Asymptotic and bootstrap inference for inequality and poverty measures

A random sample drawn from a population would appear to offer an ideal opportunity to use the bootstrap in order to perform accurate inference, since the observations of the sample are IID. In this paper, Monte Carlo results suggest that bootstrapping a commonly used index of inequality leads to inference that is not accurate even in very large samples, although inference with poverty indices is satisfactory. We find that the major cause is the extreme sensitivity of many inequality indices to the exact nature of the upper tail of the income distribution. This leads us to study two non-standard bootstraps, the m out of n bootstrap, which is valid in some situations where the standard bootstrap fails, and a bootstrap in which the upper tail is modelled parametrically. Monte Carlo results suggest that accurate inference can be achieved with this last method in moderately large samples.     

STATISTICAL INFERENCE VIA BOOTSTRAPPING FOR MEASURES OF INEQUALITY

In this paper we consider the use of bootstrap methods to compute interval estimates and perform hypothesis tests for decomposable measures of economic inequality. Two applications of this approach, using the Gini coefficient and Theil's entropy measures of inequality, are provided. Our first application employs pre- and post-tax aggregate state income data, constructed from the Panel Study of Income Dynamics. We find that although casual observation of the inequality measures suggests that the post-tax distribution of income is less equal among states than pre-tax income, none of these observed differences are statistically significant at the 10% level. Our second application uses the National Longitudinal Survey of Youth data to study youth inequality. We find that youth inequality decreases as the cohort ages, but between age-group inequality has increased in the latter half of the 1980s. The results suggest that (1) statistical inference is essential even when large samples are available, and (2) the bootstrap procedure appears to perform well in this setting. © 1997 by John Wiley & Sons, Ltd.     

EVT and tail-risk modelling: Evidence from market indices and volatility series

Value-at-Risk (VaR) has become the universally accepted risk metric adopted internationally under the Basel Accords for banking industry internal control, capital adequacy and regulatory reporting. The recent extreme financial market events such as the Global Financial Crisis (GFC) commencing in 2007 and the following developments in European markets mean that there is a great deal of attention paid to risk measurement and risk hedging. In particular, to risk indices and attached derivatives as hedges for equity market risk. The techniques used to model tail risk such as VaR have attracted criticism for their inability to model extreme market conditions. In this paper we discuss tail specific distribution based Extreme Value Theory (EVT) and evaluate different methods that may be used to calculate VaR ranging from well known econometrics models of GARCH and its variants to EVT based models which focus specifically on the tails of the distribution. We apply Univariate Extreme Value Theory to model extreme market risk for the FTSE100 UK Index and S&P-500 US markets indices plus their volatility indices. We show with empirical evidence that EVT can be successfully applied to financial market return series for predicting static VaR, CVaR or Expected Shortfall (ES) and also daily VaR and ES using a GARCH(1,1) and EVT based dynamic approach to these various indices. The behaviour of these indices in their tails have implications for hedging strategies in extreme market conditions.     

On the suitability of income inequality measures for regional analysis: Some evidence from simulation analysis and bootstrapping tests

The paper looks at the sensitivity of commonly used income inequality measures to changes in the ranking, size and number of regions into which a country is divided. During the analysis, several test distributions of populations and incomes are compared with a ‘reference’ distribution, characterized by an even distribution of population across regional subdivisions. Random permutation tests are also run to determine whether inequality measures commonly used in regional analysis produce meaningful estimates when applied to regions of different population size. The results show that only the population weighted coefficient of variation (Williamson’s index) and population-weighted Gini coefficient may be considered sufficiently reliable inequality measures, when applied to countries with a small number of regions and with varying population sizes.     

Variance Estimation for Generalized Entropy and Atkinson Inequality Indices: the Complex Survey Data Case*

We derive the sampling variances of generalized entropy and Atkinson indices when estimated from complex survey data, and show how they can be calculated straightforwardly using widely available software. We also show that, when the same approach is used to derive variance formulae for the independent and identically distributed case, it leads to estimators that are simpler than those proposed before. Both cases are illustrated with a comparison of income inequality in Britain and Germany.     

基于极值理论和Bootstrap方法的E-VaR研究和实证分析

考虑金融时间序列的厚尾特性,讨论了应用极值理论中的广义Pareto分布模型度量风险的问题。利用Bootstrap和MLE方法对参数进行点估计和区间估计,得出E-VaR的估计值,并对深证综指收益进行实证分析,探讨与尾部相关的极值风险,结果令人满意。     

Forecasting extreme financial risk: A score-driven approach

This paper develops a new class of dynamic models for forecasting extreme financial risk. This class of models is driven by the score of the conditional distribution with respect to both the duration between extreme events and the magnitude of these events. It is shown that the models are a feasible method for modeling the time-varying arrival intensity and magnitude of extreme events. It is also demonstrated how exogenous variables such as realized measures of volatility can easily be incorporated. An empirical analysis based on a set of major equity indices shows that both the arrival intensity and the size of extreme events vary greatly during times of market turmoil. The proposed framework performs well relative to competing approaches in forecasting extreme tail risk measures.     

一种离散型收入分布基尼系数精确计算方法

基尼系数问世八十多年的时间里,在度量经济不平等的基尼系数指标时,最常见的是离散型收入分布,本文设计了一种离散型收入分布基尼系数计算方法,用它可以精确地求解基尼系数。     

TRUE VERSUS MEASURED THEIL INEQUALITY

This article deals with several problems that arise when the Theil coefficient of income inequality is computed in practice.
Aggregation of income data into brackets leads to an underestimation of the true Theil inequality, which is defined as the value of the coefficient as computed from individual income data. The assumption that the individual incomes are distributed according to a linear density function within the income brackets is suggested as a method to estimate this aggregation error. Calculations show that this method approximates the true aggregation error reasonably well.
Several methods are discussed concerning the treatment of negative incomes. In particular one can construct an income bracket that contains both negative and positive incomes and which in the summation formula is weighted with zero weight. Of all methods this procedure using the assumption of a linear density function within brackets, yields the highest value of the Theil coefficient and is thus preferred to the other alternatives.     

一种新的收入差距研究的计量方法——基于分布函数的半参数化估计

收入差距扩大的成因分解方法的最新研究强调描述整个收入分布,并且对收入分布的变化进行分解,进而得到各个统计量的分解,这类方法的优点是可以显示收入差距的变化主要集中在哪些收入群体中。本文着重介绍了两类对收入分布函数进行成因分解的半参数化方法:一类是Lemieux及其合作者们发展起来的权重重置法;另一类是Machado和Mata首创的基于分位数回归的分解方法。     

Testing the significance of income distribution changes over the 1980s business cycle: a cross-national comparison

Using kernel density estimation we describe the distribution of household size-adjusted real income and how it changed over the business cycle of the 1980s in the United States and the United Kingdom. We confirm previous studies that show income inequality increased in the two countries and the middle of the distribution was squashed down. Using a series of statistical tests, however, we find that while the mass in both tails of the distribution increased significantly in both countries over the period, by far the greatest gains were in the upper tail. Copyright © 1999 John Wiley & Sons, Ltd.     

Has Federal Student Aid Contributed to Equality in Higher Education?

ABSTRACT. TO analyze the effect of federal student aid on the structure of the higher educational system, an index of inequality of access is developed It measures the differences in the distribution of income of entering students , as compared to the U S distribution of income at thirteen different categories of institutions Its application shows that structural inequality exists in the nation's system of higher education and that while some improvements occurred in the 1970s, especially at the bottom of the system, by 1986 most of these gains had been eliminated When these indices were regressed against per capita federal student financial aid , relative student costs , and variables reflecting changes in the larger economy, using time series data, the aid variables show little or no statistical effect on the structural inequality of the nation's system of higher education, as measured by the index of inequality On the other hand, the cost variable generally has strong effect The general economic variables occasionally have significant effect     

Growth and inequality: Dependence on the time path of productivity increases (and other structural changes)

This paper examines the significance of the time path of a given productivity increase on growth and inequality. Whereas the time path impacts only the transitional paths of aggregate quantities, it has both transitional and permanent consequences for wealth and income distribution. Hence, the growth-inequality tradeoff generated by a given discrete increase in productivity contrasts sharply with that obtained when the same productivity increase occurs gradually. The latter can generate a Kuznets-type relationship between inequality and per-capita income. Our results suggest that economies with similar aggregate structural characteristics may have different outcomes for income and wealth inequality, depending on the nature of the productivity growth path.     

Distribution of income and expenditures across nations

This paper outlines a premise that state social and economic planning in recent decades were the outgrowth of Marxist doctrine, even though Marxism as an economic system failed to materialize. The paper addresses the rise of the New Left with its doctrine of radical economics and its influence for state planning. For this purpose, the paper compares 126 countries grouped by income distribution and income inequality. The findings indicate that richer economies perform better than economies of lesser income in the distribution and equality of income.     

Understanding Italian Inequality Trends*

This article develops a unifying framework for analysing the effects of: (i) the changing distribution of individual incomes by main factor sources, (ii) the increasing participation of wives in the labour force, and (iii) the changing distribution of family types on the distinctive trends towards inequality in equivalent household income in Italy between 1977 and 2004. Changes in the distribution of work and pension incomes explain most of the trend. The higher average likelihood of wage‐earning wives had an unequalizing effect on households on the left tail of the income distribution. Little is explained by the changing distribution of family types.     

收入流动性对收入不平等的影响

虽然收入流动性在收入分配研究领域正在逐渐升温,但收入流动性对收入不平等的影响这一关键问题的研究不足。文章推导了Gini系数在绝对收入流动性矩阵中对应的表达式,分析了收入流动性矩阵对不平等程度的作用途径。并利用1998年-2002年的收入面板数据验证了该理论分析。实证分析表明,由于向收入高低两端流动的力量增长较快并在势头上超过了同样增长的向中等收入水平流动的力量,整体的收入不平等程度恶化了。     

Robust Methods for the Analysis of Income Distribution, Inequality and Poverty

Income distribution embeds a large field of research subjects in economics. It is important to study how incomes are distributed among the members of a population in order for example to determine tax policies for redistribution to decrease inequality, or to implement social policies to reduce poverty. The available data come mostly from surveys (and not censuses as it is often believed) and often subject to long debates about their reliability because the sources of errors are numerous. Moreover the forms in which the data are availabe is not always as one would expect, i.e. complete and continuous (microdata) but one also can only have data in a grouped form (in income classes) and/or truncated data where a portion of the original data has been omitted from the sample or simply not recorded.
Because of these data features, it is important to complement classical statistical procedures with robust ones. In tis paper such methods are presented, especially for model selection, model fitting with several types of data, inequality and poverty analysis and ordering tools. The approach is based on the Influence Function (IF) developed by Hampel (1974) and further developed by Hampel, Ronchetti, Rousseeuw & Stahel (1986). It is also shown through the analysis of real UK and Tunisian data, that robust techniques can give another picture of income distribution, inequality or poverty when compared to classical ones.     

Distributional implications of informal economy in the EU countries: Accounting for the spread of tax evasion benefits and cultural characteristics

This study provides evidence about the impact of informal economy on income inequality by using annual cross-country panel data from 28 European Union countries observed during the period 2005–2017. Particular attention is dedicated to the cultural setting which is expected to impact taxpayers’ behaviour and thereby income distribution. The study reveals a relationship between the size of the underground economy and income inequality, which is negative when approaching domestic informality and positive when international tax evasion by individuals is related to the top-bottom income disparity. Moreover, a delay of one-year is needed for these effects to occur, sustaining the hypothesis of secondary consequences of the informality. The study also shows that a set of cultural peculiarities are relevant for the nexus of international tax evasion by individuals and income polarisation. The inequality enhancing effect of the offshore activities is larger in countries with high degree of uncertainty avoidance and power distance, and low level of indulgence and long-term orientation respectively.     

De ontwikkeling van de inkomensongelijkheid gemeten volgens informatietheoretische maatstaven

Summary Some years ago Theil introduced new inequality measures, originating from information theory. He applied them 8.0. to income distributions in the United States. This inequality coefficient is - in contrast to Pareto's constant - based on the whole income distribution. Moreover, the inequality can be disaggregated into inequalities for groups of population. The numerical value of the inequality depends on the chosen class intervals; this may be inconvenient when making comparisons between countries, between different periods and the like. Hence we suggest an appropriate standardizing procedure. The inequality of income distributions has been calculated for the period 1950–1964 and a cornparison is made with Pmm's constant. The investigation relates to three groups of population: wage earners, self employed and others. A remark is made about between-province inequalities.     

An empirical welfare approach to regional income distributions: Alternative specifications and estimates of income inequality measures

The paper attempts to develop a welfare theoretical approach to the analysis and comparison of regional income differences. Three types of alternative methods are discussed, viz. (a) an a priori method of inserting parameter values in a prespecified welfare function, (b) an ex post method of deriving parameter values of a welfare function on the basis of income statistics and (c) a canonical correlation analysis on the basis of a set of underlying explanatory variables. All these three methods are used to define an income inequality measure appropriate for interregional comparisons. The methods employed are illustrated by means of empirical applications to regional income statistics in the Netherlands.