The Gini Methodology: A Primer on a Statistical Methodology

✍ Scribed by Shlomo Yitzhaki, Edna Schechtman

• Publisher: Springer
• Year: 2012
• Tongue: English
• Leaves: 550
• Series: Springer Series in Statistics, 272
• Category: Library

✦ Synopsis

Gini's mean difference (GMD) was first introduced by Corrado Gini in 1912 as an alternative measure of variability. GMD and the parameters which are derived from it (such as the Gini coefficient or the concentration ratio) have been in use in the area of income distribution for almost a century. In practice, the use of GMD as a measure of variability is justified whenever the investigator is not ready to impose, without questioning, the convenient world of normality. This makes the GMD of critical importance in the complex research of statisticians, economists, econometricians, and policy makers.

This book focuses on imitating analyses that are based on variance by replacing variance with the GMD and its variants. In this way, the text showcases how almost everything that can be done with the variance as a measure of variability, can be replicated by using Gini. Beyond this, there are marked benefits to utilizing Gini as opposed to other methods. One of the advantages of using Gini methodology is that it provides a unified system that enables the user to learn about various aspects of the underlying distribution. It also provides a systematic method and a unified terminology.

Using Gini methodology can reduce the risk of imposing assumptions that are not supported by the data on the model.  With these benefits in mind the text uses the covariance-based approach, though applications to other approaches are mentioned as well.

✦ Table of Contents

The Gini Methodology
Acknowledgement
Contents
Chapter 1: Introduction
Part I: Theory
Chapter 2: More Than a Dozen Alternative Ways of Spelling Gini
Introduction
2.1 Alternative Representations of GMD
2.1.1 Formulas Based on Absolute Values
2.1.2 Formulas Based on Integrals of the Cumulative Distributions
2.1.3 Covariance-Based Formulas
2.1.4 Lorenz Curve-Based Formulas
2.2 The GMD and the Variance
2.2.1 The Similarities Between GMD and the Variance
2.2.2 The Differences Between the GMD and the Variance: City Block vs. Euclidean
2.3 The Gini Coefficient
2.4 Adjustments Needed for Discrete Distributions
2.4.1 Inconsistencies in the Definitions of Lorenz Curves and Cumulative Distributions
2.4.2 Adjustment for a Small Number of Observations
2.5 Gini Rediscovered: Examples
2.6 Summary
Chapter 3: The Gini Equivalents of the Covariance, the Correlation, and the Regression Coefficient
Introduction
3.1 Preliminaries and Terminology
3.2 Measures of Association
3.2.1 Pearson´s Correlation Coefficient
3.2.2 Spearman Correlation Coefficient
3.2.3 Kendall´s tau
3.3 Gini Correlations
3.4 The Similarity Between the Two Gini Correlations of a Pair of Variables
3.4.1 Formal Definitions of Exchangeability
3.4.2 The Implications and Applications of Exchangeability
3.5 The Gini Regression Coefficient
3.6 Summary
Chapter 4: Decompositions of the GMD
Introduction
4.1 The Decomposition of the GMD of a Linear Combination of Variables
4.1.1 One-Step Decomposition (Marginal Decomposition)
4.1.2 Two-Step Decomposition
4.2 The Decomposition of the Variability of a Population by Subpopulations
4.2.1 The Overlapping Parameter
4.2.2 Between-Groups Component GB and Its Properties
4.2.3 ANOGI vs. ANOVA: A Summary Table
(a) Components Which Are Identical in Nature to ANOVA
(b) Additional Components
4.3 The Decomposition of Gini Covariance
4.3.1 Decomposing the OLS Regression Coefficient
4.3.2 Decomposing the Gini Regression Coefficient
4.4 Summary
Appendix 4.1
Chapter 5: The Lorenz Curve and the Concentration Curve
Introduction
5.1 The Absolute Lorenz Curve
5.2 The Lorenz Curve of the Coefficient of Variation
5.3 The Absolute Concentration Curve
5.4 The Absolute Lorenz Curve and Second-Degree Stochastic Dominance
5.5 The ACC and Marginal Conditional Stochastic Dominance
5.6 The ACC and the Monotonicity of the Correlations and the Regression Slopes
5.7 An Illustration: Labor Force Participation by Gender and Age
5.8 Summary
Chapter 6: The Extended Gini Family of Measures
Introduction
6.1 The Three Introductions
6.1.1 The Dual Approach to Moments´´ Introduction 6.1.2 TheIncome Inequality Approach´´ Introduction
6.1.3 The ``Dual Approach to Risk´´ Introduction
6.2 The Alternative Definitions
6.3 The Properties of the Extended Gini Family
6.4 Alternative Presentations of the Extended Gini Covariances and Correlations
6.5 The Decomposition of the Extended Gini
6.6 Stochastic Dominance and the Extended Gini
6.7 Summary
Appendix 6.1
Appendix 6.2
Appendix 6.3
Appendix 6.4
Chapter 7: Gini Simple Regressions
Introduction
7.1 Alternative Presentations: The Semi-Parametric Approach
7.1.1 The Ordinary Least Squares Regression Coefficient
7.1.2 The Gini Semi-Parametric Regression Coefficient
7.1.3 A Presentation Based on the Decomposition to Subpopulations
7.1.4 A Presentation Based on Concentration Curves
7.1.5 Similarities and Differences Between OLS and Gini Semi-Parametric Regression Coefficients
7.2 The Minimization Approach
7.3 The Combination of the Two Gini Approaches
7.4 Goodness of Fit of the Regression Model
7.5 A Test of Normality
7.6 The Instrumental Variable Method
7.6.1 The OLS Instrumental Variable Method
7.6.2 The Gini Instrumental Variable Method
7.6.3 The Similarities and Differences Between OLS and Gini Instrumental Variable Methods
7.6.4 An Example: The Danger in Using IV
7.7 The Extended Gini Simple Regression
7.8 Summary
Appendix 7.1
Appendix 7.2
Chapter 8: Multiple Regressions
Introduction
8.1 Multiple Regression Coefficients as Composed of Simple Regression Coefficients
8.2 Gini Regression as a Linear Approximation of the Regression Curve
8.3 Combining the Two Regression Approaches: The Multiple Regression Case
8.4 OLS and Gini Instrumental Variables
8.4.1 Two-Stage Least Squares and Instrumental Variables
8.4.2 Two-Stage and IV in Gini Regressions
8.5 Effects of Commonly Used Practices
8.6 Summary
Chapter 9: Inference on Gini-Based Parameters: Estimation
Introduction
9.1 Estimators Based on Individual Observations: The Continuous Case
9.1.1 The Gini Mean Difference and the Gini Coefficient
9.1.2 The Gini Covariance and Correlation
9.1.3 The Overlapping Index
9.1.4 The Extended Gini, Extended Gini Covariance, and Extended Gini Correlation
9.1.5 Gini Regression and Extended Gini Regression Parameters
9.1.6 Lorenz Curve and Concentration Curves
9.2 Estimators Based on Individual Observations: The Discrete Case
9.3 Individual Data, Weighted
9.3.1 Estimating the Gini Coefficient from Weighted Data
9.3.2 Estimating the Extended Gini Coefficient from Weighted Data
9.4 Estimators Based on Grouped Data
9.5 Summary
Chapter 10: Inference on Gini-Based Parameters: Testing
Introduction
10.1 The One Sample Problem
10.1.1 Inference on the GMD and the Gini Coefficient
10.1.2 Inference on Gini Correlation and Gini Regression
10.1.3 Testing for the Symmetry of the Gini Correlation
10.1.4 The Extended Gini and the Extended Gini Regression Coefficients
10.2 The Two Sample Problem
10.2.1 The Overlapping Index
10.2.2 Comparing Two GMDs and Two Gini Coefficients
10.3 Summary
Chapter 11: Inference on Lorenz and on Concentration Curves
Introduction
11.1 Inference on the Ordinates of the Lorenz Curves
11.2 Necessary Conditions for Second Order Stochastic Dominance
11.3 Testing for Intersection of Two ACCs
11.4 Summary
Part II: Applications
Chapter 12: Introduction to Applications
Chapter 13: Social Welfare, Relative Deprivation, and the Gini Coefficient
Introduction
13.1 The SWF Approach
13.1.1 Welfare Dominance: The Role of the Gini Coefficient
13.2 The Theory of Deprivation
13.3 Relative Deprivation
13.3.1 Concepts of Relativity
13.3.2 Relative Deprivation
13.3.3 The Effect of Reference Groups on Deprivation
13.4 Summary
Chapter 14: Policy Analysis
Introduction
14.1 Marginal Analysis
14.1.1 Setting the Problem: Dalton and Gini Improvement Reforms
14.1.2 Characterization of a Dalton-Improvement Reform
14.2 A Description of the Economic Model
14.2.1 The Required Data and the Distributional Characteristics
14.2.2 Marginal Efficiency Cost of Funds
14.2.3 The Characterization of the Solution
14.3 More on Distributional Characteristics: The Gini Income Elasticity
14.4 An Empirical Illustration of DI Reforms
14.4.1 Distributional Characteristics of Commodities in Indonesia
14.4.2 The Marginal Efficiency Costs of Funds
14.4.3 Simple Dalton-Improving Reforms
14.4.4 Dalton-Improving Reforms
14.4.5 Sensitivity Analysis
14.4.6 Non-neutral Reforms
14.5 Summary
Chapter 15: Policy Analysis Using the Decomposition of the Gini by Non-marginal Analysis
Introduction
15.1 Decomposition by Sources: Analyzing the Coordination Between Direct Benefits and Taxation
15.1.1 The Basic Data of Ireland and Israel
15.1.2 The Impact of Equivalence Scales
15.1.3 Decomposition of the Gini According to Income Sources
15.1.4 Estimates of the Gini Coefficient
15.1.5 A Full Decomposition of Gini by Income Sources: Empirical Findings
15.2 Decomposition by Population Subgroups
15.2.1 Background
15.2.2 Empirical Findings
15.3 Decomposition Over Time: Non-marginal Analysis: Mobility, Inequality, and Horizontal Equity
15.3.1 The Gini Index of Mobility
15.3.1.1 Definitions and Properties
15.3.1.2 The Relationship with Transition Matrices
15.3.1.3 An Empirical Illustration
15.3.2 Predicting Inequality of a Linear Combination of Variables
15.3.2.1 Definitions and Properties
15.3.2.2 An Empirical Illustration
15.3.3 Mobility and Horizontal Equity
15.3.3.1 Definitions and Properties
15.3.3.2 An Empirical Application
15.4 Summary
Appendix 15.1
Proof of (15.8)
Chapter 16: Incorporating Poverty in Policy Analysis: The Marginal Analysis Case
Introduction
16.1 Analyzing the Distributional Impact of Programs Intended to Reduce Poverty
16.2 The Usefulness of a Poverty Line
16.3 The Decompositions of the Gini Coefficient and Sen´s Poverty Index
16.4 Decompositions
16.4.1 Decomposition of the Gini Coefficient
16.4.2 The Decomposition of the (Gini) Income Elasticity
16.5 An Empirical Illustration
16.5.1 The Data and the Main Findings
16.5.2 Sensitivity Analysis
16.6 Policy Analysis
16.7 Summary
Chapter 17: Introduction to Applications of the GMD and the Lorenz Curve in Finance
Introduction
17.1 The Role of Variability in Calculating the Rate of Return
17.2 Stochastic Dominance, Lorenz Curves, and Gini for the Additive Model
17.2.1 Expected Utility, Stochastic Dominance, and Mean-Gini Rules
17.2.2 Absolute Concentration Curves and Marginal Conditional Stochastic Dominance
17.3 Risk Aversion, Extended Gini, and MCSD
17.4 Beta and Capital Market Equilibrium
17.5 Summary
Chapter 18: The Mean-Gini Portfolio and the Pricing of Capital Assets
Introduction
18.1 The Mean and Mean-Extended Gini Efficient Frontiers
18.2 Analytic Derivation of the Mean-Gini Frontier
18.3 Capital Market Equilibrium with Two Types of Investors
18.3.1 The Two-Parameter Investment Model
18.3.2 The Mean-Extended Gini Ordering Function
18.4 Equilibrium
18.5 Summary
Chapter 19: Applications of Gini Methodology in Regression Analysis
Introduction
19.1 Tracing the Curvature by Simple EG Regression: Simulated Results
19.2 Tracing the Curvature of a Simple Regression Curve by the LMA Curve
19.2.1 Definitions and Notation
19.2.2 The Simple Gini Regression Coefficient and the Concentration Curve
19.3 The Decomposition Approach
19.4 An Illustration: Labor Force Participation by Gender and Age
19.5 Data Manipulations
19.5.1 Omitting a Group of Observations
19.5.2 Substituting a Continuous Variable by a Discrete One
19.5.3 The Effect of Transformations
19.6 Summary
Chapter 20: Gini´s Multiple Regressions: Two Approaches and Their Interaction
Introduction
20.1 Gini´s Multiple Regressions
20.1.1 The Semi-Parametric Approach
20.1.2 The Minimization Approach
20.2 The Relationship Between the Two Approaches
20.3 Assessing the Goodness of Fit of the Linear Model
20.4 The LMA Curve
20.5 An Illustration: The Two Explanatory Variables Case
20.6 An Application: Assessing the Linearity of Consumption as a Function of Income and Family Size
20.6.1 The Problem to be Solved
20.6.2 Empirical Findings
20.7 Summary
Chapter 21: Mixed OLS, Gini, and Extended Gini Regressions
Introduction
21.1 Mixing Gini, Extended Gini, and OLS in the Same Regression
21.2 An Illustration of Mixed OLS and Gini Regression
21.2.1 The Indirect Way of Analyzing Nonresponse
21.2.2 Empirical Results
21.2.3 A Search for an Explanation
21.2.4 Summary of the Example
21.3 An Illustration of Mixed Gini and EG Regression
21.3.1 Non-reporting in a Household Finances Survey
21.3.2 The Data
21.3.3 Empirical Results
21.4 Summary
Chapter 22: An Application in Statistics: ANOGI
Introduction
22.1 A Brief Review of the Methodology
22.2 An Illustration of ANOGI: The Melting Pot Policy
22.2.1 Definitions
22.2.2 Data Description
22.2.3 Results
22.3 Summary
Appendix 22.1
Appendix 22.2: ANOVA
Chapter 23: Suggestions for Further Research
Introduction
23.1 Convergence to the Normal Distribution
23.2 The Use of the Gini Method in the Area of Education
23.2.1 Ranking Groups According to Average Success
23.2.2 A Gini Item Characteristic Curve
23.3 The Use of the Gini Methodology in Time-Series
23.4 The Relationship Between the GMD and Absolute Mean Deviation
23.5 A Comment on Required Software
23.6 Summary
References
Author Index
Subject Index
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