Shlomo Yitzhaki

• Ph. D.
• Professor Emeritus at Hebrew University of Jerusalem

The measurement scale of exam scores is ordinal. This ordinal measurement implies that monotonic transformations of published scales convey the same information. Hence, countries should not be ranked according to averaged scores because there may be transformations that would change the mean-based rankings. We suggest alternatives to the mean-based...

One of the basic assumptions that characterizes economic theory in Finance is the assumption of declining marginal utility of income. One of the popular methods of analyzing empirical evidence by economists is the Ordinary Least Squares Regression. This paper argues that in some cases of violation of the assumptions by the data, there may be a cont...

This paper defines the efficient exam as the one that for a given number of questions maximizes the revealed variability in the distribution of abilities of examinees. Variability is measured by the Gini coefficient which is a robust measure of inequality. Using the properties of the Gini enables the classification of abilities into different disti...

Regression is the most popular means used by economists to draw empirical conclusions. It is based on imposing a given structure on data and then estimating the parameters of this structure. The aim of this paper is to suggest a regression method that lets data determine the regression curve. The suggested methodology is based on piecewise linear s...

One of the basic assumptions that characterizes economic theory in Finance is the assumption of declining marginal utility of income. One of the popular methods of analyzing empirical evidence by economists is the Ordinary Least Squares Regression. This paper argues that in some cases of violation of the assumptions by the data, there may be a cont...

We provide a comprehensive picture of the relationship between labor market outcomes and age by gender in the 28 European countries covered by the European Statistics on Income and Living Conditions. The analysis is based on a somewhat unconventional approach that refers to concentration curves in the Gini regression framework. It allows identifica...

The average of periodic growth rates is a downwardly biased estimator of the rate of growth of a country. The higher the variance of the periodical growth rates, the higher the downward bias. The longer the business cycle, the higher the downward bias. In this short paper, we demonstrate these facts on a number of different levels, from intuitive t...

This paper uses the extended Gini inequality index to examine the sensitivity of measurements of impacts of migrant remittances on the distribution of household income by size to different value judgements when measuring inequality. The results illustrate the robustness of earlier findings that the impacts of migration on village income distributio...

This paper studies the relationships between annual and subannual inequality and mobility during the course of the year. We apply an exact decomposition framework as outlined in Wodon and Yitzhaki (Econ Bull 4:1-8, 2003), and in Yitzhaki and Wodon (Research on Economic Inequality 12:179-199, 2004). Earnings records of pension insurants in Germany s...

The Lorenz and the concentration curves play important roles in the areas of GMD and the related measures such as Gini covariance, Gini correlation, Gini regression, and more. In this chapter we introduce the curves, discuss their properties, and show their connections to the Gini world. In addition, in order to be able to analyze the parallel conc...

The main purpose of this chapter is to expose the reader to additional tools that can be helpful in analyzing the distributional impact of a governmental policy. Assuming that one accepts the Gini coefficient of after-tax income as representing the social attitude toward the income distribution then one can summarize the effects of actions taken by...

Since its development by Markowitz (1952, 1970), the mean-variance (MV) model for portfolio selection has become the standard tool by which risky financial assets are allocated. MV has gained a prominent place in finance because of its conceptual simplicity and ease of computation. Many authors, however, have challenged the model’s assumptions, pri...

The objective of this chapter is to introduce the use of the concentration curves and the Gini methodology in the areas of taxation and progressivity of public expenditures. Most of the literature in these areas considers the case of a representative individual, which means that issues of income distributions are ignored and the only issue that is...

This paper defines the efficient exam as the one that maximizes the revealed variability in the abilities of the examinnees. Variability is measured by the Gini's Mean Difference which is a robust measure of variability. Using the properties of decomposition of the Gini we identify the discriminatory power of a question as the degree of stratificat...

The Item Characteristic Curve describes the relationship between the probability of correctly answering a question and the ability presumed to be required to answer the question. Since ability is a latent variable, distributional assumptions must be imposed on it in order to estimate such relationship. In this paper we overcome the need to impose a...

Chapter 9 dealt with the estimation of the parameters based on the Gini. In this chapter we introduce methods of testing for the parameters that are based on the Gini. Most of the estimators that were derived in Chap. 9 are based on U-statistics or functions of (dependent) U-statistics. The advantage is that we can use known facts about the limitin...

Ordinary least squares (OLS) regression is based on the fact that the variance of a linear combination of random variables can be decomposed into the contributions of the individual variables and to the contributions of the correlations among them. The fact that one can imitate this decomposition (under certain conditions) when decomposing the GMD...

The basic building block in regression is the covariance between the dependent variable and the explanatory variable(s). There are two regression methods that can be interpreted as based on Gini’s Mean Difference (GMD). The first method is based on the fact that one can present the Gini-covariance between the dependent variable and the explanatory...

The population parameters based on Gini were introduced in previous chapters. One of the objectives in practice is to estimate them from a given data set. This is the main objective of this chapter. When dealing with estimation, several issues come in mind. Is the data based on individual observations or are they grouped? Is the sampling procedure...

This chapter deals with applications of the GMD and the Gini coefficient in statistics. It presents an application which replicates the ANalysis Of VAriance (ANOVA) and is referred to as ANalysis Of GIni (ANOGI).

The aim of this chapter is to elaborate on the role of the Gini coefficient in two major competing theories that dominate the theoretical considerations in the area of income distribution. The major theory that dominates the economic thinking with respect to the role of the government in the area of income distribution is the Bergson’s (1938) socia...

Given two random variables, one may be interested in the correlation or association or concordance between them (Gili & Bettuzzi 1985). This purpose can be generalized by following Daniels who stated the target as “the degree of agreement” (Daniels, 1950, p. 171) between the order and the rank-order of two variables.

The objective of this chapter is to demonstrate the usefulness of several decompositions of the Gini (and the EG) in order to analyze the strengths and the weaknesses of various policies. We concentrate on distributional issues. The other component of the problem of tax reform—the estimation of the marginal cost of taxation—is identical to the desc...

The GMD has many alternative presentations. Some of these alternative presentations can be extended into families of variability measures and the GMD can be viewed as one member of such a family. The fact that there are several alternative presentations implies that one can present the GMD and the Gini coefficient as belonging to several alternativ...

The purpose of this chapter is to illustrate the use of the mixed regression technique. The meaning of mixed regression is that some of the explanatory variables are treated according to one regression method, while the others are treated according to another method. We extend this definition and include EG regressions for which different EG parame...

The main properties of the Gini mean difference (GMD) and the extended Gini (EG) were presented in the first part of the book. We have concentrated on those properties that enable the user to replicate almost everything that can be done when relying on the variance. In some sense we can claim that (almost) every analysis that is performed when usin...

Throughout the book we have stressed several properties that distinguish between the GMD and the variance, claiming that those properties give an advantage to using the GMD over the variance, in cases where the assumption of normality is not supported by the data. Among those properties are the following:

The purpose of this part of the book is to expose the reader to applications of the Gini methodology in financial theory. Those applications are relevant whenever one is interested in decision making under risk or in reducing the incompatibility between financial theory and econometric applications. Risky situations are characterized by having to m...

Several basic methods of statistical analysis such as regression and analysis of variance are based on the properties of the decomposition of the measure of variability. In the decomposition of a measure of variability we differentiate between two kinds of decompositions:

In a pathbreaking paper, Atkinson (1970) proved several results concerning the ranking of income distributions according to expected values of all concave social welfare functions. One of the important results is that for distributions with equal means, all social welfare functions show the same order of average social welfares (i.e., the same orde...

Gini's mean difference has decomposition properties that nest the decomposition of the variance as a special case. By using it one may reveal the implicit assumptions imposed on the data by using the variance. I argue that some of those implicit assumptions can be traced to be the causes of Leamer's critique. By requiring the econometrician to repo...

We study the relationships, for the case of Germany, between inequality in the annual earnings distribution, and subannual inequality and mobility during the course of the year. The study builds on an exact decomposition framework outlined in Wodon and Yitzhaki (2003), and in Yitzhaki and Wodon (2004). The study shows that inequality in the subannu...

The aim of this article is to compare the economic well-being associated to the distribution of earnings in Italy. To this purpose, dominance techniques are extensively used to minimize the disagreement about the set of social values introduced into the analysis. We will show that even in a simple economic framework, dominance conditions do not all...

We consider decision-making under risk in which random events affect the value of the portfolio multiplicatively, rather than additively. In this case, a higher variability in the rate of return not only is associated with a higher risk, a bad property, but also engenders a higher expected return, a good property. As a result, some expected utility...

We investigate the relationship between Gini’s mean difference (GMD), the mean absolute deviation, the least absolute deviation and the absolute deviation from a quantile. The latter can all be interpreted as equivalents either to the GMD of a transformed distribution or to a between-group GMD measure, according to the particular partition of the d...

The aim of this paper is to implement distributional considerations in comparing the economic well-being of individuals over time and living in different regions in Italy. In this sense, the purpose is to implement the recommendations of the Stiglitz’s (2009) report concerning augmenting national accounts by distributional considerations in an easy...

The aim of this note is to argue that the between-group Gini mean difference (BGMD) includes the mean absolute deviation (MAD), the least absolute deviation (LAD), and the absolute deviation from a given quantile (QUAD) as special cases. Those measures can be interpreted as equivalents to BGMD measures, according to the particular partition of the...

In this paper we illustrate the potential use of an old/new methodology which combines the use of concentration curves in order to investigate the components that make up a regression coefficient. The illustration is based on examining gender differences in the effect of age on labor market participation in Italy. Women participation rate is substa...

Two regressions can be interpreted as based on Gini’s Mean Difference (GMD): a semiparametric approach, which relies on weighted average of slopes defined between adjacent observations and a minimization approach, which is based on minimization of the GMD of the residuals. The estimators obtained by the semiparametric approach have representations...

The aim of this paper is to point out an inconsistency between the way income tax progressivity is measured and the policy instruments used by the income tax authorities. This inconsistency leads to violation of Feldstein's principle of horizontal equity and causes unintentional reranking among families of different sizes. The paper provides a meth...

The main purpose of this paper is to analyze patterns of non-response in the social survey and to evaluate its effect on potential biases on satisfaction from life. An additional purpose is to apply the method of mixed regression, which combines the method of Ordinary Least Squares with Gini regression in the same estimation procedure in order to e...

The basic assumption in regression analysis is that there is a linear relationship between the dependent and the independent variable over the entire range of the independent variable. The resultant regression coefficient conveys qualitative and quantitative information about the relationship. The qualitative information is given by the sign of the...

About thirty years ago Edward Leamer criticized the credibility of empirical research in economics. Since then there were huge improvements in research design, data collection and econometric methodology. On the other hand, the huge increase in computing power has increased the number of instruments available for the use of the over-zealous researc...

The extended Gini regression is a semi parametric regression which allows the user to assign different weights to different sections of the explanatory variables. We illustrate, via examples, the ability of extended Gini (EG) regression to detect sections of monotonic increase or decrease, as well as convexity and concavity of (unknown) regression...

Polarization is a concept which is defined over the distribution of income. It is clear that it does not fit into the framework of the traditional Bergson-type Social Welfare Function. The aim of this paper is to investigate whether the concept can fit into the framework of the theory of Relative Deprivation. It is suggested that it may be incorpor...

As a two-parameter model that satisfies stochastic dominance, the mean-extended Gini model is used to build efficient portfolios. The model quantifies risk aversion heterogeneity in capital markets. In a simple Edgeworth box framework, we show how capital market equilibrium is achieved for risky assets. This approach provides a richer basis for ana...

The huge technological improvement in data processing and the globalization have increased the demand for and the supply of indices that quantify the consequences of a policy. However, there are certain cases in which quantification may be misleading in the sense that it gives the impression of an accurate measurement while in reality it is not. Ne...

The aim of this paper is to compare inequality in incomes between Ireland and Israel. We first decompose the Gini coefficient in a way that resembles the decomposition of the coefficient of variation. This enables us to estimate the Gini correlation for different types of income sources and to evaluate how the transition from one income definition...

In a relative deprivation framework, unless inequality is reduced, growth is associated with both higher satisfaction and higher deprivation. This may help explain the discontent with growth despite its benefits. As is well known in the literature, knowledge of the population's mean income and Lorenz curve is all that is needed to analyze a distrib...

We consider statistical tests concerning various relationships between two absolute concentration curves (ACCs). In particular, we consider tests for determining if the two ACCs coincide, if one is above another in a specified order, or if they do not intersect without specifying which one is above/below the other one. These problems are of interes...

The aim of this paper is to suggest a new, nonparametric multiple regression method, based on the Extended Gini (EG) measures of dispersion, which enables the user to follow the curvature of the regression curve. The method is capable of estimating a series of linear approximations of the regression curve, allowing the investigator to stress differ...

Globalization and computerization enable the wide use of comparisons of average grades achieved in standardized exams by students from different nations and the rankings of the nations, based on the average grades of their students. The results are reported in front pages of newspapers. This paper suggests a method for examining the robustness of t...

A nonparametric multiple regression method, based on the extended Gini that depends on one parameter, v, is investigated. The parameter v enables production of infinite alternative linear approximations to the regression curve which differ in the weighting schemes applied to the slopes of the curve. The method allows the investigator to stress diff...

Inequality measures for multidimensional indicators of well-being may be sensitive to the weights used for weighting the various dimensions taken into account in the overall indicator. This paper provides a general yet simple method for assessing whether the Gini index of inequality is sensitive to changes in weights. The method is applied to the H...

As a two-parameter model that satisfies stochastic dominance, the mean-extended Gini model is used to build efficient portfolios. The model quantifies risk aversion heterogeneity in capital markets. In a simple Edgeworth box framework, we show how capital market equilibrium is achieved for risky assets. This approach provides a richer basis for ana...

The “melting pot” policy intends to create a new generation of migrants that will assimilate with the native population. The instruments used by the policy makers are fiscal and educational policies, which are intended to eliminate disparities between the migrants and the native population. The aim of this paper is to examine the success of this po...

Two regression methods can be interpreted as based on Gini's mean difference (GMD). One relies on a weighted average of slopes defined between adjacent observations and the other is based on minimization of the GMD of the errors. The properties of the former approach are investigated in a multiple regression framework. These estimators have represe...

Most of the literature on presumptive taxation limits its application to the less-developed economies. In this paper I argue that presumptive taxes are well entrenched in the modern world, although usually not classified as such. Presumptive taxes can take many forms, and can be incorporated in sections of a tax law that is not generally presumptiv...

The melting pot policy intends to create a new generation of migrants that will assimilate with the native population. The instruments used by the policy makers are fiscal and educational policies, which are intended to eliminate disparities between the migrants and the native population. The aim of this paper is to examine the success of this poli...

The Mean-Variance (MV) is the most popular methodology for constructing portfolios. However, it is not consistent with the expected utility hypothesis, unless one assumes the strong assumptions of either a quadratic utility or normally distributed asset returns. The Mean-Gini (MG) is an alternative method that is consistent with expected utility th...

We clarify the relationship between β-convergence and σ-convergence in a univariate setting. In such a setting, σ-convergence implies β-convergence, but β-convergence does not imply σ-convergence. The results imply that caution should be used for the interpretation of the very concept of β-convergence in empirical work.

This paper investigates the impact of international migration on technical efficiency, resource allocation and income from agricultural production of family farming in Albania. The results suggest that migration is used by rural households as a pathway out of agriculture: migration is negatively associated with both labour and non-labour input allo...

To help assign fossil material to Recent taxa we present a methodology that offers systematic quantitative tools to explore several continuous characters. It also enables the quantitative determination of which characters amongst several alternatives give results more consistent with a given classification.The Gini Mean Difference ('Gini') is the e...

A wildly discussed shortcoming of panel surveys is a potential bias arising from selective attrition. Based on data of the German Socio-Economic Panel Study (SOEP), the authors analyze potential artifacts (level, structure, income inequality) by comparing results for two independently drawn panel subsamples started in 1984 and 2000. They apply ANOG...

The aim of this paper is re-examine Card's proximity to college Instrumental Variable estimates of the returns to schooling by using a Gini-based regression. The Gini based regression is similar in its structure to the OLS, except that variability and hence covariances and correlations are measured according to city-block rather than the Euclidean...

The aim of this paper is to empirically evaluate the effect of the length of the accounting period on indices of inequality of household income in Israel. There are three main findings: (1) The analysis of the impact of the accounting period on the Gini index of inequality can be done in a way which is identical to analyzing the effect of the accou...

The huge technological improvement in data processing and the globalization have increased the demand for and the supply of indices that quantify the consequences of a policy. However, there are certain cases that quantification may be misleading in the sense that it gives the impression of an accurate measurement while in reality it is not. This p...

The basic approach to estimating the Gini and extended Gini indices is to approximate the Lorenz curve by a number of linear segments, and then estimate the Gini coefficients as the areas (or weighted areas) between the linear segments and the 45-degree line. We show that the estimator for the extended Gini, obtained from the Lorenz curve (Chotikap...

The use of equivalence scales for the measurement of inequality and social welfare poses a difficult ethical and technical dilemma. If all individuals are weighted equally while individual welfare measures are based on income or consumption per equivalent adult, then standard properties assumed in welfare economics can easily be violated. For examp...

The aim of this paper is to empirically evaluate the effect of the length of the accounting period on indices of inequality of household income in Israel. There are three main findings: (1) The analysis of the impact of the account period on the Gini index of inequality can be done in a way which is identical to analyzing the effect of the accounti...

A centralized scheme of world redistribution that maximizes a border-neutral social welfare function, subject to the disincentive effects it would create, generates a drastic reduction in world consumption inequality, dropping the Gini coefficient from 0.69 to 0.25. In contrast, an optimal decentralized (i.e., with no cross-country transfers) redis...

A main advantage of the mean-variance (MV) portfolio frontier is its simplicity and ease of derivation. A major shortcoming, however, lies in its familiar restrictions, such as the quadraticity of preferences or the normality of distributions. As a workable alternative to MV, we present the mean-Gini (MG) efficient portfolio frontier. Using an opti...

This paper proposes a social welfare framework in which to analyze the relationships between growth, trends in inequality, mobility, and social welfare. An application of the framework to worldwide and regional data on per capita GDP suggests a lack of convergence at the world level, opposite trends in convergence in various regions of the world, a...

The aim of this paper is to survey and investigate the properties of the extended Gini family of inequality measures. The paper surveys the alternative ways of spelling the extended Gini for continuous distributions (such as via covariance, via Lorenz curves, etc), and the metrics that motivate them. It also offers the adjustments needed for a cons...

A particular shortcoming of panel surveys is potential bias arising from selective attrition. Based on data from the German Socio-Economic Panel Study (SOEP) we analyze potential artifacts (level, structure, inequality of income) by comparing results from two independently drawn panel sub-samples, started in 1984 and 2000, respectively. Both sub-sa...

In a relative deprivation framework, unless inequality is reduced, growth is associated with both higher satisfaction and higher deprivation. This may help explain the discontent with growth despite its benefits. Knowledge of the population's mean income and the Lorenz curve is all that is needed to assess the satisfaction and deprivation of each i...

This paper shows that both OLS and Gini regression estimators can be interpreted as weighted averages of slopes of the regression curve, where the weights are derived from the Absolute Lorenz Curve of the independent variable. The instrumental variable estimators, under both approaches, can also be viewed as weighted sums of the same slopes, with t...

Income inequality typically declines with the length of time taken into account for measurement. This note derives an exact analytical relationship between the accounting period and inequality as measured by the Gini index. The formal relationship is similar to the decomposition of the coefficient of variation. The methodology is illustrated with p...

This paper puts OLS and Gini regression in a common framework by showing that the coefficients can be interpreted as weighted averages of slopes between adjacent observations, where weights are derived from the Absolute Lorenz Curve of the independent variable. Instrumental Variable (IV) estimators, under both approaches, are also put in a common f...

Mobility implies initial and final distributions and a transition process linking the observations of these two distributions. An inequality index describes properties of the intitial or final distribution. A mobility index describes the transition. In most cases, mobility indices have been developed using properties of transition matrices independ...

This paper points out the similarities and differences between cost-benefit analysis and tax reform. By restricting the analysis to the margin it is shown that both areas can be handled by the same method. In both areas, there is a need to define social distributional weights and to evaluate the Marginal Efficiency Cost of Public Funds (MECF), It i...

Confronted with rising poverty after an economic crisis in 1995, the Government of Mexico changed its social policy. It terminated universal subsidies for tortilla and funded new investments in human capital through PROGRESA, an innovative program providing school stipends to poor children as well as health and nutrition benefits. After reviewing t...

The use of grouped data results in a downward bias in estimates of inequality because grouping omits intra-group inequality. This note shows that using grouped data may not lead to similar difficulties when estimating statistics such as the Gini income elasticity, which provides information on the impact on inequality of a marginal change in an inc...

The extended Gini is a family of measures of variability which is mainly used in the areas of finance and income distribution. Each index in the family is defined by specifying one parameter, which reflects the social evaluation of the marginal utility of income. The higher the parameter, the more weight is attached to the lower portion of the cumu...

Of all measures of variability, the variance is by far the most popular. This paper argues that Gini's Mean Difference (GMD), an alternative index of variability, shares many properties with the variance, but can be more informative about the properties of distributions that depart from normality. Its superiority over the variance is important when...

The extended Gini is a family of measures of variability which is mainly used in the areas of finance and income distribution. Each index in the family is defined by specifying one parameter, which reflects the social evaluation of the marginal utility of income. The higher the parameter, the more weight is attached to the lower portion of the cumu...

Tax avoidance and evasion are pervasive in all countries, and tax structures are undoubtedly skewed by this reality. Standard models of taxation and their conclusions must reflect these realities.This paper first presents theoretical models that integrate avoidance and evasion into the overall decision problem faced by individuals. Early models of...

An optimal linear world income tax that maximizes a border-neutral social welfare function provides a drastic reduction in world consumption inequality, dropping the Gini coefficient from 0.69 to 0.25. In contrast an optimal decentralized (i.e., within countries) redistribution has miniscule effect on world income inequality. Thus, the traditional...

Ranking groups (schools, regions, counties) according to the average score of their constituent parts - say, ranking schools by the academic achievements of students - is a common yardstick in evaluation and a cornerstone of any planning process. In this paper we show that under certain circumstances the ranking of groups, unlike the ranking of ind...