Business, Sauder School of
Marketing, Division of
DSpace
UBCV
Yang, Fan
Doctor of Philosophy - PhD
University of British Columbia
A prominent challenge when drawing causal inference using observational data is the ubiquitous presence of endogenous regressors. This dissertation investigates causal inference and endogeneity correction, both in methodology development and empirical analysis. The first essay (in Chapter 2) develops a new instrument-free method using copula to address the endogeneity problem. The classical econometric method to handle regressor endogeneity requires instrumental variables that must satisfy the stringent condition of exclusion restriction. We use the statistical tool copula to directly model the dependence among the regressors and the error term, and abstract information from existing regressors as a generated regressor added to the outcome regression. Our proposed 2sCOPE method extends the existing copula method to a more general setting by allowing (nearly) normal endogenous regressors and correlated exogenous regressors, and is straightforward to use and broadly applicable. We theoretically prove the consistency and efficiency of 2sCOPE, and demonstrate the performance of 2sCOPE via simulation studies and an empirical application. The second essay (in Chapter 3) further studies the causal inference and endogeneity correction methods for high-dimensional data. The more and more common high-dimensional data in the current big data era make the classical causal inference methods suffer finite-sample bias, inefficiency, or even fail to work when the dimension is larger than the sample size. In this essay, we extend the 2sCOPE method developed in Chapter 2 to the high-dimensional setting, and propose a lasso-based 2sCOPE method. We demonstrate the performance of the proposed method via simulation studies and an empirical application. The third essay (in Chapter 4) empirically studies vertical differentiation in two-sided markets, where network size plays the central role. Vertical differentiation is a common strategy in one-sided markets, but whether it is profitable for two-sided platforms is hard to say because of the network effect. In this essay, I take advantage of a unique data set from a leading ride-hailing platform, and develop a structural simultaneous demand and supply model to quantify network externalities. The result shows that besides the product intrinsic value, network value is crucial in determining the degree of product differentiation in two-sided markets.
Essays of Instrument-free Causal Inference, MachineLearning and Marketing ApplicationsbyFan YangB.S., Sun Yat-sen University, 2013Mphil, Hong Kong University of Science and Technology, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Business Administration - Marketing)The University of British Columbia(Vancouver)July 2022© Fan Yang, 2022The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the thesis entitled:Essays of Instrument-free Causal Inference, Machine Learning and Mar-keting Applicationssubmitted by Fan Yang in partial fulfillment of the requirements for the degree of Doctorof Philosophy in Business Administration - Marketing.Examining Committee:Professor Yi Qian, Sauder School of Business, The University of British ColumbiaSupervisorProfessor Joey Hoegg, Sauder School of Business, The University of British ColumbiaSupervisory Committee MemberProfessor Ralph Winter, Sauder School of Business, The University of British ColumbiaSupervisory Committee MemberProfessor Hui Xie, Health Sciences, Simon Fraser UniversitySupervisory Committee MemberProfessor Charles Weinberg, Sauder School of Business, The University of British ColumbiaUniversity ExaminerProfessor Kevin Song, Vancouver School of Economics, The University of British ColumbiaUniversity ExamineriiAbstractA prominent challenge when drawing causal inference using observational data is the ubiq-uitous presence of endogenous regressors. This dissertation investigates causal inferenceand endogeneity correction, both in methodology development and empirical analysis.The first essay (in Chapter 2) develops a new instrument-free method using copula toaddress the endogeneity problem. The classical econometric method to handle regressorendogeneity requires instrumental variables that must satisfy the stringent condition of ex-clusion restriction. We use the statistical tool copula to directly model the dependenceamong the regressors and the error term, and abstract information from existing regressorsas a generated regressor added to the outcome regression. Our proposed 2sCOPE methodextends the existing copula method to a more general setting by allowing (nearly) normalendogenous regressors and correlated exogenous regressors, and is straightforward to useand broadly applicable. We theoretically prove the consistency and efficiency of 2sCOPE,and demonstrate the performance of 2sCOPE via simulation studies and an empirical appli-cation.The second essay (in Chapter 3) further studies the causal inference and endogeneity cor-rection methods for high-dimensional data. The more and more common high-dimensionaldata in the current big data era make the classical causal inference methods suffer finite-sample bias, inefficiency, or even fail to work when the dimension is larger than the sampleiiisize. In this essay, we extend the 2sCOPE method developed in Chapter 2 to the high-dimensional setting, and propose a lasso-based 2sCOPE method. We demonstrate the per-formance of the proposed method via simulation studies and an empirical application.The third essay (in Chapter 4) empirically studies vertical differentiation in two-sidedmarkets, where network size plays the central role. Vertical differentiation is a commonstrategy in one-sided markets, but whether it is profitable for two-sided platforms is hardto say because of the network effect. In this essay, I take advantage of a unique data setfrom a leading ride-hailing platform, and develop a structural simultaneous demand andsupply model to quantify network externalities. The result shows that besides the productintrinsic value, network value is crucial in determining the degree of product differentiationin two-sided markets.ivLay SummaryCausal inference and endogeneity correction are central to social science research. Forobservational data, there are mainly three streams of approaches to correct endogeneity: in-strument variable approach, structural model approach, and instrument-free approach. Re-cently, there is a growing interest in instrument-free methods because of its simplicity thatno instruments are needed.This thesis investigates different endogeneity-correction methods in both methodol-ogy development and empirical applications. In Essay 1, I develop a new instrument-freemethod, called 2sCOPE, which is straightforward to implement and can greatly broaden theapplicability of the instrument-free methods for dealing with endogeneity issues in practice.In Essay 2, I further adapt the 2sCOPE method for high-dimensional data by combining itwith machine learning techniques. Finally, I empirically study consumer behavior in thegrowing two-sided platforms, and bring insights and suggestions for two-sided platformsbased on the estimation results using classical methods for causal inference in Essay 3.vPrefaceI am the primary author of the work presented in this Ph.D. thesis. I was responsible foridentifying the research questions, conducting the literature review, collecting and managingdata, analyzing the data, modeling and coding the estimation procedures, and preparing themanuscript. Specific contributions for each chapter are described below.Chapter 1: IntroductionI am the primary author of this chapter, with intellectual contributions from Yi Qian andHui Xie.Chapter 2 (Essay 1): Addressing Endogeneity using a Two-stage Copula GeneratedRegressor ApproachI was primarily responsible for identifying the research question, reviewing the litera-ture, conducting simulation studies, cleaning the data, developing and estimating the model,and preparing the manuscript. Yi and Hui contributed to the model development, manuscriptediting, and identifying and positioning the research question.Chapter 3 (Essay 2): Lasso-based Instrument-free Casual InferenceI was primarily responsible for identifying the research question, reviewing the liter-viature, conducting simulation studies, organizing the data, developing and estimating themodel, and preparing the manuscript. Yi and Hui contributed to the model development,manuscript editing and identifying and positioning the research question.Chapter 4 (Essay 3): Vertical Differentiation in two-sided marketsI was responsible for identifying the research question, organizing the literature, manag-ing the data, developing and estimating the model, conducting counterfactual analysis, andpreparing the manuscript.Chapter 4: ConclusionI am the primary author of this chapter, with intellectual contributions from Yi Qian andHui Xie.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Addressing Endogeneity Using a Two-stage Generated Regressor Approach . 62.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.1 Assumptions in the Existing Copula Endogeneity-Correlation Method(CopulaOrigin) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.2 Proposed Method: Two-stage Copula Endogeneity-correction(2sCOPE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.3 Multiple Endogenous Regressors . . . . . . . . . . . . . . . . . . . 282.3.4 2sCOPE for Random Coefficient Linear Panel Models with Endoge-nous Regressors . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.5 2sCOPE for Slope Endogeneity and Random Coefficient Logit Model 332.4 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.1 Case 1: Non-normal Regressors . . . . . . . . . . . . . . . . . . . 342.4.2 Case 2: Normal Regressors . . . . . . . . . . . . . . . . . . . . . . 372.4.3 Case 3: Performance Under Insufficient Non-Normality of Endoge-nous Regressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39viii2.4.4 Case 4: Multiple Endogenous Regressors . . . . . . . . . . . . . . 422.4.5 Case 5: Multiple Exogenous Control Covariates . . . . . . . . . . . 432.4.6 Case 6: Random Coefficient Linear Panel Model . . . . . . . . . . 452.4.7 Misspecification of the Error ξt . . . . . . . . . . . . . . . . . . . . 462.5 Empirical Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.6 Economic Intuition and Practical Guidance of 2sCOPE . . . . . . . . . . . 542.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 Lasso-based Instrument-free Causal Inference . . . . . . . . . . . . . . . . . 603.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.3.1 Copula Endogeneity Correction Method (2sCOPE) in Standard Case 673.3.2 2sCOPE with Lasso and post-Lasso . . . . . . . . . . . . . . . . . 713.3.3 2sCOPE with De-biased Lasso . . . . . . . . . . . . . . . . . . . . 733.4 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.5 Empirical Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814 Vertical Differentiation in Two-sided Markets: Evidence from A Ride-hailingPlatform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.3 Data and Model-Free Evidence . . . . . . . . . . . . . . . . . . . . . . . . 924.3.1 Data Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.3.2 Variables Related to Network Externalities . . . . . . . . . . . . . . 944.3.3 Data Evidence in Riders’ Choice . . . . . . . . . . . . . . . . . . . 974.3.4 Data Evidence in Drivers’ Behavior . . . . . . . . . . . . . . . . . 984.4 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.4.1 Rider’s Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.4.2 Driver’s Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.4.3 Hierarchical model . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.4.4 3SLS/2SLS and Control Function Approach . . . . . . . . . . . . . 1064.5 Estimation Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.5.1 Estimation for Network Externalities . . . . . . . . . . . . . . . . . 1094.5.2 Reduced-form Estimation . . . . . . . . . . . . . . . . . . . . . . . 1104.5.3 MCMC Estimation Results for Heterogeneous Choice Model . . . . 1124.5.4 Visualization of MCMC Estimates . . . . . . . . . . . . . . . . . . 1144.6 Platform Pricing Policy and Counterfactual Analysis . . . . . . . . . . . . 1164.6.1 Platform’s Current Pricing and Subsidy Policy . . . . . . . . . . . . 1174.6.2 Counterfactual Analysis . . . . . . . . . . . . . . . . . . . . . . . 1184.6.3 Managerial Implications . . . . . . . . . . . . . . . . . . . . . . . 1214.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125ixBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129A Chapter 2 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138A.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138A.2 Assumption 4.b in CopulaP&G . . . . . . . . . . . . . . . . . . . . . . . . 139A.3 COPE Method Development . . . . . . . . . . . . . . . . . . . . . . . . 139A.4 Proof of Theorem A1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143A.5 Proof of Theorem 2: Consistency of 2sCOPE . . . . . . . . . . . . . . . . 145A.6 Proof of Theorem 3: Nonnormality Assumption Relaxed . . . . . . . . . . 147A.7 Proof of Theorem 4: Variance Reduction . . . . . . . . . . . . . . . . . . . 148A.8 2sCOPE for Slope Endogeneity with Correlated and Normally DistributedRegressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150A.9 2sCOPE for Random Coefficient Logit Model with Correlated and Nor-mally Distributed Regressors . . . . . . . . . . . . . . . . . . . . . . . . . 152A.10 Additional Results for Smaller Sample Size . . . . . . . . . . . . . . . . . 154A.11 Misspecification of ξt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155A.12 Misspecification of Copula . . . . . . . . . . . . . . . . . . . . . . . . . . 155B Chapter 4 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158B.1 Quality Difference Between Premium and Standard . . . . . . . . . . . . . 158B.2 Data Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159B.2.1 Rider’s Expected Waiting Time . . . . . . . . . . . . . . . . . . . . 159B.2.2 Driver’s Cruising Time . . . . . . . . . . . . . . . . . . . . . . . . 162B.2.3 Construction for Outside Demand and Counterfactual Options . . . 163B.3 Bayesian MCMC Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 165xList of TablesTable 2.1 A Comparison of Copula Methods . . . . . . . . . . . . . . . . . . . . 11Table 2.2 Summary of Assumptions for the Three Methods . . . . . . . . . . . . . 30Table 2.3 Estimation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Table 2.4 Results of the Simulation Study Case 1: Non-normal Regressors . . . . . 35Table 2.5 Results of Case 2: Normal Regressors . . . . . . . . . . . . . . . . . . . 38Table 2.6 Results of the Simulation Study Case 3: Multiple Endogenous Regressors 43Table 2.7 Results of the Simulation Study Case 4: Multiple Exogenous ControlCovariates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Table 2.8 Results of the Simulation Study Case 5: Random Coefficient LinearPanel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Table 2.9 Results of the Simulation Study Case D1: Misspecification of ξt . . . . . 48Table 2.10 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Table 2.11 Estimation Results: Toothpaste Sales . . . . . . . . . . . . . . . . . . . 52Table 3.1 Simulation Results of Lasso-based 2sCOPE . . . . . . . . . . . . . . . 76Table 3.2 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Table 3.3 Estimation Results: How COVID-19 affects Happiness . . . . . . . . . . 80Table 4.1 Price Structure for Rider . . . . . . . . . . . . . . . . . . . . . . . . . . 93Table 4.2 Fare Income Structure for Driver . . . . . . . . . . . . . . . . . . . . . 94Table 4.3 Summary statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Table 4.4 How Network Externalities Influence Waiting/Cruising Time . . . . . . 110Table 4.5 Reduced-Form Estimation Results for Rider and Driver . . . . . . . . . 111Table 4.6 MCMC Estimation Results for Demand and Supply . . . . . . . . . . . 113Table 4.7 Algorithm to Solve Fixed Points . . . . . . . . . . . . . . . . . . . . . . 118Table 4.8 Counterfactual Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Table A.1 Results of the Simulation Study for Case 1 with Sample Size of 200 . . 154Table A.2 Results of the Simulation Study: Misspecification of ξt (Beta(4,4)) . . . 155Table A.3 Results of the Simulation Study: Misspecification of ξt (t(5)) . . . . . . 155Table A.4 Results of the Simulation Study Case D2: Misspecification of Copula . . 157Table B.1 Quality Difference Between the Two Types . . . . . . . . . . . . . . . . 159xiList of FiguresFigure 2.1 Estimation Bias for Different Distributions of Endogenous Regressor. . 41Figure 2.2 Log Sales and Log Retail Price of Toothpaste in Store 1. . . . . . . . . 50Figure 2.3 Histogram of Log Retail Price, Bonus and Price Reduction in Store 1 . 50Figure 3.1 Relationship Between Happiness and Stringency Index. . . . . . . . . . 78Figure 3.2 Histogram of Stringency, Happiness, and Country Characteristics in 2020 79Figure 4.1 Manhattan Neighborhoods. . . . . . . . . . . . . . . . . . . . . . . . . 95Figure 4.2 Choice Occasion between The Two Types. . . . . . . . . . . . . . . . . 98Figure 4.3 Relationship Between Market Share and Waiting Time. . . . . . . . . . 99Figure 4.4 Drivers’ Average per-minute Payment from Realized Trips. . . . . . . . 99Figure 4.5 Relationship Between Drivers’ Acceptance Rate and Cruise Time. . . . 100Figure 4.6 Relationship Between Drivers’ Acceptance Rate and Demand Size. . . . 101Figure 4.7 Distribution of Estimated Riders’ Heterogeneous Coefficients . . . . . . 115Figure 4.8 Average Predicted Reservation Utility by Hour. . . . . . . . . . . . . . 115Figure A.1 Scatter plots of Randomly Generated Pairs Up,Uw (Up,Uξ ) for Consid-ered Copulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Figure B.1 Waiting Time Distribution. . . . . . . . . . . . . . . . . . . . . . . . . 161xiiAcknowledgmentsI am deeply grateful to my advisor Dr. Yi Qian, who has taken a tremendous amount oftime and efforts in advising me and provided generous support for my doctoral study, andhas great expertise, patience and kindness in guiding my research. I am deeply honored tohave been her student. I would also like to thank my committee member Dr. Hui Xie for hisadvice and helpful discussions of my research projects. I would like to express my sinceregratitude to my supervisory committee members Dr. Joey Hoegg, and Dr. Ralph Winter fortheir constructive and insightful suggestions and advice on my research, for their kindnessand support, and for spending their valuable time serving on my committee. I would alsolike to express my sincere thankfulness to Dr. Charles Weinberg in our division for all hisenlightening advice and suggestions on my research.I extend my gratitude to faculty in the Marketing division and all other faculty memberswho have been very generous in providing support and guidance. I am also delighted thatI had such an incredible team of Ph.D. students with whom I can talk to and spend timewith. And I want to thank Ms. Elaine Cho and Ms. Florence Yen for their extraordinaryresponsiveness and approachability.Finally, I am deeply grateful to my family. They have always been supporting my pursuitof education all the way unconditionally. This long journey of graduate study would nothave been possible without their love and support.xiiiDedicationThis thesis is dedicated to my parents Fuping Yang and Liumei Wang, and to my sisters LiuYang and Xiu Wang.xivChapter 1IntroductionCausal inference is central to social science research and business practitioners. Peopleare always interested in causal effect instead of just simple descriptive correlation. Nowa-days, the rapidly available observational data and technological innovations in this digitalera make causal inference even more important to provide real-world evidence for betterdecision makings. However, one challenge for empirical researchers to draw valid causal in-ference from observational data is the presence of regressor endogeneity problem, in whichregressors are correlated with the (unobserved) structural error.In the literature, there are mainly three streams of approaches for endogeneity correctionusing observational data: instrumental variable (IV) approach, structural model approach,and instrument-free approach. The first approach, the IV approach, is to find a good IVthat satisfies two conditions: it is strongly correlated with the endogenous regressor via anexplainable and validated relationship (i.e., relevance restriction), yet uncorrelated with thestructural error (i.e., exclusion restriction). Once a good IV can be found, it can abstractthe exogenous part from the endogenous regressor and draw correct causal inference usingthe two-stage least square method. But empirical researchers usually face the challenge1of finding good IVs in practice. When using the IV approach to correct endogenous, re-searchers had better explain the validity of the IV, and show evidence of high relevance withthe endogenous regressor.The second stream of approach, the structural model approach, is to specify the eco-nomic structure that generates the observational data, including the endogenous regressors.One example is to build up the simultaneous structural supply-demand model that generatesthe endogenous marketing-mix variables. Doing so allows researchers not only to recoverparameters of interest and make causal inferences, but also to perform counterfactual analy-sis and provide managerial insights. One concern with this approach is that the performancehighly depends on model assumptions of supply side. When using this approach, the moreinformation of the supply side, the better for model identification.The third stream of approach, the instrument-free approach, has aroused increasing in-terest recently. This approach addresses the endogeneity problem by using information inthe existing variables, and the advantage is that no extra instruments are needed. One ex-ample is the copula method proposed by Park and Gupta (2012). They use the statisticaltool copula to directly model the dependence between the endogenous regressor and theerror term, and construct a generated regressor from the existing regressors to correct endo-geneity. Intuitively, they use copula to split the error term into an endogenous part and anexogenous part, and correct endogeneity by directly controlling for the endogenous part.In this thesis, I investigate different endogeneity-correction methods to draw causal in-ference. Specifically, I conduct three independent studies, two of which are about methodol-ogy development of causal inference methods, and one is an empirical application applyingthe classical causal inference methods. On the one hand, the increasingly available obser-vational data (with more complicated structure e.g., high-dimension) in this digital big dataera bring new challenges for traditional methods to draw causal inference and demand the2development of methodologies. Meanwhile, the interaction among different disciplines andknowledge combination provide researchers with fertile soil (new opportunities) for newcreative method development to address economic problems.In the first essay, I develop a new instrument-free approach using the statistical toolcopula. I extend the existing copula methods (Park and Gupta 2012, Haschka 2021) to amore general setting with much weaker assumptions. Specifically, the proposed two-stagecopula endogeneity correction (2sCOPE) method allows close-to-normal or even normalendogenous regressors, and can include exogenous regressors that are correlated with theendogenous regressor to the model. Simulation result shows that the existing copula cor-rection methods can yield biased estimates under the setting of endogenous regressors thathave insufficient non-normality or are correlated with exogenous regressors. The proposed2sCOPE can provide unbiased estimates, and can even improve estimation efficiency byup to 50%. Moreover, the 2sCOPE method uses the control function approach by addinga generated regressor derived from existing regressors to control for endogeneity, and thusis straightforward to use and broadly applicable. Overall, 2sCOPE method can greatly in-crease the ease of and broaden the applicability of instrument-free methods for dealing withregressor endogeneity in practice.In the second essay, I further adapt the instrument-free method, 2sCOPE method, pro-posed in the first essay for high-dimensional data by combining the causal inference methodwith machine learning techniques. Nowadays, with the fast growth of internet and techno-logical innovations, high-dimensional data are available everywhere (e.g., census and surveydata, complicated network data in online platforms, text, pictures, videos, genetic data etc.).I propose a lasso-based 2sCOPE method by combining the 2sCOPE method with lasso-based feature selection methods to deal with high-dimensional data for causal inference.The simulation result shows that there is a large bias of 2sCOPE method alone without fea-ture selection in high-dimensional setting, while the proposed lasso-based 2sCOPE method3gains substantial improvement in both estimation accuracy and efficiency (around 50%). Ialso apply the proposed method to an interesting data application. Specifically, I examinehow the government’s policy strictness during COVID-19 period affects citizens’ happiness.By using the lasso-based 2sCOPE method, I get a robust and more significantly negative ef-fect of policy stringency on people’s happiness, compared with the estimate using regular2sCOPE without feature selection.On the other hand, the recent technological innovations and fast growth of online andmobile platforms bring new business models that can fundamentally change consumer be-havior. These changes bring new opportunities and perspectives in understanding con-sumers. In the third essay, I empirically apply the traditional causal inference methods tostudy consumer behavior in the emerging two-sided markets (e.g., Taobao, Amazon, Airbnb,Uber, Lyft, Meituan, DoorDash). Specifically, I use structural model and IV approach tocorrect the endogeneity of network size in examining vertical differentiation in two-sidedmarkets. Network externalities, the distinct feature in two-sided markets, make the conven-tional vertical differentiation strategy more complicated. It on the one hand makes firmsbetter off in market expansion, but on the other hand further segments the market and limitsthe positive network effects, weakening the benefit of market expansion. Thus, understand-ing and quantifying the economic impact of network externalities in two-sided markets isof great importance. In this essay, I take advantage of a distinct data set from a leadingride-hailing platform in New York City, and develop a simultaneous structural demand andsupply model to accommodate both the riders’ and drivers’ decisions and quantify networkexternalities. The result shows that both the product intrinsic value and the network valueare important in determining the degree of product differentiation for two-sided platforms.The chapters proceed as follows. In the first essay, we address endogeneity using atwo-stage generated regressor approach in Chapter 2. We provide the study overview inSection 2.1, summarize the relevant literature in Section 2.2, develop a method in Section42.3, conduct simulation studies in Section 2.4 and empirical application in Section 2.5. Wefurther discuss the economic intuition and scope in Section 2.6 and conclude this chapterin Section 2.7. In the second essay, we adapt the proposed method in the first essay to thehigh-dimensional setting by combining with machine learning methods in Chapter 3. Wediscuss the study motivation and overall results in Section 3.1 and review related literaturein Section 3.2. We develop the method in Section 3.3, and conduct simulation study inSection 3.4 and data application in Section 3.5. We further conclude the chapter in Section3.6. In the third essay, we investigate vertical differentiation in two-sided markets in Chapter4. We provide the study overview in Section 4.1, review relevant literature in Section 4.2,and discuss used dataset and preliminary data evidence in Section 4.3. We further developthe structural model in Section 4.4, show the estimation results in Section 4.5, conductcounterfactual analysis in Section 4.6 and conclude the chapter in Section 4.6. Chapter 5gives a brief conclusion to the thesis.5Chapter 2Addressing Endogeneity Using aTwo-stage Generated RegressorApproach2.1 IntroductionCausal inference is central to many problems faced by academics and practitioners, andbecomes increasingly important as rapidly-available observational data in this digital erapromise to offer real-world evidence on cause-and-effect relationships for better decisionmakings. However, a prominent challenge faced by empirical researchers to draw validcausal inferences from these data is the presence of endogenous regressors that are corre-lated with the structural error in the population regression model representing the causalrelationship of interest. For example, omitted variables such as ability would cause endo-geneity of schooling when examining schooling’s effect on wages (Angrist and Krueger1991). Simultaneous equations models of supply and demand can also be subject to the6regressor endogeneity issue because both the supply and the demand are influenced by un-observed omitted common shocks.Regressor endogeneity poses great empirical challenges to researchers and demandsspecial handling of the issue in order to draw valid causal inferences. One classical methodto deal with the endogeneity issue is using instrumental variables (IV). The ideal IV has tomeet two requirements: it is correlated with the endogenous regressor via an explainableand validated relationship (i.e., relevance restriction), yet uncorrelated with the structuralerror (i.e., exclusion restriction). Although the theory of IVs is well-developed, researchersoften face the challenge of finding good IVs satisfying these two requirements. PotentialIVs often suffer from either weak correlation with endogenous regressors or challengingjustification for exclusion restriction, which hampers using IVs to correct for the underlyingendogeneity concerns (Rossi 2014).To address the lack of suitable IVs, there has been a growing interest in developingand applying IV-free endogeneity-correction methods. Several instrument-free approacheshave been developed, including identification via higher moments (Lewbel 1997, Ericksonand Whited 2002), heteroscedasticity (Rigobon 2003, Hogan and Rigobon 2003), and latentinstrumental variables (Ebbes et al. 2005). All three IV-free methods decompose the en-dogenous regressor into an exogenous part and an endogenous part. The assumption of theendogenous regressor containing an exogenous component is akin to the stringent conditionof exclusion restriction for IVs, and thus can be difficult to justify.Park and Gupta (2012) proposed an alternative instrument-free method that uses the cop-ula model (Danaher and Smith, 2011) to capture the regressor-error dependence. 1 Com-pared with the three IV-free methods above, their copula method does not impose the exo-1In statistics, a copula is a multivariate cumulative distribution function where the marginal distributionof each variable is a uniform distribution on [0,1]. Copulas permit modeling dependence without imposingassumptions on marginal distributions.7geneity assumption as it directly models the association between the structural error and theendogenous regressor via copula. Furthermore, the copula method can handle discrete en-dogenous regressors better than other IV-free methods. These features considerably increasethe feasibility of endogeneity correction, as evidenced by the rapidly increasing use of thecopula correction method (see examples of recent applications in the next section of theliterature review). However, similar to other IV-free methods, the copula correction methodalso requires the distinctiveness between the distributions of the endogenous regressor andthe structural error (Park and Gupta 2012). This means that the endogenous regressor isrequired to have a non-normal distribution for model identification with the commonly as-sumed normal structural error distribution. Furthermore, we show that the existing copulacorrection method implicitly requires all exogenous regressors to be uncorrelated with thelinear combination of copula transformations of endogenous regressors (henceforth referredto as copula control function (CCF)) used to control for endogeneity, and may yield signifi-cant bias when there are noticeable correlations between the CCF and exogenous regressors.In practical applications, both requirements of sufficient regressor non-normality andno correlation between CCF and exogenous regressors can be too strong, and pose sig-nificant challenges and limitations for applying the copula correction method. We oftenencounter endogenous regressors or include transformations of endogenous variables asregressors that have close-to-normality distributions. Examples of such regressors in eco-nomics and marketing management studies include stock market returns (Sorescu et al.,2017), corporate social responsibility (Eckert and Hohberger, 2022), the organizational in-telligent quotient (Mendelson, 2000), and the logarithm of price (see Figure 2.3 in Section5). Theoretically, the endogenous regressor and the structural error can contain a commonset of unobservables that collectively have a normal distribution, which can lead to a close-to-normal distribution of the endogenous regressor. In these situations, even if the model isidentified asymptotically, close-to-normality of endogenous regressors can cause substan-8tial estimation bias even in moderate sample size and can require a very large sample size(> 2000) to mitigate the finite-sample bias (Becker et al., 2021). Correlations between theCCF and exogenous regressors are also quite common in practical applications, especiallywhen the exogenous regressors are included to control for observed confounders. Examplesof such exogenous control variables abound in marketing and management studies, suchas customer-specific variables (age, household size, income, past purchase behaviors, etc.)when estimating the returns of consumer targeting strategies on product sales (Papies et al.,2017) and firms’ similarity when estimating the effect of competition on innovation (Aghionet al., 2005). Although regressor normality or insufficient regressor non-normality leads tomore severe identification issues, including model non-identifiability and poor finite sampleperformance (Table 2.1), correlations between CCF and exogenous regressors may occurmore frequently than close-to-normality of endogenous regressors. Thus, we consider thetwo requirements of sufficient regressor non-normality and no correlation between CCF andexogenous regressors as being equally stringent, which call for more general and flexiblecopula correction methods that relax both requirements.In this paper, we develop a generalized two-stage copula endogeneity correction method,denoted as 2sCOPE, that relaxes the above two requirements. Similar to the existing cop-ula method (Park and Gupta 2012, denoted as CopulaOrigin), 2sCOPE requires neither IVsnor the assumption of exclusion restriction. It corrects endogeneity by adding residuals ob-tained from regressing the latent copula data for each endogenous regressor on the latentcopula data for the exogenous regressors as generated regressors in the structural regres-sion model. To demonstrate the benefits of 2sCOPE, we also consider as a benchmarkmethod, denoted as COPE, which is a direct extension of CopulaOrigin and corrects endo-geneity by adding latent copula data themselves as generated regressors. The proposed2sCOPE method is straightforward to use, and it overcomes the above two key limitationsof CopulaOrigin as shown in Table 2.1. CopulaOrigin can be viewed as a special case of the92sCOPE. Importantly, we prove that the 2sCOPE can identify causal effects under muchweaker assumptions than CopulaOrigin, as summarized in Table 2.1.The contributions of this work are three folds. First, to the best of our knowledge,this work is the first in the literature to provide formal proofs of the theoretical properties ofcopula correction methods, along with clearly defined assumptions required for causal effectidentification (Table 2.1). These theoretical results are much needed because model identi-fiability is central to addressing the endogeneity issue, and timely given the rapid adoptionof copula correction methods in marketing research and elsewhere. Recent methodologicalwork notes lacking rigorous proofs of required model identification conditions and estima-tion properties (consistency and efficiency) as one main weakness of existing copula correc-tion methods (Haschka, 2021) 2, and calls for further studies of their theoretical properties(Becker et al., 2021). The theoretical results presented in this work fill in this importantknowledge gap, and contribute to a better understanding of the properties of the copulacorrection methods and guidance of their practical use.Two novel theoretical findings emerge from this study. First, we identify an implicit as-sumption required for CopulaOrigin to yield consistent estimation, and provide conditions toverify this implicit assumption to ensure consistent causal-effect estimation. This helps im-prove the effectiveness of the rapidly adopted method for addressing the endogeneity issue.An important result is that the existence of the correlations between endogenous and exoge-nous regressors alone does not automatically introduce bias to and invalidate CopulaOrigin.Instead, we show that the implicit assumption is the uncorrelatedness of the exogenous re-gressors with the CCF, the linear combination of copula transformations of endogenousregressors used to control for endogeneity. The difference between the implicit assumption2Owing to the complex form of the estimation method, Haschka (2021) notes the lack of theoretical proofsof required model identification conditions and estimation consistency as one limitation of the copula cor-rection method developed there, and thus has to rely solely on simulation studies to evaluate its empiricalproperties.10Features CopulaOrigin Haschka (2021) 2sCOPENonnormality ofRequired Required Not Required2Endogenous Regressors1No CorrelatedRequired (implicit) Not Required Not RequiredExogenous Regressors3Intercept Included4 YES NO5 YESTheoretical Proof YES NO YESEstimation Method Control Function MLE Control Function& MLEStructural Model Linear Regression LPM-FE Linear RegressionRCL LPM-FE, LPM-RE, LPM-MESlope Endogeneity RCL, Slope EndogeneityTable 2.1: A Comparison of Copula MethodsNote: 1: When required, normality of any endogenous regressor leads to non-identifiable models. Insufficientnon-normality of endogenous regressors can also cause poor finite sample performance (finite sample biasand large standard errors) and require extremely large sample size to perform well.2: Non-normality of endogenous regressors is not required as long as at least one correlated exogenousregressor is not normally distributed.3: In our paper, correlated exogenous regressors refer to those exogenous regressors correlated with the CCF(copula control function) used to control for endogeneity.4: Becker et al. (2021) shows the significance of including intercept in marketing applications, and theproblem of adding intercept using the copula method CopulaOrigin (Park and Gupta 2012).5: The approach cannot estimate the intercept term, which is removed from the panel model prior toestimation using first-difference or fix-effects transformation (Web Appendix A.8 of Haschka (2021)).LPM: Linear Panel Model; FE: Fixed Effects for individual-specific intercepts with common slopecoefficients; RE: Random Effects; ME: Mixed-Effects (including both fixed-effects and random coefficients);RCL: Random Coefficient Logitand the condition of zero pairwise correlations between endogenous and exogenous regres-sors can be substantial, especially with multiple endogenous regressors. 3 We prove that thenew 2sCOPE method yields consistent causal-effect estimates when the implicit assumptionabove is violated, which we show can cause biased causal effect estimates for CopulaOrigin.The second novel finding of our theoretical investigation is as follows. Although theexogenous regressors being correlated with the CCF require special handling for consistent3Although Haschka (2021) explains why correlated regressors can cause potential bias for CopulaOrigin,no condition of when bias can occur is given. Specifically, it is possible that with multiple endogenous re-gressors, the CCF is uncorrelated with exogenous regressors when pairwise correlations between endogenousand exogenous regressors are non-zeros. Even if there is only one endogenous regressor and CCF reducesto be proportional to the copula transformation of the endogenous regressor, the correlation coefficient is notinvariant to nonlinear transformations and thus changes after the copula transformation of the endogenousregressor.11causal-effect estimation, they can be beneficial as well by providing additional informationto help relax model identification requirements. They could help address the problem ofinsufficient regressor non-normality, and sharpen model estimates. Furthermore, we provethat when both COPE and 2sCOPE methods yield consistent estimates, 2sCOPE improvesthe efficiency (i.e., precision) of the structural model estimation by exploiting the correla-tions between the endogenous and exogenous regressors. The efficiency gain is substantialand can be up to ∼50% in our empirical evaluation, meaning that the sample size can bereduced by ∼50% to achieve the same estimation efficiency as compared with the COPEmethod, which does not exploit the correlations between endogenous and exogenous regres-sors.Second, the proposed 2sCOPE method is the first copula-correction method that simul-taneously relaxes the non-normality assumption of endogenous regressors and handles cor-related endogenous and exogenous regressors (Table 2.1). Except for Haschka (2021), ex-isting copula correction methods do not consider correlated endogenous and exogenous re-gressors, and are subject to potential bias if this correlation is present. Haschka (2021) gen-eralized Park and Gupta (2012) to fixed-effects linear panel models with correlated regres-sors by jointly modeling the structural error, endogenous and exogenous regressors usingcopulas and maximum likelihood estimation (MLE). However, as noted in Haschka (2021),Haschka’s approach still requires the non-normality of endogenous regressors. Thus, allexisting copula correction methods require sufficient non-normality assumption of endoge-nous regressors for model identification (Park and Gupta, 2012; Haschka, 2021; Beckeret al., 2021; Eckert and Hohberger, 2022); even when the model is identified, insufficientregressor non-normality can cause significant finite sample bias in the sample size of lessthan 2,000 (Haschka, 2021; Becker et al., 2021; Eckert and Hohberger, 2022). Becker et al.(2021) suggested a minimum absolute skewness of 2 for an endogenous regressor in orderto ensure good performance of Gaussian copula correction methods in sample size as small12as 200 (Figure 8 in Becker et al. 2021). These strong requirements can significantly limitthe use of copula correction methods in practical applications, despite the rapid adoption ofthese methods recently. Our proposed 2sCOPE method overcomes these important limita-tions of existing copula correction methods. First, we prove that the structural model withnormally distributed endogenous regressors can be identified using the 2sCOPE method aslong as one of the exogenous regressors correlated with endogenous ones is nonnormallydistributed, which is considerably more feasible in many practical applications. Second,consistent with the above theoretical result, our evaluation in Section 2.4.3 demonstratessuperior finite-sample performance of 2sCOPE and shows that 2sCOPE eliminates or sub-stantially reduces the significant finite sample bias problem due to insufficient regressornon-normality raised in Becker et al. (2021) and Eckert and Hohberger (2022). Overall,the proposed 2sCOPE method can greatly broaden the applicability of the instrument-freemethods for dealing with endogeneity issues in practice, as seen in Table 2.1.Finally, 2sCOPE employs generated regressors to address endogeneity. By includinggenerated regressors in the structural model to control endogeneity, 2sCOPE enjoys severalbenefits associated with using a control function (Heckman and Robb Jr 1985, Blundell andPowell 2003, Blundell and Powell 2004, Petrin and Train 2010, Blundell and Matzkin 2014,Wooldridge 2015) to address endogeneity, such as incurring little extra computational andmodeling burden to address endogeneity, broader applicability with much weaker assump-tions, and increased robustness to model misspecifications 4. The vast majority of applica-tions of the existing copula correction method have used the generated-regressor approach(Becker et al., 2021). We demonstrate that 2sCOPE retains these desirable properties of thecontrol function approach for a range of commonly used models in marketing studies, asshown in Table 2.1, while relaxing the two key assumptions of CopulaOrigin: regressor non-normality and uncorrelatedness between CCF and exogenous regressors. In many of these4As shown in Becker et al. (2021), Gaussian copula control function approach is more robust against errorterm misspecifications than the Gaussian copula MLE approach. Note that when we compare MLE withcontrol function approach, we only consider the estimation method dimension.13models, the MLE approach becomes much more difficult or computationally infeasible,but our 2sCOPE approach is straightforward. Section 2.3.4 presents an example in whichextending the MLE approach of Haschka (2021) to random coefficient linear panel mod-els with correlated endogenous and exogenous regressors requires numerically evaluatingpotentially high-dimensional integrals of complicated functions of the random effect distri-butions, whereas 2sCOPE eliminates the need to evaluate these high-dimensional integralsand can be implemented using standard software programs for random coefficient linearpanel models assuming all regressors are exogenous. Furthermore, the generated-regressorapproach facilitates studying the theoretical properties of the proposed 2sCOPE procedureand the comparison of these procedures. In this work, we provide theoretical proofs for theimplicit assumption needed to ensure consistency of CopulaOrigin, and the consistency andefficiency comparison for the proposed 2sCOPE under correlated regressors and normallydistributed regressors.The remainder of this paper unfolds as follows. Section 2.2 reviews the related lit-erature on methods for causal inference with endogenous regressors. In Section 2.3, weshow the implicit assumption of CopulaOrigin. We then propose the 2sCOPE method, pro-viding theoretical proofs for the consistency of the proposed 2sCOPE method as well asfor efficiency gain and model identifiability with normally distributed regressors under the2sCOPE method. We also summarize the estimation procedure of the proposed method. InSection 2.4, we evaluate the performance of our proposed 2sCOPE method using simula-tion studies and compare it with CopulaOrigin and its direct extension COPE under differentscenarios. In Section 2.5, e apply the proposed 2sCOPE method to estimate price elasticityusing store purchase databases in section 2.5. We discuss the economic intuition and scopeof the proposed 2sCOPE method in section 2.6, and conclude the paper in Section 2.7.142.2 Literature ReviewThe marketing, economic and statistics literature develops a rich set of methods to drawcausal inferences. The gold standard to estimate causal effects is using randomized as-signment2 such as controlled lab experiments and field experiments (Johnson et al. 2017,Anderson and Simester 2004, Godes and Mayzlin 2009). When controlled experiments arenot feasible, quasi-experimental designs such as regression discontinuity and difference indifferences are used to mimic randomized experiments and to enable the identification ofcausal effects with observational data (Hahn et al. 2001, Hartmann et al. 2011, Narayananand Kalyanam 2015, Athey and Imbens 2006, Shi et al. 2017). However, these quasi-experimental designs have special data and design requirements, and cannot cope with thegeneral issue of endogenous regressors when estimating causal effects using observationaldata.There is a large literature focusing on approaches to addressing endogenous regressorswhen inferring causal effects. Rutz and Watson (2019), Papies et al. (2017) and Park andGupta (2012) provided an overview of addressing endogeneity in marketing. Three broadclasses of solutions are discussed. The most commonly used solution is to find observedinstrumental variables to correct for endogeneity (Kleibergen and Zivot 2003, Qian 2008,Ataman et al. 2010, Van Heerde et al. 2013 and Novak and Stern 2009). Angrist and Krueger(2001) and Rossi (2014) provide a survey of literature that uses the instrumental variablesapproach. Rossi (2014) surveyed 10 years of publications in Marketing Science and Quanti-tative Marketing and Economics, which revealed that the most commonly used instrumentalvariables are lagged variables, costs, fixed effects and Hausman-style variables from othermarkets. However, the survey found that the strength of the instruments is rarely measuredand reported, which is needed to detect the weak instrument problem. Moreover, one gen-erally cannot test the exclusion restriction condition and verify the validity of instruments.15The survey also found that most papers lack a discussion of why the instruments used arevalid. In a word, though the theory of instrumental variables is well-developed, good instru-ments are difficult to find, making the IV approach hard to implement in practice. Studiesthat identify good instruments are subsequently highly valued.The second class of solutions to mitigate endogeneity is to specify the economic struc-ture that generates the observational data including endogenous regressors (e.g., a supply-side model for marketing-mix variables). Doing so allows researchers not only to recoverparameters of interest and make causal inferences, but also to perform counterfactual anal-ysis (Chintagunta et al. 2006). Some other examples of this approach in the marketing lit-erature are Berry (1994), Sudhir (2001), Dube´ et al. (2002), Yang et al. (2003), Sun (2005),Dotson and Allenby (2010) and Otter et al. (2011). The key concern with this approachis that the performance highly depends on model assumptions of the supply side. Incor-rect assumptions or insufficient information on the supply side can lead to biased estimates(Chintagunta et al. 2006, Hartmann et al. 2011)The third class of solutions in the domain of endogeneity correction is instrument-free methods. This is a more recent stream of methodological development. Three ex-tant instrument-free approaches are discussed in Ebbes et al. (2009): the higher moments(HM) approach (Lewbel 1997, Erickson and Whited 2002), the identification through het-eroscedasticity (IH) estimator (Rigobon 2003, Hogan and Rigobon 2003), the latent in-strumental variables (LIV) method (Ebbes et al. 2005). Recently Wang and Blei (2019)proposed a deconfounder approach that has some flavor to the LIV approach. All theseapproaches divide the endogenous regressor P into an endogenous and an exogenous part,P = f (Z)+ v, where f (Z) is treated as an exogenous random variable with unique struc-tures imposed for model identification in different methods. However, the assumption off (Z) being exogenous is hard to guarantee. Park and Gupta (2012) introduced anotherinstrument-free method that doesn’t require the exogeneity of f (Z). It directly models the16association between the structural error and the endogenous regressor via copula.The copula method has been rapidly adopted by researchers to deal with the endogeneityproblem because of its feasibility in that no instruments are needed. For example, the copulamethod has been used to study the effects of marketing activities such as promotion, adver-tising, and loyalty programs (Burmester et al. 2015, Datta et al. 2015, Gruner et al. 2019,Keller et al. 2019, Bombaij and Dekimpe 2020, Guitart et al. 2018, Lamey et al. 2018); tostudy product design and brand equity (Wetzel et al. 2018, Heitmann et al. 2020); to studysales force training (Atefi et al. 2018); to study healthy food consumption (Elshiewy andBoztug 2018). Haschka (2021) developed an MLE method that extends Park and Gupta(2012) to linear panel models with fixed-effect intercepts and constant slope coefficients inthe presence of correlated regressors. In our paper, we delineate the precise and verifiablecondition for CopulaOrigin to yield consistent estimates with correlated endogenous and ex-ogenous regressors. For the case when this condition fails, we develop a new two-stageendogeneity correction method using copula control functions (2sCOPE) that relaxes twokey assumptions imposed in Park and Gupta (2012): (1) all endogenous regressors musthave non-normal distributions and (2) exogenous regressors must be uncorrelated with theCCF used to control for the endogeneity. We provide proof of the theoretical propertiesof the proposed methods, including consistency and efficiency comparisons. We derive thenew procedures for a variety of types of structural models, including the random coefficientsmodels commonly used in marketing studies. As a result, the proposed 2sCOPE method isapplicable in more general settings with the capability to exploit exogenous regressors toimprove model identification and estimation.172.3 MethodsIn this section, we develop a copula-based instrument-free method to handle endogenousregressors when there exist exogenous regressors that are correlated with endogenous re-gressors. We first review the CopulaOrigin method in Park and Gupta (2012). We showthat CopulaOrigin implicitly assumes no correlations between the exogenous regressors andthe CCF, as well as how the violation of the assumption can cause bias in the structuralmodel parameter estimates for the current copula-based instrument-free method. Then wepresent a newly proposed method to deal with the problem and the detailed estimation pro-cedure. We also show how exogenous regressors correlated with endogenous regressors cansharpen structural model parameter estimates and permit the identification of the structuralmodel containing normally distributed endogenous regressors, known to cause the modelnon-identifiability issue for CopulaOrigin.2.3.1 Assumptions in the Existing Copula Endogeneity-Correlation Method(CopulaOrigin)Consider the following linear structural regression model with an endogenous regressor anda vector of exogenous regressors 5:Yt = µ+Ptα+W ′t β +ξt , (2.1)where t = 1,2, ...,T indexes either time or cross-sectional units, Yt is a (1×1) dependentvariable, Pt is a (1×1) endogenous regressor, Wt is a (k×1) vector of exogenous regressors,ξt is the structural error term, and (µ,α,β ) are model parameters. Pt is correlated with ξt ,and this correlation generates the endogeneity problem. Wt is exogenous, which means it is5Unlike Park and Gupta (2012), our model includes the intercept term. As shown in Becker et al. (2021),it is important to include the intercept term when evaluating the copula correction method.18not correlated with ξt , but can be correlated with the endogenous variable Pt .The key idea of the copula method (Park and Gupta 2012) is to use a copula to jointlymodel the correlation between the endogenous regressor Pt and the error term ξt . The ad-vantage of using copula is that marginals are not restricted by the joint distribution. Usinginformation contained in the observed data, marginals of the endogenous regressor and theerror term are first obtained respectively. Then the copula model enables researchers to con-struct a flexible multivariate joint distribution that captures the correlation between the twovariables.Let F(P,ξ ) be the joint cumulative distribution function (CDF) of the endogenous re-gressor Pt and the structural error ξt with marginal CDFs H(P) and G(ξ ), respectively. Fornotational simplicity, we may omit the index t in Pt and ξt below when appropriate. Ac-cording to Sklar’s theorem (Sklar 1959), there exists a copula function C(·, ·) such that forall P and ξ ,F(P,ξ ) =C(H(P),G(ξ )) =C(Up,Uξ ), (2.2)where Up = H(P) and Uξ = G(ξ ), and they both follow uniform(0,1) distributions. Thus,the copula maps the marginal CDFs of the endogenous regressor and the structural error totheir joint CDF, and makes it possible to separately model the marginals and correlations ofthese random variables.To capture the association between the endogenous regressor P and the error ξ , Park andGupta (2012) used the following Gaussian copula for its many desirable properties (Danaher192007; Danaher and Smith 2011):F(P,ξ ) =C(Up,Uξ ) =Ψρ(Φ−1(Up),Φ−1(Uξ ))=12π(1−ρ2)1/2∫ Φ−1(Up)−∞∫ Φ−1(Uξ )−∞exp[−(s2−2ρ · s · t+ t2)2(1−ρ2)]dsdt,(2.3)whereΦ(·) denotes the univariate standard normal distribution function andΨρ(·, ·) denotesthe bivariate standard normal distribution with the correlation coefficient ρ . Note that theabove Gaussian copula model depends on the rank-order of raw data only, and is invariantto strictly monotonic transformations of variables in (Pt ,Wt ,ξt). Thus the above Gaussiancopula model is considered general and robust for most marketing applications (Danaherand Smith, 2011). In the Gaussian copula model, ρ captures the endogeneity of the regressorP, and a non-zero value of ρ corresponds to P being endogenous.Under the above copula model for (Pt ,ξt) and the commonly-assumed normal distri-bution for the structural error ξt , Park and Gupta (2012) developed the following gen-erated regressor procedure to correct for regressor endogeneity. Let P∗t = Φ−1(Up) andξ ∗t = Φ−1(Uξ ), the above Gaussian copula assumes [P∗t ,ξ ∗t ]′ follow the standard bivariatenormal distribution with the correlation coefficient ρ as follows:P∗tξ ∗t ∼ N 00 , 1 ρρ 1 (2.4)Under the assumption that the structural error ξt follows N(0,σ2ξ ), Park and Gupta (2012)showed that the structural error can be split into two parts as follows:ξt = σξξ ∗t = σξρP∗t +σξ√1−ρ2ωt , (2.5)where the first part σξρP∗t captures the correlation between ξt and the endogenous regressor,20and the other part σξ ·√1−ρ2ωt is an independent new error term. Equation (2.1) can berewritten as follows:Yt = µ+Ptα+Wtβ +σξ ·ρ ·P∗t +σξ ·√1−ρ2 ·ωt . (2.6)Based on the above representation, Park and Gupta (2012) suggested the following gener-ated regressor approach to correcting for the endogeneity of Pt : the ordinary least square(OLS) estimation of Equation (2.6) with P∗t =Φ−1(Up) included as an additional regressorwill yield consistent model estimates. Park and Gupta (2012) also pointed out that in orderfor the above approach to work, Pt needs to have a non-normal distribution. Suppose Pt isnormally distributed, Pt = P∗t ·σp, resulting in perfect collinearity between Pt and P∗t andviolating the full rank assumption required for identifying the linear regression model inEquation (2.6). Thus, Pt should follow a different distribution from the normal error term sothat the causal effect of P that is independent of all other regressors can be identified.However, we show here that an additional and implicit assumption for the above generated-regressor approach to yield consistent model estimates is the uncorrelatedness between P∗tand Wt . For the OLS estimation to yield consistent estimation, the error term ωt in Equa-tion (2.6) is required to be uncorrelated with all the regressors on the right-hand side of theequation: Pt ,Wt ,P∗t . The theorem below shows that Wt becomes endogenous in Equation(2.6) when Wt and P∗t are correlated.Theorem 1. Assuming (1) the error term is normal, (2) a Gaussian Copula for the structuralerror term and Pt , and (3) Pt is endogenous: ρ ̸= 0, Cov(ωt ,Wt) =− ρ√1−ρ2Cov(Wt ,P∗t ) ̸= 0if P∗t and Wt are correlated.Proof: See the Appendix, Proof of Theorem 1.To summarize, the generated regressor procedure based on Equation (2.6) makes the21following set of assumptions.Assumption 1. The structural error follows a normal distribution;Assumption 2. Pt and the structural error follow a Gaussian copula;Assumption 3. Nonnormality of the endogenous regressor Pt;Assumption 4. Wt and P∗t are uncorrelated.As shown in the Appendix, Assumption 4 can be extended to Assumption 4(b) belowfor the case of multiple endogenous regressor.Assumption 4(b). When there are multiple endogenous regressors, Wt is uncorrelated withthe CCF, i.e., the linear combination of P∗t that is used to control for endogenous regressors.Specifically, Cov(Wt ,ρξ1−ρ12ρξ21−ρ212·P∗1,t +ρξ2−ρ12ρξ11−ρ212·P∗2,t) = 0 is required in the 2-endogenousregressors case. 6Assumptions 4 and 4(b) are verifiable and provide users with criteria to check whetherCopulaOrigin would provide consistent estimation when there exist exogenous regressorsthat may be correlated with the CCF. With only one endogenous regressor, one can simplycheck the correlations between the copula transformation of this endogenous regressor witheach exogenous regressor. For multiple endogenous regressors, one should check the cor-relations between the CCF (i.e., the linear combination of copula transformations of theseendogenous regressors used to control for endogeneity) in CopulaOrigin with each exoge-nous regressor. If there exists one exogenous regressor in Wt that fails Assumption 4 or4(b), CopulaOrigin yields biased estimates, and our proposed 2sCOPE method should beused, which is derived below.6It is clear that this requirement is not the same as either Cov(Wt ,P∗1,t) = 0,Cov(Wt ,P∗2,t) = 0 orCov(Wt ,P1,t) = 0,Cov(Wt ,P2,t) = 0.22In Park and Gupta (2012), all the above assumptions except Assumption (4) have beenmade explicit. Among the first three assumptions, Park and Gupta (2012) have shown rea-sonable robustness of their copula method to non-normal distributions of the error term(Assumption 1) and alternative forms of copula functions (Assumption 2), although it is notsurprising to observe the sensitivity of CopulaOrigin to gross violations of these assumptions,such as highly skewed error distributions (Becker et al., 2021). By contrast, the assumptionthat the endogenous regressor Pt follows a non-normal distribution (Assumption 3) is criti-cal. An endogenous regressor following a normal distribution can cause the structural modelto be unidentifiable regardless of sample size; a nearly normally distributed endogenous re-gressor may require a very large sample size for the method to perform well and may causethe method to have poor performance for a finite sample size. Moreover, we have shownabove that for their method to work, there should be no exogenous regressors that are cor-related with P∗t (Assumption 4). Both the Assumptions (3 and 4) can be too strong andsubstantially limit the applicability of the instrument-free copula method in practice.2.3.2 Proposed Method: Two-stage Copula Endogeneity-correction(2sCOPE)In this subsection, we propose a two-stage Copula (2sCOPE) method and show that it canrelax both the uncorrelatedness assumption between the copula-transformed endogenousregressor and the exogenous regressors (Assumption 4) and the key identification assump-tion of non-normality on the endogenous regressors (Assumption 3). The 2sCOPE methodjointly models the endogenous regressor, Pt , the correlated exogenous variable, Wt , and thestructural error term, ξt , using the Gaussian copula model, which implies that [P∗t ,W ∗t ,ξ ∗t ]23follows the multivariate normal distribution:P∗tW ∗tξ ∗t ∼ N000 ,1 ρpw ρpξρpw 1 0ρpξ 0 1 , (2.7)where P∗t =Φ−1(H(Pt)), W ∗t =Φ−1(L(Wt)), and ξ ∗t =Φ−1(G(ξt)), and H(·), L(·) and G(·)are marginal CDFs of Pt , Wt and ξt respectively.Under the above Gaussian copula model in Equation (3.4), one can develop a directextension of CopulaOrigin, which adds generated regressors P∗t and W ∗t into the structural re-gression model to correct for endogeneity bias (Appendix). The resulting method, denotedas COPE, is shown to yield consistent causal effect estimates (Theorem A1 in the Ap-pendix) without requiring Assumption 4 needed for CopulaOrigin. However, COPE requiresendogenous regressors Pt and exogenous regressors Wt to be both non-normally distributed(Theorem A1 in the Appendix). To overcome the limitations of COPE, below we derivethe 2sCOPE method that relaxes both assumptions and is shown to be more efficient thanCOPE.Under the above Gaussian copula model, we have the following system of equations thatare similar to the two-stage least square method. However, we do not require any variablethat satisfies the exclusion restriction.Yt = µ+Ptα+Wtβ +ξt (2.8)P∗t =W∗t γ+ εt , (2.9)where the two error terms εt and ξt are correlated because of the endogeneity of Pt . Underthe assumption that ξt follows a normal distribution, εt and ξt follow a bivariate normaldistribution, since they are a linear combination of tri-normal variate (ξ ∗t ,P∗t ,W ∗t ) under the24Gaussian copula assumption.The main idea of 2sCOPE is to make use of the fact that by conditioning on εt , thestructural error term ξt becomes independent of both Pt and Wt . That is, by conditioningon the component of Pt causing the endogeneity of Pt (i.e, εt here), the structural error isnot correlated with both Pt and Wt , thereby ensuring the consistency of standard estimationmethods. In this sense, εt serves as a (scaled) control function to address the endogeneitybias. To demonstrate this point, note that the Gaussian copula model in Equation (3.4) canbe rewritten as follows:P∗tW ∗tξ ∗t=1 0 0ρpw√1−ρ2pw 0ρpξ−ρpwρ pξ√1−ρ2pw√1−ρ2pξ −ρ2pwρ2pξ1−ρ2pw ·ω1,tω2,tω3,t ,ω1,tω2,tω3,t∼ N000 ,1 0 00 1 00 0 1 . (2.10)Given the above joint normal distribution for (P∗t ,W ∗t ,ξ ∗t ) and ξ ∗t = σξξt , we haveP∗t = ρpwW∗t +√(1−ρ2pw) ·ω2,t = ρpwW ∗t + εt , (2.11)25which shows γ in Equation (2.9) is ρpw and εt =√(1−ρ2pw) ·ω2,t , andYt = µ+Ptα+Wtβ +σξρpξ1−ρ2pwP∗t +−σξρpwρpξ1−ρ2pwW ∗t +σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pw ·ω3,t= µ+Ptα+Wtβ +σξρpξ1−ρ2pw(P∗t −ρpwW ∗t )+σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pw ·ω3,t ,= µ+Ptα+Wtβ +σξρpξ1−ρ2pwεt +σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pw ·ω3,t . (2.12)Equation (2.12) suggests adding the estimate of the error term εt from the first stage regres-sion as a generated regressor to the outcome regression instead of using P∗t and W ∗t . Asshown in Theorem 2, the new error term ω3,t is uncorrelated with all the regressors in Equa-tion (2.12), ensuring the consistency of model estimates. This two-step procedure, named2sCOPE, adds the first-stage residual term ε̂t to control for endogeneity and in this aspect issimilar to the control function approach of Petrin and Train (2010). However, unlike Petrinand Train (2010), 2sCOPE requires no use of instrumental variables.Theorem 2. Estimation Consistency. Assuming (1) the error term is normal, (2) the en-dogenous variable Pt or correlated regressors Wt is nonnormal, and (3) a Gaussian Copulafor the error term, Pt and Wt , Cov(ω3,t ,Wt) =Cov(ω3,t ,Pt) =Cov(ω3,t ,εt) = 0 in Equation(3.8).Proof: See the Appendix, Proof of Theorem 2.According to Theorem 2, the proposed method 2sCOPE can yield consistent estimateswhen assumptions are met. Specifically, Assumption 4 is relaxed because 2sCOPE canhandle the case when extra exogenous regressors that can be correlated with the endogenousregressor are included in the model. Theorem 3 below further shows that 2sCOPE relaxesAssumption 3 (the non-normality assumption on endogenous regressors), a critical model26identification condition required in all other copula correction methods.Theorem 3. Nonnormality Assumption Relaxed. Assuming (1) the error term is normal,(2) one of the correlated exogenous regressors Wt is nonnormal, and (3) a Gaussian Copulafor the error term, Pt and Wt , 2sCOPE estimator θ̂2 is consistent when Pt follows a normaldistribution while the COPE estimator θ̂1 is not consistent.Proof: See the Appendix, Proof of Theorem 3.Theorem 3 shows that as long as one of the exogenous regressors that are correlatedwith the endogenous regressor Pt is nonnormally distributed, 2sCOPE can correct for endo-geneity for a normal regressor Pt while COPE cannot. Intuitively, when one of Pt and Wtis normal, P∗t (or W ∗t ) becomes a linear function of Pt (or Wt) under the Gaussian copulaassumption, rendering the second stage model in COPE to fail the full rank assumption andbecome unidentified. Thus, COPE cannot deal with normal endogenous regressors. Forthe proposed 2sCOPE method in Equation (2.12), adding the first stage residual ε̂t as thegenerated regressor helps model identification. As long as not all Wts are normal, εt wouldnot be a linear function of Pt and Wt and thus the second stage model (Equation 2.12) in2sCOPE would satisfy the full rank requirement for model identification. Thus, our pro-posed method 2sCOPE can relax the nonnormality assumption on the endogenous regressorrequired in Park and Gupta (2012) as long as one of Wt is nonnormally distributed.Theorem 4 below shows that when both COPE and 2sCOPE yield consistent estimates,2sCOPE outperforms COPE, the direct extension of CopulaOrigin to more general settings,by reducing the variance of the estimates and improving estimation efficiency.Theorem 4. Variance Reduction. Assuming (1) the error term is normal, (2) the endoge-nous variable Pt and correlated regressors Wt are nonnormal, and (3) a Gaussian Copulafor the error term, Pt and Wt , Var(θ̂2)≤ Var(θ̂1), where θ̂1 and θ̂2 denote parameter esti-mates from COPE and 2sCOPE, respectively.27Proof: See the Appendix, Proof of Theorem 4.To sum up, we have proved the consistency of 2sCOPE method (Theorem 2). Theo-rems 3 and 4 further establish that the 2sCOPE method outperforms the COPE method, theextended CopulaOrigin, in terms of estimation efficiency gain and relaxing the nonnormalityassumption on the endogenous regressors required in CopulaOrigin by satisfying a very weakcondition.2.3.3 Multiple Endogenous RegressorsIn this subsection, we extend 2sCOPE to the general case of multiple endogenous regressors.Consider the following structural linear regression model with two endogenous regressors(P1,t and P2,t) that are potentially correlated with the exogenous regressor Wt :Yt = µ+P1,t ·α1+P2,t ·α2+Wtβ +ξt . (2.13)Under the multivariate Gaussian distribution assumption on (ξt ,P∗1,t ,P∗2,t ,W∗t ), the sys-tem equations of 2sCOPE method in Equation (2.8, 2.9) are readily extended to the casewith two endogenous regressors asYt = µ+P1,tα1+P2,tα2+Wtβ +ξt , (2.14)P∗1,t = ρwp1W∗t + ε1,t , (2.15)P∗2,t = ρwp2W∗t + ε2,t , (2.16)where Equations (2.15) and (2.16) can be directly derived from the Gaussian copula as-sumption; (ξt ,ε1,t .ε2,t) are a linear transformation of (ξt ,P∗1,t ,P∗2,t ,W∗t ), and thus also followa multivariate Gaussian distribution. As a result, we can decompose the structural error ξt28as additive terms for ε1,t , ε2,t and a remaining independent error term ω4,t as followsYt = µ+P1,tα1+P2,tα2+Wtβ +η1ε1,t +η2ε2,t +σξ ·m ·ω4,t , (2.17)where ε1,t = P∗1,t −ρwp1W ∗t and ε2,t = P∗2,t −ρwp2W ∗t , m is a constant depending only on thecorrelation coefficients in the Gaussian copula, and η1, η2 and ω4,t are the same as thosedefined in Equation (A.6) in the Appendix for describing COPE for multiple endogenousregressors and thus the new (scaled) error term ω4,t is independent of latent copula data(P∗1,t ,P∗2,t ,W∗t ) as well as all functions of these latent data including P1,t ,P2,t ,Wt ,ε1,t ,ε2,t .Because ω4,t is independent of all regressors on the right side of Equation (2.17), the OLSestimation of Equation (2.17) yields a consistent estimation of structural model parameters.Note that Equation (2.17) can also be obtained from Equation (A.7) in Online Appendix A.3for describing COPE for multiple endogenous regressors by noting that ε1,t = P∗1,t−ρwp1W ∗tand ε2,t = P∗2,t − ρwp2W ∗t . However, 2sCOPE adds only two residual terms (ε1,t ,ε2,t) asgenerated regressors instead of three copula transformations of regressors (P∗1,t ,P∗2,t ,W∗t )as generated regressors, as COPE does (Equation (A.7) in the Appendix). Thus, 2sCOPEadds a smaller number of generated regressors than COPE, and provides higher estimationefficiency. In addition, by adding residual terms as the generated regressors, 2sCOPE re-laxes the assumption of regressor non-normality required to COPE as long as not all Wtsare normal. The proof for the estimation consistency of 2sCOPE, estimation efficiency gainand relaxation of the regressor-nonnormality assumption for 2sCOPE can be found in theAppendix under the related Theorems 2, 3, 4.Table 2.2 summarizes the assumptions for the three methods: our proposed 2sCOPEmethod, the existing copula method CopulaOrigin and CopulaOrigin’s direct extension, COPE.Our proposed 2sCOPE method can deal with the case when there are exogenous regressorsthat are correlated with the endogenous regressors. Moreover, 2sCOPE can further relaxthe regressor-nonnormality assumption. Table 2.3 summarizes and compares the estimation29CopulaOrigin COPE 2sCOPE• The structural error follows a nor-mal distribution (Asm. 1);• Pt and the structural error follow aGaussian copula (Asm. 2);• All regressors in Pt are nonnor-mally distributed (Asm. 3);• Wt is uncorrelated with the CCF(copula control function which isthe linear combination of all P∗tused to control for endogeneity)(Asm. 4, 4(b)).• The structural error follows anormal distribution;• Pt , Wt and the structural errorfollow a Gaussian copula;• All regressors in Pt and Wt arenonnormally distributed.• The structural error follows anormal distribution;• Pt , Wt and the structural errorfollow a Gaussian copula;• Pt can be normally distributedas long as one of Wt is nonnor-mal.Table 2.2: Summary of Assumptions for the Three Methodsprocedure of COPE and 2sCOPE.COPE 2sCOPEStage 1:• Obtain empirical CDFs for each regressor in Ptand Wt , denoted as Ĥ(Pt) and L̂(Wt);• Compute P∗t = Φ−1(Ĥ(Pt)) and W ∗t =Φ−1(L̂(Wt));• Add P∗t and W ∗t to the outcome structural regres-sion model as generated regressors.• Obtain empirical CDFs for each regressor in Pt andWt , Ĥ(Pt) and L̂(Wt);• Compute P∗t =Φ−1(Ĥ(Pt)) and W ∗t =Φ−1(L̂(Wt));• Regress each endogenous regressor in P∗t separatelyon W ∗t and obtain residual ε̂t ;Stage 2:• Add ε̂t to the outcome structural regression model asgenerated regressors.• Standard errors of parameter estiamtes are estimated using bootstrap in both methods.Table 2.3: Estimation Procedure302.3.4 2sCOPE for Random Coefficient Linear Panel Models withEndogenous RegressorsWe consider the following random coefficient model for linear panel dataYit |µi,αi,βi = µ¯+µi+P′itαi+W ′itβi+ξit , (2.18)where i = 1, · · · ,N indexes cross-sectional units and t = 1, · · · ,T indexes occasions. Pit(Wit) denotes a vector of endogenous (exogenous) regressors. Pit and Wit can be correlated.The error term ξitiid∼ N(0,σ2ξ ) is correlated with Pit due to the endogeneity of Pit but is un-correlated with the exogenous regressors in Wit . The individual-specific intercept µi andindividual-specific slope coefficients (αi,βi) permit heterogeneity in both intercepts and re-gressor effects across cross-sectional units. Extant marketing studies have shown the ubiq-uitous presence of heterogeneous consumers’ responses to marketing mix variables (e.g.,price sensitivity) and substantial bias associated with ignoring such heterogeneity in slopecoefficients. Thus, it is important to permit individual-specific slope coefficients, especiallyin marketing studies.The linear panel data model as specified in Equation (2.18) is general and includes thelinear panel model with only individual-specific intercepts considered in Haschka (2021) asa special case. Specifically, Haschka (2021) fixes (αi,βi) to be the same value (α,β ) acrossall units, assuming all cross-sectional units have the same slope coefficients. In contrast,the model in Equation (2.18) relaxes this strong assumption and can generate unit-specificslope parameters, which can be used for targeting purposes.A fully random coefficient model typically assumes that (µi,αi,βi) follows a multi-variate normal distribution. When all regressors are exogenous, estimation algorithms forsuch random coefficient models are well-established and computationally feasible even fora high-dimensional vector of random effects (µi,αi,βi): with the normal conditional dis-31tribution for Yit |(µi,αi,βi) in Equation (2.18) and the multivariate normal prior distributionfor random effects (µi,αi,βi), marginally Yit follows a normal distribution with a closed-form expression containing no integrals of random effects (µi,αi,βi), leading to an easy-to-evaluate likelihood function (Greene, 2012). For instance, R function lme() can beused to obtain MLEs of population model estimates and empirical Bayes estimates of ran-dom effects. Alternatively, one can assume a mixed-effect model where µi is a fixed effectparameter with µi’s allowed to be correlated with the regressors Pit and Wit . To avoid the po-tential incidental parameter problem associated with these fix-effect parameters, one oftenuses the first-difference or fixed-effects transformation to eliminate the incidental interceptparameters as followsy˜it |αi,βi = P˜′itαi+W˜ ′itβi+ ξ˜it , (2.19)where y˜it , P˜it , W˜it and ξ˜it denote new variables obtained from the first-difference or fixed-effect transformation. Haschka (2021) considered a special case of Equation (2.19) by fixing(αi,βi) to be constants.It is straightforward to apply 2sCOPE to address regressor endogeneity in the generalrandom coefficient model for linear panel data in Equation (2.18) and the transformed onewithout intercepts in Equation (2.19). 7 Assuming (Pit ,Wit ,ξit) follow a Gaussian copula,COPE adds the generated regressor P∗it = Φ−1(Ĥ(Pit)) and W ∗it = Φ−1(L̂(Wit)) into Equa-tion (2.18) to control for regressor endogeneity. The 2sCOPE procedure adds the residualsobtained from regressing P∗it on W ∗it . Thus, 2sCOPE method can be implemented using stan-dard software programs for random coefficient linear panel models assuming all regressorsare exogenous (see Section 2.4.6 for an illustration using the R function lme()). By con-trast, the MLE approach for copula correction in the random coefficients model accounting7Similar to Haschka (2021), a GLS transformation can be applied to both sides of Equation (2.19), resultingin a pooled regression for which 2sCOPE can be directly applied.32for correlated endogenous and exogenous regressors is not available yet and would requireconstructing complicated joint likelihood on the error term, Pt and Wt , which involves newlyappearing numerical integrals of random effects and cannot be maximized by standard esti-mation algorithms for random coefficient models. 8 Finally, current applications applyingCopulaOrigin do not consider the role of exogenous regressors. Our analysis shows thatthis may yield bias if any exogenous regressor is correlated with the CCF added to controlendogeneity, for which 2sCOPE should be used to address regressor endogeneity.2.3.5 2sCOPE for Slope Endogeneity and Random Coefficient LogitModelIn the Appendix, we derive the 2sCOPE method to tackle the slope endogeneity problem andaddress endogeneity bias in random coefficient logit models with correlated and normallydistributed regressors. In these two cases, we show how to apply 2sCOPE to correct forthe endogenous bias, which can avoid the potential bias of CopulaOrigin due to the potentialcorrelations between the exogenous regressors and CCF, as well as make use of the corre-lated exogenous regressors to relax the non-normality assumption of endogenous regressors,improve model identification and sharpen model estimates. As shown there, 2sCOPE canbe implemented using standard estimation methods by adding generated regressors to con-trol for endogenous regressors. By contrast, the maximum likelihood approach can requireconstructing a complicated joint likelihood that is not what the standard estimation methoduses and thus requires separate development and significantly more computation involvingnumerical integration.8With endogenous regressors, the individual random effects parameters enter into both the density functionfor the outcome Yit |(µi,αi,βi) and the density of copula function C(Uξ ,it ,UP,it ,UW,it) via Uξ ,it , and thus cannotbe integrated out in closed form from the likelihood function even with the normal structural error term andnormal random effects. Therefore, numerical integration is required for obtaining MLEs in random coefficientmodels with endogenous regressors, which cannot be performed with standard software programs for randomcoefficient model estimation.332.4 Simulation StudyIn this section, we conduct Monte Carlo simulation studies for the following goals: (a) toassess the performance of the proposed method for correlated regressors, (b) to assess theperformance of the proposed method under regressor normality and near normality, (c) toassess generalizability and restrictions of the distributional assumptions about the endoge-nous and exogenous regressors, and (d) to compare the performance of the proposed methodwith existing methods. Following Park and Gupta (2012), we measure the estimation biasusing tbias calculated as the ratio of the absolute difference between the mean of the samplingdistribution and the true parameter value to the standard error of the parameter estimate. Asdefined above, tbias represents the size of bias relative to the sampling error. The Appendixprovides additional simulation results on the robustness of 2sCOPE to the misspecificationsof the structural error distribution and the copula dependence structure.2.4.1 Case 1: Non-normal RegressorsWe first examine the case when P and W are correlated. The specific data-generating process(DGP) is summarized below:P∗tW ∗tξ ∗t∼ N000 ,1 ρpw ρpξρpw 1 0ρpξ 0 1= N000 ,1 0.5 0.50.5 1 00.5 0 1 ,(2.20)ξt = G−1(Uξ ,t) = G−1(Φ(ξ ∗t )) =Φ−1(Φ(ξ ∗)) = 1 ·ξ ∗t , (2.21)Pt = H−1(UP,t) = H−1(Φ(P∗t )), Wt = L−1(UW,t) = L−1(Φ(W ∗t )), (2.22)Yt = µ+α ·Pt +β ·Wt +ξt = 1+1 ·Pt +(−1) ·Wt +ξt . (2.23)34where ξ ∗t and P∗t are correlated with the correlation coefficient ρpξ = 0.5, and thus ξt and Ptare correlated, generating the endogeneity problem. W ∗t is exogenous and is not correlatedwith ξ ∗t . But W ∗t and P∗t are correlated with the correlation coefficient ρpw = 0.5, andthus Wt and Pt are correlated. We consider four different estimation methods: (i) OLS, (ii)CopulaOrigin in the form of Equation (2.6), (iii) the extended method COPE in the form ofEquation (A.3), and the proposed method 2sCOPE in the form of Equation (3.8). We set thesample size T = 1000, and generate 1000 data sets as replicates using the DGP above. Inthe simulation, we use the gamma distribution Gamma(1,1) with shape and rate equal to 1for Pt and the exponential distribution Exp(1) with rate 1 for Wt . Models are estimated onall generated data sets, providing the empirical distributions of the parameter estimates.OLS CopulaOrigin COPE 2sCOPEρpw Parameters True Mean SE tbias Mean SE tbias Mean SE tbias Mean SE tbias0.5 µ 1 0.689 0.045 6.964 1.231 0.081 2.849 1.012 0.093 0.129 1.009 0.059 0.157α 1 1.571 0.036 15.75 1.055 0.069 0.791 0.985 0.072 0.213 0.986 0.070 0.197β -1 -1.259 0.031 8.236 -1.289 0.031 9.169 -0.997 0.067 0.038 -0.995 0.042 0.123ρpξ 0.5 - - - 0.570 0.047 1.504 0.505 0.055 0.090 0.504 0.038 0.097σξ 1 0.862 0.020 6.902 1.011 0.043 0.244 1.008 0.041 0.206 1.006 0.040 0.143D-error - - 0.002613 0.0016140.7 µ 1 0.730 0.041 6.629 1.307 0.076 4.037 1.011 0.085 0.124 1.005 0.053 0.088α 1 1.800 0.041 19.67 1.260 0.068 3.838 0.988 0.078 0.148 0.991 0.075 0.118β -1 -1.529 0.037 14.21 -1.567 0.037 15.36 -0.997 0.071 0.041 -0.994 0.056 0.110ρpξ 0.5 - - - 0.633 0.043 3.130 0.503 0.057 0.048 0.500 0.026 0.000σξ 1 0.799 0.018 11.18 0.980 0.044 0.468 1.007 0.041 0.160 1.003 0.040 0.084D-error - - 0.002902 0.001760Table 2.4: Results of the Simulation Study Case 1: Non-normal RegressorsNote: Mean and SE denote the average and standard deviation of parameter estimates over all the 1,000simulated samples.Table 2.4 reports estimation results. As expected, OLS estimates of both α and β are35biased (tbias = 15.75/8.24) as a result of the regressor endogeneity. The estimation resultof CopulaOrigin reduces the bias, but still shows significant bias for both the coefficientestimates of Pt and Wt . The bias of CopulaOrigin depends on the strength of the correlationbetween W and P. Stronger correlations between P∗ and W ∗ can cause a larger bias inCopulaOrigin estimates. For example, when the correlation between W ∗ and P∗ increasesfrom 0.5 to 0.7, the bias of estimated α increases by around five times (from 0.055 to 0.260in Table 2.4 under the column “CopulaOrigin”). The bias confirms our derivation in the modelsection, demonstrating that using the existing copula method may not solve the endogeneityproblem completely with correlated regressors.We next examine our proposed method. 2sCOPE provides consistent estimates withoutthe use of instruments. The average estimate of ρpξ is close to the true value 0.5 and issignificantly different from 0, implying a significant correlation between the endogeneityregressor and the error term. Moreover, the proposed method 2sCOPE shows larger effi-ciency. The standard error of α(β ) in 2sCOPE is 0.070 (0.042), which is 2.78% (37.31%)smaller than the corresponding standard errors using COPE. We further calculate the esti-mation precision of COPE and 2sCOPE using the D-error measure |Σ|1/K (Arora and Huber2001, Qian and Xie 2021), where Σ is the covariance matrix of the parameter estimates inthe regression mean function, and K is the number of these parameters. A smaller valueof D-error means greater estimation efficiency and improved estimation precision. Whenρpw = 0.5, the D-error measure is 0.002613 for COPE and 0.001614 for 2sCOPE (Table2.4), and thus 2sCOPE increases estimation precision by 38.2%, meaning that for 2sCOPEto achieve the same precision with COPE, sample size can be reduced by 38.2%. A 39.3%of efficiency gain for 2sCOPE is found for ρpw = 0.7 in Table 2.4.We perform a further simulation study for a small sample size. Specifically, we use thesame DGP as described above to generate synthetic data, except with the sample size T=200.Table A.1 in the Appendix reports the results and shows that OLS estimates have endogene-36ity bias and CopulaOrigin reduces the endogeneity bias but the significant bias remains. Ourproposed method, 2sCOPE, performs well and has unbiased estimates for the small sam-ple size of T=200. The efficiency gain of 2sCOPE relative to COPE appears to be greaterwhen the sample size becomes smaller. When the correlation between P∗ and W ∗ is 0.5,the D-error measures are 0.0166 and 0.0091 for COPE and 2sCOPE (Appendix Table A.1),respectively, meaning that 2sCOPE increases estimation precision by 1-0.0091/.0166=46%compared with COPE, and thus sample size can be reduced by almost a half (∼50%) for2sCOPE to achieve the same estimation precision as that achieved by COPE. A similarmagnitude of efficiency gain for 2sCOPE relative to COPE (∼50%) is observed when thecorrelation between P∗ and W ∗ is 0.7 (Appendix Table A.1).2.4.2 Case 2: Normal RegressorsNext, we examine the case when the endogenous regressor and (or) the correlated exogenousregressor are normally distributed. We pay special attention to this case because normalityis not allowed for endogenous regressors in Park and Gupta (2012). We use the Gaussiancopula as described in Equations (2.24) to Equations (3.17) for DGP to generate the data,except that the marginal CDFs for regressors (H(·) and L(·)) are chosen according to thedistributions listed in the first two columns in Table 2.5.Table 2.5 summarizes the estimation results. As expected, OLS estimates are biased.CopulaOrigin produces biased estimates whenever the endogenous regressor P follows a nor-mal distribution. The estimates of CopulaOrigin are biased when P follows a gamma distri-bution (first row of Table 2.5) for a different reason: P and W are correlated. Same withCopulaOrigin, the results of COPE are biased in all three scenarios when at least one of Ptand Wt is normal. When Wt is normal, β is 0.323 away from the true value -1; when Pt isnormally distributed, α is 0.684 away from the true value; when both Pt and Wt are normal,37α is 0.663 away from the true value 1 and β is 0.324 away from the true value -1. This isexpected because COPE adds P∗t and W ∗t , the copula transformation of regressors, as addi-tional regressors, and will cause perfect co-linearity and model non-identification problemswhenever at least one of these regressors is normally distributed.Distribution OLS CopulaOrigin COPE 2sCOPEP W Parameters True Mean SE tbias Mean SE tbias Mean SE tbias Mean SE tbiasGamma Normal µ 1 0.431 0.045 12.63 1.018 0.078 0.227 1.017 0.080 0.217 1.015 0.077 0.190α 1 1.569 0.037 15.40 0.979 0.070 0.302 0.979 0.070 0.296 0.985 0.070 0.212β -1 -1.259 0.030 8.619 -1.333 0.028 11.78 -1.323 0.433 0.746 -0.997 0.045 0.067ρpξ 0.5 - - - 0.640 0.039 3.556 0.589 0.141 0.631 0.506 0.036 0.151σξ 1 0.861 0.019 7.240 1.064 0.046 1.394 1.135 0.162 0.837 1.005 0.038 0.134Normal Exp µ 1 1.286 0.042 6.777 1.286 0.045 6.374 0.994 0.073 0.081 1.023 0.070 0.334α 1 1.628 0.031 20.36 1.532 0.462 1.152 1.684 0.437 1.568 1.048 0.126 0.381β -1 -1.286 0.032 8.956 -1.287 0.032 8.960 -0.992 0.066 0.127 -1.024 0.062 0.383ρpξ 0.5 - - - 0.089 0.419 0.980 -0.167 0.384 1.738 0.465 0.074 0.473σξ 1 0.829 0.018 9.492 0.940 0.151 0.394 0.981 0.151 0.129 0.980 0.063 0.318Normal Normal µ 1 1.001 0.026 0.046 1.002 0.030 0.052 1.001 0.033 0.024 1.002 0.028 0.057α 1 1.668 0.030 22.38 1.663 0.450 1.474 1.663 0.460 1.441 1.655 0.395 1.657β -1 -1.335 0.029 11.44 -1.335 0.029 11.42 -1.324 0.438 0.740 -1.328 0.197 1.668ρpξ 0.5 - - - 0.006 0.412 1.198 0.001 0.412 2.426 0.010 0.303 1.616σξ 1 0.816 0.019 9.687 0.917 0.155 0.534 1.003 0.211 0.016 0.879 0.092 1.317Table 2.5: Results of Case 2: Normal RegressorsBy contrast, the proposed 2sCOPE method provides consistent estimates as long as Ptand Wt are not both normally distributed. Both α and β are tightly distributed near thetrue value whenever Pt or Wt is nonnormally distributed. Unlike CopulaOrigin and COPE,2sCOPE adds the residual term obtained from regressing P∗t on W ∗t as the generated re-gressor. Thus, as long as Pt and Wt are not both normally distributed, the residual term isnot perfectly co-linear with the original regressors, permitting model identification. Onlywhen both Pt and Wt are normally distributed (the last scenario in Table 2.5), the residualterm added into the structural regression model becomes a linear combination of Pt and Wt ,38causing perfect co-linearity and model non-identification. Overall, this simulation studydemonstrates the advantage of the proposed 2sCOPE to relax the nonnormality assumptionin CopulaOrigin as long as one of Pt and Wt is nonnormally distributed.2.4.3 Case 3: Performance Under Insufficient Non-Normality ofEndogenous RegressorsThe above section shows that the proposed 2sCOPE can deal with normal endogenous re-gressors, while CopulaOrigin and COPE cannot. In this section, we examine the performanceof these methods in the more common situation of close-to-normal regressors. Althoughmodels are identified asymptotically (i.e., infinite sample size), appreciable finite samplebias can occur with realistic sample size commonly seen in marketing studies, if the endoge-nous regressor is too close to a normal distribution (Becker et al., 2021; Haschka, 2021).Becker et al. (2021) suggest a minimum absolute skewness of 2 for an endogenous regres-sor in order for CopulaOrigin to have good performance in a sample size as small as 200.This requirement can significantly limit the use of copula correction methods in practicalapplications. Given that the proposed 2sCOPE can handle normal endogenous regressors,we expect that 2sCOPE can handle much better the finite sample bias caused by insuffi-cient regressor non-normality than the existing copula correction methods. Thus, in thissubsection, we further explore and compare the finite sample performance of those methodswhen the distribution of the endogenous regressor has various magnitudes of closeness tonormality. Specifically, we show the performance using some commonly used distributionsin practice.We use the data-generating process (DGP) below:P∗tW ∗tξ ∗t∼ N000 ,1 ρpw ρpξρpw 1 0ρpξ 0 1= N000 ,1 0.5 0.50.5 1 00.5 0 1 ,(2.24)39ξt = G−1(Uξ ,t) = G−1(Φ(ξ ∗t )) =Φ−1(Φ(ξ ∗)) = 1 ·ξ ∗t , (2.25)Pt = H−1(UP,t) = H−1(Φ(P∗t )), Wt = L−1(UW,t) = L−1(Φ(W ∗t )), (2.26)Yt = µ+α ·Pt +β ·Wt +ξt = 1+1 ·Pt +(−1) ·Wt +ξt . (2.27)where we still use Exp(1) for W as an example. We compare the performance of CopulaOrigin,COPE and 2sCOPE methods when the endogenous regressor P is from some commonlyused distributions that are close to normal to different extents. Specifically, we use uniform,log normal, t, gamma, beta and normal distributions, and use the average absolute bias ofall the estimated coefficients to measure the performance.Figure 2.1 shows the estimation bias using different distributions of the endogenousregressor P. According to the figure, estimates of CopulaOrigin are biased for all distribu-tions, consistent with our theoretical proof. Compare the estimates of COPE and 2sCOPE,first, COPE cannot handle normal endogenous regressors. This result is consistent with ourtheoretical proof and the simulation result in Case 2. Moreover, the bias is not even decreas-ing as the sample size increases. Second, COPE suffers from finite-sample bias when theendogenous regressor follows some insufficient non-normal distributions (e.g., beta distri-bution, t distribution with a large degree of freedom). Moreover, the smaller the sample size,the larger the estimation bias of COPE would be with insufficient non-normal endogenousregressor. This tells us that COPE doesn’t perform well for close-to-normal distributions.Last but not least, the bias of COPE also indicates how close a distribution is to the normaldistribution. T distribution with a degree of freedom of 30 is closer to normal than the samedistribution with degrees of freedom of 10, 5 and 2, and beta distribution locates between t(df=5) and t(df=10). In contrast to COPE, our proposed 2sCOPE method can get consistentestimates for all normal and close-to-normal distributions.40(a) Sample Size N=200(b) Sample Size N=1000Figure 2.1: Estimation Bias for Different Distributions of Endogenous Regressor.Note: ’lognormal’ is lognormal(0,1), ’uniform’ is U[0,1], and ’gamma’ is Gamma(1,1).412.4.4 Case 4: Multiple Endogenous RegressorsIn this case, we examine the performance of our proposed method when the model hasmultiple endogenous regressors. We use the data-generating process (DGP) with two en-dogenous regressors and one exogenous regressor that is correlated with the endogenousregressor below:P∗1,tP∗2,tW ∗tξ ∗t∼ N0000,1 ρp ρwp1 ρξ p1ρp 1 ρwp2 ρξ p2ρwp1 ρwp2 1 0ρξ p1 ρξ p2 0 1= N0000,1 0.3 0.4 0.50.3 1 0.4 0.50.4 0.4 1 00.5 0.5 0 1, (2.28)ξt = G−1(Uξ ,t) = G−1(Φ(ξ ∗t )) =Φ−1(Φ(ξ ∗)) = 1 ·ξ ∗t , (2.29)P1,t = H−11 (Up1) = H−11 (Φ(P∗1,t)), P2,t = H−12 (Φ(P∗2,t)), (2.30)Wt = L−1(UW,t) = L−1(Φ(W ∗t )), (2.31)Yt = µ+α ·Pt +β ·Wt +ξt = 1+1 ·P1,t +1 ·P2,t +(−1) ·Wt +ξt , (2.32)where H−11 (·) (H−12 (·)) and L−1(·) are the inverse distribution functions of the gamma andexponential distributions used to generate these regressors. Sample size T = 1000. We gen-erate 1000 data sets, and use existing methods and our proposed method to estimate themodel. Table 2.6 shows the estimation results. Both the OLS and CopulaOrigin estimates arebiased, while our proposed method provides unbiased estimates for all parameters, indicat-ing that our proposed method performs well with multiple endogenous regressors.42OLS CopulaOrigin COPE 2sCOPEParameters True Mean SE tbias Mean SE tbias Mean SE tbias Mean SE tbiasµ 1 0.419 0.045 13.02 1.267 0.090 2.949 1.012 0.097 0.125 1.008 0.069 0.120α1 1 1.450 0.029 15.46 1.040 0.060 0.665 0.990 0.060 0.166 0.991 0.059 0.153α2 1 1.450 0.031 14.72 1.040 0.059 0.673 0.990 0.058 0.177 0.991 0.056 0.167β -1 -1.320 0.029 11.04 -1.353 0.028 12.56 -0.997 0.057 0.061 -0.995 0.040 0.134ρξ p1 0.5 - - - 0.567 0.043 1.545 0.503 0.049 0.052 0.502 0.040 0.048ρξ p2 0.5 - - - 0.568 0.042 1.625 0.503 0.047 0.073 0.503 0.038 0.075σξ 1 0.772 0.018 12.58 1.019 0.048 0.402 1.012 0.044 0.283 1.010 0.042 0.233Table 2.6: Results of the Simulation Study Case 3: Multiple Endogenous Regressors2.4.5 Case 5: Multiple Exogenous Control CovariatesWe investigate the performance of our proposed method when there exist multiple exoge-nous regressors consisting of both continuous and discrete variables. We generate the datausing the following DGP:P∗tW ∗1,tW ∗2,tξ ∗t∼ N0000,1 ρpw1 ρpw2 ρξ pρpw1 1 ρw 0ρpw2 ρw 1 0ρξ p 0 0 1= N0000,1 0.5 0.5 0.50.5 1 0.3 00.5 0.3 1 00.5 0 0 1, (2.33)43ξt = G−1(Φ(ξ ∗t )) =Φ−1(Φ(ξ ∗)) = 1 ·ξ ∗t , (2.34)Pt = H−1(Φ(P∗t )), W1,t = L−1(Φ(W ∗1,t)), (2.35)W2,t =1, if Φ(W ∗2,t)≥ 0.50. if Φ(W ∗2,t)< 0.5, (2.36)Yt = µ+α ·Pt +β1 ·W1,t +β2 ·W2,t +ξt = 1+1 ·Pt +(−1) ·W1,t +(−1) ·W2,t +ξt ,(2.37)where H−1(·) and L−1(·) are the inverse distribution functions of the gamma and expo-nential distributions. W2,t is a binary variable that follows a Bernoulli distribution. Weset sample size T = 1000 and generate 1000 data sets to estimate parameters using OLSand copula methods. We follow the approach of Park and Gupta (2012) to generate la-tent copula data for discrete variables. Specifically, for a discrete regressor Wt , such asthe binary exogenous regressor W2,t , we define UW,t , uniformly distributed on [0,1], asthe CDF for a latent variable W ∗t that determines the discrete value of Wt . We then re-late UW,t to Wt through the following inequality: K(Wt − 1) < UW,t < K(Wt), where K(·)is the CDF of Wt and can be directly estimated from the frequencies of the observed data.The above inequality implies the following relationship between W ∗t =Φ−1(UW,t) and KW,t :Φ−1(K(Wt −1))<W ∗t <Φ−1(K(Wt)).The estimation results for the multiple-exogenous-regressor case with both discrete andcontinuous ones are summarized in Table 2.7. The OLS and CopulaOrigin estimates arebiased because of endogeneity and correlated exogenous regressors, respectively. The pro-posed 2sCOPE method performs well and provides consistent estimates for all parameters.This indicates that our proposed method performs well with multiple exogenous correlatedregressors. Moreover, correcting for endogeneity using our proposed method does not re-quire every exogenous correlated regressor to be informative (i.e., continuously distributed).44OLS CopulaOrigin COPE 2sCOPEParameters True Mean SE tbias Mean SE tbias Mean SE tbias Mean SE tbiasµ 1 0.701 0.046 6.452 1.281 0.083 3.394 1.007 0.115 0.057 1.005 0.061 0.085α 1 1.573 0.038 15.10 1.037 0.071 0.532 0.985 0.073 0.208 0.987 0.072 0.180β1 -1 -1.225 0.041 5.523 -1.220 0.039 5.584 -0.990 0.069 0.140 -0.992 0.048 0.161β2 -1 -1.096 0.075 1.273 -1.202 0.073 2.758 -1.006 0.115 0.051 -1.003 0.080 0.042ρpξ 0.5 - - - 0.589 0.045 1.976 0.503 0.061 0.053 0.504 0.038 0.097σξ 1 0.862 0.020 7.066 1.023 0.044 0.532 1.011 0.040 0.264 1.006 0.040 0.115Table 2.7: Results of the Simulation Study Case 4: Multiple Exogenous Control Co-variates2.4.6 Case 6: Random Coefficient Linear Panel ModelWe investigate the performance of our proposed 2sCOPE method in a random coefficientlinear panel model. We use the copula and marginal distributions for [Pit ,Wit ,ξit ] as specifiedin Case 1 (Equations 2.24-3.16). We assign ρpw = 0.7 as an example. We then generate theoutcome Yit using the following standard random coefficient linear panel model:Yit = µ¯+µi+Pit(α¯+ai)+Wit(β¯ +bi)+ξit = 1+µi+Pit(1+ai)+Wit(−1+bi)+ξit ,where [µi,ai,bi]∼ N(0, I3), t = 1, ...,50 indexes occasions for repeated measurements, andi = 1, ...,500 indexes the individual units. The above random coefficients model permits in-dividual units to have heterogeneous baseline preferences (µi) and heterogeneous responsesto regressors (ai,bi). Such random coefficient models are frequently used in marketing stud-ies to capture individual heterogeneity and to profile and target individuals. The correlationbetween ξit and Pit creates the regressor endogeneity problem in the random coefficientmodel, which can cause biased estimates for standard linear random coefficient estimationmethods ignoring the regressor-error correlation. We generate individual-level panel data45as described above 1000 times and use the data for estimation. Estimation results are inTable 2.8. LME is the standard estimation method for linear mixed models assuming allregressors are exogenous, as implemented in the R function lme(). LME and CopulaOriginare biased because of endogeneity and correlated exogenous regressors, respectively. Ourproposed method 2sCOPE provides unbiased estimates that are tightly distributed aroundthe true values for all parameters.LME CopulaOrigin COPE 2sCOPEParameters True Mean SE tbias Mean SE tbias Mean SE tbias Mean SE tbiasµ¯ 1 0.722 0.046 6.052 1.314 0.049 6.399 1.001 0.054 0.016 1.004 0.048 0.091α¯ 1 1.853 0.045 18.83 1.293 0.045 6.469 1.000 0.045 0.009 1.000 0.046 0.008β¯ -1 -1.557 0.045 12.39 -1.598 0.044 13.56 -0.996 0.048 0.079 -1.000 0.044 0.005σµ 1 0.985 0.033 0.459 0.982 0.033 0.547 0.985 0.033 0.463 0.984 0.031 0.522σα 1 0.988 0.036 0.326 0.987 0.034 0.397 0.986 0.035 0.403 0.989 0.035 0.316σβ 1 0.993 0.031 0.235 0.992 0.033 0.249 0.992 0.031 0.264 0.992 0.033 0.248ρpξ 0.5 - - - 0.646 0.009 16.33 0.509 0.012 0.757 0.507 0.005 1.365σξ 1 0.794 0.004 57.71 0.957 0.010 4.439 0.985 0.009 1.689 0.985 0.009 1.640Table 2.8: Results of the Simulation Study Case 5: Random Coefficient Linear PanelModelNote: σµ ,σα ,σβ are standard deviations of µi,ai,bi.2.4.7 Misspecification of the Error ξtSimilar to CopulaOrigin, we assume the structural error ξt to be normally distributed. Thoughthe normality of ξt is a reasonable and commonly used assumption in marketing and eco-nomics literature, the true distribution of ξt is often unknown, resulting in possible misspec-ifications. In this simulation study, we examine the robustness of the proposed method to thedepartures from the normality of ξt . We generate 1,000 data sets using the same multivariate46normal distribution as in Equation (2.24). The rest of DGP is:ξt = G−1(Uξ ,t) = G−1(Φ(ξ ∗t )), (2.38)Pt = H−1(Up,t) = H−1(Φ(P∗t )), Wt = L−1(Uw,t) = L−1(Φ(W ∗t )), (2.39)Yt = µ+α ·Pt +β ·Wt + ξt = 1+1 ·Pt +(−1) ·Wt +ξt . (2.40)where we set Pt ∼ Gamma(1,1) and Wt ∼ Exp(1) in the simulation. We check the robust-ness of the structural error ξt using different distributions (e.g., a uniform distribution, betadistribution and t distribution) instead of a normal distribution. We assume the normality ofξt and use the OLS estimator, CopulaOrigin and the proposed method for estimation.Table 2.9 reports estimation results. OLS and CopulaOrigin estimates are still biased, con-sistent with the normal error case. Importantly, Becker et al. (2021) showed that CopulaOriginhas estimation bias for misspecification of ξt even when no Wts are included, the case inwhich CopulaOrigin should be consistent. This indicates that CopulaOrigin is not robust inthe misspecification of the error term. In contrast, 2sCOPE can recover the true parame-ter values despite the misspecification of ξt , demonstrating the robustness of the proposed2sCOPE method to the normal error assumption. See more robustness check results fordifferent distributions of the error term in the Appendix.2.5 Empirical ApplicationIn this section, we apply our method to a real marketing application. We illustrate theproposed method to address the price endogeneity issue using store-level sales data of thetoothpaste category in Chicago over 373 weeks from 1989 to 1997 9. To control for prod-uct size, we select data of toothpaste with the most common size, which is 6.4 oz. Retail9We obtained the data from https://www.chicagobooth.edu/research/kilts/datasets/dominicks.47OLS COPE 2sCOPEDistribution of ξt Parameters True Mean SE tbias Mean SE tbias Mean SE tbiasU[-0.5,0.5] µ 1 0.912 0.013 6.808 1.004 0.025 0.144 1.002 0.017 0.105α 1 1.160 0.010 16.41 0.996 0.017 0.263 0.996 0.017 0.233β -1 -1.072 0.009 8.033 -1.000 0.019 0.023 -0.998 0.011 0.147ρpξ 0.5 - - - 0.497 0.049 0.054 0.495 0.035 0.155σξ 0.289 0.251 0.004 9.018 0.291 0.009 0.287 0.290 0.008 0.197Beta(0.5,0.5) µ 1 0.896 0.016 6.461 1.005 0.031 0.166 1.003 0.020 0.145α 1 1.190 0.012 15.72 0.994 0.019 0.343 0.994 0.018 0.318β -1 -1.086 0.011 7.763 -0.999 0.022 0.043 -0.998 0.014 0.183ρpξ 0.5 - - - 0.483 0.050 0.339 0.481 0.033 0.593σξ 0.354 0.311 0.005 9.046 0.357 0.009 0.355 0.356 0.009 0.258t (df=3) µ 1 0.504 0.082 6.071 0.972 0.198 0.142 0.983 0.127 0.135α 1 1.903 0.089 10.13 1.026 0.227 0.113 1.024 0.217 0.110β -1 -1.410 0.064 6.448 -1.003 0.129 0.020 -1.012 0.109 0.111ρpξ 0.5 - - - 0.449 0.088 0.577 0.454 0.069 0.676σξ 1.732 1.503 0.231 0.992 1.701 0.246 0.124 1.698 0.244 0.141Table 2.9: Results of the Simulation Study Case D1: Misspecification of ξtprice is usually considered endogenous. The endogeneity of retail price can come from un-measured product characteristics or demand shocks that can influence both consumers’ andretailers’ decisions. Since these variables are unobserved by researchers, they are absorbedinto the structural error, leading to the endogeneity problem. Prices of different stores arecorrelated and often used as an IV for each other. This allows us to test the performanceof the proposed 2sCOPE method in an empirical setting where a good IV exists. Besidesthe endogenous price, two promotion-related variables, bonus promotion and direct pricereduction, would also affect demand. Following Park and Gupta (2012), we treat the pro-48motion variables as exogenous regressors. We focus on category sales in two large storesin Chicago (referred to as Stores 1 and 2). We convert retail price, in-store promotion andsales from the UPC level to the aggregate category level. They are computed as weeklymarket share-weighted averages of UPC-level variables. The correlation between log retailStore 1 Store 2Variables Mean SD Max Min Mean SD Max MinSales (Unit) 115 52.8 720 35 165.7 93.7 1334 26Price ($) 2.06 0.20 2.48 1.46 2.10 0.21 2.48 1.47Bonus 0.18 0.20 0.80 0.00 0.16 0.19 0.79 0.00PriceRedu 0.10 0.19 0.72 0.00 0.10 0.19 0.73 0.00Table 2.10: Summary Statisticsprice and bonus promotion in Store 1 (Store 2) is -0.30 (-0.15), and the correlation betweenlog retail price and price reduction promotion in Store 1 (Store 2) is -0.23 (-0.35). Boththe correlations are significantly different from zero. The appreciable correlations betweenprice and promotion variables provide a good setting for testing our method and examiningthe impact that our proposed method can make in the setting of correlated endogenous andexogenous regressors. Summary statistics of key variables are summarized in Table 3.2.We estimate the following linear regression model:log(Salest) = β0+ log(Retail Pricet) ·β1+W ′t β2+ξt ,where t = 1,2, ...,T indexes week. The vector Wt includes all exogenous regressors, whichare two promotion variables, bonus promotion and price reduction, in this application.Figure 2.2 shows log sales and log retail prices of toothpaste at store 1 over time (store2 is very similar). To control for the possible trend of retail price over time, we use the49(a) Store 1 log sales (b) Store 1 log retail priceFigure 2.2: Log Sales and Log Retail Price of Toothpaste in Store 1.de-trended log retail prices (as instrumental variables as well) for estimation below.(a) detrended log price (b) bonus (c) price reductionFigure 2.3: Histogram of Log Retail Price, Bonus and Price Reduction in Store 1Figure 2.3 shows the histograms of detrended log retail prices and the two promotionvariables. All three variables are continuous. Moreover, except for log retail price, whichis a bit close to the normal distribution, the other two regressors, bonus and price reduction,are both nonnormally distributed. Therefore, we expect that the proposed 2sCOPE methodcan exploit these additional features of exogenous regressors correlated with the endogenousregressor for model identification and estimation even if the endogenous regressor is close tonormal distribution. We estimate the model using the OLS, two-stage least squares (TSLS),CopulaOrigin and our two proposed 2sCOPE method.50We use the IV-based TSLS estimator as a benchmark to test the validity of our proposedmethod. Following Park and Gupta (2012), we use retail price at the other store as an instru-ment for the endogenous price in the focal store. This variable can be a valid instrument asit satisfies the two key requirements. First, retail prices across stores in the same market canbe highly correlated because wholesale prices are usually offered the same (or very close).The Pearson correlation between the detrended log retail prices at Stores 1 and 2 is 0.79,providing strong explanatory power on the endogenous price. The correlation is comparableto that in Park and Gupta (2012). Second, some unmeasured product characteristics suchas shelf-space allocation, shelf location and category location are determined by retailersand are usually not systematically related to wholesale prices (exclusion restriction). Forthe three copula-based methods, we make use of information from the existing endogenousand exogenous regressors and no extra IVs are needed. In CopulaOrigin, we add the copulatransformation of the detrended log price, logP∗ = Φ−1(Ĥ(logP)), as a “generated regres-sor” to the outcome regression. For the COPE method, we add another two “generated re-gressors”, copula transformation of bonus and price reduction (Bonus∗ =Φ−1(L̂1(Bonus)),PriceRedu∗ = Φ−1(L̂2(PriceRedu))). For the 2sCOPE method, we first regress logP∗ onBonus∗ and PriceRedu∗, and then add the residual as the only “generated regressor” to theoutcome regression. Ĥ(·), L̂1(·), L̂2(·) are all estimated CDFs using the univariate empiricaldistribution for each regressor. Standard errors of parameter estimates are obtained usingbootstrap.Table 3.3 reports the estimation results. Beginning with the results from Store 1, OLSestimates are significantly different from TSLS estimates, indicating that the price endo-geneity issue occurs. Instrumenting for retail price changes the price coefficient estimatefrom -0.767 to -1.797, implying that there is a positive correlation between unobservedproduct characteristics and the price. The estimates of ρ in the three IV-free copula-basedmethods, representing the correlation between the endogenous regressor Pt and the error51term, are all significantly positive, further confirming our previous conclusion. This di-rection of correlation is consistent with previous empirical findings (e.g., Villas-Boas andWiner 1999, Chintagunta et al. 2005).OLS TSLS CopulaOrigin COPE 2sCOPEStore Parameters Est SE t-value Est SE t-value Est SE t-value Est SE t-value Est SE t-valueStore 1 Constant 1.301 1.197 0.25 -2.993 1.646 1.82 -8.526 2.619 3.26 -8.569 2.820 3.04 -3.908 2.314 1.69Price -0.767 0.288 2.66 -1.797 0.396 4.54 -3.082 0.620 4.97 -3.111 0.664 4.69 -2.014 0.555 3.63Bonus 0.371 0.122 3.31 0.104 0.141 0.74 0.415 0.115 3.61 0.522 0.288 1.81 0.064 0.171 0.37PriceRedu 0.498 0.115 4.33 0.285 0.125 2.28 0.544 0.111 4.90 1.033 0.211 4.90 0.275 0.143 1.92ρ - - - - - - 0.521 0.098 5.32 0.662 0.117 5.66 0.297 0.089 3.34Store 2 Constant -3.898 1.246 3.13 0.763 1.943 0.39 1.107 3.404 0.33 1.324 3.430 0.39 0.001 2.702 0.00Price -1.982 0.300 6.61 -0.864 0.467 1.85 -0.799 0.807 0.99 -0.783 0.811 0.96 -1.048 0.648 1.62Bonus 0.062 0.116 0.53 0.286 0.148 1.93 0.032 0.117 0.27 -0.819 0.426 1.92 0.239 0.151 1.58PriceRedu 0.283 0.111 2.55 0.540 0.137 3.94 0.275 0.110 2.5 0.540 0.194 2.78 0.467 0.152 3.07ρ - - - - - - -0.319 0.177 1.80 -0.358 0.164 2.18 -0.188 0.109 1.72Table 2.11: Estimation Results: Toothpaste SalesThe price elasticity estimates from the CopulaOrigin, the extension COPE and the pro-posed method 2sCOPE are -3.082, -3.111 and -2.014, respectively. Among the three es-timates, the estimate of -2.014 from the proposed 2sCOPE is close to the estimate of -1.797 from the TSLS method, whereas the existing copula and the COPE yield substantiallysmaller price elasticity estimates. We confirm in the literature that the TSLS and 2sCOPEestimates are reasonable because the price elasticity of the toothpaste category in the liter-ature is around -2.0 (Hoch et al. 1995, Mackiewicz and Falkowski 2015). Comparing theestimates of ρ from the three IV-free copula-based methods, our proposed 2sCOPE providesa much smaller estimate of ρ (0.297 for 2sCOPE vs 0.521 for CopulaOrigin and 0.662 forCOPE in Table 3.3), consistent with the over-correction in both CopulaOrigin and COPE.Reasons for the substantial difference in the estimates from the CopulaOrigin include (1)it’s ignoring correlated endogenous and exogenous regressors which can lead to inconsis-tent estimates, and (2) the unimodal close-to-normality distribution for the logarithm of the52price variable leading to potentially poor finite sample performance. In fact, the correlationsbetween logP∗ and the exogenous regressors are -0.44 for Bonus and -0.26 for PriceRedu,both of which are substantially larger than the corresponding correlations (-0.30 and -0.15,respectively) between logP and the exogenous regressors. The p-value for the null hypoth-esis of these correlations being zeros are significantly less than 0.05 (< 0.001), indicating aviolation of Assumption 4 required for CopulaOrigin to yield consistent estimates.Reasons for the substantial difference in the estimates from the COPE method include(1) a uni-modal close-to-normality distribution for the price variable leading to potentiallypoor finite sample performance of COPE, and (2) loss of estimation precision manifesteddue to a larger standard error of estimates as compared with those from 2sCOPE. By con-trast, the proposed 2sCOPE can relax the non-normality assumption of the endogenousregressor, and yield consistent and efficient estimates even if the endogenous regressor fol-lows a normal or nearly normal distribution. Moreover, 2sCOPE provides estimates withsmaller standard error than COPE, which confirms Theorem 4 showing that using two-stagecopula estimation reduces estimation variance.Unlike Store 1, the results from Store 2 indicate that the retail price is not endogenous.First, The estimates of ρ , which is the correlation between price and the error term, arenot significantly different from 0 for both CopulaOrigin and 2sCOPE (t-value ≤ 1.96 undercolumns “CopulaOrigin” and “2sCOPE” for Store 2 in Table 3.3), and only slightly signifi-cantly different from 0 for COPE (a t-value of 2.18, slightly larger than 1.96 under Column“COPE” in Table 3.3). The estimate of ρ for COPE, however, is questionable because ofthe limitations of COPE mentioned in the paragraph above. Second, the estimated pricecoefficient of OLS is -1.982, which is very close to the estimates of TSLS and 2sCOPE instore 1 and further confirms no endogeneity of price in store 2. Overall, the price elasticityestimates from TSLS and the three IV-free copulas-based methods are close to each otherfor Store 2, and the observed differences between them and the OLS estimate can be at-53tributed to estimation variability incurred from using more complicated models instead ofthe presence of endogeneity.In sum, the convergence of results between TSLS and the proposed method 2sCOPEin both stores supports the validity of the proposed method in addressing the endogeneityissue. Moreover, the difference between the estimates in COPE and 2sCOPE in store 1shows the advantages of 2sCOPE in terms of relaxing the non-normality assumption of theendogenous regressor and estimation efficiency gain by exploiting additional informationfrom correlated exogenous regressors.2.6 Economic Intuition and Practical Guidance of2sCOPEIn this section, we intuitively discuss how our proposed 2sCOPE works for correcting en-dogeneity, and the practical guidance of 2sCOPE.Intuitively, the proposed 2sCOPE method divides the error term into two parts, one beingan endogenous part, which is the first-stage residual that is correlated with the endogenousvariable P, and the other being an exogenous part. The 2sCOPE method corrects endogene-ity by directly controlling for the endogenous part in the error term. Suppose we are using2sCOPE to estimate the effect of price on quantity demanded, a classical marketing ques-tion. P is the endogenous price, W contains all observed variables that are correlated withprice and would affect demand (e.g., observed product attributes), and Y is the observeddemand. The remaining unobserved variables that also influence demand are all containedin the error term, and some that are correlated with the price can cause the endogeneityproblem. Those unobservables can be product attributes that are observed by customersbut not researchers (i.e., the smell of perfume products). Then in the first-stage regressionby regressing P∗ on W ∗, W ∗ helps to explain part of the price that influences demand, and54the residual can be interpreted as containing all the unobserved variables (e.g., unobservedproduct attributes) that cause the endogeneity plus the errors. Adding the residual as a gen-erated regressor in the demand model can be interpreted as controlling for all the unobservedvariables that cause the correlation between price and error term. Here, we take the effectof price on demand as an example. If supply-side data are observed, 2sCOPE can also beused to estimate the effect of price on supply.Our proposed 2sCOPE method empirically provides sound causal inference once theobservational data are given. Data is not limited to cross-sectional data, 2sCOPE can alsocorrect endogeneity for panel data. For panel data analysis, researchers use fixed effects tocontrol for endogeneity. But adding fixed effects might not be enough, and our 2sCOPEmethod can use copula to further correct endogeneity besides panel fixed effects. Moreover,2sCOPE is straightforward to be applied to a wider range of commonly used marketingmodels, including linear regression models, linear panel models with mixed effects, randomcoefficient logit models and slope endogeneity. Any model that can use the control functionapproach can in principle apply our method for correcting endogeneity.We have listed in the literature review different approaches for correcting endogeneity.Each method has its own assumptions. 2sCOPE method requires the normality assumptionon the error term (Villas-Boas and Winer 1999, Yang et al. 2003, Ebbes et al. 2005), andthe Gaussian copula relationship among regressors and the error term. Compare 2sCOPEwith IV approach, IV corrects endogeneity by using extra exogenous shock from the otherside while 2sCOPE identifies the model by controlling the endogenous part from exist-ing regressors themselves. The exclusion restriction is difficult to achieve in IV, while theGaussian copula structure required in 2sCOPE, which results in the normality assumptionof the first stage residual, is very general and robust (Danaher and Smith 2011). Eckert andHohberger (2022) further use simulations to explore the practical usefulness of GaussianCopula approach and compare it with OLS, IV regression, and Higher Moments (HM) esti-55mator. They find that the performance of Gaussian Copula approach is as good as a strongIV case when assumptions are met. Generally speaking, our 2sCOPE method can be usedin practice when it’s hard to find a strong instrumental variable or when experiments arenot available to conduct. But future research is needed for comparing more methods andproviding clearer guidance on when to use our 2sCOPE method.2.7 ConclusionCausal inference lies at the center of social science research, and observational studies of-ten beg rigorous post-study designs and methodologies to overcome endogeneity concerns.In this paper, we focus on the instrument-free copula method to handle the problem ofendogenous regressors. We propose a generalized two-stage copula endogeneity correc-tion (2sCOPE) method that overcomes two key limitations of the existing copula-basedmethod in Park and Gupta (2012) (CopulaOrigin), and extends CopulaOrigin to more generalsettings. Specifically, 2sCOPE allows exogenous regressors to be correlated with endoge-nous regressors and relaxes the nonnormality assumption on the endogenous regressors. Todemonstrate the benefits of 2sCOPE, we compare it with the direct extension of CopulaOriginmethod, called COPE. Similar to the CopulaOrigin, 2sCOPE method corrects endogeneity byadding “generated regressors” derived from the existing regressors and is straightforward touse. COPE is a direct extension to CopulaOrigin by adding latent copula transformation ofexisting regressors, while 2sCOPE has two stages and adds the residuals from regressinglatent copula data for the endogenous regressor on the latent copula data for the exogenousregressors as a “generated regressor” in the structural regression model. We theoreticallyprove that 2sCOPE can yield consistent causal-effect estimates when exogenous regres-sors are correlated with the endogenous regressors, which can cause biased estimates in themethod of Park and Gupta (2012). Moreover, the 2sCOPE method can further relax thenonnormality assumption on the endogenous regressors and improve estimation efficiency.56We conduct simulation studies and use an empirical marketing application to empiricallyverify the performance of our proposed method. The simulation results show that 2sCOPEyields consistent estimates under relaxed assumptions. Moreover, 2sCOPE method outper-forms COPE in terms of dealing with normal endogenous regressors and improving estima-tion efficiency. Endogenous regressors are allowed to be normally distributed as long as oneof the exogenous regressors is nonnormally distributed, which is a very weak assumption.The efficiency gain is substantial and can be up to ∼50%, meaning that the sample size canbe reduced by ∼50% to achieve the same estimation efficiency as compared with COPEmethod which does not exploit the correlations between endogenous and exogenous regres-sors. Last but not least, our robustness checks show that the proposed method 2sCOPE isreasonably robust to the structural error distributional assumption and non-Gaussian copulacorrelation structure (see details in Appendix). We further apply our 2sCOPE method to acommonly used public dataset in marketing. When dealing with endogenous price, we findthat the estimated price coefficient using our proposed 2sCOPE is very close to the TSLSestimate, while OLS and CopulaOrigin show large biases. Moreover, the difference betweenthe results of 2sCOPE and COPE demonstrates the advantage of 2sCOPE in dealing with(nearly) normal endogenous regressors and improving estimation efficiency.These findings have rich implications for guiding the practical use of copula-basedinstrument-free methods to handle endogeneity. A known critical assumption for CopulaOriginis the non-normality of endogenous regressors. The users of the method in the literature haveall been practicing the check and verification of this assumption. However, our work showsthat this is insufficient: one also needs to check Assumption 4 for the one-endogenous-regressor case, and Assumption 4(b) for the multiple-endogenous-regressors case. Notethat neither assumption is the same as checking the pairwise correlations between the en-dogenous and exogenous regressors. Assumption 4 evaluates pairwise correlations involv-ing copula transformation of the endogenous regressor, which, as shown in our empirical57application, can be substantially different from the pairwise correlations using the regres-sor itself (Danaher and Smith 2011). Assumption 4(b) evaluates the correlations betweenexogenous regressors and the linear combination of generated regressors, which are evenmore different from checking pairwise correlations on the regressors themselves. When theabove assumptions are satisfied, CopulaOrigin is preferred to our proposed 2sCOPE method,since the simpler and valid model outperforms more general but more complex models.If any endogenous regressor fails to have sufficient departure from being normally dis-tributed, or any exogenous regressor violates Assumption 4 or 4(b), our proposed 2sCOPEmethod should be used instead of CopulaOrigin. The 2sCOPE is straightforward to extend tomany other settings, and we have derived 2sCOPE for a range of commonly used marketingmodels, including linear regression models, linear panel models with mixed-effects, randomcoefficient logit models and slope endogeneity. The 2sCOPE method proposed here can beapplied to these cases and many other cases not studied here, while accounting for corre-lations between exogenous and endogenous regressors and exploiting the correlations formodel identification in the presence of insufficient non-normality of endogenous regressors.Although the proposed 2sCOPE contributes to the literature by relaxing key assump-tions of the existing copula method CopulaOrigin and extending it to more general settings,it is not without limitations. For the 2sCOPE to work best, the distributions of the en-dogenous regressors need to contain adequate information. The condition is violated whenthe endogenous regressors follow Bernoulli distributions or discrete distributions with smallsupport, as noted in Park and Gupta (2012). The proposed 2sCOPE method does not addressthis limitation. Developing instrument-free methods to handle such inadequately distributedendogenous regressors is an important topic for future research. The simplicity of 2sCOPEhinges on the normal structural error and Gaussian copula dependence structure. Although2sCOPE demonstrates reasonable robustness to departures from these assumptions as shownin the Appendix, future research is needed for more flexible methods testing and relaxing58these assumptions. Despite these limitations, we expect that the proposed 2sCOPE will pro-vide a useful alternative to a broad range of empirical problems when instruments are notavailable.59Chapter 3Lasso-based Instrument-free CausalInference3.1 IntroductionCopula-based instrument-free method to correct endogeneity and draw causal inference hasaroused wide interest because of its simplicity to implement as no instruments are needed.Yang et al. (2022) proposed a general two-stage copula method called 2sCOPE to accountfor endogeneity by adding a generated regressor, which is the first-stage residual, to thestructural regression model, analogous to the control function approach. The statistical toolcopula is a cumulative distribution function used to model multiple variables jointly. Yanget al. (2022) used copula to model the joint distribution among the endogenous regressors,exogenous regressors and the error term, and obtain the first-stage residual by regressinglatent copula data for each endogenous regressor on the latent copula data for the exogenousregressors. Intuitively, the residual in the first stage contains all the unobserved variables thatmake the endogenous regressor and the error term correlated. By controlling the residual,60endogeneity can be corrected because all the unobservables that cause the endogeneity arecontrolled for. Thus, the residual plays a central role in correcting for endogeneity using2sCOPE, and the wrong estimation of the residual can cause bias in the estimation of theendogenous regressor.A problem empirical researchers face when using the conditional-on-residual identifica-tion strategy (i.e., control function approach) is that the residual is unknown and is relyingon knowing what variables to include in the first-stage regression. The problem is inevitable,especially for high-dimensional data, which are becoming increasingly common in this bigdata era. Belloni et al. (2014a) pointed out that high-dimensional data can arise throughthe inherent high-dimensional features. Examples are conventional data such as censusand survey data, scanner data and genomics data (Peng et al. 2010) , and unconventionaldata that is too high-dimensional for standard estimation methods, including image (Zhanget al. 2017) and language/text data (Amado et al. 2018) that we conventionally had not eventhought of as data we can work with (Mullainathan and Spiess 2017). We cannot avoidhigh-dimensional data in empirical analysis. Sometimes economic intuition would help insuggesting a set of variables that might be important, but it cannot identify exactly whatvariables are important. With high-dimensional data and too many irrelevant variables, thetraditional endogeneity-correction methods may have poor performance with finite-samplebias. Moreover, the dimension of variables can even be larger than the sample size, makingthe traditional estimation methods infeasible.In this paper, we combine the causal inference method with machine learning techniquesto deal with the high-dimensional problem in drawing causal inference. Machine learning(ML) is a powerful tool for data analysis, especially for large or high-dimensional datasets. Researchers are recently trying to apply the powerful machine learning tool to diverseareas, such as Economics (Athey and Imbens 2019), Marketing (Cui et al. 2006, Ascarza2018, Ngai and Wu (2022)) and operation research (Feldman et al. 2022). However, unlike61the single-equation prediction setting in most papers that apply the ML tool, our causalinference setting, which takes a two-stage structure, makes the application of ML differentand more complicated. Several papers recently have tried to apply ML to some economicmodels for causal inference. For example, Belloni et al. (2012) applied ML instrumentalvariable models, and Belloni et al. (2014a) and Belloni et al. (2014b) used ML to studytreatment effects.Our paper studies the two-stage instrument-free method 2sCOPE to correct endogene-ity with machine learning techniques. Specifically, we use lasso-based feature selectionmethods for the first-stage estimation of 2sCOPE because the copula relationship amongthe endogenous and exogenous regressors in the first stage are linearly correlated. Lassomethod has been widely used for feature selection in the literature (Tibshirani 1996, Belloniet al. 2012, Belloni et al. 2014a, Belloni et al. 2014b, Belloni and Chernozhukov 2013, Baiand Ng 2008, Bai and Ng 2009, Javanmard and Montanari 2014a). But since Lasso selectsimportant variables by adding a penalty term of non-zero coefficients to the least squares op-timization function, the Lasso estimator has estimation bias. After noticing this drawback,researchers are devoted to methodology development in alleviating or even eliminating thebias. For example, Belloni et al. (2012) proposed a post-Lasso method to alleviate the biasof the Lasso estimator, and Javanmard and Montanari (2014a) proposed a de-biased lassomethod. In this chapter, we propose a method combining the 2sCOPE method with dif-ferent Lasso-based methods (Lasso, post-Lasso, de-biased Lasso) in the estimation of first-stage residual, and examine how Lasso-based methods work for correcting endogeneity anddrawing causal inference for high-dimensional data. We demonstrate the performance of theproposed lasso-based 2sCOPE, compared with the regular 2sCOPE method without featureselection methods in the first stage, via simulation studies and real-data application. Thesimulation result shows that using Lasso-based methods in the first stage can improve theestimation accuracy and efficiency by around 50%, as compared with the regular 2sCOPE62method in high-dimensional data. We further apply our lasso-based 2sCOPE method toreal data, examining how governments’ policy stringency in the COVID-19 period affectscitizens’ happiness. We use country-level cross-sectional data for analysis. The dimensionof country-level characteristics is relatively large, around half of the sample size. The es-timation result shows that using Lasso-based methods in the first stage makes the effectof policy stringency on happiness more significantly negative compared with the regular2sCOPE method without feature selection. Moreover, the estimates using different Lasso-based methods are quite robust.The remainder of this paper unfolds as follows. Section 3.2 reviews the related litera-ture on causal inference with feature selection. In Section 3.3, we propose a method thatcombines 2sCOPE with lasso-based methods. In Section 3.4, we evaluate the performanceof our proposed lasso-based 2sCOPE method using simulation studies and compare the per-formance with the 2sCOPE method without feature selection. In Section 3.5, we apply theproposed lasso-based 2sCOPE method to examine the effect of policy strictness of COVID-19 on citizens’ happiness using high-dimensional country-level cross-sectional databases.We conclude the paper and discuss future work in Section 3.6.3.2 Literature ReviewThere is a large literature in marketing, economics and statistics, focusing on approaches toaddressing endogeneity when inferring causal effects. Among them, the most commonlyused approach is the instrumental variable approach (Angrist and Krueger 2001, Rossi2014). Though the theory of instrumental variables is well-developed, good instrumentsare extremely difficult to find in practice. Rossi (2014) summarized the most frequentlyused instrumental variables in the literature, and pointed out that many of them are weakIVs with poor performance or even fail the validity (exclusion restriction) condition. This63forces researchers to seek for new methodologies to correct endogeneity and draw causalinference.One stream of methodologies called the instrument-free method has recently arousedwide interest. Literally speaking, instrument-free method means no instrumental variablesare needed. Park and Gupta (2012) proposed a method using the statistic tool copula tocorrect endogeneity by directly modeling the association between the endogenous variableand the error term via copula. By doing so, it can address the lack of suitable instrumentsissue using information in the existing regressors themselves without any extra information.After that, the copula method has been rapidly adopted by researchers to deal with the endo-geneity problem because of its feasibility (Burmester et al. 2015, Datta et al. 2015, Gruneret al. 2019, Keller et al. 2019, Bombaij and Dekimpe 2020, Guitart et al. 2018, Lameyet al. 2018, Wetzel et al. 2018, Heitmann et al. 2020, Atefi et al. 2018, Elshiewy and Boztug2018). Regarding methodology development, Yang et al. (2022) and Haschka (2021) furtherextend the method to a more general setting. Both the papers allow exogenous regressorsthat are correlated with the endogenous regressor to be included in the model, while theoriginal paper doesn’t allow that. But compared with Haschka (2021), the method proposedin Yang et al. (2022) has weaker assumptions on the regressors, is simpler to implementand can be applied to a wider range of models because it uses the control function approachby adding a generated regressor to the outcome regression. However, none of the papersstudies causal inference when high-dimensional data are present, which is quite common inthis big data era. The performance of the method relies on the first-stage residual (in thecontrol function approach), conditional on which the endogeneity can be corrected. But theresidual is unknown and needs estimation. With too many irrelevant controls in the firststage, the performance of first-stage regression can be poor with finite-sample bias, thuscausing amplifying bias in the estimation of the endogenous variable.Moreover, sometimesthe dimension of controls can even be larger than the sample size, making the traditional64estimation methods infeasible. To solve the high-dimensional problem, we propose a com-bined 2sCOPE method in Yang et al. (2022) with lasso-based feature selection methods thathelp to select important control variables in the first stage Our paper contributes to the lit-erature on instrument-free approach to addressing the endogeneity problem by combiningwith machine learning methods for estimating the first-stage regression on high-dimensionalexogenous control variables.Another related literature is feature selection in high-dimensional data for causal infer-ence. There is a large and growing literature on Lasso-based methods for feature selection(Tibshirani 1996, Belloni et al. 2012, Belloni et al. 2014a, Belloni et al. 2014b, Belloni andChernozhukov 2013, Bai and Ng 2008, Bai and Ng 2009, Javanmard and Montanari 2014a).Recently, researchers start to consider applying Lasso-based methods to economic modelsfor causal inference. For example, Belloni et al. (2012) proposed a Lasso-based method forinstrument selection in linear instrumental variables models with many instruments. Belloniet al. (2014b) and Belloni et al. (2014a) used Lasso methods to study the treatment effectswith high-dimensional controls in a partially linear model. But the challenge is that whenthe goal is causal inference, model selection can be problematic. The model selection pro-cedures are originally designed for forecasting (prediction). When doing model selectionfor causal inference, it’s a two-stage model instead of just a one-equation prediction, andthe model selection method is acting directly on the endogenous variable in the first stage.One cannot guarantee that exactly all the variables with nonzero coefficients are perfectlyselected, and the omission of some variables with small effects can contaminate causal in-ference results based on the selected set of variables (Leeb and Po¨tscher 2008a, Leeb andPo¨tscher 2008b). As mentioned previously, several papers have tried applying machinelearning methods to draw the causal inference. Belloni et al. (2012) studied instrument se-lection in instrumental variables models with many instruments. Belloni et al. (2014b) andBelloni et al. (2014a) studied the treatment effects with high-dimensional controls.65In this paper, we address the endogeneity problem using an instrument-free approach2sCOPE with high-dimensional exogenous variables. We apply Lasso-based methods inthe first stage to select important and meaningful controls in explaining the endogenousvariable and use the estimated residual to draw causal inference. It’s natural to use theLasso-based methods because of the linear relationship among the copula-transformed re-gressors in the first stage. But one drawback of the lasso method is that it has shrinkage bias.Belloni et al. (2012) proposed a post-Lasso method to alleviate the bias, and Javanmardand Montanari (2014a) developed a de-biased Lasso method. But no paper has examinedhow those lasso-based methods work in the instrument-free model to correct endogeneitywith high-dimensional data. Our paper contributes to the feature selection for causal in-ference literature by being one of the first to study feature selection in the copula-basedinstrument-free method. We propose a combined lasso-based method with 2sCOPE, andcompare the performance using different Lasso-based methods to draw causal inferences inthe instrument-free approach setting.3.3 MethodsIn this section, we develop a copula-based instrument-free method that is combined withmachine learning methods to handle endogenous regressors when the dimension of the ex-ogenous variables is huge, relative to the sample size. Specifically, we build on the two-stageCopula (2sCOPE) method that deals with endogeneity (Yang et al. 2022) and then use theLasso-based methods to select important features among the exogenous variables in the firststage to better explain the endogenous variable and control the endogeneity.663.3.1 Copula Endogeneity Correction Method (2sCOPE) in StandardCaseWe first introduce the concept of copula and review the 2sCOPE method (Yang et al. 2022)in the standard low-dimension case. Consider the following linear structural regressionmodel with an endogenous regressor and a vector of exogenous regressors:Yt = µ+Ptα+W ′t β +ξt (3.1)where t = 1,2, ...,T indexes either time or cross-sectional units, Yt is a (1×1) dependentvariable, Pt is a (1×1) endogenous regressor, Wt is a (k×1) vector of exogenous regressors,ξt is the structural error term, and (µ,α,β ) are model parameters. Pt is correlated with ξt ,which generates the endogeneity problem.The key idea of the 2sCOPE method is to divide the error term into an endogenous andan exogenous part by jointly modeling the relationship among the endogenous regressorPt , exogenous regressors Wt and the error term ξt using copula, and then control for theendogenous part, the first-stage residual, to correct endogeneity. The statistical tool copulais a cumulative distribution function used to model multiple variables jointly. It outperformsthe traditional multivariate distribution in that the marginal distributions are not needed tobe from the same distribution family. (ξt ,Pt ,Wt) is assumed to follow Gaussian Copula forits many desirable properties (Danaher 2007; Danaher and Smith 2011).In statistics, a copula is a multivariate cumulative distribution function that models thedependence among variables without imposing assumptions on marginal distributions. LetF(P,W,ξ ) be the joint cumulative distribution function (CDF) of the endogenous regressorPt and the structural error ξt with marginal CDFs H(P), L(W ) and G(ξ ), respectively. Fornotational simplicity, we may omit the index t in Pt and ξt below when appropriate. Ac-cording to Sklar’s theorem (Sklar 1959), there exists a copula function C(·, ·) such that for67all P, W and ξ ,F(P,W,ξ ) =C(H(P),L(W ),G(ξ )) =C(Up,Uw,Uξ ), (3.2)where Up = H(P), Uw = L(W ) and Uξ = G(ξ ), and they all follow uniform(0,1) distribu-tions. Thus, the copula maps the marginal CDFs of the endogenous regressor, exogenousregressors and the structural error to their joint CDF, and makes it possible to separatelymodel the marginals and correlations of these random variables.To capture the association between the endogenous regressor P, Wt and the error ξ , weuse the Gaussian copula (Park and Gupta 2012, Haschka 2021, Yang et al. 2022) for itsmany desirable properties (Danaher and Smith 2011):F(P,W,ξ ) =C(Up,Uw,Uξ ) =ΨΣ(Φ−1(Up),Φ−1(Uw),Φ−1(Uξ ))=1√(2π)3|Σ|∫ Φ−1(Up)−∞∫ Φ−1(Uw)−∞∫ Φ−1(Uξ )−∞exp[−[s, t,q]′Σ−1[s, t,q]2]dsdtdq,(3.3)whereΦ(·) denotes the univariate standard normal distribution function andΨΣ(·, ·) denotesthe multivariate standard normal distribution with the covariance matrix Σ. In the Gaussiancopula model, Σ captures the correlation among variables.The Gaussian copula assumption on (Pt ,Wt ,ξt) is equivalent to that [P∗t ,W ∗t ,ξ ∗t ] followsthe multivariate normal distribution:P∗tW ∗tξ ∗t ∼ N000 ,1 ρpw ρpξρpw 1 0ρpξ 0 1 , (3.4)where P∗t = Φ−1(H(Pt)), W ∗t = Φ−1(L(Wt)), and ξ ∗t = Φ−1(G(ξt)). H(·), L(·) and G(·)68are marginal CDFs of Pt , Wt and ξt respectively. We call P∗t ,W ∗t ,ξ ∗t the Gaussian copulatransformations of Pt Wt , ξt , respectively.The above multivariate distribution can be rewritten as follows:P∗tW ∗tξ ∗t=1 0 0ρpw√1−ρ2pw 0ρpξ−ρpwρ pξ√1−ρ2pw√1−ρ2pξ −ρ2pwρ2pξ1−ρ2pw ·ω1,tω2,tω3,t ,ω1,tω2,tω3,t∼ N000 ,1 0 00 1 00 0 1 . (3.5)Then, we have a linear relationship between P∗ and W ∗, and here comes the first-stageregression,P∗t = ρpwW∗t + εt (3.6)and the structural error in Equation (3.1) can be re-expressed asξt = σξ ·ξ ∗t =σξρpξ1−ρ2pwP∗t +−σξρpwρpξ1−ρ2pwW ∗t +σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pwω3,t ,=σξρpξ1−ρ2pw(P∗t −ρpwW ∗t )+σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pw ·ω3,t ,=σξρpξ1−ρ2pwεt +σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pw ·ω3,t (3.7)In this way, the structural error term ξt is split into two parts: one part is the first-stageresidual, which is a function of P∗t and W ∗t that captures the endogeneity of Pt and theassociation of Wt with ξt |Pt 1, and the other part is an independent new error term. Then, we1Although the exogenous regressor Wt and ξt are uncorrelated, Wt and ξt |Pt (the error component in ξtremaining after removing the effect of the endogenous regressor Pt ) can be correlated.69substitute Equation (A.2) into the main model in Equation (3.1), and obtain the followingregression equation:Yt = µ+Ptα+Wtβ +σξρpξ1−ρ2pwεt +σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pw ·ω3,t . (3.8)Equation (3.8) suggests adding the estimate of the residual εt from the first stage regres-sion (Equation 3.6) as a generated regressor to the outcome regression. Adding the firststage helps model identification with extra information, the correlation between P∗t and W ∗t ,making the new error term ω3,t to be uncorrelated with all the regressors in Equation (3.8)and thus consistent estimates can be obtained. This two-step 2sCOPE method adds thefirst-stage residual term ε̂t to control for endogeneity and in this aspect is similar to thecontrol function approach of Petrin and Train (2010), while requires no use of instrumentalvariables.Intuitively, the usage of the copula function allows us to directly model the relationshipamong the endogenous regressors, the observed exogenous regressors and the structuralerror. The correlated exogenous regressors added in the first stage help to capture and ex-plain part of the endogenous regressor. By controlling the residual εt obtained from the firststage, we actually control all the unobserved variables that cause the endogeneity problem,and thus can correct endogeneity and get consistent estimates.However, it becomes problematic when we have a large dimension of Wt , and we don’tknow which ones in Wts are important in explaining the endogenous regressor. This problemis similar to the many-instruments problem (Belloni et al. 2012). A naive approach is toinclude all the possible exogenous variables in the first stage (regular 2sCOPE). However,doing so would make the performance of finite-sample prediction worse and inefficient.In the section below, we propose a lasso-based method to select important features in thefirst-stage of 2sCOPE in helping explain the endogenous regressor.703.3.2 2sCOPE with Lasso and post-LassoIn this subsection, we propose a method to deal with the case when the dimension of ex-ogenous regressors is huge, but only a few are correlated with the endogenous regressor.This is actually very common in practice. Specifically, we combine the 2sCOPE method indealing with endogeneity with a feature selection machine learning method in the first stageto select important Wts when the dimension of Wt is large relative to the sample size. If toomany irrelevant variables are included in the first stage to explain the endogenous regressor,it would affect both the finite-sample estimation accuracy and efficiency.To eliminate the too-many-control issue, we use a Lasso method in the first stage ofthe 2sCOPE method in Equation (3.6). Under the Gaussian copula assumption in Equation(3.4), we have a linear relationship between P∗t and all the W ∗t s,P∗t =W∗t γ+ εtwhere P∗t and W ∗t are the Gaussian copula transformations of Pt and Wt , respectively. Con-sider the usual least squares optimization function:Qˆ(γ) :=12n∥P∗−W ∗γ∥22, (3.9)Since the dimension of W is large, which can even be larger than the sample size, we need toselect important features to obtain better estimation in the first stage, and thus improve theestimation accuracy and efficiency of the endogenous regressor to draw the causal inference.Because of the linear relationship between P∗ and W ∗ and the sparsity of important W s, it’svery natural to use Lasso regression to penalize too many parameters. Lasso solves forregression coefficients by minimizing the sum of the usual least squares objective functionand a penalty for model size. The Lasso estimator (Tibshirani 1996) is defined as a solution71of the optimization program below:γˆl(P∗,W ∗;λ ) = arg minγ∈RkQˆ(γ)+λ∥γ∥1, (3.10)where λ is the penalty level. In the estimation procedure, we use cross validation to tunethe penalty level. Specifically, we set λ = λmin, for which we can get the minimum meancross-validated error. Using Lasso method to form first-stage prediction provides an ef-fective approach to obtain efficiency gains or even accuracy for finite sample estimation.However, since the Lasso estimator reduces the dimension of variables and estimates thefirst-stage regression coefficients by adding a penalty of non-zero coefficients based on theusual least squares objective function, the estimates of Lasso would be biased. Thus, wefurther use a post-lasso method to alleviate the bias. The post-Lasso estimator discards theLasso estimates, but will take advantage of variables selected by Lasso and then refit thefirst-stage regression using the selected variables via OLS estimation.Denote Il the variables selected among W ∗ in the first stage using the Lasso method.The post-Lasso estimator is defined as the ordinary least squared estimator among variablesIpl ⊇ Il . Ipl contains the variables in Il and some meaningful variables according to real dataapplication (Im). The post-Lasso estimator γˆpl isγˆpl ∈ arg minγ∈Rk:γ(Iˆl∪Im)c=0Qˆl(γ) (3.11)The Lasso and post-Lasso methods are motivated by the desire to predict the first-stageresidual well without overfitting, and thus gain more estimation accuracy and efficiency forcorrecting the endogeneity. It can also deal with the case when the dimension of variablesis even larger than the sample size, which would cause a problem for OLS estimation. Oncewe get estimates using Lasso and post-Lasso in the first stage, we calculate the residualand then add it as a generated regressor to the outcome regression model. In this way, we72combine the 2sCOPE method with the lasso-based machine learning methods to deal withthe high-dimension case.3.3.3 2sCOPE with De-biased LassoAs mentioned above, though the Lasso method is powerful in reducing the dimension ofvariables, the Lasso estimator is biased. The post-Lasso method introduced above can al-leviate the bias to some extent by running OLS estimation after variables are selected byLasso. However, the post-Lasso estimator can still have bias. In this section, we intro-duce the de-biased Lasso method proposed by Javanmard and Montanari (2014a) that caneliminate the bias using Lasso-based methods, and combine it with the 2sCOPE method toaddress the endogeneity problem with high-dimensional exogenous controls.The de-biased Lasso estimator γˆdl of the first-stage regression is,γˆdl = γˆl(λ )+1nM(W ∗)T (P∗−W ∗γˆl(λ )). (3.12)where γˆl(λ ) is the Lasso estimator in Equation (3.10), and M is a chosen matrix depend-ing on the P∗. We can find that the second part 1nM(W∗)T (P∗−W ∗γˆl(λ )) is proportionalto a subgradient of the l1 norm at the Lasso estimator, (W ∗)T (P∗−W ∗γˆl(λ ))/(nλ ). Sointuitively, the de-biased lasso estimation γˆdl can compensate the bias introduced by the l1penalty in the lasso by adding a term proportional to the subgradient.The quality of the debiasing procedure depends on the choice of M. Javanmard andMontanari (2014b) suggests the choice M = cΣ−1 with Σ = E{P∗(P∗)T} being the pop-ulation covariance matrix and c a positive constant. Javanmard and Montanari (2014a)73suggested setting M = (m1,m2, ...,mk)T by solving a convex program below.minimize mT Σˆmsubject to ∥Σˆm− ei∥∞ ≤ µ, (3.13)where ei ∈ Rk is the vector with one at the ith position and zero everywhere else, and Σˆ =((P∗)T P∗)/n. The benefit of this choice M is that the method requires weaker assumptionson P∗. It can be applied to general covariance structures Σ, while Javanmard and Montanari(2014b) can only be applied to sparse Σ. In this paper, we use the de-biased lasso estimatorin Javanmard and Montanari (2014a) (Equation 3.12) with M solved by Equation 3.13 forthe first-stage estimation in 2sCOPE in Equation (3.6).3.4 Simulation StudyIn this section, we conduct Monte Carlo simulation studies for the following goals: (a) toassess the performance of the proposed method for high-dimensional correlated regressors,and (b) to compare the performance of the proposed methods with existing methods.We examine the case when the dimension of W s is large relative to the sample size, andonly a few of them play important roles in explaining endogeneity. Specifically, we generatethe sample size N = 1000 and the dimension of W , Nw = 600. The specific data-generatingprocess (DGP) is summarized below:P∗tW ∗tξ ∗t∼ N000 ,1 ρpw ρpξρpw 1 0ρpξ 0 1 (3.14)74ξt = G−1(Uξ ,t) = G−1(Φ(ξ ∗t )) =Φ−1(Φ(ξ ∗)) = 1 ·ξ ∗t , (3.15)Pt = H−1(UP,t) = H−1(Φ(P∗t )), Wt = L−1(UW,t) = L−1(Φ(W ∗t )), (3.16)Yt = µ+α ·Pt +β ·Wt +ξt = 1+1 ·Pt +(−1) ·Wt +ξt . (3.17)where ξ ∗t and P∗t are correlated with the correlation coefficient ρpξ = 0.2, and thus ξt andPt are correlated, generating the endogeneity problem. W ∗t s are exogenous and are notcorrelated with ξ ∗t . W ∗t and P∗t are set to be correlated with different correlation levels,which determines how important each W ∗ is in explaining the endogenous P. Specifically,we set two different levels of the correlation, ρpw ∈ {0.1,0.3}. Among all the possible W ∗s,only 13 are important ones that are correlated with P∗ with non-zero ρpw, among which threeof them are the most important ones with correlation ρpw = 0.3. For simplicity, we assumethat there is no correlation among W ∗s. We consider six different estimation methods:(i) OLS, (ii) 2sCOPE with all W ∗t s included in the first stage (2sCOPEAll), (iii) 2sCOPEwith selected W ∗t s using Lasso with penalty parameter λ = λmin (Lasso), (iv) 2sCOPE withselected W ∗t s using post-Lasso (post-Lasso), (v) 2sCOPE with selected W ∗t s using De-biasedLasso (De-biased Lasso), and (vi) 2sCOPE with optimal W ∗t s (Golden).We generate 1000 data sets as replicates using the DGP above. In the simulation, weuse the gamma distribution Gamma(1,1) with shape and rate equal to 1 for Pt and theexponential distribution Exp(1) with rate 1 for Wt . Models are estimated on all generateddata sets, providing the empirical distributions of the parameter estimates.Table 3.1 reports estimation results. In the table, we only list the first three (β1,β2,β3) ofall the 600 Wt coefficients to save space. As expected, OLS estimates of all µ , α and β arebiased as a result of the regressor endogeneity. However, unexpectedly, the estimation resultof 2sCOPEAll has a significant bias, especially for the estimated coefficient of Pt (0.118 awayfrom the true value). This tells us that, for finite sample estimation, including too manyirrelevant or unimportant variables in the first stage of the model would make the estimation75OLS 2sCOPEAll Lasso post-Lasso De-biased Lasso GoldenParameters True Mean SE Mean SE Mean SE Mean SE Mean SE Mean SEµ 1 1.166 1.099 1.088 1.105 1.187 1.085 1.061 1.072 1.045 1.078 1.013 1.078α -1 -0.749 0.060 -0.882 0.082 -1.019 0.105 -1.006 0.104 -0.981 0.100 -1.009 0.102β1 1 0.931 0.049 0.966 0.051 0.988 0.054 1.002 0.052 0.992 0.054 0.999 0.055β2 1 0.934 0.050 0.968 0.050 0.990 0.052 0.996 0.049 0.994 0.053 1.001 0.053β3 1 0.933 0.052 0.967 0.054 0.989 0.055 0.995 0.050 0.993 0.056 1.000 0.056MSE(α) 0.0668 0.0207 0.0114 0.0108 0.0104 0.0105MSE(ALL) 0.00420 0.00399 0.00385 0.00384 0.00383 0.00380Table 3.1: Simulation Results of Lasso-based 2sCOPENote: Mean and SE denote the average and standard deviation of parameter estimates over all the 1,000simulated samples.biased. This actually demonstrates the importance of using feature selection in the firststage to get unbiased causal estimates in the high-dimensional case. We next examine somelasso-based machine learning methods. All three types of lasso methods (Lasso, post-Lassoand De-biased Lasso) can significantly reduce estimation biases. Moreover, post-Lassocan reduce more bias and get better performance in estimation accuracy than Lasso. Theestimated coefficient α using post-Lasso is -1.006, very close to the true value. The meansquared error (MSE) is 0.0108, which is 47.8% smaller than the MSE using 2sCOPEAll . Itmeans that using post-Lasso in the first stage can improve estimation efficiency by around47.8%, compared with the naive approach of including all variables (2sCOPEAll). De-biasedLasso shows comparable performance with post-Lasso. The estimated coefficient α usingDe-biased Lasso is -0.981, and the mean squared error (MSE) is 0.0104, which is 49.8%smaller than the MSE using 2sCOPEAll .763.5 Empirical ApplicationIn this section, we apply our method to a real data application. We are interested in howCOVID-19 changes individual life. Specifically, we want to examine how governments’policies to contain the spread of COVID-19 affect citizens’ happiness. We use the Strin-gency Index 2 constructed by Oxford Coronavirus Government Response Tracker (Ox-CGRT) to measure the strictness of policy. The COVID-19-related data is from the OurWorld in Data team and the Covid-19 Data Repository by the Center for Systems Scienceand Engineering (CSSE) at Johns Hopkins University (JHU) 3. The data contains daily con-firmed COVID-19 cases, daily stringency level, etc. We further collect happiness data fromthe World Happiness Report 4, and country-level characteristics from the World Bank 5 andFraser Institute 6. Country-level characteristics include economy (GDP, inflation), popula-tion, gender, labor market-related, governance-related, legal system characteristics and soon.Government policies in response to COVID-19 can be endogenous. The endogeneitycan come from unmeasured country characteristics such as ideology, democracy, and free-dom that can influence both people’s happiness and governments’ decisions. Since thesevariables are unobserved by researchers, they are absorbed into the structural error, leadingto the endogeneity problem. Besides the endogenous stringency index, many country-levelexogenous variables are included. In total, there are 41 country characteristics and 102country observations. Table 3.2 summarizes the statistics of several key variables.2The nine metrics used to calculate the Stringency Index are: school closures; workplace closures; cancel-lation of public events; restrictions on public gatherings; closures of public transport; stay-at-home require-ments; public information campaigns; restrictions on internal movements; and international travel controls.https://ourworldindata.org/covid-stringency-index3COVID-19 data: https://github.com/owid/covid-19-data/tree/master/public/data.4https://worldhappiness.report;5https://databank.worldbank.org/source/world-development-indicators6https://www.fraserinstitute.org/economic-freedom/map?geozone=worldpage=mapyear=2019.77Variables Mean SD Max MinHappiness 0.49 0.13 0.67 0.08HappinessGrowth 0.07 0.03 0.19 -0.03Stringency/10 6.12 1.23 8.84 1.51LogCases/million 8.87 1.83 11.01 2.11LogPopuden 4.34 1.42 8.98 0.68LogGDP 9.66 1.09 11.50 7.17LogGDP Growth -0.04 0.04 0.06 -0.23Table 3.2: Summary StatisticsFigure 3.1 shows data evidence of the relationship between happiness and stringencylevel. It shows that both the happiness and the happiness growth in the year 2020 are de-creasing with the stringency level. The stricter the policy in response to COVID-19, the lesshappy people will be.(a) Happiness (b) Happiness GrowthFigure 3.1: Relationship Between Happiness and Stringency Index.78We then consider the following linear regression model:HappyGrowthi = β0+StringencyChangei ·β1+W ′i β2+ξi, (3.18)where i = 1,2, ..., I indexes country. Wi includes all country characteristics, such as changeof confirmed COVID-19 cases, GDP growth, population, freedom and legal-system-relatedcharacteristics in this application. We are interested in examining how the stringency changeduring the COVID-19 period and country characteristics affect people’s happiness growth.(a) Stringency Index (b) Happiness (c) HappinessGrowth(d) LogCases permillion (e) Log Populationdensity (f) Log GDPFigure 3.2: Histogram of Stringency, Happiness, and Country Characteristics in 2020Figure 3.2 shows the histograms of the Stringency index and some country-level char-acteristics. All the variables are continuous. Different country characteristics can havedifferent correlations with the endogenous stringency index, which means they might havedifferent importance levels in explaining the endogenous variable. For example, the cor-relation between stringency and log number of confirmed COVID-19 cases/million is 0.2479(p-value = 0.017). The correlation between stringency and logGDP growth is -0.27 (p-value = 0.006). Both the correlations are significantly different from zero. These are twocharacteristics that have high correlations with the endogenous variable, and many charac-teristics don’t have high pairwise correlations with the endogenous stringency. Importantly,whether a variable is important in explaining the endogenous variable is hard to determine.It is complicatedly affected by all variables together, and the simple pairwise correlationmay not fully measure the importance. That’s why we use the lasso-based methods to fullymeasure and select important features in explaining the endogenous stringency in the firststage.The many exogenous variables and the different correlations between the endogenousvariable and exogenous variables provide a good setting for examining the performanceof our proposed lasso-based 2sCOPE method. We estimate the linear regression model inEquation (3.18) using the OLS, the 2sCOPE method with all exogenous variables includedin the first stage (2sCOPEAll), the 2sCOPE method using Lasso in the first stage (Lasso),and the 2sCOPE method using De-biased Lasso in the first stage (De-biased Lasso).OLS 2sCOPEAll Lasso post-Lasso De-biased LassoParameters Est SE t-value Est SE t-value Est SE t-value Est SE t-value Est SE t-valueConstant 0.226 0.153 1.479 0.219 0.158 1.385 0.266 0.167 1.597 0.249 0.157 1.590 0.242 0.153 1.581Stringency -0.025 0.010 -2.394 -0.023 0.012 -1.866 -0.031 0.014 -2.152 -0.031 0.013 -2.331 -0.034 0.014 -2.494LogCases -0.009 0.007 -1.314 -0.009 0.007 -1.304 -0.007 0.008 -0.926 -0.008 0.007 -1.018 -0.007 0.007 -0.987LogGDP 0.207 0.067 3.109 0.208 0.067 3.090 0.200 0.068 2.943 0.198 0.068 2.905 0.200 0.067 2.995ρ -0.0207 0.378 0.258 0.465Table 3.3: Estimation Results: How COVID-19 affects HappinessTable 3.3 reports the estimation results. The estimated OLS coefficient of the Stringencyis -0.025, which means that increasing Stringency by 10 units would decrease happinessgrowth by 0.025. The estimated coefficient is significantly different from zero. Look at theestimates of 2sCOPEAll , the 2sCOPE method with all the exogenous variables included in80the first stage, the stringency coefficient is -0.023, which is very close to the OLS estimateand becomes even less significant. This can result from adding too many irrelevant variablesin the first stage. When using Lasso-based methods to select important variables in the firststage, the estimated coefficients of stringency are -0.031 for Lasso, -0.031 for post-Lasso,and -0.034 for De-biased Lasso, all significantly different from zero. Those estimates arequite close, indicating the robustness of different Lasso-based methods in obtaining causalinference using 2sCOPE under this specific setting. Overall speaking, using Lasso-basedmethods to select important features in the first stage changes the stringency coefficientestimate from -0.023 to up to -0.034, implying that there is a positive correlation betweenunobserved country characteristics and stringency. For example, unobserved ideologicalcharacteristics such as conservative level can be positively correlated with both stringencyand happiness. The positive estimates of ρ in the three Lasso-based 2sCOPE methods alsoconfirm the positive correlation.In sum, including too many irrelevant variables in the first stage in the high-dimensionalcase would weaken the power of 2sCOPE in correcting endogeneity and achieving accuratecausal inference. After the feature selection procedure using Lasso-based methods in thefirst stage of 2sCOPE, we obtain a more negative and significant coefficient of stringencythan using 2sCOPE with all observed exogenous variables included in the first stage. Theestimated coefficient is -0.034, which means that increasing Stringency by 10 units woulddecrease happiness growth by 0.034. Moreover, the estimated coefficients are quite closeusing different Lasso-based methods, showing the robustness of the estimates.3.6 ConclusionCausal inference lies at the center of social science research, but the more and more com-mon high-dimensional data in this big data era might make the traditional causal inference81methods fail to work. On the one hand, too many (irrelevant) variables added might bringfinite-sample bias, reduce estimation efficiency, or even make the traditional regression in-feasible when the dimension of variables is larger than the sample size. That’s why featureselection machine learning methods are needed for causal inference methods. On the otherhand, the more complicated structure of causal inference methods (e.g., two-stage estima-tion) other than a single-equation regression provides a distinct setting and a different per-spective for feature selection methods. Adaption of traditional causal inference methods forhigh-dimensional data is much needed.In this paper, we are interested in the copula instrument-free causal inference methodfor correcting endogeneity, and adapting the method to a high-dimensional setting. Thecopula method, first proposed by Park and Gupta (2012), has aroused wide interest becauseof its feasibility that no instruments are needed. After that, many empirical researchers haveapplied the copula method for real-data analysis (Burmester et al. 2015, Datta et al. 2015,Gruner et al. 2019, Keller et al. 2019, Bombaij and Dekimpe 2020, Guitart et al. 2018,Lamey et al. 2018, Wetzel et al. 2018, Heitmann et al. 2020, Atefi et al. 2018, Elshiewyand Boztug 2018). We consider a two-stage copula method, called 2sCOPE in Yang et al.(2022), that extends the original one to a much more general setting, and adapt it to thehigh-dimensional setting. Specifically, 2sCOPE corrects endogeneity by adding a generatedregressor, which is the first-stage residual, to the outcome regression, just like what thecontrol function approach does. By controlling the residual estimated in the first stage,endogeneity can be corrected. Thus, the estimation of the first-stage residual plays a centralrole. We propose a method combining the generalized two-stage copula method (2sCOPE)with some lasso-based methods (lasso, post-lasso, de-biased lasso) in selecting importantvariables in the first stage. We call it the lasso-based 2sCOPE method.We conduct simulation studies and use an empirical real-data application to empiricallyverify the performance of our proposed method. The simulation results show that 2sCOPE82without feature selection suffers the finite-sample bias when the dimension is relativelylarge. Moreover, the lasso-based 2sCOPE method can substantially improve estimation ac-curacy and efficiency, by around 50%, as compared with the 2sCOPE without feature selec-tion. We further apply our method to a current data set about COVID-19, and examine theeffect of governments’ policy strictness in response to COVID-19 on citizens’ happiness.We use country-level cross-sectional data. The dimension of country characteristics is rela-tively large, around half of the sample size. The result shows that the estimated coefficientof the endogenous policy stringency level from the 2sCOPE with all controls included in thefirst stage is very close to the OLS estimate and becomes even insignificant. In contrast, ourproposed lasso-based 2sCOPE method makes the effect of policy strictness on happinessmore significantly negative. Moreover, the estimates using different lasso-based methodsare quite robust in this application. This application result confirms the findings in the sim-ulation study. Under the lasso-based 2sCOPE estimation, the application tells us that policystrictness has a significant negative effect on people’s happiness during the COVID-19 pe-riod. Increasing policy strictness by 10 units would decrease happiness growth by 0.034,which is more than one standard deviation of happiness growth.In the above analysis, we only consider using feature selection methods in the first stageof 2sCOPE, which means that we assume what variables are going to have effects on themain outcome are known. It is worth exploring and incorporating cases where one hasless certainty on important controls in the outcome regression. Our next step is to furtherexplore how to apply feature selection methods in the main regression model as well, whichwe label as double lasso-based 2sCOPE method for correcting endogeneity. Moreover, forthe current simulation analysis on 2sCOPE combined with the lasso-based methods, weonly conduct simulation studies for one single setting. In reality, lasso can be sensitive tovariations in real data. For future research, we will conduct more simulation studies in termsof parameter variation, and theoretically prove what are the requirements for our method to83work.84Chapter 4Vertical Differentiation in Two-sidedMarkets: Evidence from A Ride-hailingPlatform4.1 IntroductionTwo-sided (or, more generally, multi-sided) markets are roughly defined as markets in whichone or several platforms enable interactions between end-users and try to get the two (ormultiple) sides “on board” by appropriately charging each side (Rochet and Tirole (2006)).Leading examples of such platforms include Uber and Lyft for rides; Airbnb for accom-modation; and eBay and Taobao for E-commerce. Statistics show that consumers spend$65 billion on Uber 1, and $3.5 trillion on goods on E-commerce sites 2 in 2019. Giventhe growing economic significance of such platforms, it is important to understand the two-sided markets—what makes them different from traditional markets, and whether traditional1https://www.businessofapps.com/data/uber-statistics/2https://www.statista.com/topics/871/online-shopping/85business strategies are still applicable to them.One distinct feature of two-sided markets is network externalities, which can exist bothwithin and across sides. The demand (supply) of a product is dependent on the demand (sup-ply) from other users considering that product. In addition, demand will attract more serviceproviders (supply) to join the platform, and meanwhile, more supply will further make theplatform more attractive to consumers. Since users’ utilities from the product depend onother users’ decisions, network externalities create and provide users with another productquality dimension, which I call the network value. That is, in markets where products aresubject to network externalities, the number of users also determines, at least partially, theperceived quality of the product. For example, users of a ride-sharing platform value shortwaiting time, which comes from a large scale of drivers available on the platform, whereasdrivers value short cruising time, which results from a large demand base from the riderside.Vertical differentiation is a business strategy commonly used by firms in both conven-tional markets and two-sided markets. By designing products with different qualities, afirm can attract and satisfy consumers with heterogeneous preferences and thus expand themarket. For example, Uber designs Uber Black to target high-end customers and UberXto target low-end customers. Several studies have examined whether price discriminationis profitable for a monopoly. For example, Gabszewicz et al. (1986), Maskin and Riley(1984), Mussa and Rosen (1978), Salant (1989) and Stokey (1979) discussed vertical dif-ferentiation in a one-sided market scenario. In particular, Salant (1989) and Stokey (1979)identify conditions under which the monopolist finds it optimal to price discriminate. Thesepapers are developed in the context of conventional markets without network externalities.Jing (2007) further investigates how network externalities affect a firm’s decision to pricediscriminate in a one-sided market. They find that network externality is a factor that makesfirms favor extending a product line. Moreover, they find that the monopolist employs two86qualities of products for rather different purposes: the low-end product is used mainly toexpand its network size, and the high-end product is its primary source of profits. However,the paper only considers vertical differentiation with network externalities in one-side mar-kets. In a two-sided market, both consumers (demand side) and service providers (supplyside) are affected by network externalities. Moreover, the network externalities can existboth within and across sides. All these make vertical differentiation more complicated intwo-sided markets.In the two-sided markets, it is unclear whether network externalities still make firmsfavor extending a product line. On the one hand, according to the literature on one-sidedmarkets, firms can be better off by designing vertically differentiated products to expandthe market, especially when network externalities exist. On the other hand, offering ver-tically differentiated products in a two-sided market might be less optimal than offering ahomogeneous product if the further segmented demand and supply limit positive networkeffects. Firms might want to unite markets from different products into one scaled-size mar-ket to achieve large positive network effects. Thus, there is a tradeoff when using verticaldifferentiation in two-sided markets. The main goal of this paper is to examine whether ver-tical differentiation is better than a homogeneous product design in two-sided markets. Thispaper contributes to the literature by being one of the first to empirically examine verticaldifferentiation in two-sided markets.I use a distinct dataset from a leading ride-hailing company located in New York Cityto empirically analyze and answer the research question above. On this platform, there aretwo types of differentiated products with different qualities of car make and service quality,one is called premium and the other is called standard. The data is from January 2016 toMay 2016, and include all individual order requests from riders and drivers’ correspondingresponses about whether to pick up the riders. I build up a structural model to simulta-neously estimate demand and supply, and use the Bayesian Markov chain Monte Carlo87(MCMC) method for model estimation. The results show that both intrinsic product qualityand network value are significant components for riders’ and drivers’ decisions, and thusfor determining the degree of vertical differentiation. I also conduct counterfactual analysesbased on the estimation results. In counterfactual analysis, I evaluate alternative strategiesby using the estimated parameters to solve the fixed points and obtain the equilibrium in thenew setting. Specifically, I compare the current vertical differentiation case with a homo-geneous product case. The result shows that using vertical differentiation under the currentpricing strategy is better than a standard-only homogeneous product in terms of both mar-ket size expansion and profit maximization, which means that the positive effect of marketexpansion from vertical differentiation can offset the loss from network segmentation. Bycomparing different pricing strategies under the same vertical differentiation setting, I findthat the roles of the two vertically differentiated products are different. Premium product ismore profitable, but also costs the firm more to maintain the network because of the smallernetwork size (value). Low-end product is not profitable, but is valuable to the firm in enlarg-ing user size. In a word, network externalities play important roles in two-sided markets,and two-sided platforms should understand users’ preferences over both the network valueand the product value when determining the optimal product strategy and pricing strategies.The remainder of the paper is organized as follows. In section 4.2, I reviewed the market-ing and economics literature relevant to vertical differentiation and ride-hailing platforms.Section 4.3 describes the data used in the empirical implementation and presents model-free evidence of network externalities and how network externalities influence riders’ anddrivers’ behavior. Section 4.4 describes the model and estimation method used for esti-mating the simultaneous demand and supply model and quantifying network externalities.Section 4.5 presents the detailed estimation results. Section 4.6 further conducts counter-factual analysis and provides managerial implications. Section 4.7 concludes the paper withkey results and suggestions for future work.884.2 Literature ReviewThis paper is broadly related to three streams of literature. I first discuss the general verticaldifferentiation literature, then link vertical differentiation to two-sided markets, and discussthe relevant work on the ride-hailing industry in the end.Vertical differentiation research has attracted considerable attention in both economicsand marketing (Lancaster (1990), Mussa and Rosen (1978), Shaked and Sutton (1982),Sridhar Moorthy (1984)). In a vertically differentiated product space, more attributes willattract consumers with different preferences, who may have different willingness to pay.Vertical differentiation has already been widely discussed in one-sided markets. For exam-ple, Mussa and Rosen (1978), Sridhar Moorthy (1984), Salant (1989) and Stokey (1979)studied vertical differentiation for monopolists. In particular, Salant (1989) and Stokey(1979) identified conditions under which the monopolist finds it optimal to price discrimi-nate. Shaked and Sutton (1982) built up a theoretical model for vertical differentiation ina competitive setting. These papers focused on one-dimensional product differentiation,many papers also discussed multidimensional product differentiation. For example, Van-denbosch and Weinberg (1995) studied product and price competition in a two-dimensionalvertical differentiation model, whereas product qualities are differentiated into two productattributes.All those papers are developed in the context of conventional markets without networkexternalities. Jing (2007) further investigated how network externalities affect a firm’s de-cision to price discriminate in a one-sided market. They find network externalities are afactor that favors extending a product line. Specifically, they find that monopolists employthe two qualities for rather different purposes: The low-end product is used mainly to ex-pand its network size, which is why the monopolist would lower the price of the low-end,and the high-end product is its primary source of profits with a rather higher price than in89conventional markets. However, the paper only considers the effect of network externalitiesin one-side markets. That is, they assume network externalities only affect the demand sideand assume supply is sufficient.Recently, two-sided markets have raised wide concern in academia. In a two-sided mar-ket setting, not only consumers are affected by network externalities, but service providers(supply) are also affected. The within- and cross-network externalities would make verticaldifferentiation more complicated. Only a few papers in the literature have studied verticaldifferentiation in two-sided markets. For example, Liu and Serfes (2013) theoretically com-pared the effect of vertical differentiation on competition in two-sided markets with that ina one-sided market. They find that price discrimination in a two-sided market may actu-ally soften competition. Gabszewicz and Wauthy (2014) developed a theoretical model anddemonstrated that a unique vertical differentiated pricing equilibrium exists for two firmscompeting in a two-sided market. They also assume that products are vertically differen-tiated only by network size. While in this paper, I focus on vertical differentiation withina platform and products are vertically differentiated not only by network value but also byproduct intrinsic quality. Lin (2020) studied vertical differentiation in a monopoly casefor media platforms. These papers theoretically study vertical differentiation in two-sidedmarkets, while this paper studies vertical differentiation using empirical analysis within atwo-sided platform. There is also some empirical work in two-sided markets. For exam-ple, Zervas et al. (2017) and Li and Srinivasan (2019) studied the impact of the two-sidedplatform Airbnb’s entry on hotels. However, they didn’t study vertical differentiation. Thispaper is one of the first papers to empirically analyze vertical differentiation in the two-sidedmarket context.With the increasing scale of ride-hailing platforms, it has attracted wide attention fromresearchers. Most papers in ride-hailing literature focus on driver-side behavior. They studythe effect of the new ride-hailing technology on drivers. For example, Chen et al. (2019)90studied the value of ride-hailing platforms in creating flexible work for drivers. Wang et al.(2019) studied the impact of mobile hailing technology on taxi driving behaviors. Frechetteet al. (2019) studied matching frictions and technologies in the taxi industry. Guda andSubramanian (2019) studied the effect of surge pricing on worker incentives with flexi-ble work. Few papers have studied vertical differentiation. Lin (2020) studied verticaldifferentiation in a monopoly case for media platforms. They theoretically develop a two-sided media model and examine how a monopoly ad-financed media can price discriminatethrough versioning using membership-based pricing. While our paper empirically studiesusage-based pricing strategy for a non-monopoly firm. Another related paper is Bryan andGans (2019). It theoretically studied vertical differentiation in a ride-hailing setting. Butthey assume that the number of drivers is endogenously determined by the platform. In thisway, they only model how the network size of drivers would influence riders through wait-ing time, but didn’t consider the effect of rider size on drivers’ decisions. While this papermodels demand and supply simultaneously. Moreover, they assume products are verticallydifferentiated only in network size, while I consider the case when products are verticallydifferentiated in two quality dimensions, product intrinsic quality and network value.In a word, the contribution of this paper to the literature on vertical differentiation intwo-sided markets is mainly twofold. First, I study two-dimension vertical differentiationin two-sided markets. That is, products are assumed to be vertically differentiated in twodimensions, product intrinsic quality and network value, and both are important in deter-mining the level of vertical differentiation. Second, this paper is one of the first papers toempirically study vertical differentiation in two-sided markets and quantify network exter-nalities by modeling demand and supply simultaneously.914.3 Data and Model-Free EvidenceIn this section, I will introduce the data, and provide data patterns and evidence about whynetwork externalities are important in riders’ and drivers’ choices towards vertically differ-entiated products.4.3.1 Data BackgroundThe global ride-hailing market is valued at USD 113 billion in 2020. Overall, Didi, Uber,Lyft, and Grab are major players who have prominent market share globally. UBER andLyft have a prominent share in the North American region, whereas, in China, Didi ChuxingTechnology Co 3. contributes to the high market share.The focal platform under study is a leading on-demand ride-hailing company located inNew York City, where the major competitors are yellow/green taxi, Uber and Lyft. Riderson this platform order a car by using the company’s location-based smartphone app. Theycan place order requests at any time and anywhere they want. Once a rider placed an orderrequest, drivers nearby who are active on the platform can receive the offer notificationsfrom the platform, and then decide whether to accept the offer.This platform provides two main ride types that riders can choose from, which are calledstandard and premium. Standard type serves affordable everyday rides and premium typeserves a bit more expensive high-end rides, which are similar to UberX and UberBlack.Riders are charged fees for each realized trip, and drivers earn income from trip fares. Thefares for the two types of products are different for both riders and drivers. Note that thefare structure for drivers varies in peak and off-peak time, and also changes once during thetime window, while remaining the same for riders over time. Tables 4.1 and 4.2 show the3https://www.mordorintelligence.com/industry-reports/ride-hailing-market92detailed fare structure for rider and driver side separately. On the rider side, the standardtype charges a fixed fee of $8.98 for each trip, and the premium type charges a two-parttariff consisting of a one-time fixed fee and a duration-based fee. The rate is $8 for the first15 minutes, $0.5 per minute between 15-29 minutes, $0.75 per minute above 30 minutes,and $1 per minute above 40 minutes.Table 4.1: Price Structure for RiderTime Price Structure Standard PremiumJan – May 2016 Base fee $ 8.98 $8Per additional minute (15-29 mins) 0 $0.5Per additional minute (30-39 mins) 0 $0.75Per additional minute (≥ 40 mins) 0 $1On the driver side, earnings all follow a two-part tariff, and a significant fare changewas implemented in February 2016. Before February 14, 2016, drivers earn the same fixedfee, $8, for both types, while the number of minutes to earn the base fee is shorter (13 <15) and the per-minute payment for an additional minute is higher for the premium type($0.62 > $0.55). Throughout the whole period, there was a surcharge of $0.75 for premiumand $0.9 for standard in peak times during 7:00 a.m-9:00 a.m, 6:00 p.m.- 8:00 p.m., andnight hours between 8:00 p.m. and 3:00 a.m on weekends. After February, the platformchanges the fixed fee to $7 and also lowers per-minute earnings for both types of productsin both peak and off-peak hours. The above shows the fare structures for rider and driver, andthe platform further takes a fixed rate of fares as commission from drivers. The commissionrate of Uber is 25% while that is 20% on this platform. All order requests from riders and allthe drivers’ corresponding responses in the Manhattan area from January 9 to May 15, 2016,a total of 18 weeks, are observed. The raw data comprises 98,342 riders who request at leastone ride and 3,509 drivers available during the period. On average, each rider requested 8times, for a total of 760,298 requests from all riders.93Table 4.2: Fare Income Structure for DriverStandard Premiumoff-peak peak off-peak peakBefore Feb. 14, 2016 Base fee ($) 8Minutes to earn minimum fare (min) 15 11 13 9Per additional minute ($) 0.55 0.75 0.62 0.9After Feb. 14, 2016 Base fee ($) 7Minutes to earn minimum fare (min) 13 13 14 11Per additional minute ($) 0.54 0.6 0.52 0.724.3.2 Variables Related to Network ExternalitiesI first clean the raw data, and construct key variables that are related to network externalitiesin both rider (demand) and driver (supply) sides for further analysis.Rider’s Expected Waiting TimeOn the rider side, the rider’s waiting time for each type of product, a direct indicator ofnetwork effect, is a key factor to affect their choices. In this ride-hailing platform, thoughthe time riders have to wait is unobserved, it’s reasonable to assume that they can formexpectations on the waiting time because they can observe the real traffic environment.Below shows the procedure of how I construct the rider’s expected waiting time for eachtype.• Define and calculate the rider’s waiting time of each trip as the total time from ordercreation to estimated driver’s arrival, both of which are observed by researchers;• Divide the total area into smaller markets for constructing expected waiting time ata specific time and location. Specifically, I first divide the Manhattan area into 29smaller locations, according to a common demarcation of Manhattan4, which is visu-alized in Figure 4.1. Then define a market at a week-hour-location level. That is, order4https://data.cityofnewyork.us/City-Government/Neighborhood-Tabulation-Areas-NTA-/cpf4-rkhq94requests at a specific location in a specific hour and week are in the same market. Inthe end, there are 5,672 markets in total.• After the market is defined, I construct the rider’s ex-ante expected waiting time foreach type by averaging the waiting time of all same-type trips in the same market.Figure 4.1: Manhattan Neighborhoods.Driver’s Cruising TimeOn the driver side, cruising time for a trip is essential for their decisions on whether to takethe trip. Cruising time is defined as the estimated time to pick up the rider once an offeris received. The driver’s cruising time is not observed directly by the driver, but the rider’slocation and the distance from the rider at the time the driver receives an offer are observed.And researchers can observe the cruising time for drivers to pick up riders for realized trips.We assume drivers can approximate the cruise time, and fill in the missing cruising timefor unrealized trips by using the observed distance from riders of all trips and the observedcruising time for realized trips (see more details in Appendix). It’s reasonable to approx-imate the cruising time in this way because drivers must be familiar with the geographic95information once the location of the rider is revealed. The reason I want to construct cruis-ing time instead of using distance alone is two folds. First, drivers care about both distanceand time spent. Second, distance itself cannot fully reveal the dynamic traffic environment,while cruising time can because traffic environment can be different in different times andareas because of the real-time network effect. For example, two trips with the same distancefrom riders, one in rush hour while the other not, can have exact different cruising times.Table 4.3: Summary statisticsVariables Unit Mean SD Max MinPremium Type Trip fare $ 10.76 4.15 24.44 5.38Rider waiting time min. 7.54 4.47 24.37 0.20Driver wage $ 13.6 7.68 62.8 7.0Driver wage (no cruise) $/min 0.73 0.25 3 0.28Driver wage (incl cruise) $/min 0.54 0.19 4.57 0.10Trip duration min. 18.8 9.0 69.6 4.2Driver cruise time min. 5.81 2.60 20.35 0.40Demand size Units 4.98 4.78 50.5 1.0Supply size Units 3.30 2.98 29.75 0.25Standard Type Trip fare $ 8.98 0.23 24.00 8.00Rider waiting time min. 7.38 4.35 24.37 0.20Driver wage $ 14.4 7.03 48.5 7.0Driver wage (no cruise) $/min 0.68 0.23 3.7 0.39Driver wage (incl cruise) $/min 0.53 0.18 5.15 0.10Trip duration min. 21.4 9.74 69.8 4.2Driver cruise time min. 5.44 2.09 19.50 0.32Demand size Units 8.28 7.04 55.75 1.0Supply size Units 5.72 5.07 31.0 0.25After the construction of riders’ expected waiting time and drivers’ cruise time for eachtrip, I further calculate the network size of both demand and supply in each market to ana-lyze network externalities. Here, I define demand as the number of total trip requests fromriders, and measure supply using the number of trips that drivers accept. Until now, I havefinished the construction of rider’s expected waiting time, driver’s expected cruising time,demand size and supply size for each type. I further clean the data by removing erroneousobservations and outliers, and summarize the descriptive statistics of key variables in Ta-96ble 4.3. It shows that on average, the premium type has a higher trip fare with $10.76than the standard type with $8.98 for riders, and provides a higher per-minute payment with$0.73 than the standard type with $0.68 for drivers. However, both riders’ waiting timeand drivers’ cruising time are longer for the premium type. After accounting for cruisingtime, drivers’ per-minute payments from the two types become quite close. The statisticsprovide some key factors that might influence drivers’ and riders’ choices towards the twotypes, which I will show detailed evidence in the next section. In the end, 304,044 trips with63,260 available riders and 3,509 drivers on this platform are surviving for further analysis.4.3.3 Data Evidence in Riders’ ChoiceIn this section, I will show data evidence about how riders choose between the two types,and what factors would affect their choice decisions.Figure 4.2 shows riders’ aggregate choices. Around two-thirds of riders choose thestandard type and one third choose the premium. What are the key factors or the differencesbetween the two types that differentiate riders’ preferences? What do high-end riders payhigher prices for?First, the products’ intrinsic qualities are different, which is the core to help differenti-ate in traditional one-sided markets. The quality difference between the two types mainlycomes from car quality and service quality. Data shows that cars are more luxury and ratingis higher for premium (see details about the quality difference between the two types inAppendix B.1). This quality difference provides different product intrinsic value to ridersthat help to differentiate riders with heterogeneous preferences 5.Second, besides product intrinsic quality, waiting time is another important factor that5Note that some proportion of standard drivers in this platform is allowed to serve both the premium andstandard riders. No need to worry, as the market is still differentiated. See more details in Appendix B.197Figure 4.2: Choice Occasion between The Two Types.plays an important role in riders’ choice decisions. Figure 4.3 shows the relationship be-tween market share and waiting time. The line with the negative slope indicates that themarket share of the premium type decreases with the waiting time difference between pre-mium and standard type. That is, the longer the waiting time for premium type comparedwith that for standard type, the smaller the market share of premium type would be. Thismodel-free evidence demonstrates that riders prefer shorter waiting time, and waiting timeis an important factor that affects riders’ preferences over the two types of products. In aword, data evidence shows that product intrinsic quality and waiting time are two importantattributes for product differentiation in two-sided markets.4.3.4 Data Evidence in Drivers’ BehaviorIn this subsection, I further explore what would affect driver behavior.First, drivers care about earnings from the received offer. Since time is money for98Figure 4.3: Relationship Between Market Share and Waiting Time.(a) Payment without Cruise Time (b) Payment with Cruise TimeFigure 4.4: Drivers’ Average per-minute Payment from Realized Trips.drivers, per-minute payment should be a better measure for drivers’ earnings instead ofa total fare from a trip. Figure 4.4(a) shows the average per-minute payment drivers canget from the two types of products across hours. This per-minute payment is calculatedby dividing drivers’ earnings by the driving time with passengers. It shows that generallyspeaking, drivers receive higher payment from the premium than from the standard-typetrips. However, not only does the driving time which creates value matter, but the cruising99time for drivers to pick up the rider also matters. Since drivers can observe the location ofthe rider, cruising time can be another key factor that would affect drivers’ decision aboutwhether to accept the current offer. Figure 4.4(b) shows drivers’ per-minute payment afteraccounting for the cruising time to pick up riders. It shows that after accounting for cruisingtime, drivers’ average per-minute payment decreases from around $0.7 to $0.5, a more than20% decrease, which indicates that cruising time accounts for non-negligible importance.Moreover, drivers’ per-minute payments from the two types become surprisingly close. Thistells that though premium has higher payment, cruising time for drivers to fulfill premiumis also longer. Thus, cruising time can be an important factor that would affect drivers’ de-cisions. Figure 4.5 further shows the relationship between drivers’ average acceptance ratewithin each hour and the cruising time in that hour. The negative slope tells us that driversfavor trips with shorter cruising times.Figure 4.5: Relationship Between Drivers’ Acceptance Rate and Cruise Time.Second, besides the earning from the current offer, whether there exist other possibleopportunities can also be an important factor that would affect drivers’ decisions on whetherto accept the current offer. Drivers can be picky about offers if there are many options forthem. Figure 4.6 shows that drivers’ acceptance rate of offers is negatively correlated with100the total demand size in that market. It tells us that once drivers have more options (demand),they will be more selective about offers and less likely to accept the current offer. This isactually one way how network externalities play roles in affecting drivers’ decisions. Alarger demand size provides more opportunities to drivers, which will further affect drivers’decisions.Figure 4.6: Relationship Between Drivers’ Acceptance Rate and Demand Size.4.4 ModelIn this section, I develop a structural model to capture both riders’ and drivers’ choicessimultaneously, together with the IV and control function approach to correct endogeneity,and use the Bayesian MCMC method for estimation.1014.4.1 Rider’s DecisionAn individual rider needs to decide which car type to choose before placing an order request.Premium type and the standard type are different in prices and some quality-level attributes.The demand function is specified as a multinomial logit model. Suppose requests froma group of riders (i = 1,2, . . . , I) for k types of choices (k = 1,2) across m markets (m =1,2, . . . ,M) are observed. The utility rider i receives from type k product at market m isspecified as:Uikm = β d0i+βd1iPikm+βd2iEwaitkm+Xdmβd +ξ dkm+ εikm, (4.1)In the case of choosing the outside option, I denote it as k = 0 and the associated utilityfunction is:Ui0tm = εi0m. (4.2)where Pikm is the price charged for that ride, Ewaitkm is riders’ expected waiting time fortype k at market m, which is the key variable that indicates network externalities in thecurrent market, and Xdkm is a vector of market-level exogenous controls for each type k.Ewaitkm depends on the current total demand and the total supply of drivers for that type inthe same market.I further model riders’ expected waiting time as a function of current demand and supply.Ewaitkm = γdk0+ γdk1Dkm+ γdk2Skm+ εdkm. (4.3)β0i, β1i and β2i are individual-level response coefficients, and γk0, γk1,γk2 are market-levelresponse coefficients. Dkm and Skm are the logarithms of demand and supply size for typek of market m. εikm and εkm are independent individual-level and market-level unobservederror terms separately. I make assumptions on the error terms and response coefficients as102the following:εikm ∼ extreme value (0,1), (4.4)εkm ∼ N(0,σ21 ), (4.5)θi = (β0i,β1i,βi)∼MV N(θ¯i,Σθ i). (4.6)According to Equations (4.1) and (4.3), I can easily find that riders’ expected waiting timewill affect their choice of which type of product to request, and meanwhile, the number oftotal requests for that type will in turn influence riders’ waiting time for that type. Thus, thisis a system of simultaneous equations. However, this system of simultaneous equations isdifferent from traditional ones, as the demand is modeled non-linearly using a choice model.The type I extreme value specification of εikm leads to a standard logit choice probability forrider i choosing type k at market m,Pr(yikm = 1) =exp(uikm)1+ exp(ui1m)+ exp(ui2m),(k = 1,2). (4.7)whereuikm = β d0i+βd1iPikm+βd2iEwaitkm+Xdtmβd +ξ dkm. (4.8)and the predicted demand using this choice model will further influence Ewaitkm. I lateruse 3SLS estimation and the control function approach to estimate this nonlinear systemof simultaneous equations. After riders make their choice decisions, the platform will sendthose order requests to nearby drivers. Then drivers will make decisions on whether toaccept the offer.1034.4.2 Driver’s DecisionOnce a driver receives an offer from a rider, some offer-specific information such as the typeof the offer, price, and time they have to cruise around to pick up the rider (distance awayfrom the rider) is revealed to the driver. He/she might also obtain some information about thecurrent market, such as the expected demand and supply. Based on all those observed andinferred information, they make decisions on whether to accept the offer. I model drivers’behavior in the following way. After receiving an offer, a driver will decide whether toaccept the offer by comparing the earning from the current trip with his reservation wage inthe current market. He will accept the offer if and only if the per-minute payoff from thetrip is larger than or equal to his reservation wage. That is, I assume the driver j will acceptthe offer f in market m if and only if his expected per-minute pay exceeds his reservationwage at market m.Y j f m = 1 if Payment j f m ≥ R jm,0 otherwise.Then the utility of driver j by accepting the offer f in market m is specified asVj f m = β s1 jWage j f mRideDur j f m+Cruise j f m−R jm+ εsj f m, (4.9)where Wage j f m is the total payment from offer f , RideDur j f m and Cruise j f m are ride du-ration and cruising time of offer f . R jm is individual and market-level reservation wage,which depends on the driver’s specific preference and market environment, and indicatesthe possible opportunities for the driver j in the market m. The reservation wage of driver jat market m is modeled asR jm = β s0 j +βs2 jD1m+βs3 jS1m+βs4 jD2m+βs5 jS2m+βs6 jXsm+ξsm+ εsjm. (4.10)104where Xm includes all market-specific variables such as week, hour, and location dummies.Dkm,Skm (k=1,2) are network sizes in rider and driver sides for each product separately.Here, I model that network externalities would affect drivers’ decisions by also influencingtheir reservation wages (opportunities) in the focal market. This model setting is reason-able because, besides the payment from the current offer, the driver also cares about otherpossible opportunities in the current market.I make similar assumptions on the error terms and response coefficients for drivers asthe following:εsj f m ∼ extreme value (0,1) (4.11)εsm ∼ N(0,σ22 ) (4.12)θ j = (β s0 j,βs1 j, ...,βs6 j)∼MV N(θ¯ j,Σθ j) (4.13)The type I extreme value specification of ε j f m leads to a standard logit choice probabilityfor driver j choosing to accept offer f at market m,Pr(Yj f m = 1) =exp(v j f m)1+ exp(v j f m). (4.14)whereVj f m = v j f m+ εsj f m. (4.15)4.4.3 Hierarchical modelAfter modeling drivers’ and riders’ decisions, I further add a hierarchical layer to estimateindividual parameters. The individual parameters, (θi ≡ (β d0i,β d1i,β di ),θ j ≡ (β s0i,β s1i,β si )),105are determined by a vector of individual-related Wi,Wj separately as follows:θi = GdWi+ηi,ηi ∼ N(0,Σθ i),θ j = GsWj +η j,η j ∼ N(0,Σθ j).(4.16)where Gd(Gs) is a matrix of parameters that indicate how each demographic variable influ-ences the individual parameter θ . Σθ i(Σθ j) is the variance-covariance matrix capturing theinterdependence among θi(θ j). The Wi(Wj) includes riders’ (drivers’) demographic vari-ables that will influence their decision-making process. In the current estimation, I includeintercepts only in Wi(Wj).After model setup, I next introduce the endogeneity issue and some exogenous variablesin this setting that could be used as IVs to deal with the endogeneity of the simultaneous-equation system.4.4.4 3SLS/2SLS and Control Function ApproachIn this simultaneous-equation system, riders’ expected waiting time and demand (supply)are endogenous variables. On the one hand, riders’ expected waiting time for a specifictype of product will affect their decisions of whether to choose that type. On the otherhand, in this dynamic request and pick-up process, each rider’s choice will influence supplyand demand and in turn affect riders’ waiting time. For example, suppose there’s a suddendemand shock (e.g., a concert is just over), many people are requesting for rides and thusthe waiting time in that place must be quite long because of no enough supply. This wouldcause the waiting time and demand to be positively correlated, giving us the wrong andconfusing intuition that riders prefer long waiting time. Similar to the driver side. networksize would simultaneously influence drivers’ decisions.106I use the instrumental variable approach to correct the endogeneity of expected wait-ing time (expected cruising time) and demand (supply). Specifically, I use the expectedwaiting time in the last week at the same hour and location as an instrumental variable forendogenous waiting time. Last-period waiting time will not influence the current demand,but has a strong correlation with the waiting time in the current period. For the endogenousdemand and supply in Equations (4.3) and (4.10), I use last-hour demand and supply in thesame week at the same location as instrument variables. Another important IV for supply isthe exogenous subsidy level, which is defined as the difference between driver’s wage andrider’s price. On this platform, riders’ price structure remains the same over time, whiledrivers’ wage structure changes a couple of times. This exogenous variation can help tocorrect endogeneity. To sum up, I use 3-stage (2-stage) lease squares to correct endogeneityon the rider (driver) side.Stage 1: In the first stage, I use lag demand and lag supply as IV to correct endogeneity inwaiting time and reservation wage.Demandkm(h) = δdk0+δdk1Demandkm(h−1)+ εdkm, (4.17)Supplykm(h) = δs0k +δsk1Supplykm(h−1)+δsk2Subsidykm(h)+ εskm. (4.18)The IVs of demand and supply are valid that on the one hand, the last-hour demand (supply)is correlated with the current demand (supply). On the other hand, the IV satisfies theexclusion restriction that last-hour demand (supply) should not affect the current waitingtime directly, but instead will affect waiting time only by affecting the current demand(supply).Stage 2: Then I substitute the predicted demand and supply into the waiting time stage onthe rider side (reservation wage on the driver side), and add the waiting time from last week107at the same hour and location as IV to correct endogeneity in the choice stage.Ewaitkm = γdk0+ γdk1̂Demandkm+ γdk2 ̂Supplykm+ γdk3Ewaitk(w−1)m+µdktm. (4.19)Stage 3: Since the choice stage is not linear, I cannot simply substitute the predicted Ewaitinto the third stage. Here, I use the control function approach. Following Petrin and Train(2010), I use the simplest version of the control function, which is a linear function ofsecond-stage residual.CF(µ;λ ) = λµ, (4.20)where µ is residual from the waiting time stage, and λ is the parameter to be estimated.Then utility with the control function isUikm = uikm+λµkm+σmηkm+ εikm. (4.21)where uikm is the mean utility in Equation (4.8), and ηkm is i.i.d standard normal. In theestimation below, I use the random coefficient model so that σm here acts as the standarddeviation of the intercept. I use a similar way on the supply side. The only difference isthat there are only 2 stages (i.e., stages 1 & 3) on the supply side. No cruising time stage isneeded as cruising time is observed by drivers. I replace network size in reservation wagewith predicted network size to deal with the simultaneous endogeneity issue. After I set upmodels and address the endogeneity problem, I use Bayesian MCMC for model estimation(see more details about the estimation procedure in Appendix).4.5 Estimation ResultIn the sections above, I focus on the model-free evidence of network externalities andthe structural model development. In this section, I will show both the reduced-form and108MCMC estimation results to quantify the network externalities and product intrinsic valuein riders’ and drivers’ choice decisions.4.5.1 Estimation for Network ExternalitiesI first show estimation results in terms of how network sizes affect riders’ waiting time anddrivers’ cruising time. Note that, the cruising time stage is not used for estimation becausedrivers can observe cruising time directly. However, the estimates of the cruising time stagewill be used in the following counterfactual analysis.I use IVs in section 3.2 to correct endogeneity in each stage. Table 4.4 shows the 2SLSestimation results in terms of how network size on each side influences riders’ waiting timeand drivers’ cruising time. The dependent variables of the four columns are riders’ waitingtime, drivers’ cruising time, the demand size, and the supply size separately. The firstcolumn shows that riders’ waiting time is influenced by both the direct and indirect networkexternalities. The waiting time is increasing with demand (direct network externalities), andis decreasing with supply (indirect network externalities). For example, a 10% increase insupply would decrease waiting time by 0.38 (=log(1.1)∗3.97) minutes, and a 10% increasein demand would increase waiting time by 0.357 (=log(1.1) ∗ 3.75) minutes. Similarly,drivers’ cruising time is influenced by both the direct and indirect network externalities aswell. It is decreasing with demand (indirect network externalities) and is increasing withsupply (direct network externalities). In a word, Table 4.4 shows that there exist both directand indirect network externalities on both rider and driver sides. Next, I will show both thereduced-form and MCMC estimation results in riders’ and drivers’ choice decisions.109Table 4.4: How Network Externalities Influence Waiting/Cruis-ing TimeWait2SLS Cruise2SLS IVDemand IVSupply8.01 4.77 -0.17 -0.30Intercept (0.29)*** (0.17)*** (0.07)* (0.06)***3.75 -1.09Log demand (0.11)*** (0.12)***-3.97 1.11Log supply (0.14)*** (0.13)***0.10LagWeek wait (0.008)***0.15LagWeek cruise (0.009)***0.59 0.28Laghour log demand (0.01)*** (0.01)***0.20 0.29Laghour log supply (0.02)*** (0.01)***0.41Subsidy level (0.01)***Fixed EffectsWeek YES YES YES YESHour YES YES YES YESLocation YES YES YES YESR2 0.49 0.55 0.83 0.81Num. obs. 5,672 5,672 5,672 5,672***p<0.001, **p<0.01, *p<0.05Note: Dependent variables are market-level waiting time, cruising time,log of demand and log of supply respectively. Standard errors are inparentheses.4.5.2 Reduced-form EstimationPrevious model-free evidence in Figure 4.3 shows that riders favor short waiting time, andthe section above already shows how direct and indirect network externalities affect riders’waiting time. In this section, I further show reduced-form evidence about how the waitingtime influences riders’ choices toward the two types of products on this platform. Moreover,besides waiting time, whether there exist other factors that also help to determine the degreeof vertical differentiation.Table 4.5 shows the reduced-form estimation results. On the demand side, after addingIVs to control for endogeneity (Model 2), waiting time adds a significant negative value to110Table 4.5: Reduced-Form Estimation Results for Rider and DriverRider Side Model Driver Side ModelModels Model 1 Model 2 Model 3 Model 40.22 0.19PremiumDummy (0.006)*** (0.007)***-3.02 -3.02LogPrice (0.01)*** (0.02)***-0.015 -0.10Wait (0.002)*** (0.01)***-2.81 -2.75Intercept (0.05)*** (0.06)***6.33 6.35Wage (0.03)*** (0.03)***-0.43 -0.64LogDemandp (0.02)*** (0.05)***0.09 0.26LogSupplyp (0.02)*** (0.06)***-0.59 -0.53LogDemands (0.02)*** (0.05)***0.65 0.61LogSupplys (0.02)*** (0.05)***Num. Obs. 391,642 391,642 388,296 388,296Log Likelihood -183,242 -183,195 -233,874 -232,307IV NO YES NO YES***p<0.001, **p<0.01, *p<0.05Note: DV of Model 1 and 2 is a binary choice of whether the rider chooses thepremium type; DV of Model 3 and 4 is a binary choice of whether the driveraccepts the offer; Wage is driver’s per-minute payment after considering cruisingtime to pick up rider; Standard errors are in parentheses.riders, and premium product adds significant positive value to riders. This result is consis-tent with model-free evidence in Figure 4.3. In a word, both product intrinsic quality andnetwork externalities are crucial in affecting riders’ choices. On the driver side, accordingto the result of Model 4 in Table 4.5, network externalities would affect drivers’ decisionsin two ways. First, drivers prefer offers with higher wages, which is the per-minute wagethat takes cruising time into consideration, and thus drivers favor offers with shorter cruis-ing time. Second, network size on both sides would affect drivers’ decisions by influencingdrivers’ opportunities to get other offers. More demand indicates more opportunities, andwould decrease drivers’ probability to accept the current offer; While more supply indicatesmore competitors and fewer opportunities, and would increase driver’s probability to acceptthe current offer.111The reduced-form estimation results show that network externalities are key componentsto affect riders’ and drivers’ choices. Both the product intrinsic quality and the networkexternalities play important roles in determining the degree of vertical differentiation intwo-sided markets. Next, I will add outside demand from Green/Yellow taxi in NYC, anduse Bayesian MCMC to estimate the model with heterogeneous parameters.4.5.3 MCMC Estimation Results for Heterogeneous Choice ModelThe above reduced-form estimation results show that riders’ waiting time and drivers’ cruis-ing time are affected by network size, both within and across sides. Next, I will show theMCMC estimation result, and quantify the value of each component for riders’ and drivers’decisions. Before estimation, I first add outside options for riders. When a rider decides toorder a ride, he might also consider outside options such as taxi and Uber. I include pub-lic taxi data in New York City as the outside option to control the total market size in thefollowing analysis. The taxi data in New York City is public online, including all realizedtrips with trip start time, trip fare, distance and geographic information. Please find detailsabout how I match outside demand with the data and how I construct alternative optionsfor those who choose the outside option in the Appendix. The utility for the outside optionis normalized to zero. Then I follow the estimation procedure in the model section anduse the MCMC method to estimate the random coefficient model. I run a total of 300,000MCMC iterations and report the posterior distribution of the parameters based on the last100,000 draws. Table 4.6 shows the estimation results of the mean and standard deviationof parameters among the population.Table 4.6 shows the MCMC estimation results on both the rider and driver sides. Let’sfirst look at the estimation result on the demand (rider) side. First, product intrinsic valuematters for riders when vertical differentiation is used. Choosing a premium trip would112Table 4.6: MCMC Estimation Results for Demand and SupplyVariables Mean Parameter Standard Deviation σDemand Side Intercept 9.10 (8.83, 9.36) 7.08 (6.90, 7.26)Premium Dummy 0.08 (0.06, 0.11) 1.77 (1.74, 1.80)Log Price -2.93 (-2.99, -2.88) 2.33 (2.27, 2.40)Avg wait -0.13 (-0.15, -0.10) 0.45 (0.43, 0.47)Supply Side Intercept -3.76 (-3.98, -3.59) 2.68 (2.46, 2.83)Payment ($/min) 8.00 (7.78, 8.23) 4.66 (4.41,4.87)Log Demandp -0.53 (-0.69, -0.33) 0.81 (0.74, 0.93)Log Supplyp 0.29 (0.09, 0.50) 0.82 (0.71, 0.92)Log Demands -0.23 (-0.37, -0.09) 0.71 (0.68, 0.75)Log Supplys 0.22 (0.09, 0.34) 0.73 (0.65, 0.83)Note: Numbers in parentheses indicate the 2.5 and 97.5 percentiles; Avg wait is the expectedwaiting time in that market; Subscript p represents the premium network and s represents thestandard network.make riders gain 0.08 more utility than choosing a standard trip. Second, I quantify net-work value to riders. The result shows that riders’ waiting time, the indicator of networkexternalities, provides negative values to riders. One minute increase in waiting time woulddecrease riders’ utility by 0.13, which is much larger than the utility gain if choosing thepremium type. I further link the network externalities with waiting time. According to thewaiting time stage result in Table 4.4, a 10% increase in supply productivity, keeping allothers equal, can achieve a 0.38-minute (=log(1.1) ∗ 3.97) decrease in waiting time, andfinally increase the utility of choosing the option by 0.05. Thus, besides product intrin-sic value, network externality is a crucial factor in affecting riders’ choices by influencingriders’ waiting time.On the driver side, network externalities would affect drivers’ decisions in two ways.First, network externalities can influence drivers’ earnings by affecting the time driversneed to cruise around to pick up riders. Drivers prefer higher per-minute wages, which willincrease when cruising time decreases. A $0.01 per-minute wage increase would increasethe odds of accepting the offer by 8.33% = (e8∗0.01-1), which means the probability of113accepting the offer increases from 50% to 52% if fixing the original acceptance rate to50%. Second, besides the effect on cruising time, network size would also affect drivers’probability of accepting the current offer by directly affecting their potential opportunitiesto get other offers. A 10% increase in premium demand would cause the odds of acceptingthe current offer to decrease by 5.2%, and a 10% increase in standard demand would causethe odds to decrease by 2.3%.4.5.4 Visualization of MCMC EstimatesVisualization of Rider HeterogeneityIn the analysis above, I have used a heterogeneous model to estimate and quantify networkvalue in riders’ decision-making process. In this section, I plot the distribution of those esti-mates among riders, and get some visual insights about rider heterogeneity on the platform.Figure 4.7 shows the distribution of the estimated coefficients of some key variables amongriders. First, I find that riders, on average, have negative price coefficients (Figure 4.7(a)),and the price coefficients of standard riders are more negative than those of premium riders.This tells us that the riders who choose standard type are more sensitive to price than riderswho choose premium type. Second, the coefficient of the premium dummy is much largerfor premium riders (Figure 4.7(b)), which means that riders who choose the premium typeindeed value the high intrinsic quality of the premium type. Third, the coefficient of waitingtime is more negative for standard riders (Figure 4.7(c)), which means standard riders aremore sensitive to waiting time than premium riders. In a word, the estimated coefficientsmainly tell us that the premium riders on the platform are less sensitive to price than stan-dard riders. They are willing to pay higher prices for the premium type because they valuemostly on the high intrinsic quality of the premium type, and can bear a bit longer waitingtime.114(a) Price Coefficient (b) Premium Intrinsic Value (c) Waiting Time CoefficientFigure 4.7: Distribution of Estimated Riders’ Heterogeneous CoefficientsVisualization of Driver’s Reservation WageIn the analysis above, I have used model estimation to quantify network value in both riders’and drivers’ decision-making processes. I further use estimates to calculate and visualizedrivers’ reservation utility to test the rationality of the model.Figure 4.8: Average Predicted Reservation Utility by Hour.Figure 4.8 shows drivers’ average reservation utility across hours. First, on average,premium drivers have higher reservation utility than lower-tier drivers do. Second, for bothtypes of drivers, they have higher reservation utility during rush hours and midnight periods.115The prediction results are quite reasonable. During rush hours, drivers have more opportu-nities because of more demand requests and thus have higher reservation utility. During themidnight period, drivers have high reservation utility because most of them would not goout for work and are used to staying at home for rest.In this section, I estimate the model to quantify the network value and product intrinsicvalue to riders and drivers, and I also visualize drivers’ reservation wages using the esti-mates. Next, I will do some counterfactual analysis to get managerial implications.4.6 Platform Pricing Policy and Counterfactual AnalysisPreviously, I build up a structural model to estimate and evaluate riders’ and drivers’ behav-ior under the platform’s current pricing strategy. Drivers’ (riders’) choices toward the twotypes are determined by the earnings (prices), product value and the network value from thetwo types of product. The network size in equilibrium and the firm’s profit will be fullydetermined by the firm’s product strategy (vertical differentiation vs. homogenous project)and the price strategies. Once the two-sided platforms know the preferences of their tar-get users (via data analysis like what I did above or via other approaches), they can designoptimal strategies.From the analysis above, I have quantified users’ preferences over product value andnetwork value, and users’ price (earning) sensitivities on this platform. In this section,I will introduce more details about the platform’s current strategy, and further do somecounterfactual analysis using the model estimates above to examine whether there are betterstrategies for the platform and get some managerial implications.1164.6.1 Platform’s Current Pricing and Subsidy PolicyAccording to Tables 4.3, 4.1 and 4.2, the platform’s current strategy is that it charges differ-ent prices for different types. Moreover, the prices charged from riders and the earnings theplatform provides to drivers are not the same.Table 4.3 shows that there’s a significant gap between payment from riders and paymentgiven to drivers, especially for standard-type rides. That is, under the current strategy, theplatform is not simply giving drivers what riders pay, but is actually subsidizing drivers,especially for drivers who serve standard-type trips. The data shows that 53.4% of pre-mium trips offered subsidies to drivers, and 62.9% of the standard-type trips offered sub-sidies. Ride-sharing platforms usually charge a fixed commission rate from drivers basedon drivers’ earnings from trip fares. For example, Uber charges a 25% commission fromdrivers. The commission rate that the platform charges is 20% during the time window. Onaverage, a premium trip provides a $0.12 subsidy to drivers, while a standard trip providesa $2.54 subsidy.The possible reason why the platform wants to subsidize drivers is that they want toattract more users to gain enough network value. Once the platform gets a large scale, it canstart to earn profit from the two types. Comparing the subsidy to premium and standard, thefirm is subsidizing more to the standard type, which can be interpreted as that the platformis ’earning’ more from premium trips than from standard trips. This strategy is actuallyconsistent with the theoretical work in one-sided markets (Jing (2007)) that when there existnetwork externalities, firms can provide a low-quality product to enlarge the market size andexpand the market, and earn profit from the high-end product. But is the current strategyoptimal? Specifically, is vertical differentiation better than a homogeneous product? If yes,is there any better pricing strategy? Next, I will do counterfactual analyses to address thesequestions.1174.6.2 Counterfactual AnalysisAccording to the summary of the platform’s current strategy above, the platform is usingvertical differentiation and ’earns’ more profit from the premium type. The network size ofthe standard type seems to be stably around two times the size of the premium product underthe current pricing strategy. In this section, I use the above MCMC estimates and do coun-terfactual analyses regarding what strategy would be better for the platform. Specifically, Idiscuss several counterfactual scenarios below.Base Case. The platform uses vertical differentiation under the current pricing strategy;Case 1. The platform provides a homogeneous product, and only focuses on the low-endmarket, which is the standard product;Case 2. The platform provides no subsidy to drivers. That is, earnings for drivers are set tobe equal to what riders pay for each trip.To predict demand and supply in the new counterfactual settings, I have to solve fixedpoints in the simultaneous system by substituting predicted waiting time and cruising timeas a function of demand and supply into the choice stage for both rider and driver sides.Table 4.7 shows the detailed algorithm for solving fixed points.Table 4.7: Algorithm to Solve Fixed PointsSteps:1. Start with initial demand, d0, and supply, s0 in each market;2. Calculate estimated waiting time w1 and cruising time c1 using d0 and s0;3. Calculate new demand, d1, and supply, s1 using updated w1 and c1 in the choice model;4. Repeat steps 2-3 until demand and supply converge.That is max(|dn−dn−1|, |sn− sn−1|)≤ ε.Table 4.8 shows the counterfactual results, listing the pricing strategies for both sides118of each product and the predicted number of realized trips and profit in each counterfactualscenario. I discuss the three counterfactual cases one by one.Table 4.8: Counterfactual ResultsBase Case 1 Base2 Case 12 Case 2Price (premium) 13.15 - 13.15 - 13.15(4.16) (4.16) (4.16)Wage (premium) 16.6 - 16.6 - 13.15(7.89) (7.89) (4.16)Price (standard) 9.0 9.0 9.0 9.0 9.0(0.23) (0.23) (4.16) (4.16) (4.16)Wage (standard) 14.2 14.2 14.2 14.2 9.0(6.44) (6.44) (6.44) (6.44) (4.16)# realized trips 189,872 173,150 175,324 173,872 110,941Total profit ($) -504,353 -444,413 -298,607 -390,796 129,622Profit/trip ($) -2.65 -2.57 -1.70 -2.25 1.17I first compare the current vertical differentiation strategy with a homogeneous productcase. Under the platform’s current pricing strategy (Base case), the average premium price is$13.15 and the standard price is a constant, $9.0. And the average corresponding earning fordrivers is $16.6 from a premium trip and $14.2 from a standard trip. For the homogeneousstandard-only case (Case 1), I use the same pricing for the standard product. Compare Case1 with the Base case, I find that the market size is expanded using vertical differentiation(189,872 > 173,150). However, it seems more costly for the platform to maintain twonetworks as the per-trip cost is higher for vertical differentiation than the homogeneousproduct case ($2.65>$2.57). After further exploration, I find it’s actually not the case. Thereis a tricky part of the platform’s current strategy. According to the current pricing strategy,the price for the standard type is fixed, no matter how long the trip is, while the price for thepremium type is positively correlated with trip duration. In this way, we find riders actuallystrategically choose the premium when their rides are short and choose standard when theirrides are long. This can explain why the cost to maintain the two products is higher thanthe homogeneous case, as the platform extracts less surplus from riders because of their’smart’ behavior. This result also tells us that for ride-hailing companies, it’s not wise to set119constant prices, especially when there are multiple choices for riders. Using constant pricewould undermine the benefit of vertical differentiation for usage-based platforms.To answer whether it’s better to use vertical differentiation, I next use a clean setting inBase2 and Case 12, in which the price for the standard ride is set to be equal to the price forthe premium type minus $4, instead of using a constant price. In this way, we can somehowavoid riders’ ’smart’ behavior because the price of the premium is strictly higher than that ofthe standard type, regardless of what duration the trip has. Under this clean setting, I find themarket size is still expanded when using vertical differentiation (175,324 > 173,872). Thistells us that the size gain from market expansion by using vertical differentiation can offsetthe loss from network size segmentation. Moreover, profit from vertical differentiation isalso higher than the homogeneous product case (-$1.7 > -$2.25). This tells us that by usingvertical differentiation, the platform can extract more surplus from high-end customers.Next, I further explore how the network size of the two products would change withthe platform’s pricing strategy. Specifically, I examine what would happen if no subsidy isprovided to drivers when using vertical differentiation. In case 2, I set drivers’ earnings fromtrips equal to the prices charged from riders. Compare Case 2 with the base case in Base2, byproviding subsidy to drivers, the platform in total spend $(298,607 + 129,622) = $428,229and attract (175,324 − 110,941) = 64,383 more realized trips. Thus, the cost of attractingan extra realized trip is 428,22964,383 = $6.7. I further calculate the cost of attracting an extrapremium or standard realized trip separately. The result shows that the cost of attracting anextra premium realized trip is (6,857 + 72,200) / (34,792 − 26,536) = $9.6, while the costof attracting an extra standard realized trip is (291,750 + 57,422) / (140,532 − 84,405) =$6.2. The cost of attracting an extra premium trip (marginal cost) is higher than that of astandard trip, $9.6>$6.2. Thus, it’s more costly for the platform to expand and maintainthe premium network. The network size of the standard is around three times the size ofthe premium, and the corresponding cost of attracting an additional standard realized trip120is only 64.6% (=6.2/9.6) of the cost of attracting an additional premium realized trip. Thistells us the value of network size (scale effect). A large network size would generate a largepositive network value, while the expansion of the product with a relatively small networksize might be restrained.4.6.3 Managerial ImplicationsI further get managerial implications from the above counterfactual analyses. First, for ride-hailing platforms (usage-based), prices charged from riders should be positively correlatedwith trip duration instead of being constant, especially when multiple options are offeredto riders. Otherwise, it would undermine the platforms’ benefit from using vertical differ-entiation. Second, extending a product line by offering another high-end product is betterthan just providing a low-end product for ride-hailing platforms under the competitive en-vironment in New York City. That is, network externalities still make firms favor productdifferentiation in a two-sided market setting. On the one hand, market size can be expanded,as the effect of market expansion from vertical differentiation can offset the loss of a sin-gle network size advantage. On the other hand, firms can make more profit by extractingmore surplus from high-end customers. Finally, even though high-end customers are moreprofitable, the high-end network is more costly to maintain and harder to expand becauseof its relatively smaller network value (smaller user group). In a word, two-side platformsshould pay high attention to the network externalities when deciding whether to use verticaldifferentiation, and if yes, what price surcharges between products on each side to use.4.7 ConclusionThe distinct feature in two-sided markets, network externalities, makes the conventionalvertical differentiation strategy more complicated. On the one hand, platforms might be121better off designing vertically differentiated products in terms of attracting customers withheterogeneous preferences and expanding the market. On the other hand, offering multi-ple differentiated products might be less optimal than offering a homogeneous product ifthe segmented demand and supply limit the positive network effects in two-sided markets.Platforms might want to unite the markets from different products into one larger market toachieve greater positive network effects. Thus, understanding and quantifying the economicimpact of network externalities on both demand and supply in two-sided markets is of greatimportance to both industry practitioners and the academic audience. This paper is amongthe first to empirically examine how vertical differentiation works in two-sided markets. Iuse a distinct data set from a two-sided ride-hailing platform that provides two verticallydifferentiated types of car service. I develop a structural model to simultaneously modelthe demand and supply, and use the Bayesian MCMC method to estimate and quantifyusers’ preferences over intrinsic product value and network value. I present the main resultsconcerning riders’ and drivers’ behavior, the counterfactual analyses, and the managerialimplications for the platform below.On the rider side, both the intrinsic product value and the network value are crucial in af-fecting riders’ choices toward vertically differentiated products. According to the evidencefrom the data, riders who choose the premium product can get a better quality car and possi-bly better service quality, but has to on average wait longer because of the smaller networksize and pay more in trip fare. Overall, one-third of riders are willing to order a premiumtrip. According to the estimation results in Tables 4.6 and 4.4, both intrinsic product valueand network externalities are important for riders’ decisions. First, the premium productadds value for riders. The utility obtained from the premium product is 0.08 larger thanthat from the standard product. Second, a large network size also provides value to riders.Specifically, a 10% increase in supply productivity would decrease riders’ waiting time by0.38 minutes on average and result in a 0.05 utility gain for riders.122On the driver side, network externalities also play important roles in affecting drivers’decisions about whether to accept the offer. Specifically, network externalities affect drivers’decisions in two ways. First, network externalities can influence drivers’ earnings by af-fecting the time drivers need to cruise around to pick up riders. Data statistics show that,although a driver’s per-minute earning from a premium trip is $0.05 more than that froma standard trip, the cruising time for the premium trip is 0.34 minutes longer, making theearnings from the two types, taking cruising time into account, in equilibrium surprisinglyclose, around $0.53 per minute. According to the estimation results in Tables 4.6 and 4.4, a10% increase in demand would decrease cruising time by 0.10 minutes, which will furtherdriver’s per-minute earning. The exact increase in earnings is not determined, as it dependson other information about the offer, such as the price and trip duration. Suppose the de-mand variation can bring a $0.01 per-minute wage increase, it would increase drivers’ oddsof accepting the offer by 8.33%. Second, network externalities would also affect drivers’decisions of whether to accept the current offer by influencing the driver’s potential op-portunity to receive other offers. Estimation results show that a 10% increase in premiumdemand would cause the odds of accepting the current offer to decrease by 5.2%, and a 10%increase in standard demand would cause the odds to decrease by 2.3%. More opportunitiesto receive other offers would decrease drivers’ probability of accepting the current receivedoffer. Thus, network externalities play important roles in both riders’ and drivers’ behavior,and the intrinsic product value and network value together determine the degree of verticaldifferentiation in equilibrium.For two-sided platforms, it is crucial to understand users’ preferences for both intrinsicproduct quality and network value when deciding what strategy to use. Once users’ pref-erences are revealed using model analysis, the platform can further decide whether to usevertical differentiation, and if yes, what pricing strategies on each side for different productsto use. Taking the focal ride-hailing platform as an example, I conduct several counterfactual123studies based on the estimation results to answer those questions. Specifically, I comparethe platform’s current strategy with two counterfactual settings, one being a standard-onlyhomogenous product case and the other being vertical differentiation with a different pricingstrategy, to obtain managerial implications about what can be better strategies for the firm.These counterfactual analyses yield several managerial implications. First, for ride-hailing(usage-based) platforms, prices charged from riders should be monotonically increasingwith trip duration. The platforms should never use a constant price, especially when multi-ple options are offered. Second, extending the product line by offering a second high-endproduct is better for the ride-hailing platform than just providing a low-end product in thecompetitive environment of New York City. That is, network externalities still lead firms tofavor product differentiation in two-sided markets. The benefit from the market expansionby using vertical differentiation can offset the loss from market segmentation. Moreover,firms can make more profit by extracting more surplus from high-end customers. Finally,even though high-end customers are more profitable, the high-end network is more costlyto maintain and harder to expand because of its smaller network value (smaller user group).Two-sided platforms should take both the benefit and the cost of vertical differentiationinto consideration, and if using vertical differentiation, carefully decide the optimal pricesurcharges between products on both demand and supply sides.The findings should be interpreted based on the limitations inherent to this context.Though my research context offers a clean and detailed setting to observe riders’ individualdecisions towards two vertically differentiated products, it is also a setting lacking otherride-hailing competitors such as Uber. I try to make up by treating the green and yellowtaxis in New York City as the main competitors and assume the three choices—the premiumand standard types on the focal platform and the outside option—satisfy the IIA assumptionin my model. I leave it to future research when more information about other ride-hailingcompetitors is available.124Chapter 5ConclusionThis dissertation addresses the endogeneity problem for causal inference, and applies causalinference methods based on observational data to some interesting marketing and economictopics. For methodology development, I propose a new instrument-free method for correct-ing endogeneity. The method requires no instrument variables or even extra information.It corrects endogeneity using a control function approach by adding a generated regressorconstructed on existing variables. I also extend the method for high-dimensional data. Be-sides the methodology development, I empirically apply causal inference methods to learnconsumer behavior and get managerial insights.The first essay develops a generalized two-stage copula endogeneity correction (2sCOPE)instrument-free method to correct endogeneity. I theoretically prove that 2sCOPE can yieldconsistent and efficient causal-effect estimates under a much weaker assumption than theproposed copula methods in Park and Gupta (2012) and Haschka (2021). The simulationresults show that 2sCOPE yields consistent estimates under relaxed assumptions and im-proves estimation efficiency by up to 50%. Moreover, simulation studies also show that the2sCOPE method can deal with normal endogenous variables, while other copula methods125in Park and Gupta (2012) and Haschka (2021) cannot even handle close-to-normal endoge-nous variables such as t distribution with the degree of freedom equals to 10. The estimationresults in data application also confirm the performance of 2sCOPE because the estimatedprice coefficient using 2sCOPE is very close to the TSLS estimate, while OLS and previouscopula method in Park and Gupta (2012) show large biases. Finally, the 2sCOPE is straight-forward to implement, and can be widely applied to many other models, including linearregression models, linear panel models with mixed effects, random coefficient logit models,slope endogeneity, etc.The second essay further studies the 2sCOPE method and extends it to the high dimen-sional setting. As described in detail above, 2sCOPE corrects endogeneity by adding thefirst-stage residual as a generated regressor to the outcome regression. Thus, the estimationof the first-stage residual plays a central role. With high-dimensional data (dimension caneven be larger than sample size), estimation of the residual using the traditional method (i.e.,OLS estimation) would be problematic. To address the high-dimension problem, I proposea method combining the 2sCOPE with some lasso-based methods (lasso, post-lasso, de-biased lasso) to select important variables in the first stage. I call it the lasso-based 2sCOPEmethod. Simulation results show that 2sCOPE without feature selection in the first stage suf-fers a large finite-sample bias when the dimension is relatively large, while the lasso-based2sCOPE method can improve substantial estimation accuracy and efficiency, by around50%. I further apply the method to examine the effect of the government’s policy strin-gency in response to COVID-19 on citizens’ happiness. I use country-level cross-sectionaldata with a large dimension of country characteristics (i.e., around half of the sample size).The result shows that the estimated coefficient of the proposed lasso-based 2sCOPE methodmakes the effect of policy strictness on happiness more significantly negative. Increasingpolicy strictness by 10 units would decrease happiness growth by 0.034, which is larger thanone standard deviation of happiness growth.126The above two essays focus on methodology development in correcting endogeneity forcausal inference. The third essay empirically applies causal inference methods to solve theendogeneity problem in an interesting marketing scenario, two-sided markets. Specifically,I apply structural modeling with IV approach to correct the endogeneity of network sizein examining vertical differentiation in two-sided markets. Data is from a leading ride-hailing platform that provides two vertically differentiated types of car service. I buildup a simultaneous structural model together with the IV approach on both demand andsupply sides, use the Bayesian MCMC method to quantify users’ preferences over intrinsicproduct value and network value, and also conduct counterfactual analyses to get managerialinsights. The result shows that both intrinsic product value and network externalities areimportant in determining the degree of vertical differentiation. Users favor high intrinsicproduct quality and hate long waiting (cruising) time affected by network size. For two-sided platforms, it is crucial to understand users’ preferences over both intrinsic productquality and network value when deciding what product strategy (i.e., vertical differentiationvs. homogeneous product) and price strategies to use.In the end, I wish to point out a few general limitations and research directions regardingthe dissertation. First, in terms of methodology development to correct endogeneity, the dis-sertation proposes an instrument-free method and also adapts it to the high-dimensional set-ting combined with machine learning techniques. Though the method relaxes some key as-sumptions in the literature, it still relies on some assumptions. For example, for the 2sCOPEin the first essay to work best, the distributions of the endogenous regressors need to containadequate information, which is violated for Bernoulli distributions or discrete distributionswith small support. Moreover, 2sCOPE hinges on the normal structural error and Gaus-sian copula dependence structure. Future research is needed for more flexible methods thattest and relax these assumptions. Second, it is interesting to apply our proposed method(2sCOPE or lasso-based 2sCOPE) for more classical empirical applications. For example,127we could further apply the proposed instrument-free method to quantify the endogenousnetwork value in the third essay in the future.128BibliographyAghion, P., Bloom, N., Blundell, R., Griffith, R., and Howitt, P. (2005). Competition andinnovation: An inverted u relationship. Quarterly Journal of Economics, 120:701–728.→ page 9Amado, A., Cortez, P., Rita, P., and Moro, S. (2018). Research trends on big data inmarketing: A text mining and topic modeling based literature analysis. EuropeanResearch on Management and Business Economics, 24(1):1–7. → page 61Anderson, E. T. and Simester, D. I. (2004). Long-run effects of promotion depth on newversus established customers: three field studies. Marketing Science, 23(1):4–20. →page 15Angrist, J. D. and Krueger, A. B. (1991). Does compulsory school attendance affectschooling and earnings? The Quarterly Journal of Economics, 106(4):979–1014. →page 6Angrist, J. D. and Krueger, A. B. (2001). Instrumental variables and the search foridentification: From supply and demand to natural experiments. Journal of Economicperspectives, 15(4):69–85. → pages 15, 63Arora, N. and Huber, J. (2001). Improving parameter estimates and model prediction byaggregate customization in choice experiments. Journal of Consumer Research,28(2):273–283. → page 36Ascarza, E. (2018). Retention futility: Targeting high-risk customers might be ineffective.Journal of Marketing Research, 55(1):80–98. → page 61Ataman, M. B., Van Heerde, H. J., and Mela, C. F. (2010). The long-term effect ofmarketing strategy on brand sales. Journal of Marketing Research, 47(5):866–882. →page 15Atefi, Y., Ahearne, M., Maxham III, J. G., Donavan, D. T., and Carlson, B. D. (2018).Does selective sales force training work? Journal of Marketing Research,55(5):722–737. → pages 17, 64, 82Athey, S. and Imbens, G. W. (2006). Identification and inference in nonlineardifference-in-differences models. Econometrica, 74(2):431–497. → page 15129Athey, S. and Imbens, G. W. (2019). Machine learning methods that economists shouldknow about. Annual Review of Economics, 11:685–725. → page 61Bai, J. and Ng, S. (2008). Forecasting economic time series using targeted predictors.Journal of Econometrics, 146(2):304–317. → pages 62, 65Bai, J. and Ng, S. (2009). Boosting diffusion indices. Journal of Applied Econometrics,24(4):607–629. → pages 62, 65Becker, J.-M., Proksch, D., and Ringle, C. M. (2021). Revisiting gaussian copulas tohandle endogenous regressors. Journal of the Academy of Marketing Science, pages1–21. → pages 9, 10, 11, 12, 13, 18, 23, 39, 47Belloni, A., Chen, D., Chernozhukov, V., and Hansen, C. (2012). Sparse models andmethods for optimal instruments with an application to eminent domain. Econometrica,80(6):2369–2429. → pages 62, 65, 66, 70Belloni, A. and Chernozhukov, V. (2013). Least squares after model selection inhigh-dimensional sparse models. Bernoulli, 19(2):521–547. → pages 62, 65Belloni, A., Chernozhukov, V., and Hansen, C. (2014a). High-dimensional methods andinference on structural and treatment effects. Journal of Economic Perspectives,28(2):29–50. → pages 61, 62, 65Belloni, A., Chernozhukov, V., and Hansen, C. (2014b). Inference on treatment effectsafter selection among high-dimensional controls. The Review of Economic Studies,81(2):608–650. → pages 62, 65Berry, S. T. (1994). Estimating discrete-choice models of product differentiation. TheRAND Journal of Economics, pages 242–262. → page 16Blundell, R. and Matzkin, R. L. (2014). Control functions in nonseparable simultaneousequations models. Quantitative Economics, 5(2):271–295. → page 13Blundell, R. and Powell, J. L. (2003). Endogeneity in nonparametric and semiparametricregression models. Econometric society monographs, 36:312–357. → page 13Blundell, R. W. and Powell, J. L. (2004). Endogeneity in semiparametric binary responsemodels. The Review of Economic Studies, 71(3):655–679. → page 13Bombaij, N. J. and Dekimpe, M. G. (2020). When do loyalty programs work? themoderating role of design, retailer-strategy, and country characteristics. InternationalJournal of Research in Marketing, 37(1):175–195. → pages 17, 64, 82Bryan, K. A. and Gans, J. S. (2019). A theory of multihoming in rideshare competition. J.Econ. Manag. Strateg., 28(1):89–96. → page 91Burmester, A. B., Becker, J. U., van Heerde, H. J., and Clement, M. (2015). The impact ofpre-and post-launch publicity and advertising on new product sales. InternationalJournal of Research in Marketing, 32(4):408–417. → pages 17, 64, 82130Chen, M. K., Rossi, P. E., Chevalier, J. A., and Oehlsen, E. (2019). The value of flexiblework: Evidence from uber drivers. Journal of Political Economy, 127(6):2735–2794. →page 90Chintagunta, P., Dube´, J.-P., and Goh, K. Y. (2005). Beyond the endogeneity bias: Theeffect of unmeasured brand characteristics on household-level brand choice models.Management Science, 51(5):832–849. → page 52Chintagunta, P., Erdem, T., Rossi, P. E., and Wedel, M. (2006). Structural modeling inmarketing: review and assessment. Marketing Science, 25(6):604–616. → page 16Cui, G., Wong, M. L., and Lui, H.-K. (2006). Machine learning for direct marketingresponse models: Bayesian networks with evolutionary programming. ManagementScience, 52(4):597–612. → page 61Danaher, P. (2007). Modeling page views across multiple websites with an application tointernet reach and frequency prediction. Marketing Science, 26:422–437. → pages19, 67Danaher, P. and Smith, M. (2011). Modeling multivariate distributions using copulas:Applications in marketing (with discussion and rejoinder). Marketing Science, 30:4–21.→ pages 7, 20, 55, 58, 67, 68Datta, H., Foubert, B., and Van Heerde, H. J. (2015). The challenge of retaining customersacquired with free trials. Journal of Marketing Research, 52(2):217–234. → pages17, 64, 82Dotson, J. P. and Allenby, G. M. (2010). Investigating the strategic influence of customerand employee satisfaction on firm financial performance. Marketing Science,29(5):895–908. → page 16Dube´, J.-P., Chintagunta, P., Petrin, A., Bronnenberg, B., Goettler, R., Seetharaman, P. S.,Sudhir, K., Thomadsen, R., and Zhao, Y. (2002). Structural applications of the discretechoice model. Marketing letters, pages 207–220. → page 16Ebbes, P., Wedel, M., and Bo¨ckenholt, U. (2009). Frugal iv alternatives to identify theparameter for an endogenous regressor. Journal of Applied Econometrics,24(3):446–468. → page 16Ebbes, P., Wedel, M., Bo¨ckenholt, U., and Steerneman, T. (2005). Solving and testing forregressor-error (in) dependence when no instrumental variables are available: With newevidence for the effect of education on income. Quantitative Marketing and Economics,3(4):365–392. → pages 7, 16, 55Eckert, C. and Hohberger, J. (2022). Addressing endogeneity without instrumentalvariables: An evaluation of the gaussian copula approach for management research.Journal of Management, DOI: 10.1177/01492063221085913. → pages 8, 12, 13, 55131Elshiewy, O. and Boztug, Y. (2018). When back of pack meets front of pack: How salientand simplified nutrition labels affect food sales in supermarkets. Journal of PublicPolicy & Marketing, 37(1):55–67. → pages 17, 64, 82Erickson, T. and Whited, T. M. (2002). Two-step gmm estimation of the errors-in-variablesmodel using high-order moments. Econometric Theory, 18(3):776–799. → pages 7, 16Feldman, J., Zhang, D. J., Liu, X., and Zhang, N. (2022). Customer choice models vs.machine learning: Finding optimal product displays on alibaba. Operations Research,70(1):309–328. → page 61Frechette, G. R., Lizzeri, A., and Salz, T. (2019). Frictions in a competitive, regulatedmarket: Evidence from taxis. American Economic Review, 109(8):2954–92. → page 91Gabszewicz, J. J., Shaked, A., Sutton, J., and Thisse, J.-F. (1986). Segmenting the market:The monopolist’s optimal product mix. Journal of economic theory, 39(2):273–289. →page 86Gabszewicz, J. J. and Wauthy, X. Y. (2014). Vertical product differentiation and two-sidedmarkets. Economics Letters, 123(1):58–61. → page 90Godes, D. and Mayzlin, D. (2009). Firm-created word-of-mouth communication:Evidence from a field test. Marketing science, 28(4):721–739. → page 15Greene, W. H. (2012). Econometric Analysis. 7th Edition. Prentice Hall, Upper SaddleRiver, NJ. → page 32Gruner, R. L., Vomberg, A., Homburg, C., and Lukas, B. A. (2019). Supporting newproduct launches with social media communication and online advertising: sales volumeand profit implications. Journal of Product Innovation Management, 36(2):172–195. →pages 17, 64, 82Guda, H. and Subramanian, U. (2019). Your uber is arriving: Managing on-demandworkers through surge pricing, forecast communication, and worker incentives.Management Science, 65(5):1995–2014. → page 91Guitart, I. A., Gonzalez, J., and Stremersch, S. (2018). Advertising non-premium productsas if they were premium: The impact of advertising up on advertising elasticity andbrand equity. International journal of research in marketing, 35(3):471–489. → pages17, 64, 82Hahn, J., Todd, P., and Van der Klaauw, W. (2001). Identification and estimation oftreatment effects with a regression-discontinuity design. Econometrica, 69(1):201–209.→ page 15Hartmann, W., Nair, H. S., and Narayanan, S. (2011). Identifying causal marketing mixeffects using a regression discontinuity design. Marketing Science, 30(6):1079–1097. →pages 15, 16132Haschka, R. E. (2021). Express: Handling endogenous regressors using copulas: Ageneralization to linear panel models with fixed effects and correlated regressors.Journal of Marketing Research, First appeared online on Dec 18, 2021. → pages3, 10, 11, 12, 14, 17, 31, 32, 39, 64, 68, 125, 126Heckman, J. J. and Robb Jr, R. (1985). Alternative methods for evaluating the impact ofinterventions: An overview. Journal of econometrics, 30(1-2):239–267. → page 13Heitmann, M., Landwehr, J. R., Schreiner, T. F., and van Heerde, H. J. (2020). Leveragingbrand equity for effective visual product design. Journal of Marketing Research,57(2):257–277. → pages 17, 64, 82Hoch, S. J., Kim, B.-D., Montgomery, A. L., and Rossi, P. E. (1995). Determinants ofstore-level price elasticity. Journal of marketing Research, 32(1):17–29. → page 52Hogan, V. and Rigobon, R. (2003). Using unobserved supply shocks to estimate the returnsto education. Unpublished manuscript. → pages 7, 16Javanmard, A. and Montanari, A. (2014a). Confidence intervals and hypothesis testing forhigh-dimensional regression. The Journal of Machine Learning Research,15(1):2869–2909. → pages 62, 65, 66, 73, 74Javanmard, A. and Montanari, A. (2014b). Hypothesis testing in high-dimensionalregression under the gaussian random design model: Asymptotic theory. IEEETransactions on Information Theory, 60(10):6522–6554. → pages 73, 74Jing, B. (2007). Network externalities and market segmentation in a monopoly. EconomicsLetters, 95(1):7–13. → pages 86, 89, 117Johnson, G. A., Lewis, R. A., and Nubbemeyer, E. I. (2017). Ghost ads: Improving theeconomics of measuring online ad effectiveness. Journal of Marketing Research,54(6):867–884. → page 15Keller, W. I., Deleersnyder, B., and Gedenk, K. (2019). Price promotions and popularevents. Journal of Marketing, 83(1):73–88. → pages 17, 64, 82Kleibergen, F. and Zivot, E. (2003). Bayesian and classical approaches to instrumentalvariable regression. Journal of Econometrics, 114(1):29–72. → page 15Lamey, L., Deleersnyder, B., Steenkamp, J.-B. E., and Dekimpe, M. G. (2018). Newproduct success in the consumer packaged goods industry: A shopper marketingapproach. International Journal of Research in Marketing, 35(3):432–452. → pages17, 64, 82Lancaster, K. (1990). The Economics of Product Variety: A Survey. Mark. Sci.,9(3):189–206. → page 89Leeb, H. and Po¨tscher, B. M. (2008a). Can one estimate the unconditional distribution ofpost-model-selection estimators? Econometric Theory, 24(2):338–376. → page 65133Leeb, H. and Po¨tscher, B. M. (2008b). Guest editors’editorial: Recent developments inmodel selection and related areas. Econometric Theory, 24(2):319–322. → page 65Lewbel, A. (1997). Constructing instruments for regressions with measurement error whenno additional data are available, with an application to patents and r&d. Econometrica:journal of the econometric society, pages 1201–1213. → pages 7, 16Li, H. and Srinivasan, K. (2019). Competitive dynamics in the sharing economy: ananalysis in the context of airbnb and hotels. Marketing Science, 38(3):365–391. → page90Lin, S. (2020). Two-sided price discrimination by media platforms. Marketing Science,39(2):317–338. → pages 90, 91Liu, Q. and Serfes, K. (2013). Price Discrimination in Two-Sided Markets. J. Econ.Manag. Strateg., page 19. → page 90Mackiewicz, R. and Falkowski, A. (2015). The use of weber fraction as a tool to measureprice sensitivity: a gain and loss perspective. Advances in Consumer Research, 43. →page 52Maskin, E. and Riley, J. (1984). Monopoly with incomplete information. The RANDJournal of Economics, 15(2):171–196. → page 86Mendelson, H. (2000). Organizational architecture and success in the informationtechnology. Management Science, 46:513–529. → page 8Mullainathan, S. and Spiess, J. (2017). Machine learning: an applied econometricapproach. Journal of Economic Perspectives, 31(2):87–106. → page 61Mussa, M. and Rosen, S. (1978). Monopoly and product quality. Journal of Economictheory, 18(2):301–317. → pages 86, 89Narayanan, S. and Kalyanam, K. (2015). Position effects in search advertising and theirmoderators: A regression discontinuity approach. Marketing Science, 34(3):388–407.→ page 15Ngai, E. W. and Wu, Y. (2022). Machine learning in marketing: A literature review,conceptual framework, and research agenda. Journal of Business Research, 145:35–48.→ page 61Novak, S. and Stern, S. (2009). Complementarity among vertical integration decisions:Evidence from automobile product development. Management Science, 55(2):311–332.→ page 15Otter, T., Gilbride, T. J., and Allenby, G. M. (2011). Testing models of strategic behaviorcharacterized by conditional likelihoods. Marketing Science, 30(4):686–701. → page 16134Papies, D., Ebbes, P., and Van Heerde, H. J. (2017). Addressing endogeneity in marketingmodels. In Advanced methods for modeling markets, pages 581–627. Springer. → pages9, 15Park, S. and Gupta, S. (2012). Handling endogenousregressors by joint estimation using copulas. Marketing Science, 31(4):567–586.→ pages2, 3, 7, 8, 9, 11, 12, 15, 16, 17, 18, 19, 20, 21, 23, 27, 34, 37, 44, 48, 51, 56, 58, 64, 68, 82, 125, 126, 139, 151, 153Peng, J., Zhu, J., Bergamaschi, A., Han, W., Noh, D.-Y., Pollack, J. R., and Wang, P.(2010). Regularized multivariate regression for identifying master predictors withapplication to integrative genomics study of breast cancer. The annals of appliedstatistics, 4(1):53. → page 61Petrin, A. and Train, K. (2010). A control function approach to endogeneity in consumerchoice models. Journal of marketing research, 47(1):3–13. → pages 13, 26, 70, 108Qian, Y. (2008). Impacts of entry by counterfeiters. Quarterly Journal of Economics,123:1577–1609. → page 15Qian, Y. and Xie, H. (2021). Simplifying bias correction for selective sampling: A unifieddistribution-free approach to handling endogenously selected samples. MarketingScience, Forthcoming, available at https://www.nber.org/papers/w28801. → page 36Rigobon, R. (2003). Identification through heteroskedasticity. Review of Economics andStatistics, 85(4):777–792. → pages 7, 16Rochet, J.-c. and Tirole, J. (2006). Two-sided markets : a progress report. RAND J. Econ.,37(3):645–667. → page 85Rossi, P. E. (2014). Even the rich can make themselves poor: A critical examination of ivmethods in marketing applications. Marketing Science, 33(5):655–672. → pages7, 15, 63Rutz, O. J. and Watson, G. F. (2019). Endogeneity and marketing strategy research: Anoverview. Journal of the Academy of Marketing Science, 47(3):479–498. → page 15Salant, S. W. (1989). When is inducing self-selection suboptimal for a monopolist? TheQuarterly Journal of Economics, 104(2):391–397. → pages 86, 89Shaked, A. V. and Sutton, J. (1982). Relaxing Price Competition Through ProductDifferentiation. Rev. Econ. Stud., page 11. → page 89Shi, H., Sridhar, S., Grewal, R., and Lilien, G. (2017). Sales representative departures andcustomer reassignment strategies in business-to-business markets. Journal of Marketing,81(2):25–44. → page 15Sklar, M. (1959). Fonctions de repartition an dimensions et leurs marges. Publ. inst.statist. univ. Paris, 8:229–231. → pages 19, 67135Sorescu, A., Warren, N. L., and Ertekin, L. (2017). Event study methodology in themarketing literature: An overview. Journal of the Academy of Marketing Science,45:186–207. → page 8Sridhar Moorthy, K. (1984). Market Segmentation, Self-Selection, and Product LineDesign. Technical Report 4. → page 89Stokey, N. L. (1979). Intertemporal price discrimination. The Quarterly Journal ofEconomics, pages 355–371. → pages 86, 89Sudhir, K. (2001). Competitive pricing behavior in the auto market: A structural analysis.Marketing Science, 20(1):42–60. → page 16Sun, B. (2005). Promotion effect on endogenous consumption. Marketing science,24(3):430–443. → page 16Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of theRoyal Statistical Society: Series B (Methodological), 58(1):267–288. → pages62, 65, 71Van Heerde, H. J., Gijsenberg, M. J., Dekimpe, M. G., and Steenkamp, J.-B. E. (2013).Price and advertising effectiveness over the business cycle. Journal of MarketingResearch, 50(2):177–193. → page 15Vandenbosch, M. and Weinberg, C. (1995). Product and Price Competition in aTwo-Dimensional Vertical Differentiation Model. Mark. Sci., page 26. → page 89Villas-Boas, J. M. and Winer, R. S. (1999). Endogeneity in brand choice models.Management science, 45(10):1324–1338. → pages 52, 55Wang, Y. and Blei, D. (2019). The blessings of multiple causes. Journal of the AmericanStatistical Association, 114(528):1574–1596. → page 16Wang, Y., Wu, C., and Zhu, T. (2019). Mobile hailing technology and taxi drivingbehaviors. Marketing Science, 38(5):734–755. → page 91Wetzel, H. A., Hattula, S., Hammerschmidt, M., and van Heerde, H. J. (2018). Buildingand leveraging sports brands: evidence from 50 years of german professional soccer.Journal of the Academy of Marketing Science, 46(4):591–611. → pages 17, 64, 82Wooldridge, J. M. (2015). Control function methods in applied econometrics. Journal ofHuman Resources, 50(2):420–445. → page 13Yang, F., Qian, Y., and Xie, H. (2022). Addressing endogeneity using a two-stage copulagenerated regressor approach. Technical report, National Bureau of Economic Research.→ pages 60, 64, 65, 66, 67, 68, 82Yang, S., Chen, Y., and Allenby, G. M. (2003). Bayesian analysis of simultaneous demandand supply. Quantitative marketing and economics, 1(3):251–275. → pages 16, 55136Zervas, G., Proserpio, D., and Byers, J. W. (2017). The rise of the sharing economy:Estimating the impact of airbnb on the hotel industry. Journal of marketing research,54(5):687–705. → page 90Zhang, S., Lee, D., Singh, P. V., and Srinivasan, K. (2017). How much is an image worth?airbnb property demand estimation leveraging large scale image analytics. AirbnbProperty Demand Estimation Leveraging Large Scale Image Analytics (May 25, 2017).→ page 61137Appendix AChapter 2 AppendicesAppendix: ProofsA.1 Proof of Theorem 1Under the Gaussian copula assumption for structural error term ξt and the endogenous re-gressor Pt , and the normality assumption of ξt , the outcome regression becomes (Equation2.6)Yt = µ+Ptα+Wtβ +σξ ·ρ ·P∗t +σξ ·√1−ρ2 ·ωt .Because of the exogeneity assumption of Wt in linear model (Equation 2.1), Cov(Wt ,ξt) = 0,Cov(Wt ,ξt) =Cov(Wt ,σξ ·ρ ·P∗t +σξ ·√1−ρ2 ·ωt)= σξ ·ρ ·Cov(Wt ,P∗t )+σξ ·√1−ρ2 ·Cov(Wt ,ωt) = 0.138Thus, whenever Wt and P∗t is correlated, the covariance between Wt and P∗t isCov(Wt ,ωt) =− ρ√1−ρ2Cov(Wt ,P∗t ) ̸= 0,and Wt would be correlated with the new error term ωt . Theorem proved.A.2 Assumption 4.b in CopulaP&GAccording to Park and Gupta (2012), under a Gaussian copula model for (P1,t ,P2,t ,ξt), thestructural model in Equation (2.13) with two endogenous regressors can be re-expressed asYt =µ+P1,tα1+P2,tα2+Wtβ +σξρξ1−ρ12ρξ21−ρ212·P∗1,t +σξρξ2−ρ12ρξ11−ρ212·P∗2,t+σξ ·√1−ρ2ξ1−(ρξ2−ρ12ρξ1)21−ρ212·ωt . (A.1)where P∗1,t =Φ−1(H1(P1,t)), P∗2,t =Φ−1(H2(P2,t)), and H1(·) and H2(·) are CDFs of P1,t andP1,t , respectively, ρ12 is the correlation between P∗1,t and P∗2,t , ρξ1 is the correlation betweenξ and P∗1,t , ρξ2 is the correlation between ξ and P∗2,t , and ωt is a standard normal randomvariable that is independent of P∗1,t and P∗2,t . For the OLS estimation of Equation (A.1)to yield consistent estimates, Wt need also be uncorrelated with ωt , which requires thatCov(Wt ,σξ ·√1−ρ2ξ1−(ρξ2−ρ12ρξ1)21−ρ212·ωt) =−Cov(Wt , ρξ1−ρ12ρξ21−ρ212 ·P∗1,t +ρξ2−ρ12ρξ11−ρ212·P∗2,t) =0 (Assumption 4(b) in the main text) whereρξ1−ρ12ρξ21−ρ212·P∗1,t +ρξ2−ρ12ρξ11−ρ212·P∗2,t is the CCF usedto control for endogeneity in CopulaP&G.A.3 COPE Method DevelopmentUnder the Gaussian copula model for the endogenous regressor, Pt , the correlated exoge-nous regressor, Wt , and the structural error term, ξt in Equation (3.5), the structural error in139Equation (2.1) can be re-expressed asξt = σξ ·ξ ∗t =σξρpξ1−ρ2pwP∗t +−σξρpwρpξ1−ρ2pwW ∗t +σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pwω3,t . (A.2)In this way, the structural error term ξt is split into two parts: one part as a function of P∗tand W ∗t that captures the endogeneity of Pt and the association of Wt with ξt |Pt 1, and theother part as an independent new error term. Then, we substitute Equation (A.2) into themain model in Equation (2.1), and obtain the following regression equation:Yt = µ+Ptα+Wtβ+σξρpξ1−ρ2pwP∗t +−σξρpwρpξ1−ρ2pwW ∗t +σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pw ·ω3,t . (A.3)Given P∗t and W ∗t as additional regressors, ω3,t is not correlated with all regressors on theright-hand side of Equation (A.3) as proved in Theorem ?? below, and thus we can con-sistently estimate the model using the least squares estimator. The regressors P∗t and W ∗tcan be generated from the nonparametric distribution of Pt and Wt as P∗t =Φ−1(Ĥ(Pt)) andW ∗t = Φ−1(L̂(Wt)), where Ĥ(Pt) and L̂(Wt) are the empirical CDFs of Pt and Wt , respec-tively.Theorem A1. Estimation Consistency. Assuming (1) the error term is normal, (2) the en-dogenous regressor Pt and exogenous regressors Wt are non-normally distributed, and (3) aGaussian Copula for the error term, Pt and Wt , Cov(ω3,t ,Wt)=Cov(ω3,t ,Pt)=Cov(ω3,t ,W ∗t )=Cov(ω3,t ,P∗t ) = 0 and thus the OLS estimation of Equation (A.3) yields consistent estimatesof model parameters.Proof: See Online Appendix, Proof of Theorem A11Although the exogenous regressor Wt and ξt are uncorrelated, Wt and ξt |Pt (the error component in ξtremaining after removing the effect of the endogenous regressor Pt ) can be correlated as seen by the correlationbetween Wt and ωt in Figure ?? (b).140As shown in Theorem A1, the proposed COPE method does not require the uncorre-latedness between P∗t and Wt for consistent model estimation, an assumption needed forCopulaP&G. In fact, CopulaP&G can be obtained as a special case of the COPE: when Wtis uncorrelated with Pt (i.e. ρpw = 0) and also uncorrelated with P∗t under the joint copulamodel,−σξρpwρpξ1−ρ2pw W∗t in Equation (A.3) vanishes and COPE based on Equation (A.3) reducesto CopulaP&G base on Equation (2.6). This broader applicability of COPE is a merit ofCOPE. However, similar to CopulaP&G, COPE requires the non-normality of the endoge-nous regressor Pt to fulfill the full-rank identification assumption. Moreover, a correlationbetween endogenous regressor P and the exogenous regressors W will cause CopulaP&Gto transfer the endogeneity from P to W ; the correction for the induced endogenous regres-sor W should have the same non-normality assumption for model identification as with P.In the next subsection, we will develop a novel two-stage COPE method that relaxes theregressor non-normality assumption. We further extend the model to incorporate multipleendogenous regressors in the section 4.4.COPE in Multiple Endogenous Regressors Case Under the Gaussian Copula assumptionthat [P∗1,t ,P∗2,t ,W∗t ,ξ ∗t ] follows a multivariate normal distribution:P∗1,tP∗2,tW ∗tξ ∗t∼ N0000,1 ρp ρwp1 ρξ p1ρp 1 ρwp2 ρξ p2ρwp1 ρwp2 1 0ρξ p1 ρξ p2 0 1,141we have:P∗1,tP∗2,tW ∗tξ ∗t=1 0 0 0ρp√1−ρ2p 0 0ρwp1ρwp2−ρpρwp1√1−ρ2p√1−ρ2wp1−(ρwp2−ρpρwp1)21−ρ2p 0ρξ p1ρξ p2−ρpρξ p1√1−ρ2p−ρwp1ρξ p1−(ρwp2−ρpρwp1)(ρξ p2−ρpρξ p1)1−ρ2p√1−ρ2wp1−(ρwp2−ρpρwp1)21−ρ2pm·ω1,tω2,tω3,tω4,t,ω1,tω2,tω3,tω4,t∼ N0000,1 0 0 00 1 0 00 0 1 00 0 0 1, (A.4)where m is a function of all the ρs. Under the Gaussian Copula assumption above, we canderive ξ ∗t as a function of Pt and Wt . After simplification, the structural error in Equation(2.13) can be decomposed asξt = σξξ ∗t = η1P∗1,t +η2P∗2,t − (η1ρwp1+η2ρwp2)W ∗t +σξ ·m ·ω4,t . (A.5)whereη1 =σξρξ p1(1−ρ2wp2)−σξρξ p2(ρp−ρwp1ρwp2)1−ρ2p−ρ2wp1−ρ2wp2+2ρpρwp1ρwp2,η2 =σξ (ρwp1ρwp2ρξ p1+ρξ p2−ρpρξ p1−ρ2wp1ρξ p2)1−ρ2p−ρ2wp1−ρ2wp2+2ρpρwp1ρwp2. (A.6)The COPE method with one endogenous regressor in Equation (A.3) is then extended toYt = µ+P1,tα1+P2,tα2+Wtβ +η1P∗1,t +η2P∗2,t − (η1ρwp1+η2ρwp2)W ∗t +σξ ·m ·ω4,t .(A.7)142In Equation (A.7), the new error term ω4,t is uncorrelated with all the regressors on theright-hand side of Equation (A.7). Thus, the OLS estimation of Equation (A.7) providesconsistent estimates of structural regression model parameters (µ,α1,α2,β ).A.4 Proof of Theorem A1Under the Gaussian copula model for (Pt ,ξt) and the normality assumption of the error termξt , we can divide ξt into an endogenous and exogenous part and our proposed COPE methodis based on the OLS estimation of the regression below (Equation A.3) by adding P∗t andW ∗t as generated regressors.Yt = µ+Ptα+Wtβ +σξρpξ1−ρ2pwP∗t +−σξρpwρpξ1−ρ2pwW ∗t +σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pw ·ω3,tWe want to prove that the new error term ω3,t is uncorrelated with all terms of the right-handside. Since ω1,t , ω2,t and ω3,t follow a standard multivariate Gaussian distribution (Equation3.5), they are independent. According to the same equation, W ∗t and P∗t are linear functionsof ω1,t and ω2,t . Thus, P∗t and W ∗t are normally distributed and are independent of ω3,t .Since functions of independent variables are still independent, Pt (Wt), as a function of P∗t(W ∗t ), would be uncorrelated with ω3,t and thus ω3,t is not correlated with Pt ,P∗t ,Wt and W ∗ton the right-hand side of Equation (A.3). Since Pt and Wt are nonnormal distributed, the fullrank assumption is satisfied and thus COPE yields consistent estimates. Theorem proved.Next we show that this result can be readily extended to the multi-dimension Wt case.We first derive the regression of the COPE method. Here we take 2-dimension Wt as anexample. When there are one endogenous regressor Pt and two exogenous regressors Wt ,143the linear regression is:Yt = β0+β1Pt +β2W1,t +β3W2,t +ξt (A.8)Under the Gaussian Copula assumption,P∗tW ∗1,tW ∗2,tξ ∗t∼ N0000,1 ρ1 ρ2 ρξρ1 1 ρw 0ρ2 ρw 1 0ρξ 0 0 1(A.9)The multivariate normal distribution can be written as follows:P∗tW ∗1,tW ∗2,tξ ∗t=1 0 0 0ρ1√1−ρ21 0 0ρ2 ρw−ρ1ρ2√1−ρ21√1−ρ22 − (ρw−ρ1ρ1)21−ρ210ρξ−ρ1ρξ√1−ρ21(ρw−ρ1ρ2)ρ1ρξ1−ρ21−ρ2ρξ√1−ρ22−(ρw−ρ1ρ2)21−ρ21γ·ω1,tω2,tω3,tω4,t,where ωk,t ∼N(0,1),k= 1,2,3,4, γ =√√√√√1−ρ2ξ − ρ21ρ2ξ1−ρ21 −( (ρw−ρ1ρ2)ρ1ρξ1−ρ21−ρ2ρξ√1−ρ22−(ρw−ρ1ρ2)21−ρ21)2. Structuralerror ξt can then be written as a function of P∗t and W ∗t ,ξt = σξξ ∗t =σξρξ (1−ρ2w)1−ρ21 −ρ22 +2ρ1ρ2ρw+ρ2w(P∗t −ρ1−ρ2ρw1−ρ2wW ∗1,t −ρ2−ρ1ρw1−ρ2wW ∗2,t)+σξ γ ·ω4,t .(A.10)Thus, our COPE method in 2-W case becomes:Yt = β0+β1Pt +β2W1,t +β3W2,t +β4P∗t +β5W∗1,t +β6W∗2,t +σξ γ ·ω4,t (A.11)144whereβ4 =σξρξ (1−ρ2w)1−ρ21 −ρ22 +2ρ1ρ2ρw+ρ2wβ5 =−σξρξ (1−ρ2w)1−ρ21 −ρ22 +2ρ1ρ2ρw+ρ2w· ρ1−ρ2ρw1−ρ2wβ6 =−σξρξ (1−ρ2w)1−ρ21 −ρ22 +2ρ1ρ2ρw+ρ2w· ρ2−ρ1ρw1−ρ2w.Since ω4,t is independent of P∗t , W ∗1,t and W∗2,t , it would also be uncorrelated with any func-tional form of P∗t , W ∗1,t and W∗2,t , and thus ω4,t is uncorrelated with any other terms in Equa-tion (A.11). The COPE method can easily be extended to the case with multiple endogenousregressors by adding copula transformation of each regressor as generated regressors intothe outcome regression, and the proof of estimation consistency is similar.A.5 Proof of Theorem 2: Consistency of 2sCOPEWe have shown the derivation of 2sCOPE method in Section 3. The system of equationsused in 2sCOPE method (Equations 3.1, 2.9) leads to the following equationsYt = µ+Ptα+Wtβ +σξρpξ1−ρ2pwεt +σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pw ·ω3,t ,P∗t = ρpwW∗t + εt .Since ω3,t is independent of P∗t and W ∗t , it would also be uncorrelated with any functionalform of P∗t and W ∗t , and thus ω3,t is uncorrelated with Pt , Wt and εt . Once Pt or Wt isnonnormal, εt is not a linear function of Pt and Wt , satisfying the full rank condition requiredfor model identification using the 2sCOPE method. Theorem proved.Next we show that this result can be easily extended to the multi-dimension Wt case.145We first derive the system of equations of the 2sCOPE method. Here we take 2-dimensionWt as an example. Because of the Gaussian relationship among P∗t and W ∗t we assumed inEquation (A.9), the first stage regression becomesP∗t =ρ1−ρ2ρw1−ρ2wW ∗1,t +ρ2−ρ1ρw1−ρ2wW ∗2,t +√1−ρ21 −(ρ2−ρ1ρw)21−ρ2wω3,t=ρ1−ρ2ρw1−ρ2wW ∗1,t +ρ2−ρ1ρw1−ρ2wW ∗2,t + ε2,t= γ1W ∗1,t + γ2W∗2,t + ε2,t . (A.12)The structural error ξt in Equation (A.8) and the first-stage error term ε2,t are linear transfor-mations of the Gaussian data (ξt ,P∗t ,W ∗1,t ,W∗2t ) and thus follow a bivariate normal distribu-tion. Thus, ξt can be decomposed to a sum of one term containing ε2,t and an independentnew error term, resulting in the following regression equation:Yt = β0+β1Pt +β2W1,t +β3W2,t +β4ε2,t +σξ γ ·ω4,t . (A.13)whereβ4 =σξρξ (1−ρ2w)1−ρ21 −ρ22 +2ρ1ρ2ρw+ρ2w.Since ω4,t is independent of P∗t , W ∗1,t and W∗2,t , it is uncorrelated with any functional formof P∗t , W ∗1,t and W∗2,t , and thus ω4,t is uncorrelated with Pt , W1,t , W2,t and ε2,t in Equation(A.13). Thus, 2sCOPE that performs OLS regression of Equation (A.13) yields consistentmodel estimates. Without loss of generality, the result can be extended to cases with anydimension of Wt .146A.6 Proof of Theorem 3: Nonnormality AssumptionRelaxedIn this section, we prove that our proposed 2sCOPE method can relax the nonnormalityassumption on the endogenous regressors imposed in CopulaP&G, while COPE does not.We first examine the COPE method in Equation (A.3),Yt = µ+Ptα+Wtβ +σξρpξ1−ρ2pwP∗t +−σξρpwρpξ1−ρ2pwW ∗t +σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pw ·ω3,t .If the endogenous regressor Pt is normally distributed, Pt = Φ−1σp (Φ(P∗t )) = σpP∗t and thusP∗t and Pt would be fully collinear, violating the full rank assumption and making the modelunidentified.We then examine the 2sCOPE method in Equation (3.8).Yt = µ+Ptα+Wtβ +σξρpξ1−ρ2pwεt +σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pw ·ω3,t ,εt = P∗t −ρpwW ∗t .When the endogenous regressor Pt is normally distributed, Pt =Φ−1σp (Φ(P∗t )) = σpP∗t . Sincewe add the residual εt from the first stage to the outcome regression instead of adding eachP∗t and W ∗t , εt would not be perfectly collinear with Pt and Wt as long as one of the W scorrelated with Pt is not normally distributed. Theorem proved.147A.7 Proof of Theorem 4: Variance ReductionAccording to the COPE method in Equation (A.3),Yt = µ+Ptα+Wtβ +σξρpξ1−ρ2pwP∗t +−σξρpwρpξ1−ρ2pwW ∗t +σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pw ·ω3,t .Note that all the regressors on the right-hand side of the equation above are not endogenous.The coefficients of P∗t and W ∗t follows a linear relationship. Denote δ3 and δ4 the coefficientsof P∗t and W ∗t respectively. Then,δ4+ρpwδ3 = 0.With the two-stage estimation in 2sCOPE (Equation 3.8), ρpw is estimated in the first stageand is thus treated as a known parameter in the main regression. That is, 2sCOPE can beviewed as the COPE method with a linear restriction. The linear restriction is,δ4+ ρˆpwδ3 = 0. (A.14)In this case, the two-stage copula method (2sCOPE) can be viewed as one kind of restrictedleast square estimation based on COPE. We next prove that restricted least square canachieve reductions in standard errors. Suppose we simplify the regression expression inEquation (A.3) asy = Xθ + ε,148where ε ∼ N(0,σ2I), X ≡ (1,Pt ,Wt ,P∗t , W ∗t ), and θ = (µ,α,β ,δ3,δ4). The restriction inEquation (A.14) becomesRθ = 0,whereR = (0,0,0, ρˆpw,1).Thus, the 2sCOPE yields the least square estimates θˆ2 of Equation (A.3) subject to the aboverestriction, whereas COPE yields the unrestricted least square estimates, θˆ1, as follows.θˆ1 ∼ N(θ ,σ2(X ′X)−1),θˆ2 ∼ N(θ ,σ2M(X ′X)−1M′).where according to restricted least square theory, M = I−(X ′X)−1R′(R(X ′X)−1R′)−1R. Letus compare the variance of θˆ1 and θˆ2. Note that,M(X ′X)−1M′=(I− (X ′X)−1R′(R(X ′X)−1R′)−1R)(X ′X)−1(I−R′(R(X ′X)−1R′)−1R(X ′X)−1)=(X ′X)−1− (X ′X)−1R′(R(X ′X)−1R′)−1R(X ′X)−1.Therefore,Var(θˆ1)−Var(θˆ2) = σ2{(X ′X)−1−M(X ′X)−1M′}= σ2(X ′X)−1R′(R(X ′X)−1R′)−1R(X ′X)−1 ≥ 0.Since the matrix Var(θˆ1)−Var(θˆ2) is positive semi-definite, all the diagonal elementsshould be greater than or equal to zero. Thus, the imposition of the linear restriction bringsabout a variance reduction. Theorem proved.We have proved that there would be variance reduction when there exist restriction of149parameters. When the exogenous regressor Wt is a scalar, the linear restriction is shown inEquation (A.14). We next show that when Wt is extended to a multi-dimension vector, thereare still linear restrictions and variance reduction of 2sCOPE. We take a 2-dimension Wt asan example below. According to the 2sCOPE method with 2-dimension Wt in Equations(A.12, A.13), 2sCOPE is equivalent to adding two restrictions to COPE in Equation (A.11).The two restrictions are:β5+ γˆ1β4 = 0β6+ γˆ2β4 = 0where γˆ1 and γˆ2 are estimated and obtained in the first stage in Equation (A.12). Thus,compared with COPE, we still have variance reduction using 2sCOPE in the 2-W case.Without loss of generality, this result can be extended to cases with any dimension of Wt .A.8 2sCOPE for Slope Endogeneity with Correlated andNormally Distributed RegressorsIn this section, we describe the 2sCOPE approach to addressing slope endogeneity withcorrelated regressors in the following model:Yt = µ+Ptαt +W ′t βt +ηt , where αt = α¯+ξt , (A.15)αt ,βt are individual-specific regression coefficients and α¯ is the mean of αi, ξt ∼ N(0,σ2ξ ).The normal error term ηi is uncorrelated with the regressors Pt and Wt and thus causes no en-dogeneity concern. However, the random coefficient ξt can be correlated with the regressorPt , causing the problem of “slope endogeneity”. Pt and Wt can be correlated. Assuming that(Pt ,Wt ,αt) follows a Gaussian copula model, the COPE approach to addressing the slope150endogeneity problem is derived as follows.Yt = µ+Pt(α¯+σξρpξ1−ρ2pwP∗t +−σξρpwρpξ1−ρ2pwW ∗t +σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pwω3,t)+W ′t βt +ηt= µ+Ptα¯+σξρpξ1−ρ2pwPt ×P∗t +−σξρpwρpξ1−ρ2pwPt ×W ∗t +W ′t βt +σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pw Pt ×ω3,t +ηt . (A.16)Given both Pt×P∗t and Pt×W ∗t in Equation (A.16), the unobserved variable w3,t is indepen-dent of all regressors (Pt ,Wt ,P∗t ,W ∗t ) and uncorrelated with functions of these regressors.Thus, Equation (A.16) can be estimated using standard methods for random-effects modelswith ω3,t as the random effect and (Pt ×P∗t , Pt ×W ∗t ) as generated regressors. The methodof Park and Gupta (2012) adds only Pt ×P∗t as a generated regressor, and may fail to yieldconsistent estimates when Pt and Wt are correlated, resulting in the correlation between therandom effect in their method and the regressor Wt .The 2sCOPE for addressing the slope endogeneity problem with correlated regressors isderived as followsYt = µ+Pt(α¯+σξρpξ1−ρ2pwεt +σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pw ·ω3,t)+W ′t βt +ηt= µ+Ptα¯+σξρpξ1−ρ2pwP∗t × εt +W ′t βt +σξ√√√√1−ρ2pξ − ρ2pwρ2pξ1−ρ2pw Pt ×ω3,t +ηt ,(A.17)where only one generated regressor, P∗t × εt , is needed, given which the random effect ω3,tis independent of all regressors in Equation (A.17).Both COPE and 2sCOPE can be implemented using the standard methods for randomeffects models by simply adding generated regressors to control for endogenous regressors.By contrast, the maximum likelihood approach requires constructing a complicated joint151likelihood of (ξt ,ηt ,P∗t ,W ∗t ), which is not what the standard random effects method uses andthus requires separate development and significantly more computation involving numericalintegration.A.9 2sCOPE for Random Coefficient Logit Model withCorrelated and Normally Distributed RegressorsWe next consider endogeneity bias in the following random utility model with correlatedendogenous and exogenous regressors:uh jt = ψh j +P′jtαh+W′jtβh+ξ jt + εh jt , j = 1, · · · ,J,uh0t = εh0t , j = 0 if no purchase,where uh jt denotes the utility for household h = 1, · · · ,nh at occasion t = 1, · · · ,T withj = 1, · · · ,J alternatives and j = 0 denotes the option of no purchase. In the utility function,ψh j is the individual-specific preference for choice j with ψhJ normalized to be zero foridentification purpose, (Pjt ,Wjt) include the choice characteristics, and (αh,βh) denote theindividual-specific random coefficients. These individual-specific coefficients (ψh j,αh,βh)permit heterogeneity in both intercepts and regressor effects across cross-sectional units,such as consumers or households. In this model, the association between regressors inPjt and the unobserved common shock ξ jt causes endogeneity bias. We further allow Pjtand Wjt to be correlated. The term εh jt is the idiosyncratic error uncorrelated with all re-gressors. An individual at any occasion chose the alternative with the largest utility, i.e.,Yh jt = 1 iff uh jt > uh j′t ∀ j′ ̸= j. When εh jt follows an i.i.d Type I extreme value distribu-tion, the choice probability follows the random-coefficient multinomial logit model.The 2sCOPE approach can be used to address the endogeneity issue using the following152two-step procedure. In the first step, we estimate the modeluh jt = δ jt + ψ˜h j +P′jtah+W′jtbh+ εh jt ,where δ jt = µ j +P′jtα¯+W ′jt β¯ +ξ jt , (µ j, α¯, β¯ ) is the mean of random effects (ψh j,αh,βh),ψ˜h j =ψh j−µ j,ah = αh− α¯ and bh = βh− β¯ . δ jt is treated as occasion- and choice-specificfixed-effect parameters in this model. Since the regressors are uncorrelated with the errorterm εhi j, there is no endogeneity bias in the model. In the second step, we estimate theequation below.δ̂ jt = µ j +P′jtα¯+W′jt β¯ +ξ jt +η jt , (A.18)where δ̂ jt denotes the estimate of the fix-effect δ jt ; η jt denotes the estimation error of δ̂i jand is approximately normally distributed. In the second-step model, the structural erroris correlated with Pjt , leading to endogenous bias. We then apply 2sCOPE to correct forthe endogenous bias, which can avoid the potential bias of CopulaP&G due to the potentialcorrelations between P and W , as well as make use of this correlation to relax the non-normality assumption of Pit , improve model identification and sharpen model estimates.The above development is for individual-level data. Park and Gupta (2012) also derivedtheir copula method for addressing endogeneity bias in random coefficient logit modelsusing aggregate-level data. It is straightforward to extend the 2sCOPE to the setting withcorrelated regressors and (nearly) normal regressor distributions.153Appendix: Additional ResultsA.10 Additional Results for Smaller Sample SizeIn previous simulation studies in section 4, we use sample size N=1000. We want to furthercheck the robustness of sample size. That is, whether our proposed methods can be appliedto smaller sample size. We simulate sample N=200 for T=1000 times, and use the sameDGP for continuous endogenous and exogenous regressors as in case 1. Table A.1 showsthat 2sCOPE has unbiased estimates for small sample size N=200. Hence, our proposedmethod is robust and can be applied to very small sample size.OLS CopulaP&G COPE 2sCOPEρpw Parameters True Mean SE tbias Mean SE tbias Mean SE tbias Mean SE tbias0.5 µ 1 0.683 0.097 3.264 1.228 0.191 1.194 1.020 0.223 0.091 0.999 0.137 0.005α 1 1.583 0.079 7.388 1.048 0.178 0.271 0.990 0.184 0.056 0.996 0.175 0.023β -1 -1.265 0.068 3.902 -1.291 0.068 4.293 -1.019 0.166 0.116 -1.004 0.101 0.044ρpξ 0.5 - - - 0.559 0.122 0.489 0.493 0.139 0.048 0.489 0.097 0.109σξ 1 0.857 0.044 3.224 1.016 0.107 0.148 1.018 0.100 0.176 1.001 0.094 0.013D-error - - 0.016598 0.0090690.7 µ 1 0.723 0.091 3.050 1.304 0.175 1.740 1.006 0.197 0.031 0.983 0.114 0.153α 1 1.817 0.095 8.583 1.255 0.161 1.584 1.032 0.182 0.175 1.044 0.174 0.253β -1 -1.539 0.084 6.388 -1.574 0.086 6.686 -1.045 0.180 0.250 -1.033 0.131 0.251ρpξ 0.5 - - - 0.624 0.103 1.200 0.490 0.135 0.077 0.480 0.067 0.297σξ 1 0.796 0.039 5.156 0.988 0.105 0.116 0.999 0.096 0.011 0.982 0.090 0.205D-error - - 0.016245 0.008867Table A.1: Results of the Simulation Study for Case 1 with Sample Size of 200154A.11 Misspecification of ξtTable 2.9 reports estimation results of misspecification of ξt using uniform[-0.5,0.5], beta(0.5,0.5)and t(df=3) distributions. We provide estimation results of beta and t distributions with dif-ferent distribution parameters below.OLS CopulaP&G COPE 2sCOPEParameters True Mean SE tbias Mean SE tbias Mean SE tbias Mean SE tbiasµ 1 0.948 0.008 6.928 1.036 0.013 2.909 1.000 0.014 0.022 1.000 0.010 0.009α 1 1.095 0.006 16.61 1.011 0.011 1.024 0.999 0.011 0.057 1.000 0.010 0.044β -1 -1.043 0.005 8.149 -1.048 0.005 9.267 -1.000 0.011 0.004 -1.000 0.007 0.030ρpξ 0.5 - - - 0.565 0.046 1.414 0.499 0.053 0.010 0.499 0.037 0.025σξ 0.167 0.144 0.003 7.969 0.168 0.006 0.136 0.167 0.006 0.077 0.167 0.006 0.011Table A.2: Results of the Simulation Study: Misspecification of ξt (Beta(4,4))OLS CopulaP&G COPE 2sCOPEParameters True Mean SE tbias Mean SE tbias Mean SE tbias Mean SE tbiasµ 1 0.603 0.059 6.723 1.270 0.115 2.343 0.992 0.126 0.066 0.997 0.080 0.039α 1 1.727 0.053 13.65 1.095 0.114 0.836 1.006 0.118 0.053 1.006 0.113 0.057β -1 -1.328 0.043 7.642 -1.366 0.043 8.554 -0.997 0.090 0.030 -1.002 0.067 0.037ρpξ 0.5 - - - 0.549 0.059 0.829 0.483 0.065 0.254 0.486 0.047 0.292σξ 1.291 1.118 0.049 3.506 1.293 0.074 0.030 1.292 0.071 0.008 1.289 0.070 0.032Table A.3: Results of the Simulation Study: Misspecification of ξt (t(5))A.12 Misspecification of CopulaIn the proposed methods, we use the Gaussian copula to capture the dependence structureamong the regressors and error term (Up, Uw and Uξ ). In practice, the dependence might155come from an economic mechanism (such as marketing strategic decisions) and thus mightbe different from what the Gaussian copula generates. In this section, we examine the ro-bustness of the Gaussian copula in simulated data. Specifically, we generate the dependenceamong Up, Uw and Uξ using copula models other than the Gaussian copula. Specifically, weconsider the following T copula models which provide flexible random general generationfrom arbitrary and heterogeneous correlation structures among more than two variables:C(Up,Uw,Uξ ) =∫ t−1ν (Up)−∞∫ t−1ν (Uw)−∞∫ t−1ν (Uξ )−∞Γ(ν+d2 )Γ(ν2 )√(πν)d|Σ|(1+x′Σ−1xν)dx, (A.19)where t−1ν denotes the quantile function of a standard univariate tν distribution. We set thedegree of freedom ν=2, and the dimension of the copula d=3 in this example. Σ is covari-ance matrix capturing correlations among variables. The data-generating process (DGP) oft copula is summarized below:P∗tW ∗tξ ∗t∼ tdν000 ,1 ρpw ρpξρpw 1 0ρpξ 0 1= tdν000 ,1 0.5 0.50.5 1 00.5 0 1 .(A.20)Figure A.1 shows the scatter plots of randomly generated (Up,Uw,Uξ ) pairs from theabove copulas, as well as the Gaussian copula with the same correlation of 0.5. The fig-ure shows disparate dependence structures between Up and ξt (Up and Uw) for these twocopulas.We then use the following process to generate Pt ,Wt and ξt :ξt = G−1(Uξ ) =Φ−1(Uξ ), (A.21)Pt = H−1(Up),Wt = L−1(Uw), (A.22)Yt = 1+1 ·Pt +(−1) ·Wt +ξt . (A.23)156(a) T Copula (ρ = 0.5) (b) Gaussian Copula (ρ = 0.5)Figure A.1: Scatter plots of Randomly Generated Pairs Up,Uw (Up,Uξ ) for Consid-ered Copulas.where H(·) is a gamma distribution and L(·) is an exponential distribution. We set T =1000, generate 1000 data sets and estimate the parameters using the OLS estimator and theproposed 2sCOPE method.Table A.4 summarizes the estimation results. OLS estimates are still biased for all pa-rameters. By contrast, estimates from the proposed COPE and 2sCOPE methods are cen-tered closely around the true values. Therefore, the proposed methods based on the Gaus-sian copula are reasonably robust to the misspecification of the copula dependence structureamong the regressors and the structural error.OLS COPE 2sCOPEParameters True Mean SE tbias Mean SE tbias Mean SE tbiasµ 1 0.710 0.530 5.463 1.002 0.127 0.016 0.988 0.077 0.156α 1 1.580 0.044 13.13 1.030 0.115 0.257 1.029 0.116 0.250β -1 -1.289 0.047 6.142 -1.033 0.127 0.262 -1.017 0.070 0.248ρpξ 0.5 - - - 0.463 0.085 0.435 0.458 0.067 0.622σξ 1 0.864 0.026 5.236 0.993 0.054 0.133 0.988 0.054 0.230Table A.4: Results of the Simulation Study Case D2: Misspecification of Copula157Appendix BChapter 4 AppendicesB.1 Quality Difference Between Premium and StandardOne thing to note is that on this platform, each driver is not just belongs to a single type.Among the 3,509 drivers, 1,074 drivers with luxury cars are assigned to serve premiumriders only, which I call the high-tier drivers. However, the remaining 2,435 drivers withrelatively lower quality cars are allowed to serve both premium and standard riders, andI call them low-tier drivers. One possible reason why the platform allows the mixture ofdriver type is that the platform wants to make better use of each driver’s time to providemore supply. Table B.1 shows the car make and service quality difference between the twotypes of drivers. Both the car price and review rate of high-tier drivers are significantlyhigher than that of low-tier drivers.Though there is a quality mixture of the low- and high-quality cars for premium riders,it shouldn’t be a big concern for analyzing vertical differentiation. Vertical differentiationcan be interpreted from a different perspective. On the rider side, a premium ride can be158Table B.1: Quality Difference Between the Two TypesHigh-tier Driver Low-tier DriverCar Make Bentley, BMW etc. Toyota, Honda, Ford etc.Average Car Price $62,445 $29,833Review Rate (without 0 rate) 4.68 4.60Review Rate (with 0 rate) 1.26 0.85viewed as the potential to get a high-quality car, while a standard ride can be viewed asa type to get a low-quality car for sure. In this way, riders still get higher quality valuefrom premium type, and the market can still be differentiated. On the supply side, the twoproducts can be viewed as competing for drivers’ available time instead of the number ofdrivers. The larger the demand size, the more time a mixed driver would spend serving thattype because of the larger network value. In this way, this setting can still be a good settingto study vertical differentiation and quantify the network externalities of the two productsby analyzing riders’ and drivers’ choices.B.2 Data PreparationB.2.1 Rider’s Expected Waiting TimeI first construct rider’s expected waiting time. Because of the distinct feature network ex-ternalities in two-sided markets, network size would add value to riders by affecting thetime they have to wait for a car. Though riders cannot observe waiting time directly, theycan observe the real traffic environment and would form expectations on the time they haveto wait. In this data set, the time when the order is placed by rider and the time when itis confirmed by driver, if available, are observed by researchers. Once an order request isconfirmed by a driver, the estimated driver arrival time is further observed. I use those timepoints to construct and approximate the rider’s expected waiting time in a specific time andlocation.159There are mainly three steps for the construction of rider’s waiting time. First, there’s asignificant number of reorders occur in this platform, and I will first find a way to combinethem. Reorder is defined as orders placed by the same rider in the same location within avery short period. There can be multiple reasons why those reorders occur. For example, therider might fill in wrong information; the order request might expire without any responsefrom the driver, etc. Since waiting time is a key quality dimension that differentiate ridersto choose different type of rides, it’s crucial to capture rider’s actual waiting time for a ride.Figure B.1(a) shows the distribution of time from order creation to confirmation. There’sa clear cutoff at 3 minutes, which means order requests would expire after 3 minutes andwould be cancelled automatically if no response is received from drivers within this shortperiod. Then riders might place an order again or leave. I observe that reorders account for27% of all orders in this data set. If I just treat each order request as an independent trip,waiting time for the trip would be downward biased. To solve the problem, I combine thosereorders into one trip and use the aggregate time from the creation of the first request to thefinal driver estimated arrival time as the rider’s total waiting time. I set several criteria todefine reorders. An order is defined as a reorder ifa. the rider’s last order request was not realized andb. the time interval from the last request is shorter than 15 minutes andc. the pick-up location the rider provides is the same with last request.After I change requests to trip-level requests, I calculate rider’s waiting time for a trip asthe aggregate time of all requests from order creation to driver estimated arrival time if thetrip is finally confirmed. Figure B.1(b) shows the distribution of the total waiting time fromorder creation to driver estimated arrival in a combined trip. After combining to trip-levelrequest, total number of orders decreases from 1.2 million to 766,471, but percentage ofrealized trips increases from 37.7% to 59.4%. Moreover, trips with more than one request160(a) Order-level Wait (b) Payment with Cruise TimeFigure B.1: Waiting Time Distribution.only accounts for 27.3% of all trips, and among those trips with multiple requests, 84% ofthem don’t include requests for both the types of requests, which means that most riderswouldn’t change their idea of what type to choose when they make multiple requests. Thisgives us a clear setting to calculate aggregate waiting time for each trip.Second, after the construction of waiting time above, another problem is that the waitingtime of trips that are not confirmed by any driver is missing. I approximate and fill inthose missing waiting time using trips with available waiting time in a similar environment.Below shows the steps how I construct a market and fill in the missing waiting time with theobserved waiting time in the same market.1. I first divide Manhattan area into 29 smaller locations. The 29 neighborhoods aredivided according to a common demarcation of Manhattan, which is visualized inFigure 4.1;2. I assume traffic environment in a specific hour in weekdays are similar, and definemarket in the location-week-hour level. That is, I treat observations in the same weekand at the same location and hour from Monday to Thursday as in the same market.This is a reasonable assumption, as both the price patterns and people’s behavior are161the same from Monday to Friday, while are very different on Friday, Saturday, andSunday. For example, traffic in 7:00 am of Monday and Tuesday should be similar,and traffic at the same in Monday and Saturday should be quite different. Thus, tokeep it clean, I define market in the above way and just use observations from Mondayto Thursday for analysis. In the end, there are 5,672 markets in total;3. Once I group together trips with similar environment, I fill in the missing waiting timeof a trip with the maximum waiting time among same-type trips in the same market.After this step, all trips have waiting time;Third, after we obtain waiting time for each trip, we further constructed rider’s expectedwaiting time. The waiting time constructed above is ex post waiting time, which is obtainedafter the trip is fulfilled. However, when riders make order request, they cannot observewaiting time in advance, and thus I continue this step to further construct ex ante expectedwaiting time for each type of ride by averaging the waiting time of all trips with the sametype in the same market.B.2.2 Driver’s Cruising TimeOnce a driver receives an offer, he will observe the distance from the rider and thus can inferhow long he has to cruise to pick up the rider. In this section, I focus on how to constructcruising time using observed distance.What I can observe in data is drivers’ distance from rider for all received offers, anddriver’s expected arrival time for offers that are accepted by drivers. I first get drivers’cruising time for those realized trip by calculate the time between driver’s expected arrivaltime and the time he accepted the offer. Then I regress the cruising time on distance andmarket fixed effects such as week, hour and location, and then use the regression result to162predict cruising time for those unrealized trips. The construction of drivers’ cruising time issimpler than rider’s waiting time. For cruising time, I don’t need to calculate market-levelexpected cruising time, as drivers can observe the location of the rider and the distance uponreceiving each offer.B.2.3 Construction for Outside Demand and Counterfactual OptionsOn this platform, I can only observe order requests from riders who choose this platform,and cannot observe options from those who choose outside options. In real situation, ridersmay consider more options besides the two options in this platform. Moreover, for thosewho choose this platform, we researchers can only observe the information of the chosenoption, while the information of the one not chosen is not available. But riders can observeinformation for each type at the time when placing an order.Thus, to better capture and model rider’s behavior, I include taxi data in New YorkCity as outside option for estimation to control market size, and estimate rider’s choicedecision taking outside option into consideration. The source of taxi data is public online 12. Meanwhile, I fill in the missing information of those options not chosen for all possibledemand, including the outside demand. The taxi data in New York City is public online. Itincludes all realized trips with trip start time, trip fare, distance and geographic information.I first match taxi demand with the demand using geographic location to get demand size ineach market, and then simulate the price of the two options in this platform for those outsidedemands. To construct the counterfactual choice for all demand, I fill in the missing order-level variable (e.g., price) of the option not chosen. According to the price structure listedin Table 4.1, price for standard is a fixed price $8.98, while price for premium depends ontrip duration and trip distance. In the dataset, I can observe trip duration for realized trips,1https://data.cityofnewyork.us/dataset/2016-Yellow-Taxi-Trip-Data/uacg-pexx;2https://data.cityofnewyork.us/Transportation/2016-Green-Taxi-Trip-Data/hvrh-b6nb.163and origin and destination for all trips, thus can get relationship between trip price, durationand distance using realized trip and then predict trip price for counterfactual option. I listthe steps how I get outside demand in each specific market and how I predict and simulateprice for counterfactual option below.1. Download Green and Yellow taxi data in 2016 at New York City online, in whichinformation such as price, trip duration and trip distance are listed for each realizedtrip;2. Get the market index for each taxi trip using trip time and geographic information (lat-itude and longitude), and calculate taxi demand in each market (week-hour-location).To save time for MCMC code running, I randomly sample 10% taxi demand as theoutside demand for estimation;3. For unrealized trips in this platform, I approximate trip distance and duration usingtrips with same origin and destination in taxi data. I define two trips to have the sameorigin and destination if the rounded latitude and longitude with 2 decimal places arethe same.4. Regress price on trip distance and duration using the realized trips in this platform,and get the relationship between price and trip distance and duration;5. Use the relationship in step 4 to predict price for the option not chosen or trips withmissing price;6. Fill in waiting time for the option not chosen with market-level waiting time of thattype.164B.3 Bayesian MCMC EstimationIn my model, I have individual specific coefficients and unobserved demand and supplyshocks. For model estimation, I use Bayesian method and treat those random coefficientsand unobserved shocks as latent variables instead of using frequentist approach and inte-grating those parameters out. Analysis proceeds by iteratively sampling from the full condi-tional distributions of all model parameters, including those latent variables. The estimationprocess can be written in hierarchical form:Y d|Xd,Ewaitkm,θi,ζkm,λi,µw,ε Observed demand, (B.1)Y s|X s,D,S,θ j,ζm,λ j,µc,ε Observed supply, (B.2)Ewaitkm|Dkm,Skm,γ,µw,µc Market-average waiting time (B.3)θi,θ j|θ¯ , Ii, I j,Σθ Heterogeneity, (B.4)ε, Extreme value error (B.5)where θi = (β di ), θ j = (βsj ) are rider and driver heterogeneous parameters, and Ii, I j aretheir demographic information. Observed demand and supply are dependent on rider’s anddriver’s coefficients (θi,θ j), demand and supply shock ζ , unobserved error (ε) and explana-tory variables. The conditional demand and supply density, given θ and explanatory vari-ables is multinomial-logit probabilities (see Equation (4.7)). Rider and driver coefficientsare specified as random coefficients and depend on demographics. The joint density of allmodel parameters is thenf (θi,θ j,ζ , θ¯ ,Σθ |Yd,Ys,X , I)∝ΠiΠ jΠtΠmprob(Yd,Ys|X ,θi,θ j)π1(θi,θ j|I, θ¯ ,Σθ )π2(θ¯ ,Σθ )(B.6)Estimation is carried out using a Markov chain Monte Carlo procedure that involves gener-ating a sequence of draws from the full conditional distributions of the model (see Gelfand165and Smith, 1990; Gelfand et al., 1990).Estimation steps:1. Generate θi,θ j for (i = 1,2,. . . , I; j = 1,2,. . . , J)[θi,θ j|∗] ∝ΠiΠ jprob(Yd,Ys|X ,θi,θ j,ζm)(θi ∼MV N(θ¯i,Σθ i))(θ j ∼MV N(θ¯ j,Σθ j))I use Metropolis-Hastings algorithm with a random walk chain to generate draws ofθi,θ j.2. Generate θ¯ ,Σθ using Gibbs sampling.Note that random-coefficients for rider and driver are generated and updated sepa-rately. Here I only update coefficients for rider, θ¯iΣθ i, as an example.[θ¯i|θi,Di,Σθ i]∼ N((D˜i′D˜i+A)−1(D˜i′θ˜i+Aβ0),(D˜i′D˜i+A)−1))D˜i =U−1′Diθ˜i =U−1′θiΣθ i =U ′U[Σθ i|θ¯i,θi]∼ Inverted Wishart(ΣIi=1(θ ′i − θ¯iDi)(θ ′i − θ¯iDi)′+V0, I+n0)suppose the priors of θ¯ ,Σθ are θ¯ ∼ N(β0,A−1),Σθ ∼ IW (V0,n0).3. Generate γ using Gibbs sampling.166
Thesis/Dissertation
eng
Business Administration - Marketing
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Qian, Yi
Essays of instrument-free causal inference, machine learning and marketing applications
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