{"id":184,"date":"2024-08-05T09:04:10","date_gmt":"2024-08-05T09:04:10","guid":{"rendered":"https:\/\/www.endogeneity.net\/?page_id=184"},"modified":"2025-04-10T08:27:50","modified_gmt":"2025-04-10T08:27:50","slug":"gaussian-copulas-approach","status":"publish","type":"page","link":"https:\/\/www.endogeneity.net\/?page_id=184","title":{"rendered":"Gaussian Copulas Approach"},"content":{"rendered":"

Implementing the Gaussian Copulas Approach<\/strong><\/p>\n

The Gaussian copulas approach addresses omitted variables, simultaneity, and measurement error with the assumption that a Gaussian copula removes the endogenous part of the predictor (Haschka 2022; Park and Gupta 2012, 2024). Sufficient sample size, enough variation in predictor, nonnormality of predictor (Becker, Proksch, and Ringle (2022), and normality of residual are required. First, the inverse normal of the cumulative density function for the predictor needs to be calculated. Second, the inverse normal of the cumulative density function is included as a control variable, and bootstrapping is used to calculate standard errors.<\/p>\n

The following code can be used to implement the gaussian copula approach from Park and Gupta (2012) in Stata<\/strong>.<\/p>\n

\/\/sort the data
sort Predictor<\/p>\n

\/\/generate empirical CDF (percentile rank)
gen C_Predictor = (_n – 0.5) \/ _N<\/p>\n

\/\/adjust values exactly at 0 or 1 to avoid issues with inverse normal
replace C_Predictor = 0.0000001 if C_Predictor == 0
replace C_Predictor = 0.9999999 if C_Predictor == 1<\/p>\n

\/\/build the copula term using the inverse normal (probit transformation)
gen C_Term = invnormal(C_Predictor)<\/p>\n

\/\/residualize the copula term if controls are present
regress C_Term Controls
predict C_Res, residuals<\/p>\n

\/\/estimate the regression with controls and obtain bootstrapped 95% standard errors based on 250 bootstrap samples
regress Outcome Predictor Controls C_Res, vce(bootstrap, reps(250) dots(1))<\/p>\n

 <\/p>\n

The following code can be used to implement the gaussian copula approach of from Park and Gupta (2012) in R<\/strong>.<\/p>\n

#load the boot package
library (boot)<\/p>\n

#building the copula correction term
Dataset$C_Function <- stats::ecdf(Predictor, data = Dataset)
Dataset$C_Predictor <- C_Function(Predictor)
Dataset$C_Predictor <- ifelse(C_Predictor==0, 0.0000001, C.function)a
Dataset$C_Predictor <- ifelse(C_Predictor==1, 0.9999999, C.function)a<\/p>\n

#building the copula correction term without controls in the model
Dataset$C_Term <- stats::qnorm(C_Predictor)<\/p>\n

#building the copula correction term if controls are in the model
Dataset$C_Res <- residuals(lm(C_Term ~ Controls, data = Dataset))<\/p>\n

#estimate the regression without controls
model_C <- lm(Outcome ~ Predictor + C_Term, data = Dataset)<\/p>\n

#estimate the regression with controls
model_C <- lm(Outcome ~ Predictor + Controls + C_Res, data = Dataset)<\/p>\n

#obtain bootstrapped 95% standard errors based on 250 bootstrap samples
results_model_C <- Boot(model_C, R=250)
summary(results_model_C)
confint(results_model_C, level=.95)<\/p>\n

 <\/p>\n

References<\/strong><\/p>\n

Becker, Jan-Michael, Dorian Proksch, and Christian M. Ringle (2022), \u201cRevisiting Gaussian Copulas to Handle Endogenous Regressors,\u201d Journal of the Academy of Marketing Science, 50 (1), 46-66.<\/p>\n

Haschka, Rouven E. (2022), \u201cHandling Endogenous Regressors using Copulas: A Generalization to Linear Panel Models with Fixed Effects and Correlated Regressors,\u201d Journal of Marketing Research, 59(4), 860-881.<\/p>\n

Park, Sungho, and Sachin Gupta (2012), \u201cHandling Endogenous Regressors by Joint Estimation using Copulas,\u201d Marketing Science, 31(4), 567-586.<\/p>\n

Park, Sungho, and Sachin Gupta (2024), \u201cA Review of Copula Correction Methods to Address Regressor\u2013Error Correlation,\u201d Impact at JMR. https:\/\/www.ama.org\/marketing-news\/a-review-of-copula-correction-methods-to-address-regressor-error-correlation\/<\/p>\n<\/div><\/div><\/div><\/div><\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"100-width.php","meta":{"footnotes":""},"class_list":["post-184","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.endogeneity.net\/index.php?rest_route=\/wp\/v2\/pages\/184","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.endogeneity.net\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.endogeneity.net\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.endogeneity.net\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.endogeneity.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=184"}],"version-history":[{"count":2,"href":"https:\/\/www.endogeneity.net\/index.php?rest_route=\/wp\/v2\/pages\/184\/revisions"}],"predecessor-version":[{"id":243,"href":"https:\/\/www.endogeneity.net\/index.php?rest_route=\/wp\/v2\/pages\/184\/revisions\/243"}],"wp:attachment":[{"href":"https:\/\/www.endogeneity.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=184"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}