Pareto–lognormal distributions: Inequality,poverty, and estimation from grouped income data

The (double) Pareto–lognormal is an emerging parametric distribution for income that has a sound underlying generating process, good theoretical properties, and some limited favorable evidence of its fit to data. We extend existing results for this distribution in 3 directions. (1) We derive closed form formula for its moment distribution functions, and for various inequality and poverty measures. (2) We show how it can be estimated from grouped data using GMM. (3) Using grouped data from ten countries, we compare its performance with that of another leading 4-parameter income distribution, the generalized beta-2 distribution. The results confirm that, when using grouped data, both distributions provide a good fit, with the double Pareto–lognormal distribution outperforming the beta distribution in 4 out of 10 cases.     

INCOME DISTRIBUTION STATISTICS IN LATIN AMERICA AND THEIR RELIABILITY

This paper is addressed to the question of how far income distribution statistics currently available in Latin America can be relied upon, either to assess the degree of inequality in the national distribution of income or to undertake comparisons between countries or over time. It gives a summary account of research carried out on Latin American data.
The sources available in Latin America for estimating income distributions are discussed. Concentrating the attention on household surveys conducted in various Latin American countries, an inventory of such surveys and their characteristics is offered, along with a detailed exposition of survey methods and income concepts used for estimating household income. Methods used for assessing the representativeness of samples are summarily reviewed. The case for comparing income data from household surveys and population censuses with national accounts estimates is put forward, along with the procedures and assumptions used for carrying out such comparisons. The relative discrepancy between the two sources is taken as indicative of the degree of underestimation of each type of income in each survey. An analysis of such discrepancies across the set of surveys considered gives clues on possible underestimation biases in measuring each type of income and total household income in different types of survey and in population censuses.
Differential effects on comparability of survey results call for appropriate methods of adjusting income distribution estimates to account for the missing incomes. A method for carrying out such an adjustment is applied to income distributions from a selected number of Latin American surveys. The results obtained provide an indication of how much difference it makes to use unadjusted or adjusted data to assess income concentration or to carry out comparisons over time or space.     

Interpolating the Lorenz Curve: Methods to Preserve Shape and Remain Consistent with the Concentration Curves for Components

C¹‐class interpolation methods that preserve monotonicity and convexity and are thus suitable for the estimation of the Lorenz curve from grouped data are not widely known. Instead, parametric models are usually applied for such estimation. Parametric models, however, have difficulty in accurately approximating every part of income/expenditure distributions. This paper proposes two types of C¹‐class shape‐preserving interpolation methods. One is a piecewise rational polynomial interpolation (proposed independently by Stineman and Delbourgo) that enables consistent interpolation of the concentration curves for income/expenditure components, attaining approximately the same accuracy as that of the existing methods when applied to decile‐grouped data or to more detailed aggregation. Another is a Hybrid interpolation that employs pieces of curves derived from parametric models on end intervals. Empirical comparisons show that the Hybrid interpolation (with the assistance of parametric models for class‐boundary estimation) outperforms the existing methods even when applied to quintile‐grouped data without class boundaries.     

Calculating Poverty Measures from the Generalised Beta Income Distribution

Data for measuring poverty are frequently available in a summary form that describes the proportion of income or expenditure for each of a number of population proportions. While various discrete poverty measures can be applied directly to data in this limited form, they typically require an arbitrary approach to within‐group interpolation. This problem can be overcome by fitting either a parametric income distribution or a Lorenz curve to the grouped data and computing the required quantities from estimated parameters. The Lorenz curve approach is widely used by the World Bank, but can encounter problems. As an alternative, in this article we show how to calculate several poverty measures from parameters of the generalized beta income distribution, and its popular special cases. An analysis of poverty changes in countries from South and Southeast Asia is used to illustrate the methodology.     

Modeling of Income and Indicators of Poverty and Social Exclusion Using the Generalized Beta Distribution of the Second Kind

There are three reasons why estimation of parametric income distributions may be useful when empirical data and estimators are available: to stabilize estimation; to gain insight into the relationships between the characteristics of the theoretical distribution and a set of indicators, e.g. by sensitivity plots; and to deduce the whole distribution from known empirical indicators, when the raw data are not available. The European Union Statistics on Income and Living Conditions (EU‐SILC) survey is used to address these issues. In order to model the income distribution, we consider the generalized beta distribution of the second kind (GB2). A pseudo‐likelihood approach for fitting the distribution is considered, which takes into account the design features of the EU‐SILC survey. An ad‐hoc procedure for robustification of the sampling weights, which improves estimation, is presented. This method is compared to a non‐linear fit from the indicators. Variance estimation within a complex survey setting of the maximum pseudo‐likelihood estimates is done by linearization (a sandwich variance estimator), and a simplified formula for the sandwich variance, which accounts for clustering, is given. Performance of the fit and estimated indicators is evaluated graphically and numerically.     

Averaging Income Distributions

Various inequality and social welfare measures often depend heavily on the choice of a distribution of income. Picking a distribution that best fits the data involves throwing away information and does not allow for the fact that a wrong choice can be made. Instead, Bayesian model averaging utilizes a weighted average of the results from a number of income distributions, with each weight given by the probability that a distribution is ‘correct’. In this study, prior densities are placed on mean income, the mode of income and the Gini coefficient for Australian income units with one parent (1997–8). Then, using grouped sample data on incomes, posterior densities for the mean and mode of income and the Gini coefficient are derived for a variety of income distributions. The model‐averaged results from these income distributions are obtained.     

Bivariate income distributions for assessing inequality and poverty under dependent samples

As indicators of social welfare, the incidence of inequality and poverty is of ongoing concern to policy makers and researchers alike. Of particular interest are the changes in inequality and poverty over time, which are typically assessed through the estimation of income distributions. From this, income inequality and poverty measures, along with their differences and standard errors, can be derived and compared. With panel data becoming more frequently used to make such comparisons, traditional methods which treat income distributions from different years independently and estimate them on a univariate basis, fail to capture the dependence inherent in a sample taken from a panel study. Consequently, parameter estimates are likely to be less efficient, and the standard errors for between-year differences in various inequality and poverty measures will be incorrect. This paper addresses the issue of sample dependence by suggesting a number of bivariate distributions, with Singh–Maddala or Dagum marginals, for a partially dependent sample of household income for two years. Specifically, the distributions considered are the bivariate Singh–Maddala distribution, proposed by Takahasi (1965), and bivariate distributions belonging to the copula class of multivariate distributions, which are an increasingly popular approach to modelling joint distributions. Each bivariate income distribution is estimated via full information maximum likelihood using data from the Household, Income and Labour Dynamics in Australia (HILDA) Survey for 2001 and 2005. Parameter estimates for each bivariate income distribution are used to obtain values for mean income and modal income, the Gini inequality coefficient and the headcount ratio poverty measure, along with their differences, enabling the assessment of changes in such measures over time. In addition, the standard errors of each summary measure and their differences, which are of particular interest in this analysis, are calculated using the delta method.     

中国农村的收入差距与健康

随着收入差距的扩大,收入分配对健康和健康不平等的影响日益受到关注。本文利用中国健康营养调查(CHNS)1997年和2000年农村的面板数据回答两个问题:收入差距对健康的影响以及影响健康的方式;收入差距的扩大是否会导致健康不平等的加剧,尤其是低收入人群的健康是否受到更为不利的影响。研究发现,首先收入差距对健康的影响存在滞后效应;其次,收入差距对健康的影响呈现“倒U”型,在收入差距较高时,收入差距对健康的影响主要为负向的,一个可能的原因是收入差距影响到公共卫生设施的供给。再次,收入差距的扩大会加强收入效应,其含义是如果低收入人群的收入更容易受到负向冲击,那么收入差距对低收入人群的健康更为不利。     

Entropy-based China income distributions and inequality measures

ABSTRACT

We use information theoretic information recovery methods, on a 2005 sample of household income data from the Chinese InterCensus, to estimate the income distribution for China and each of its 31 provinces and to obtain corresponding measures of income inequality. Using entropy divergence methods, we seek a probability density function solution that is as close to a uniform probability distribution of income (with the least inequality), as the data will permit. These entropy measures of income inequality reflect how the allocation and distribution systems are performing, and we show the advantages of investigating province variation in income inequality using entropy measures rather than Gini coefficients. Finally, we use a sample of data from the China Family Panel Study to recover an estimate of the 2010 and the 2016 to investigate possible directions of inequality changes using these different additional data sources, given that the 2015 Inter-Census is not yet available.     

Growth still is good for the poor

Average incomes in the poorest two quintiles on average increase at the same rate as overall average incomes. This is because, in a global dataset spanning 121 countries over the past four decades, changes in the share of income of the poorest quintiles are uncorrelated with changes in average income. The variation in changes in quintile shares is also small relative to the variation in growth in average incomes, implying that the latter accounts for most of the variation in income growth in the poorest quintiles. In addition, we find little evidence that changes in the bottom quintile shares are correlated with country-level factors that are typically considered as important determinants for growth in average incomes or for changes in inequality. This evidence confirms the central importance of economic growth for improvements in living standards at the low end of the income distribution. It also illustrates the difficulty of identifying specific macroeconomic policies that are significantly associated with the growth rates of those in the poorest quintiles relative to everyone else.     

TOP INCOMES AND NATIONAL SAVINGS

The relationship between income inequality and national savings is theoretically ambiguous, and past empirical studies have delivered mixed results. We revisit the question using a newly available source of data on inequality: the income share of the richest 10 percent and the richest 1 percent. Combining this with historical data on national savings rates, we are able to investigate the relationship for 11 developed countries over the period 1921–2002. We find no consistent relationship between lagged top income shares and current savings rates, and our standard errors are small enough that we are able to reject more than modest effects in either direction. We view this as suggesting that inequality at the top end of the distribution is not a major driver of national savings rates.     

Characterizing life-cycle wealth distributions using statistical inference and dominance criteria

This paper examines how the distribution of household wealth in Canada varies with age over the life cycle. The wealth distribution is characterized in terms of decile means and decile shares for each of six age groups, and comparisons between age-specific distributions are based on first-and second-order stochastic dominance criteria. It is found that (i) mean wealth levels and wealth distributions increase significantly with age in concave quadratic fashion until near-retirement and then decline, and (ii) wealth inequality declines in convex fashion with age, at first steeply and then not significantly. This joint pattern in mean and inequality of wealth holdings across age groups presents a challenge for basic theories to explain.     

The Atkinson Inequality Measure and its Sampling Properties: Bayesian and Classical Approaches

This paper examines several Bayesian methods of obtaining posterior probability density functions of the Atkinson inequality measure and its associated social welfare function, in the context of grouped income distribution data. The methods are compared with asymptotic standard errors. The role of the number of income classes is investigated using a simulated distribution. If only a small number of groups is available in published data, there is a clear gain from generating the posterior probability density function when using an explicit income distribution assumption. Even with a small number of groups, the Bayesian approach gives results that are close to the sample values obtained using the corresponding individual observations.     

How large is income inequality in China: assessment on different estimates of Gini coefficient

The rising income inequality in China has attracted social attentions, especially since SWUFE published the unbelievably high Gini coefficient in December 2012. In order to answer the question how large income inequality in current China is, this paper compares existing different Gini coefficients estimated from four different household surveys, which are the NBS household survey, the CHIP survey, the CHFS of SWUFE, and the CFPS of Peking University, and then assesses these household surveys themselves. The relevant evidences indicate that the national Gini coefficient in current China is between 0.47~0.52. The sampling defects of the CHFS are significantly major and that the national Gini coefficient of 0.61 published by SWUFE is seriously overestimated.     

The tale of the tails: Canadian income inequality in the 1980s and 1990s

Abstract.  We present new evidence on levels and trends in after-tax income inequality in Canada between 1980 and 2000. We argue that existing data sources may miss changes in the tails of the income distribution, and that many of the changes in the income distribution have been in the tails. For this reason, we turn to an alternative source. In particular, we construct data on after-tax and transfer income using Census files augmented with predicted taxes based on information available from administrative tax data. Using these data, we find that Canadian after-tax inequality levels are substantially higher than has been previously recognized, primarily because income levels are lower at the bottom of the distribution than in commonly used survey data. We also find larger long-term increases in after-tax income inequality and far more variability over the economic cycle. This raises interesting questions about the role of the tax and transfer system in mitigating both trends and fluctuations in market income inequality.     

Finance and inequality: a study of Indian states

Although a large literature exists on finance and economic growth, few studies have empirically examined the relationship between finance and inequality. Using grouped national household sample survey data on monthly household consumption expenditure at the sub-national level for the years 1999–2000 to 2006–2007, we examine the relationship between Financial Development (FD) and rural and urban inequality in India. The results indicate that FD is associated with a reduction in inequality, but only in the urban areas. Further, inequality is found to be higher in the richer states compared to less developed and low income states, and as state income increases, inequality also increases both in the rural and urban areas. Finally, our results show that increase in population per bank branch leads to higher inequality in urban areas but decline in rural areas.     

A Different Perspective on the Evolution of UK Income Inequality

This paper scrutinizes the conventional wisdom about trends in UK income inequality and also places contemporary inequality in a much longer historical perspective. We combine household survey and income tax data to provide better coverage of all income ranges from the bottom to the very top (and make our estimates available to other researchers). We make a case for studying distributions of income between tax units (i.e. not assuming the full income sharing that goes with the use of the household as the unit of analysis) for reasons of principle as well as data harmonization. We present evidence that income inequality in the UK is as least as high today as it was just before the start of World War 2.     

Inequality,Redistribution, and the Financial Crisis: Evidence from Distributional National Accounts for Austria

In this paper, we provide novel findings regarding the distributional effects of the global financial and economic crisis and how redistribution operated during this time, using detailed data for Austria. We construct distributional national accounts for the period 2004–2016 by combining survey data, tabulated tax data, and detailed national accounts data. The comprehensive data set allows us to analyze the distribution of macroeconomic income growth across the income distribution and to explore the evolution of income inequality over time. Our results suggest that as the distribution of growth changed over time, this had considerable repercussions for inequality, which started to decline at the very beginning of the economic and financial crisis, but increased again after 2012. We find that capital income largely determined both the level and the dynamics of income inequality. Government spending was found to play a key role for redistributive effects across the income distribution. In particular, in-kind transfers redistributed pre-tax income to a large extent. Our results show further that individuals with lower educational levels and younger individuals faced negative growth in pre-tax income over the years and also benefited considerably from redistribution.     

Measuring lnequality in the Middle East 1990–2016: The World’s Most Unequal Region?

In this paper we combine household surveys, national accounts, income tax data and wealth data in order to estimate income concentration in the Middle East for the period 1990–2016. According to our benchmark series, the Middle East appears to be the most unequal region in the world, with a top decile income share as large as 64 percent, compared to 37 percent in Western Europe, 47 percent in the US and 55 percent in Brazil (see Alvaredo et al. 2018). This is due both to enormous inequality between countries (particularly between oil‐rich and population‐rich countries) and to large inequality within countries (which we probably under‐estimate, given the limited access to proper fiscal data). We stress the importance of increasing transparency on income and wealth in the Middle East, as well as the need to develop mechanisms of regional redistribution and investment.