Flexible Estimation Methods for Multivariate Fractional Outcomes Open Access

Abstract

Multivariate fractional outcomes are defined as vectors where each component is bounded to the unit interval and together they add up to 1. This dissertation expands the available toolkit for analyzing both univariate and multivariate fractional outcomes as well as their applications to economics and other fields. As these variables arise naturally in several areas of applied microeconomics, the focus is on cross-sectional and panel data. Emphasis is placed on providing methods that are flexible and robust while exploring several approaches to modeling of these outcomes in a variety of settings. In each chapter a different facet of multivariate fractional outcomes is studied. The first chapter presents a semiparametric extension of a quasi-likelihood estimator that is heavily used in applications with a univariate fractional outcome. As documented in the chapter, large biases can arise when the nonlinear link function is misspecified, which can be countered by the use of our extension. The second chapter provides a unified estimation methodology using copulas for multivariate fractional outcomes with a conditional mean specification. This methodology satisfies the fractional and unit-sum constraints of the outcomes, allows for cross-equation restrictions that are crucial in structural estimation, and can handle variable selection. The final chapter extends both the existing and newly proposed methods to a panel data setting, focusing on several robust alternatives and their numerical implementations. All chapters use simulation exercises and applications to showcase the performance of the proposed methods.

Table of Contents

Contents

Introduction                                                                                                                                                   1

1      Semiparametric Quasi Maximum Likelihood Estimation of the Fractional Response Model            4

1.1  Estimator and Asymptotic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1   Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5

1.1.2   Asymptotic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2  Monte Carlo Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3  Empirical Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

 

Appendices 13

1.A Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.B Computational Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.C Empirical Application Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

 

2      Copula Estimation and Variable Selection with Multivariate Fractional Outcomes 17

2.1  Methodological Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.1   Likelihood and Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.2   Frequentist Estimation and Asymptotic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2  Priors and Variable Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3  Monte Carlo Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.3.1   Reduced Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3.2   Demand Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.4  Empirical Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.5  Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

 

Appendices 73

2.A Proof of Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.B Regularity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.C Additional Numerical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

 

3      Multivariate Fractional Panel Data Methods 89

3.1  Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.1.1   Maximum Likelihood Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.1.2   Probit Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.1.3   Bayesian Latent Variable Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.2  Numerical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.2.1   Copula Data-Generating Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.2.2   Probit Data-Generating Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.2.3   Censored Data-Generating Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

3.3  Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

 

Appendices 118

3.A Details on Integration Methods for MLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.A.1  Adaptive Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.A.2  Nonadaptive Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.A.3  Pruning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.B Derivatives for MLE and Probit Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.B.1   Scores for Independent and Pooled MLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.B.2   Score and Hessian for Probit NLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

 

Bibliography 122

 

List of Figures

1.1      QQ plot for Estimators of β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

 

2.1      Dependence Patterns in Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2      Trace Plot of Bayesian Chains in a Reduced Form Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.3      Density Plot of Bayesian Chains in a Reduced Form Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.4      Trace Plot of APE Chains in a Reduced Form Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.5      Density Plot of APE Chains in a Reduced Form Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.6      Frequentist LASSO in a Reduced Form Model with a Gaussian Copula and Beta Marginals . . . . 53

2.7      Trace Plot of Coefficient Chains in a Reparameterized Bayesian AID System . . . . . . . . . . . . . . . 67

2.8      Density Plot of Coefficient Chains in a Reparameterized Bayesian AID System . . . . . . . . . . . . . . 69

2.9      Trace Plot of Elasticity Chains in an Extended Bayesian AID System . . . . . . . . . . . . . . . . . . . . . . . 87

2.10  Density Plot of Elasticity Chains in an Extended Bayesian AID System . . . . . . . . . . . . . . . . . . . . . 88

 

3.1      Trace Plot of Coefficients for Latent Dependent Variable Model . . . . . . . . . . . . . . . . . . . . . . . . 116

3.2      Density Plot of Coefficients for Latent Dependent Variable Model . . . . . . . . . . . . . . . . . . . . . . . 117

 

List of Tables

1.1        Ratios of Root Mean Squared Errors (RMSE) and Standard Errors for Estimators of β . . . . . . . 10

1.2        Empirical Results with Additional Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.C.1   Replication of Papke and Wooldridge (1996) with additional methods on restricted sample . 16

 

2.1        RMSE for Coefficients in a Reduced Form Model from a Gaussian Copula with Beta Marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2        RMSE for Coefficients in a Reduced Form Model from a FGM Copula with Beta Marginals . . . 46

2.3        RMSE for Coefficients in a Reduced Form Model from a Dirichlet . . . . . . . . . . . . . . . . . . . . . . . 47

2.4        Bayesian and Frequentist Estimates for a Reduced Form Model . . . . . . . . . . . . . . . . . . . . . . . . 50

2.5        Bayesian Estimates and Inference of APEs for a Reduced Form Model . . . . . . . . . . . . . . . . . . . 50

2.6        Bayesian APEs and Selection for an Extended Reduced Form Model . . . . . . . . . . . . . . . . . . . . . 54

2.7        RMSE for Coefficients in a Structural Demand Model from a Gaussian Copula with Beta Marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.8        RMSE for Coefficients in a Structural Demand Model from a Gaussian Distribution . . . . . . . . . 58

2.9        RMSE for Coefficients in an Extended Structural Demand Model from a Gaussian Copula with Beta Marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.10    RMSE for Coefficients in an Extended Structural Demand Model from a Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.11    Summary Statistics for Data in Chang and Serletis (2014) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.12    MLE Estimates of AID System using the Copula Y Estimator with Different Copulas and Beta Marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.13    Bayesian Estimates of a Reparameterized AID System using the Copula Y Estimator with a Gaussian Copula and Beta Marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.14    Elasticity Estimates and Inference from a Bayesian AID System . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.15    Selection f Polynomial Terms in an Extended Bayesian AID System . . . . . . . . . . . . . . . . . . . . . . 71

2.16    Elasticity Estimates and Inference from an Extended Bayesian AID System . . . . . . . . . . . . . . . . 72

2.C.1   Estimates and Standard Errors in a Reduced Form Model from a Gaussian Copula with Beta Marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.C.2   Estimates and Standard Errors in a Reduced Form Model from a FGM Copula with Beta Marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.C.3   Estimates and Standard Errors in a Reduced Form Model from a Dirichlet . . . . . . . . . . . . . . . . 81

2.C.4   Estimates and Standard Errors in a Structural Demand Model from a Gaussian Copula with Beta  Marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

2.C.5   Estimates and Standard Errors in a Structural Demand Model from a Gaussian Distribution . 83

2.C.6   Estimates and Standard Errors in an Extended Structural Demand Model from a Gaussian Copula with Beta Marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

2.C.7   Estimates and Standard Errors in an Extended Structural Demand Model from a Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

2.C.8   Bayesian Point Estimates and Inference for an Extended Reduced Form Model . . . . . . . . . . . . 86

 

3.1        RMSE for Coefficients in a from a Gaussian Copula with Beta Marginals and Multinomial Logit Link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.2        Coefficients from a Multinomial Logit Link in a Gaussian Copula with Beta Marginals . . . . . . 111

3.3        RMSE for Coefficients from a Multivariate Nonlinear Least Squares with Probit Link . . . . . . . 112

3.4        Coefficients from a Multivariate Nonlinear Least Squares with Probit Link . . . . . . . . . . . . . . . 113

3.5        Coefficients from a Bayesian Latent Dependent Variable Model . . . . . . . . . . . . . . . . . . . . . . . . 114

About this Dissertation

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School	Laney Graduate School
Department	Economics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Economics, General Economics, Theory
Keyword	Semiparametrics Panel data Copulas Variable selection Bayesian estimation Quasi-maximum likelihood Multivariate fractional outcomes
Committee Chair / Thesis Advisor	David Jacho-Chavez, Emory University
Committee Members	Christoph Breunig, Emory University Elena Pesavento, Emory University

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