sizes, tribution with θ = 20, Farlie-Gumbel-Morgenstern distribu- erroneously indicating an absence of endogeneity. tion with θ = 1, Clayton copula with θ = 2, and Frank copula with θ = 2). Comparing the Gaussian copula maximum likelihood approach to the control function approach, we find that the remaining biases (in both directions) are larger in the maximum likelihood approach. This These thresholds become more restrictive for higher power levels. suggest that the Gaussian copula control function approach is slightly For example, to accomplish a 90% power level, the Gaussian copula more robust against error term misspecifications. approach requires more than 600 (2,000) observations at a skew- With respect to the copula model misspecification, both the ness level exceeding 1.974 (0.998). Similarly, the Anderson–Darling approaches (i.e., maximum likelihood and control function) show (Cramer-von Mises) test statistic requires a value of more than 67.875 indistinguishable patterns of bias. This suggests that both approaches (12.246) for sample sizes equal to and smaller than 2,000 and a value are equally affected. of 46.832 (7.994) for sample sizes larger than 2,000 observations. 1 3 Journal of the Academy of Marketing Science Fig. 7 Bias of the endogenous regressor for different error term distributions checked by assessing the regression residual (we find prom- Discussion ising results in this regard, which we report in Web Appen- dix 5), the correlation structure with the unobservable error Researchers estimating models with intercept (which is the term is inherently unobservable, and therefore solely subject standard use case in marketing research) should not only test to assumptions made by the researcher. If these assumptions the endogenous regressor’s nonnormality carefully, but they are violated, the method may experience a strong remaining should also ensure the Gaussian copula approach’s addi- bias, not correct any bias at all, or even overcorrect the initial tional assumptions. While the error term’s normality can be 1 3 Journal of the Academy of Marketing Science bias in the other direction. Hence, the copula model might which may have occurred unintentionally in the past. In not perform better than the original endogenous model. doing so, we contribute to the rigor of regression models’ application and to the accurate presentation and interpreta- tion of marketing research. Summary of key findings In our five studies, we reveal that several factors affect the Gaussian copula approach’s performance. We focus on Researchers in marketing and other disciplines are increas- the interplay between the regression intercept and sample ingly taking advantage of the IV-free Gaussian copula size, as P&G examined regression models without intercept approach to identify and correct endogeneity problems in and, only to a limited extent, the sample size. Our literature regression models. The method’s increasing relevance moti- review reveals that these two factors play an important role vates a closer examination of its adequate performance on when applying the Gaussian copula approach. First, almost the basis of simulation studies. This research replicates and all researchers include an intercept in their model or mean- extends P&G’s initial simulation studies with several new center their data (66 of 69 studies in our literature review and important simulation factors that are highly relevant or 95.7%). Second, our literature review provides indica- in research applications. The results reveal critical issues tions that sample size, and, therefore, the statistical power, and limitations when using the Gaussian copula approach are more important for the results than originally expected. to identify and correct regressions models with intercept. Consequently, our simulation studies shed light on the role The method is not as straightforward and easy to use as pre- of the sample size and the statistical power when using the viously assumed. At the same time, our simulation results Gaussian copula approach to identify and correct endogene- allow us to provide recommendations that are essential to ity problems. ensure that researchers use the Gaussian copula approach In accordance with P&G, our Studies  1 and  2 con- appropriately and obtain valid results on which they can base firm the method’s high performance in regression models their findings and conclusions. Table  3 summarizes our find- without intercept, even in a wider range of sample sizes. ings and provides guidelines to take advantage of the IV-free A very different picture emerges when researchers use Gaussian copula approach while avoiding misapplications, regression models with intercept or mean-centered data, Table 3 Summary of conclusions • Gaussian copula models with intercept are subject to several additional considerations and constraints than those without intercept:   - There is a substantial remaining bias for smaller sample sizes in models with intercept   - There is only low statistical power to identify endogeneity in small samples, especially regarding the control function approach, but to a lesser degree also regarding the maximum likelihood approach   - Beside the sample size, the endogenous regressor’s nonnormality and the error correlation’s size are also important factors that influence performance   - In multilevel (or panel) models, the total sample size imposes the same performance restriction as in cross-sectional models, while only within or between sample sizes are less relevant   - The method is much less robust against misspecifications of the error distribution and the copula structure, resulting in remaining biases that do not vanish when the sample size is increased • Estimating a model without intercept is usually not an option without strong prior assumptions, as this estimation would also induce substantial bias when the true intercept is not zero and the copula approach cannot correct for omitted intercepts • All disclosed limitations concern finite (small) sample sizes, but the method works well within the limits if there is sufficient information to identify the model and the error term and copula are not misspecified • Based on simulation Study 4, we propose the following guidelines as a rough orientation for applying the Gaussian copula approach:   - In general, researchers should consider applying far more conservative nonnormality tests to ensure sufficient (and not only significant) non- normality, especially with sample sizes < 5,000   - The Anderson–Darling and Cramer-van Mieses nonnormality tests are conceptually closest to the copula approach and yield the best cor- respondence   - Sample sizes equal to or less than 200 observations should always be avoided   - For sample sizes below 1,000 observations, only a few very nonnormal distributions (e.g., skewness larger than 2 or Anderson–Darling test statistics above 20) yield sufficient power to identify endogeneity   - All these recommendations are based on continuous distributions. Discrete distributions, such as Poisson or Likert-scale survey data, might require even larger sample sizes, as they contain less information • The Gaussian copula approach is not free of assumptions and researchers need to be very careful when applying the method, especially to smaller sample sizes. Ultimately, researchers will always need to argue that the underlying assumptions have been fulfilled. Some of these assumptions (like the copula correlation structure) are inherently unobservable 1 3 Journal of the Academy of Marketing Science which is common in marketing studies. The Gaussian observations. In contrast, the nonnormality is still impor- copula approach has far less power and higher remaining tant for sample sizes above 1,000 observations, but to a endogeneity bias in these regression models, especially lesser degree. These boundary conditions of factors of key when using the estimation method that most researchers relevance for the Gaussian copula approach’s valid use in prefer: the control function approach (i.e., adding addi- regression models with intercept (i.e., the required sam- tional copula terms as new variables to the regression ple size, the endogenous regressor’s nonnormality, and model). In such models, the Gaussian copula approach’s its identification with suitable nonnormality tests) allow identification and correction of endogeneity problems researchers to effectively identify and correct endogeneity requires a much larger sample size. If this requirement problems. is not met, the approach may not identify an endogene- Using the maximum likelihood approach might be a ity problem even though it is present and has substantial potential solution to remedy some of these concerns, as it endogeneity bias. This finding is of central relevance, has slightly larger power to identify endogeneity (but has because our literature review reveals that most studies the same remaining bias). However, the control function apply the Gaussian copula approach to regression mod- approach has several advantages: (1) It is much faster and els that include an intercept or mean-centered data. It easier to implement in models that go beyond the simple is therefore very likely that studies with smaller sample linear regression model (e.g., panel models, binary choice sizes do not always identify significant copula terms due models, etc.) that might make deriving the appropriate likeli- to their insufficient power, although endogeneity prob- hood function more complex or even impossible, and (2) it lems are present (Fig. 1). Consequently, researchers may allows for including more than one copula term and, there- come to the false conclusion that endogeneity problems fore, for treating several endogenous regressors simultane- do not affect their studies’ results and present invalid ously. For these reasons, the maximum likelihood approach findings and conclusions. might not be a practical solution in many situations, and the In Study 3, we show that the findings of Studies 1 and 2 gains in power are also limited. extend to multilevel models, in which the endogeneity is Finally, Study 5 sheds light on the misspecification of present at the within-cluster level (i.e., the correlation the error term and the copula structure when regression between a within-cluster predictor and the structural error) models are estimated with intercept. This study’s results when the total sample size is taken into account. In our underscore concerns about the method’s estimation accu- literature review, most studies with multilevel data have racy when an intercept is present, and contradict find- rather larger sample sizes, but a few also have total sam- ings about its robustness as presented in P&G's original ple sizes in the range for which we identify the Gaussian study. Researchers should ensure both the presence of an copula approach’s reduced performance. Consequently, in appropriate Gaussian copula correlation structure and a respect of multilevel (or panel) models, the same recom- normally distributed error term. While the analysis of mendations apply regarding a sufficient sample size and the regression residual allows an assessment of the error nonnormality as do for the simpler, cross-sectional regres- term, the correlation structure is inherently unobserv- sion model. able and therefore only subject to untestable theoretical Study 4 aims at helping researchers apply the Gaussian considerations. copula approach appropriately and exploit its advantages We have two recommendations for research that does not effectively. More specifically, in Study 4, we extend the satisfy the boundary conditions identified in this research: simulations to include several additional factors that are first , researchers should carefully assess whether the data relevant for regression analyses, such as the endogenous and model might be prone to empirical identification issues. regressor’s nonnormality, the explained variance (R They can do so by, for example, carefully checking whether level), and the error correlation. Study 4 derives boundary the endogenous regressor has sufficient nonnormality and conditions for these factors to guide the Gaussian copula checking for multicollinearity issues after including the approach’s appropriate use in studies. The findings sub - Gaussian copula, as well as testing the regressions’ residual stantiate that for sample sizes below 1,000 observations, for normality. Second, and more importantly, researchers only a few very nonnormal distributions (e.g., skewness should avoid using the Gaussian copula approach to test above 2 or Anderson–Darling test statistics above 20) lead for endogeneity (i.e., concluding that endogeneity is not to sufficiently high power when using the copula term in a problem due to insignificant copula terms), but should regression models (i.e., larger than 80%). Nevertheless, revert to traditional ways of handling endogeneity problems, none of our considered distributions has sufficiently such as using IVs or other means of identifying the causal large power for sample sizes equal to or less than 200 mechanism. 1 3 Journal of the Academy of Marketing Science application has gained the method the reputation of being an Conclusions and future research easy-to-use add-on in any study that has a potential endoge- neity problem, our results highlight that researchers should The Gaussian copula approach is valuable for identifying use the Gaussian copula approach more cautiously, espe- and correcting endogeneity issues in regression models cially when sample sizes are small and the model includes a when the assumptions are fulfilled. However, when the regression intercept (or mean-centered data). Figure 8 sum- regression models contain an intercept, the method is much marizes our studies’ findings and conclusions in a decision more constrained than initially thought. It is less robust flowchart by illustrating the path of choices that researchers against deviations of the error term’s normality, the Gauss- need to consider when deciding whether to apply the Gauss- ian copula correlation structure between the error and the ian copula approach. Given these new recommendations, regressor, and the regressor’s nonnormality. Even if these researchers might far less often conclude that the Gaussian preconditions are met, the approach requires large sample copula approach is a recommended method for dealing with sizes to perform well in models with intercept. However, endogeneity problems. constraining the intercept to zero is usually not an option, These recommendations represent approximate thresh- because this would also induce substantial bias as high- olds based on the results of our simulation studies that lighted in our Study 2. While the Gaussian copula’s simple Fig. 8 Flowchart for the decision on the application of the Gaussian copula approach 1 3 Journal of the Academy of Marketing Science provide researchers with an indication of whether the copula which the Gaussian copula approach is not applicable. We can be successfully applied. However, they do not replace therefore call for further research on comparing the methods careful theoretical consideration of the nature of endogene- under varying conditions to provide researchers with bet- ity and the fulfillment of the Gaussian copula approach’ s ter guidelines on which method to use when. However, all general assumptions (i.e., the nonnormality of the endog- IV-free approaches demand fulfillment of certain identifica- enous regressor, the normality of the error term, and the tion requirements, which are often untestable. Applying any Gaussian copula correlation structure). Moreover, these rec- of these methods blindly may provide no better results than ommendations are based on models with a single continu- merely ignoring endogeneity problems does. Consequently, ous endogenous regressor variable. It is likely that multiple it is important that researchers are aware of these approaches’ endogenous regressors or discrete variables will increase limitations, because ultimately, they always need to carefully the requirements for identifying the copula and, therefore, argue that the underlying assumptions have been fulfilled. for the method’s successful application. More research is Supplementary Information The online version contains supplemen- needed to extend the recommendations in respect of these tary material available at https://doi. or g/10. 1007/ s11747- 021- 00805-y . areas. Future research should therefore extend our findings by Acknowledgements The authors presented some of the research results adding simulation studies that, for instance, analyze the depicted in this paper at the 2020 AMA Winter Academic Confer- Gaussian copula’s performance with additional endogenous ence, San Diego, CA, February 14-16, 2020, but without publication in the conference proceedings. The authors thank Peter Ebbes, HEC regressor distributions (i.e., additional nonnormality levels) Paris, France, and Dominik Papies, University of Tübingen, Germany, to further substantiate our thresholds. Furthermore, we are for their useful comments on our research presentation. Moreover, the currently not aware of possibilities to test the assumption authors like to thank Edward E. Rigdon, Georgia State University, that the error term and the endogenous variable follow a United States of America, for his helpful comments to improve an ear- lier version of the manuscript. For their simulation studies, the authors Gaussian copula correlation structure. However, a misspeci- used the high-performance computing cluster of Hamburg University fication potentially leads to invalid results as our simula- of Technology, Germany. tion results show. Thus, creating a test for this assumption would greatly enhance confidence in the method’ s accuracy. Funding This research was partly funded by the Deutsche Forschun- gsgemeinschaft (DFG, German Research Foundation)—VO 1555/1-1. In addition, future studies should analyze discrete distribu- tions further (i.e., P&G show that discrete distributions Open Access This article is licensed under a Creative Commons Attri- suffer even more from identification problems and that bution 4.0 International License, which permits use, sharing, adapta- thresholds might therefore be much higher in such cases) tion, distribution and reproduction in any medium or format, as long and revert to more complex regression models with mul- as you give appropriate credit to the original author(s) and the source, tiple endogenous regressors. Additional knowledge about provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are these factors’ relevance will help researchers use the method included in the article's Creative Commons licence, unless indicated adequately to derive valid inferences for marketing decision otherwise in a credit line to the material. If material is not included in making. Moreover, future research should address the core the article's Creative Commons licence and your intended use is not issue that the Gaussian copula approach’s usability is limited permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a regarding finite (small) sample sizes, but works well in the copy of this licence, visit http://cr eativ ecommons. or g/licen ses/ b y/4.0/ . limit when sufficient information is available to identify the model. Methodological research should aim at developing a solution for this limitation. 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Journal

Journal of the Academy of Marketing ScienceSpringer Journals

Published: Oct 11, 2021

Keywords: Endogeneity; Gaussian copula; Intercept; Linear regression; Multilevel models; Sample size; Simulation

References