Identification and Estimation of Distributional Impacts of Interventions Using Changes in Inequality Measures

This paper presents estimators of distributional impacts of interventions when selection to the program is based on observable characteristics. Distributional impacts are calculated as differences in inequality measures of the marginal distributions of potential outcomes of receiving and not receiving the treatment. The estimation procedure involves a first non‐parametric estimation of the propensity score. In the second step weighted versions of inequality measures are computed using weights based on the estimated propensity score. Consistency, semi‐parametric efficiency and validity of inference based on the percentile bootstrap are shown for the estimators. Results from Monte Carlo exercises show its good performance in small samples. Copyright © 2015 John Wiley & Sons, Ltd.     

Statistical Inference in Nonparametric Frontier Models: The State of the Art

Efficiency scores of firms are measured by their distance to an estimated production frontier. The economic literature proposes several nonparametric frontier estimators based on the idea of enveloping the data (FDH and DEA-type estimators). Many have claimed that FDH and DEA techniques are non-statistical, as opposed to econometric approaches where particular parametric expressions are posited to model the frontier. We can now define a statistical model allowing determination of the statistical properties of the nonparametric estimators in the multi-output and multi-input case. New results provide the asymptotic sampling distribution of the FDH estimator in a multivariate setting and of the DEA estimator in the bivariate case. Sampling distributions may also be approximated by bootstrap distributions in very general situations. Consequently, statistical inference based on DEA/FDH-type estimators is now possible. These techniques allow correction for the bias of the efficiency estimators and estimation of confidence intervals for the efficiency measures. This paper summarizes the results which are now available, and provides a brief guide to the existing literature. Emphasizing the role of hypotheses and inference, we show how the results can be used or adapted for practical purposes.     

New Model-assisted Estimators for the Distribution Function Using the Pseudo Empirical Likelihood Method

This paper proposes using a model-assisted approach based on the pseudo empirical likelihood method to construct estimators for the finite population distribution function. It shows that the proposed sample-based estimators are genuine distribution functions that exhibit several attractive features, such as the fact that they do not depend on unknown parameters, and good performance at any argument is expected to be obtained. Consequently, estimation of other measures, such as quantiles, is a problem that is efficiently addressed by the proposed methodology and applications in various areas are therefore derived. Simulation studies based upon real and artificial populations show that the proposed estimators perform better than the existing ones. A practical situation in which the proposed estimators can be applied is also described.     

Shrinkage Estimation Strategies in Generalised Ridge Regression Models: Low/High-Dimension Regime

In this study, we suggest pretest and shrinkage methods based on the generalised ridge regression estimation that is suitable for both multicollinear and high-dimensional problems. We review and develop theoretical results for some of the shrinkage estimators. The relative performance of the shrinkage estimators to some penalty methods is compared and assessed by both simulation and real-data analysis. We show that the suggested methods can be accounted as good competitors to regularisation techniques, by means of a mean squared error of estimation and prediction error. A thorough comparison of pretest and shrinkage estimators based on the maximum likelihood method to the penalty methods. In this paper, we extend the comparison outlined in his work using the least squares method for the generalised ridge regression.     

Estimating the Kullback–Liebler risk based on multifold cross‐validation

This paper concerns a class of model selection criteria based on cross‐validation techniques and estimative predictive densities. Both the simple or leave‐one‐out and the multifold or leave‐m‐out cross‐validation procedures are considered. These cross‐validation criteria define suitable estimators for the expected Kullback–Liebler risk, which measures the expected discrepancy between the fitted candidate model and the true one. In particular, we shall investigate the potential bias of these estimators, under alternative asymptotic regimes for m. The results are obtained within the general context of independent, but not necessarily identically distributed, observations and by assuming that the candidate model may not contain the true distribution. An application to the class of normal regression models is also presented, and simulation results are obtained in order to gain some further understanding on the behavior of the estimators.     

Poverty Orderings

This paper reviews the literature of partial poverty orderings. Partial poverty orderings require unanimous poverty rankings for a class of poverty measures or a set of poverty lines. The need to consider multiple poverty measures and multiple poverty lines arises inevitably from the arbitrariness inherent in poverty comparisons. In the paper, we first survey the ordering conditions of various individual poverty measures for a range of poverty lines; for some measures necessary and sufficient conditions are identified while for others only some easily verifiable sufficient conditions are established. These ordering conditions are shown to have a close link with the stochastic dominance relations which are based on the comparisons of cumulative distribution functions. We then survey the ordering conditions for various classes of poverty measures with a single or a set of poverty lines; in all cases necessary and sufficient conditions are established. These conditions again rely on the stochastic dominance relations or their transformations. We also extend the relationship between poverty orderings and stochastic dominance to higher orders and explore the possibility and the conditions of increasing the power of poverty orderings beyond the second degree dominance condition.     

Consistent moment estimators of regression coefficients in the presence of errors in variables

This paper examines the possibilities of moment estimators of regression coefficients in the errors-in-variables problem suggested by Geary (1942) and others [Scott (1950) and Drion (1951)]. This approach yields consistent estimators of regression coefficients based on uni- and bi-variate moments (or cumulants) of third or higher order. These are computationally simple and need milder assumptions than the standard techniques, viz., ML and IV estimation. After a review of past investigations, this paper proposes new moment estimators and compares the asymptotic efficiencies of six estimators proposed earlier or here and of the OLS estimator. The case where the true regressor is lognormally distributed receives considerable attention in this communication.     

Robust growth-equity decomposition of change in poverty: The case of Iran (2000–2009)

This paper examines a robust nonparametric methodology for decomposition of change in poverty into growth and redistribution components. The decomposition is exact, symmetric and free of residual terms. It is equivalent to the Shapley value decomposition in this two-component case. We avoid parametric assumptions about the underlying distributions and Lorenz functions. All of the currently popular poverty measures may be decomposed as suggested in this paper. We identify the issues that arise with parametric approaches to decomposition. An empirical application is given based on recent data on real consumption in rural and urban areas of Iran in 2000, 2004 and 2009 (covering the country's third and fourth five-year development plans). We find that both ‘pure growth’ and ‘redistribution’ components are present in a striking change in poverty, especially among rural households. It would appear that stochastic dominance rankings of the consumption distributions make poverty analyses and decompositions robust to the choice of a poverty line, or poverty measure.     

A REVIEW OF STOCHASTIC DOMINANCE METHODS FOR POVERTY ANALYSIS

Stochastic dominance techniques have been mainly employed in poverty analyses to overcome what it is called the multiplicity of poverty indices problem. Moreover, in the multidimensional context, stochastic dominance techniques capture the possible relationships between the dimensions of poverty as they rely upon their joint distribution, unlike most multidimensional poverty indices, which are only based on marginal distributions. In this paper, we first review the general definition of unidimensional stochastic dominance and its relationship with poverty orderings. Then we focus on the conditions of multivariate stochastic dominance and their relationship with multidimensional poverty orderings, highlighting the additional difficulties that the multivariate setting involves. In both cases, we focus our discussion on first‐ and second‐order dominance, though some guidelines on higher order dominance are also mentioned. We also present an overview of some relevant empirical applications of these methods that can be found in the literature in both univariate and multivariate contexts.     

Simplified Jackknife Variance Estimates for Fuzzy Measures of Multidimensional Poverty

In this paper, we present a practical methodology for variance estimation for multi‐dimensional measures of poverty and deprivation of households and individuals, derived from sample surveys with complex designs and fairly large sample sizes. The measures considered are based on fuzzy representation of individuals' propensity to deprivation in monetary and diverse non‐monetary dimensions. We believe this to be the first original contribution for estimating standard errors for such fuzzy poverty measures. The second objective is to describe and numerically illustrate computational procedures and difficulties in producing reliable and robust estimates of sampling error for such complex statistics. We attempt to identify some of these problems and provide solutions in the context of actual situations. A detailed application based on European Union Statistics on Income and Living Conditions data for 19 NUTS2 regions in Spain is provided.     

Aggregate Poverty Measures

The way poverty is measured is important for an understanding of what has happened to poverty as well as for anti-poverty policy evaluation. Sen's (1976) pathfinding work has motivated many researchers to focus on the way poverty should be measured. A poverty measure, argued by Sen, should satisfy certain properties or axioms and the desirability of a poverty measure should be evaluated by these axioms. During the last two decades, many researchers have adopted the axiomatic approach pioneered by Sen to propose additional axioms and develop alternative poverty measures. The objective of this survey is to provide a clarification on the extensive literature of aggregate poverty measures. In this survey, we first examine the desirability of each axiom, the properties of each poverty measure, and the interrelationships among axioms. The desirability of an axiom cannot be evaluated in isolation, and some combination of axioms may make it impossible to devise a satisfactory poverty measure; some axioms can be implied by other axioms combined and so are not independent; some others are ad hoc and are disqualified as axioms for poverty measurement. Based on the interactions among axioms, we identify the ‘core’ axioms which together have a strong implication on the functional form of a poverty measure. We then review poverty measures that have appeared in the literature, evaluating the interrelationships among different measures, and examining the properties of each measure. The axioms each measure satisfies/violates are also summarized in a tabular form. Several ‘good’ poverty measures, which have not been documented by previous surveys, are also included.     

On Some Statistical Methods for Modelling the Incidence of Poverty

This paper analyzes a common method of estimating the pattern of individual characteristics associated with poverty – the logit/probit technique applied to a dummy variable identifying those individuals who are below a poverty line. We show that this procedure is seriously flawed by a fundamental logical difficulty and further statistical complications when the poverty line is estimated or when generalized poverty measures are used. An alternative approach is proposed, based on a semi-parametric series expansion appproximation to the conditional income distribution. Estimation, testing and summary procedures are established and applied to Hungarian survey data.     

A regularization approach to the many instruments problem

This paper focuses on the estimation of a finite dimensional parameter in a linear model where the number of instruments is very large or infinite. In order to improve the small sample properties of standard instrumental variable (IV) estimators, we propose three modified IV estimators based on three different ways of inverting the covariance matrix of the instruments. These inverses involve a regularization or smoothing parameter. It should be stressed that no restriction on the number of instruments is needed and that all the instruments are used in the estimation. We show that the three estimators are asymptotically normal and attain the semiparametric efficiency bound. Higher-order analysis of the MSE reveals that the bias of the modified estimators does not depend on the number of instruments. Finally, we suggest a data-driven method for selecting the regularization parameter. Interestingly, our regularization techniques lead to a consistent nonparametric estimation of the optimal instrument.     

Distribution-sensitivity of rank-dependent poverty measures

We propose a criterion to rank poverty measures on the basis of distribution-sensitivity. The criterion compares reactions to ‘lossy’ transfers among the poor. We focus on the class of rank-dependent poverty measures and provide distribution-sensitivity rankings of the poverty gap ratio, the first and second Sen measures, the Thon measure, the Shorrocks measure, and the Thon, Kakwani and S-Gini classes of measures. Moreover, we discuss the relationship between the proposed criterion and two alternative distribution-sensitivity criteria based on the Arrow–Pratt theory of risk aversion. Finally, we provide an empirically tractable necessary and sufficient condition for unanimous poverty rankings by all continuous and replication invariant rank-dependent poverty measures that exhibit a predetermined minimum degree of distribution-sensitivity.     

How to improve the performances of DEA/FDH estimators in the presence of noise?

In frontier analysis, most nonparametric approaches (DEA, FDH) are based on envelopment ideas which assume that with probability one, all observed units belong to the attainable set. In these “deterministic” frontier models, statistical inference is now possible, by using bootstrap procedures. In the presence of noise, envelopment estimators could behave dramatically since they are very sensitive to extreme observations that might result only from noise. DEA/FDH techniques would provide estimators with an error of the order of the standard deviation of the noise. This paper adapts some recent results on detecting change points [Hall P, Simar L (2002) J Am Stat Assoc 97:523–534] to improve the performances of the classical DEA/FDH estimators in the presence of noise. We show by simulated examples that the procedure works well, and better than the standard DEA/FDH estimators, when the noise is of moderate size in term of signal to noise ratio. It turns out that the procedure is also robust to outliers. The paper can be seen as a first attempt to formalize stochastic DEA/FDH estimators.      

Inference by linearization for Zenga’s new inequality index: a comparison with the Gini index

Zenga’s new inequality curve and index are two recent tools for measuring inequality. Proposed in 2007, they should thus not be mistaken for anterior measures suggested by the same author. This paper focuses on the new measures only, which are hereafter referred to simply as the Zenga curve and Zenga index. The Zenga curve Z(α) involves the ratio of the mean income of the 100α % poorest to that of the 100(1?α)% richest. The Zenga index can also be expressed by means of the Lorenz Curve and some of its properties make it an interesting alternative to the Gini index. Like most other inequality measures, inference on the Zenga index is not straightforward. Some research on its properties and on estimation has already been conducted but inference in the sampling framework is still needed. In this paper, we propose an estimator and variance estimator for the Zenga index when estimated from a complex sampling design. The proposed variance estimator is based on linearization techniques and more specifically on the direct approach presented by Demnati and Rao. The quality of the resulting estimators are evaluated in Monte Carlo simulation studies on real sets of income data. Finally, the advantages of the Zenga index relative to the Gini index are discussed.     

MEASURING POVERTY – WHAT HAPPENED TO COPENHAGEN?

Absolute poverty can be thought of as a condition of 'insufficiency', i.e. the inability to acquire the basic necessities of life. Relative poverty can be thought of as a condition of 'inequality'. At the World Summit on Social Development in Copenhagen in 1995, all participants made a commitment to produce official measures of both absolute and relative poverty and to strive to eradicate absolute poverty within a reasonable time frame. Despite these commitments, measures of absolute poverty are rare in the developed world. This paper concludes that both kinds of measures are needed for intelligent discussions and good policy-making.     

Statistical inference on seemingly unrelated varying coefficient partially linear models

This paper is concerned with the statistical inference on seemingly unrelated varying coefficient partially linear models. By combining the local polynomial and profile least squares techniques, and estimating the contemporaneous correlation, we propose a class of weighted profile least squares estimators (WPLSEs) for the parametric components. It is shown that the WPLSEs achieve the semiparametric efficiency bound and are asymptotically normal. For the non‐parametric components, by applying the undersmoothing technique, and taking the contemporaneous correlation into account, we propose an efficient local polynomial estimation. The resulting estimators are shown to have mean‐squared errors smaller than those estimators that neglect the contemporaneous correlation. In addition, a class of variable selection procedures is developed for simultaneously selecting significant variables and estimating unknown parameters, based on the non‐concave penalized and weighted profile least squares techniques. With a proper choice of regularization parameters and penalty functions, the proposed variable selection procedures perform as efficiently as if one knew the true submodels. The proposed methods are evaluated using wide simulation studies and applied to a set of real data.     

The Effect of Income Distribution on the Ability of Growth to Reduce Poverty: Evidence from Rural and Urban African Economies

The present study examines the extent to which income distribution affects the ability of economic growth to reduce poverty, based on 1990s data for a sample of rural and urban sectors of African economies. Using the basic‐needs approach, an analysis‐of‐covariance model is derived and estimated, with the headcount, gap, and squared gap poverty ratios serving as the respective dependent variables, and the Gini coefficient and PPP‐adjusted incomes as explanatory variables. The study finds that the responsiveness of poverty to income growth is a decreasing function of inequality, albeit at varying rates for the three poverty measures: lowest for the headcount, followed by the gap and fastest for the squared gap. The ranges for the income elasticity in the sample are estimated at: 0.02–0.68, 0.11–1.05, and 0.10–1.35, respectively, for these poverty measures. Furthermore while, on average, the responsiveness of poverty to income growth appears to be the same between the rural and urban sectors, there are substantial sectoral differences across countries. The results suggest the need for country‐specific emphases on growth relative to inequality.     

Benchmarked Estimators for a Small Area Mean Under a Onefold Nested Regression Model

In this paper, we modify small area estimators, based on the unit‐level model, so that they add up to reliable higher‐level estimates of population totals. These modifications result in benchmarked small area estimators. We consider two benchmarking procedures. One is based on augmenting the unit‐level model with a suitable variable. The other one uses the calibrated weights of the direct estimators that are reliable at the higher levels. These weights are used in estimators that are based on the aggregation of the unit‐level model for each small area. The mean squared error estimators of the proposed benchmarked estimators are obtained by suitably modifying those associated with the corresponding non benchmarked estimators. The properties of the estimators are evaluated via simulation.