1. Endogeneity in PLS-SEM: The Gaussian Copula Approach
Hult, G. T. M., Hair, J. F., Proksch, D., Sarstedt, M., Pinkwart, A., and Ringle, C. M. (2018). Addressing Endogeneity in International Marketing Applications of Partial Least Squares Structural Equation Modeling, Journal of International Marketing, forthcoming.

2. What’s the Endogeneity Problem? (I)
Endogeneity can have various roots such as measurement errors, simultaneous causality, common method variance, and (un)observed heterogeneity.
Often endogeneity problems arise from omitted variables that correlate with one or more independent variable(s) and the dependent variable(s) in the regression model.
Omitting such variables induces a correlation between the corresponding independent variables and the dependent variables’ error term.
That is, the independent variables then not only explain the dependent variable, but also the error in the model. 2

3. What’s the Endogeneity Problem? (II)
Consider the following regression model where y represents the dependent variable, x1 and x2 are independent variables, β0 the intercept, β1 and β2 the regression coefficients of x1 and x2, and ε the error term:
y = β0+ β1x1+β2x2+ ε.
Let us assume that the independent variable x2 is uncorrelated with ε (i.e., x2 is exogenous), whereas x1 is endogenous since it is correlated with the error term ε (i.e., Cov(x1,ε)≠0).
The coefficient estimates from standard regression analyses are biased and inconsistent, thereby becoming causally uninterpretable and potentially triggering type I and type II errors.
corr. ≠ 0 x1 1 y ε
Bias: Is the difference between this estimator's expected value and the true value of the parameter being estimated.
Consistency: Consistent estimators are asymptotically unbiased, i.e., they converge to the correct value as sample size increases to infinity. x2 2
corr. = 0 3

4. On the Relevance Assessment in PLS-SEM
Dealing with endogeneity has been extensively discussed in the literature, especially with respect to different forms of regression and panel models, as well as conjoint analysis.
While several studies have discussed endogeneity in the context of factor-based SEM, there is a paucity of research on this topic in PLS-SEM.
Some researchers even claim that PLS-SEM does not allow for addressing endogeneity at all. This assertion is astonishing and inaccurate given that PLS-SEM is grounded in regression analysis, for which numerous approaches for handling endogeneity exist.
Hence, research must address when and how to address endogeneity in PLS-SEM. 4

5. Comparison of Approaches to Deal with Endogeneity
Methods
Criteria
Control variable approach
Instrumental variable (IV) approach
Instrumental variable-free approaches
Gaussian copula
Latent instrumental variable (LIV)
Number of variables
Data on additional variables must be collected
Instrumental variables have to be identified and data has to be collected
No additional variables needed
No additional variables needed
Distribution of variables
No assumptions required
Endogenous variables have to be non-normally distributed
Endogenous variables have to be non-normally distributed
Nature of dependent variable
Discrete or continuous
Continuous
Statistical tests
Not necessary
Test for significance and relevance
Test for significance
Acceptance in scientific community
Widely accepted and commonly used
Relatively new and therefore rarely used
Implementation in software
No additional implementation necessary
Supported by, for example, SPSS, STATA, and R software packages
The REndo (Gui et al. 2017) package supports the Gaussian copula approach
The REndo (Gui et al. 2017) package supports LIV model with one dependent and one independent variables 5

6. PLS-SEM Endogeneity Assessment Procedure
Does the Gaussian copula approach detect endogeneity issues?
Assumptions of the Gaussian copula approach fulfilled?
Yes
Requirements Check
Model Analysis 6

7. 7

8. 8

9. The Gaussian Copulas Approach
Park and Gupta (2012) introduced the Gaussian copula approach, which controls for endogeneity by directly modeling the correlation between the endogenous variable and the error term by means of a copula.
To determine whether endogeneity is at a critical level, researchers need to assess the significance of the copula coefficient. A significant coefficient indicates a critical level of endogeneity.
As c* is an estimated quantity, the standard errors of the OLS regressions are not correct. Park and Gupta (2012) suggest a bootstrapping approach
The Gaussian copula approach requires the endogenous variable to be nonnormally distributed.
𝑐 ∗ = Φ −1 (𝐻 𝑥 1
Inverse normal cumulative density function
Empirical cumulative density function (ECDF)
 𝑦= 𝛽′′ 0 + 𝛽′′ 1 𝑥 1 + 𝛽′′ 2 𝑥 2 + 𝛽′′ 3 𝑐 ∗ + 𝜀′′ 9

10. Download and Save the Files of the Example
… e.g. to C:\endo 10

11. Create and estimate the PLS path model
Data Preparation
Create and estimate the PLS path model
Generate the data file
Requirements Check
Prepare the R code for the requirements check
Run the requirements check in R and analyze the results
Model Analysis
Prepare the R code for the Gaussian copula analysis
Run the Gaussian copula analysis in R and interpret the results 11

12. Simple Corporate Reputation Model Example
The book on PLS-SEM explains how to create and estimate the simple corporate reputation model example using SmartPLS 3:
Hair, J. F., Hult, G. T. M., Ringle, C. M., and Sarstedt, M. (2017). A Primer on Partial Least Squares Structural Equation Modeling (PLS-SEM), Thousand Oaks, CA: Sage. 12

13. Run the Simple Corporate Reputation Model Example in SmartPLS
Missing value marker: -99
Case wise deletion
Path weighting scheme 13

14. Create and estimate the PLS path model
Data Preparation
Create and estimate the PLS path model
Generate the data file
Requirements Check
Prepare the R code for the requirements check
Run the requirements check in R and analyze the results
Model Analysis
Prepare the R code for the Gaussian copula analysis
Run the Gaussian copula analysis in R and interpret the results 14

15. Copy Latent Variable Scores to Excel and Save as CSV15

16. CRP_dataset_std.csv
… e.g. save to C:\endo 16

17. Create and estimate the PLS path model
Data Preparation
Create and estimate the PLS path model
Generate the data file
Requirements Check
Prepare the R code for the requirements check
Run the requirements check in R and analyze the results
Model Analysis
Prepare the R code for the Gaussian copula analysis
Run the Gaussian copula analysis in R and interpret the results 17

18. Open CRP_KS-test_code.r with Text Editor
or Notepad++ with
highlighting
… e.g. from c:\endo 18

19. Prepare the R Code (1)
# R code for the Kolmogorov–Smirnov test with Lilliefors correction
# Set directory -> REPLACE WITH THE DIRECTORY INCLUDING THE EXAMPLE CSV FILE
# ON YOUR COMPUTER
setwd ("C:/endo")
# Load required libraries -> PLEASE INSTALL THE "KScorrect" PACKAGE IF YOU HAVE NOT
# ALREADY. SEE
# FOR INSTRUCTIONS HOW TO INSTALL A PACKAGE
library(KScorrect)
# Read data (extracted standardized latent variable scores from PLS model)
CRDdata = read.csv2("CRP_dataset_std.csv", header=TRUE, sep=";", dec=".", stringsAsFactors=FALSE);
LIKE <- CRDdata[,"LIKE"]
COMP <- CRDdata[,"COMP"]
CUSA <- CRDdata[,"CUSA"]
#Run the he Kolmogorov–Smirnov test with Lilliefors correction
LcKS(LIKE, "pnorm", nreps = 4999)
LcKS(COMP, "pnorm", nreps = 4999)
LcKS(CUSA, "pnorm", nreps = 4999)
Path to the working directory of this R session (i.e., folder where you saved the data file)
Name of the csv file including the latent variable scores
Get the data of the latent variable scores. In this example of the three independent variables when regressing CUSL on LIKE, COMP, and CUSA.
Define the variables for which you like to run the KS test 19

20. Create and estimate the PLS path model
Data Preparation
Create and estimate the PLS path model
Generate the data file
Requirements Check
Prepare the R code for the requirements check
Run the requirements check in R and analyze the results
Model Analysis
Prepare the R code for the Gaussian copula analysis
Run the Gaussian copula analysis in R and interpret the results 20

21. Download, Install and Run the Statistical Software R 21

22. Run R and Copy & Paste the Code Into the Console22

23. Results
If the p-value is below 0.05, the variable does not follow a normal distribution.
Scroll up (and down) to see the p-values of the other variables 23

24. Requirements Check Before Running Gaussian Copulas
Before initiating the Gaussian copula approach to meet its assumptions, we first verify if the variables, which potentially exhibit endogeneity, are non-normally distributed.
We do so by running the Kolmogorov–Smirnov test with Lilliefors correction (Sarstedt and Mooi 2014) on the standardized composite scores of COMP, LIKE, and CUSA, which the PLS path model estimation provides.
If the p-value is below 0.05, the variable does not follow a normal distribution.
The results show that none of the constructs has normally distributed scores, which allows us to consider them endogenous in the Gaussian copula analysis. 24

25. Create and estimate the PLS path model
Data Preparation
Create and estimate the PLS path model
Generate the data file
Requirements Check
Prepare the R code for the requirements check
Run the requirements check in R and analyze the results
Model Analysis
Prepare the R code for the Gaussian copula analysis
Run the Gaussian copula analysis in R and interpret the results 25

26. Open CRP_copula_code.r with Text Editor
or Notepad++ with
highlighting
… e.g. from c:\endo 26

27. Load the R package “car”
Prepare the R Code (I)
# R code for correcting for endogeneity in the Corporate Reputation Data PLS model
# using Gaussian Copula Approach as descripted in Park and Gupta (2012) #
# PLEASE CITE AS:
# Hult, G. T. M., J. F. Hair, D. Proksch, M. Sarstedt, A. Pinkwart, & C. M. Ringle (2018).
# Addressing Endogeneity in International Marketing Applications of Partial
# Least Squares Structural Equation Modeling. Journal of International Marketing,
# forthcoming.
# Set directory -> REPLACE WITH THE DIRECTORY INCLUDING THE EXAMPLE CSV FILE
# ON YOUR COMPUTER
setwd ("C:/endo")
# Load required libraries -> PLEASE INSTALL THE "CAR" PACKAGE IF YOU HAVE NOT
# ALREADY. SEE
# FOR INSTRUCTIONS HOW TO INSTALL A PACKAGE
library(car)
Path to the working directory of this R session (i.e., folder where you saved the data file)
Load the R package “car” 27

28. Prepare the R Code (II) # Function to create Gaussian Copula
# From Gui, Raluca, Markus Meierer, and Rene Algesheimer (2017),
# "R Package REndo: Fitting Linear Models with Endogenous Regressors using
# Latent Instrumental Variables (Version 1.3),"
createCopula <- function(P){
H.p <- stats::ecdf(P)
H.p <- H.p(P)
H.p <- ifelse(H.p==0, ,H.p)
H.p <- ifelse(H.p==1, ,H.p)
U.p <- H.p
p.star <- stats::qnorm(U.p)
return(p.star) }
# Function to calculate corrected p-values for regression based on bootstrapped standard errors
bootstrapedSignificance <- function(dataset, bootstrapresults, numIndependentVariables, numCopulas){
for (i in 1:nrow(summary(bootstrapresults))){
t <- summary(bootstrapresults)[i, "original"] / summary(bootstrapresults)[i, "bootSE"]
# df = n (number of observations) - k (number of independent variables + copulas) - 1
pvalue <- 2 * pt(-abs(t),df=nrow(dataset)-numIndependentVariables-numCopulas-1)
cat("Pr(>|t|)", rownames(summary(bootstrapresults))[i], ": ", pvalue, "\n") 28

29. Prepare the R Code (III)
# Read data (extracted standardized latent variable scores from PLS model)
CRDdata = read.csv2("CRP_dataset_std.csv", header=TRUE, sep=";", dec=".", stringsAsFactors=FALSE);
CUSL <- CRDdata[,"CUSL"]
LIKE <- CRDdata[,"LIKE"]
COMP <- CRDdata[,"COMP"]
CUSA <- CRDdata[,"CUSA"]
# Calculate standard regression
stdModel <- lm (CUSL ~ LIKE + COMP + CUSA)
summary(stdModel);
# Calculate copulas for independent variables within model
LIKE_star <- createCopula(LIKE)
COMP_star <- createCopula(COMP)
CUSA_star <- createCopula(CUSA)
Name of the csv file including the latent variable scores
Get the data of the latent variable scores
Define the regression model (i.e., regress CUSL on LIKE, COMP, and CUSA
Compute the Gaussian copulas of the independent variables in the regression model which may be subject to endogeneity issues 29

30. Prepare the R Code (IV) # Set bootstrapping rounds
# FOR TESTING PURPOSE, WE RECOMMEND SETTING THIS VALUE TO 100; FOR REPORTING THE FINAL
# RESULTS WE RECOMMEND SETTING IT TO 10000
bootrounds = 10000
# Calculate Results
# Include Copula for COMP (Model 1)
# Normal regression
copulaResults1 <- lm (CUSL ~ COMP + LIKE + CUSA + COMP_star + 0)
summary(copulaResults1)
# Bootstrap Standard Errors
bootCopulaResults1 <- Boot(copulaResults1, R=bootrounds)
summary(bootCopulaResults1)
# Calculate corrected p-values based on bootstrapped standard errors
bootstrapedSignificance(CRDdata, bootCopulaResults1, 3, 1)
Set the number of bootstraps (e.g., 10,000)
Run the regression model which includes the Gaussian copula for COMP
CHECK / UPDATE VALUES!
3 = Number of independent variables in the regression model
1 = Number of Gaussian copulas in the regression model 30

31. Prepare the R Code (V) # Include Copula for LIKE (Model 2)
# Normal copula regression
copulaResults2 <- lm (CUSL ~ COMP + LIKE + CUSA + LIKE_star + 0)
summary(copulaResults2)
# Bootstrap standard errors
bootCopulaResults2 <- Boot(copulaResults2, R=bootrounds)
summary(bootCopulaResults2)
# Calculate corrected p-values based on bootstrapped standard errors
bootstrapedSignificance(CRDdata, bootCopulaResults2, 3, 1)
# Include Copula for CUSA (Model 3)
copulaResults3 <- lm (CUSL ~ COMP + LIKE + CUSA + CUSA_star + 0)
summary(copulaResults3)
bootCopulaResults3 <- Boot(copulaResults3, R=bootrounds)
summary(bootCopulaResults3)
bootstrapedSignificance(CRDdata, bootCopulaResults3, 3, 1)
Run the regression model which includes the Gaussian copula for LIKE
CHECK / UPDATE VALUES!
3 = Number of independent variables in the regression model
1 = Number of Gaussian copulas in the regression model
Run the regression model which includes the Gaussian copula for CUSA
CHECK / UPDATE VALUES!
3 = Number of independent variables in the regression model
1 = Number of Gaussian copulas in the regression model 31

32. Prepare the R Code (VI) # Include Copula for LIKE and COMP (Model 4)
# Normal copula regression
copulaResults4 <- lm (CUSL ~ COMP + LIKE + CUSA + COMP_star + LIKE_star + 0)
summary(copulaResults4)
# Bootstrap standard errors
bootCopulaResults4 <- Boot(copulaResults4, R=bootrounds)
summary(bootCopulaResults4)
# Calculate corrected p-values based on bootstrapped standard errors
bootstrapedSignificance(CRDdata, bootCopulaResults4, 3, 2)
# Include Copula for LIKE and CUSA (Model 5)
copulaResults5 <- lm (CUSL ~ COMP + LIKE + CUSA + LIKE_star + CUSA_star + 0)
summary(copulaResults5)
bootCopulaResults5 <- Boot(copulaResults5, R=bootrounds)
summary(bootCopulaResults5)
bootstrapedSignificance(CRDdata, bootCopulaResults5, 3, 2)
Run the regression model which includes the Gaussian copula for COMP and LIKE
CHECK / UPDATE VALUES!
3 = Number of independent variables in the regression model
2 = Number of Gaussian copulas in the regression model
Run the regression model which includes the Gaussian copula for LIKE and CUSA
CHECK / UPDATE VALUES!
3 = Number of independent variables in the regression model
1 = Number of Gaussian copulas in the regression model 32

33. Prepare the R Code (VII)
# Include Copula for COMP and CUSA (Model 6)
# Normal copula regression
copulaResults6 <- lm (CUSL ~ COMP + LIKE + CUSA + COMP_star + CUSA_star + 0)
summary(copulaResults6)
# Bootstrap standard errors
bootCopulaResults6 <- Boot(copulaResults6, R=bootrounds)
summary(bootCopulaResults6)
# Calculate corrected p-values based on bootstrapped standard errors
bootstrapedSignificance(CRDdata, bootCopulaResults6, 3, 2)
# Include Copula for LIKE, COMP and CUSA (Model 7)
copulaResults7 <- lm (CUSL ~ COMP + LIKE + CUSA + COMP_star + LIKE_star + CUSA_star + 0)
summary(copulaResults7)
bootCopulaResults7 <- Boot(copulaResults7, R=bootrounds)
summary(bootCopulaResults7)
bootstrapedSignificance(CRDdata, bootCopulaResults7, 3, 3)
Run the regression model which includes the Gaussian copula for COMP and CUSA
CHECK / UPDATE VALUES!
3 = Number of independent variables in the regression model
2 = Number of Gaussian copulas in the regression model
Run the regression model which includes the Gaussian copula for COMP, LIKE, and CUSA
CHECK / UPDATE VALUES!
3 = Number of independent variables in the regression model
3 = Number of Gaussian copulas in the regression model 33

34. Create and estimate the PLS path model
Data Preparation
Create and estimate the PLS path model
Generate the data file
Requirements Check
Prepare the R code for the requirements check
Run the requirements check in R and analyze the results
Model Analysis
Prepare the R code for the Gaussian copula analysis
Run the Gaussian copula analysis in R and interpret the results 34

35. Run R and Copy & Paste the Code Into the Console35

36. R Results of Regression Model 1 in the Example (i. e
R Results of Regression Model 1 in the Example (i.e. the regression model includes the copula for COMP)
> # Bootstrap Standard Errors
> bootCopulaResults1 <- Boot(copulaResults1, R=bootrounds)
Loading required namespace: boot
> summary(bootCopulaResults1)
Number of bootstrap replications R = 10000
original bootBias bootSE bootMed
COMP
LIKE
CUSA
COMP_star
> # Calculate corrected p-values based on bootstrapped standard errors
> bootstrapedSignificance(CRDdata, bootCopulaResults1, 3, 1)
Pr(>|t|) COMP :
Pr(>|t|) LIKE : e-08
Pr(>|t|) CUSA : e-25
Pr(>|t|) COMP_star :
ATTENTION
As bootstrapping is based on random sampling, your results will slightly differ. Also, the results will slightly differ each time you perform the analysis. 36

37. R Results of Regression Model 7 in the Example (i. e
R Results of Regression Model 7 in the Example (i.e. the regression model includes copulas for COMP, LIKE, and CUSA)
> # Bootstrap standard errors
> bootCopulaResults7 <- Boot(copulaResults7, R=bootrounds)
> summary(bootCopulaResults7)
Number of bootstrap replications R = 10000
original bootBias bootSE bootMed
COMP
LIKE
CUSA
COMP_star
LIKE_star
CUSA_star
> # Calculate corrected p-values based on bootstrapped standard errors
> bootstrapedSignificance(CRDdata, bootCopulaResults7, 3, 3)
Pr(>|t|) COMP :
Pr(>|t|) LIKE : e-05
Pr(>|t|) CUSA : e-16
Pr(>|t|) COMP_star :
Pr(>|t|) LIKE_star :
Pr(>|t|) CUSA_star :
ATTENTION
As bootstrapping is based on random sampling, your results will slightly differ. Also, the results will slightly differ each time you perform the analysis. 37

38. Results Table Including the Outcomes of Model 1
Original model
Gaussian copula Model 1 (endogenous variable: COMP)
Gaussian copula Model 2 (endogenous variable: LIKE)
Gaussian copula Model 3 (endogenous variable: CUSA)
Variable
Value
p-value
COMP
0.016
0.746
0.014
0.850
0.017
0.763
0.021
0.707
LIKE
0.331
< 0.01
0.370
CUSA
0.509
0.511
0.582
cCOMP
0.002
0.973
cLIKE
-0.033
0.245
cCUSA
-0.041
0.063 38

39. Results Table Including the Outcomes of Model 7
Gaussian copula Model 4 (endogenous variables:
LIKE, COMP)
Gaussian copula Model 5 (endogenous variables:
LIKE, CUSA)
Gaussian copula Model 6 (endogenous variables:
COMP, CUSA)
Gaussian copula Model 7 (endogenous variables:
LIKE, COMP, CUSA)
Variable
Value
p-value
COMP
-0.006
0.939
0.021
0.705
-0.027
0.717
-0.037
0.644
LIKE
0.381
0.000
0.341
0.333
0.362
CUSA
0.509
0.580
0.592
0.589
cCOMP
0.019
0.737
0.039
0.432
0.047
0.395
cLIKE
-0.041
0.283
-0.008
0.790
-0.024
0.505
cCUSA
-0.039
0.084
-0.049
0.045
-0.047
0.056 39

40. Results of the Gaussian Copulas Endogeneity Assessment
The results show that only one Gaussian copula (i.e., cCUSA) is significant (p < 0.1) when treating one endogenous variable, which points to a potential endogeneity issue.
Including the significant Gaussian copula in the model changes the effect of CUSA on CUSL by units (from to 0.582), which points to a potential endogeneity problem of CUSA (Model 3).
Similarly, cCUSA is also significant in the CUSA models in combination with LIKE (Model 5, p < 0.1), COMP (Model 6, p < 0.05), and LIKE and COMP (Model 7, p < 0.1).
This confirms the possibility of CUSA being endogenous.
You may use the results of model 3 to treat the identified endogeneity problem. 40

41. Further Research? http://journals.ama.org/doi/10.1509/jim.17.0151
Hult, G. T. M., J. F. Hair, D. Proksch, M. Sarstedt, A. Pinkwart, & C. M. Ringle (2018). Addressing Endogeneity in International Marketing Applications of Partial Least Squares Structural Equation Modeling. Journal of International Marketing, forthcoming. 41

42. Gaussian copula, graphical representation of solution – part 142

43. Gaussian copula, graphical representation of solution – part 2?
First, we calculate the area of the empirical density function at the value 5.
Second, we identify the value for which the area of the normal density function is the same as the just identified area. 43

44. Explanation of Gaussian Copula approach – Part 1
Key idea: estimate the joint density of the structural error and the endogenous regressor
Sklar’s theorem states: every joint distribution can be written as a function of its margins and the other way around
A copula is used to map the two cumulative distribution functions (CDFs) to a joint CDF. We assume that they have a joint normal distribution (therefore Gaussian Copula is used)
We assume that 𝑋 consist of 𝑥1 (an endogenous variable) and 𝑥2 (one or more exogenous regressor)
We assume that the CDF of 𝜀 is a normal distribution with mean 0 and variance 𝜎 𝜀 2
We use the Gaussian copula to get the joint CDF 𝐺 𝑥 1 , 𝜀 =𝑁( 𝑥 1 ∗ , 𝜀 ∗ ), with 𝑥 1 ∗ = 𝜙 −1 𝐹 𝑥 𝑥 1 and 𝜀 ∗ = 𝜙 −1 𝐹 𝜀 𝜀 and 𝑁 is the bivariate standard normal distribution with correlation coefficient 𝜌.
We differentiate 𝐺 𝑥 1 , 𝜀 : Joint probability density function: 𝑔 𝑥 1 , 𝜀 = 𝛿𝛿𝐺 𝑥 1 , 𝜀 𝛿 𝑥 1 𝛿𝜀 𝑓 𝑥 𝑓 𝑦
We could use the density to obtain the likelihood function and then consistently estimate the coefficient of 𝑥1 using maximum likelihood estimation (see Park and Gupta, 2012 for the likelihood function). 44

45. Explanation of Gaussian Copula approach – Part 2
We could instead include 𝑥 1 ∗ in the equation based on the following reasoning:
The copula model implies that 𝑥 1 ∗ and 𝜀 ∗ follow the standard bivariate normal distribution with correlation coefficient 𝜌 𝑥 1 ∗ 𝜀 ∗ = 1 0 𝜌 1− 𝜌 𝑣 1 𝑣 2 , 𝑣 1 and 𝑣 2 being independent random variables drawn from a standard normal distribution
The structural error is: 𝜀= 𝐹 𝜀 −1 𝜙 𝜀 ∗ = 𝜙 𝜎 𝜀 2 −1 𝜙 𝜀 ∗ = 𝜎 𝜀 𝜀 ∗
We can rewrite the original equation by: 𝑦= 𝑥 1 𝛽 1 + 𝑋 2 𝛽 2 + 𝜎 𝜀 (𝜌 𝑥 1 ∗ +( 1− 𝜌 2 ) 𝑣 2 )
We therefore split the structural error in two parts, the first part is correlated with 𝑥 1 , the second part is uncorrelated
We can estimate the first part 45