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Conditions for the most robust multidimensional poverty comparisons using counting measures and ordinal variables

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Abstract

A natural concern with multivariate poverty measures, as well as with other composite indices, is the robustness of their ordinal comparisons to changes in the indices’ parameter values. Applying multivariate stochastic dominance techniques, this paper derives the distributional conditions under which a multidimensional poverty comparison based on the popular counting measures, and ordinal variables, is fully robust to any values of the indices’ parameters. As the paper shows, the conditions are relevant to most of the multidimensional poverty indices in the literature, including the Alkire–Foster family, upon which the UNDP’s “Multidimensional Poverty Index” (MPI) is based. The conditions are illustrated with an example from the EU-SILC data set.

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Notes

  1. Ravallion (2010), among others, discusses the pros and cons of each option.

  2. For a comparative discussion of approaches to measuring multidimensional poverty, see Atkinson (2003). For an alternative approach based on a multidimensional poverty line see Duclos et al. (2006, 2007) and Merz and Rathjen (2012).

  3. For instance, if we consider 10 dimensions of wellbeing, a multidimensional deprivation cut-off of 5 means that a person is considered multidimensionally poor if the person is deprived in 5 or more of the 10 dimensions.

  4. According to the union approach, any person deprived in at least one dimension is considered multidimensionally poor. On the other extreme, the intersection approach considers as multidimensionally poor only people who are deprived in every dimension.

  5. Alkire and Roche (2011), Alkire and Seth (2013), Battiston et al. (2013).

  6. For a recent articulation of this concern see, for instance, Ravallion (2010).

  7. In the counting approach with continuous variables, the individual poverty functions can be discontinuous over their domain. Therefore their partial and cross-partial derivatives, which are necessary to characterize the social poverty functions for which the dominance conditions are relevant, require generalized function theory for their derivation. Further elaboration can be provided upon request.

  8. Examples include educational levels, indicators of self-reported health, access to water and/or sanitation services of different qualities, etc.

  9. From a vector of poverty lines, \(Z:\left( z_{1},\ldots ,z_{d},\ldots ,z_{D}\right) \). All vectors \(Z\) belong to a set \(Z^X\) of admissible poverty lines which is bounded by the number of categories of all variables.

  10. From a vector of weights, \(W{:}\left( w_{1},\ldots ,w_{d},\ldots ,w_{D}\right) \). Weights are a necessary element for the construction of deprivation scores in the counting approach. Deprivations with higher weights have greater importance vis-a-vis others in identifying the poor and in contributing toward the overall breadth of deprivations. In policy, weights can also signal priorities in poverty reduction by highlighting the contribution of certain dimensions with higher weights. Several ways of determining weights in practice have been discussed and implemented. The reader is referred to Alkire and Santos (2010, Sect. 2.5) and Decanq and Lugo (2013).

  11. \(\varphi _{d}\left( i,\ldots ,l,\ldots ,m\right) \equiv \varphi \left( i,\ldots ,l,\ldots ,m\right) -\varphi \left( i,\ldots ,l-1,\ldots ,m\right) \). Similar definitions apply to \(g _{d}( i,\ldots ,l,\ldots ,m)\) and \(p _{d}( i,\ldots ,l,\ldots ,m)\).

  12. Their conditions are also applicable to poverty functions of the form: \(R=h\left( P\right) \), where \(h^{\prime }>0.\) This is true because, for countries \(A\) and \(B\), \(R^{A}>R^{B}\leftrightarrow P^{A}>P^{B}\). Some of the multidimensional poverty functions in the literature are of the \(R\) form (e.g. Chakravarty and D’Ambrosio 2006; Bossert et al. 2009).

  13. In the table, \(h() \) denotes an implicit function that depends on the arguments within the parenthesis.

  14. Summation by parts is also known as Abel’s lemma. For an explanation see Guenther and Lee (1988).

  15. The derivations of the sums are available from the author upon request.

  16. Specific examples can be provided upon request.

  17. In consumption theory ALEP complementarity means that the cross-partial derivative of the individual utility function with respect to a pair of goods is positive; i.e. it has the same sign as that of the partial derivatives with respect to each good. By contrast ALEP substitutability is defined as a negative cross-partial derivative. In the context of poverty measurement, since the partial derivatives of the individual poverty function are non-positive, ALEP substitutability is deemed to occur when the pairwise cross-partial derivative is non-negative. In that case the two goods can attenuate each other’s impact on poverty. By contrast, if the cross-partial derivative is non-positive then the two goods are considered ALEP complements, thereby enhancing each other’s impact on poverty. The acronym ALEP stands for Auspitz–Lieben–Edgeworth–Pareto. For more on ALEP concepts see Kannai (1980).

  18. Note that in the bivariate case it is also possible that either \(w_{1}<k<w_{2}\) or \(w_{2}<k<w_{1}\). In both situations the variables are ALEP neutral, i.e. \(\varphi _{12}=0\).

  19. The result (9) has been shown by Crawford (2005, “A nonparametric test of stochastic dominance in multivariate distributions” unpublished) for social welfare functions, and it was alluded to by Hadar and Russell (1974). The result (10) is an ordinal-variable, multivariate extension of the tri-variate derivation by Anderson (2008), also for social welfare functions.

  20. Because it involves \(\varphi _{12}\left( i,j\right) \), which depends not only on the two differentiation variables. The formula appears in Appendix 1.

  21. Here additive separability is used in the same sense as Gorman (1968).

  22. Note that in the intersection approach \(g=1\) and inequality in deprivations among the poor is irrelevant.

  23. Own calculations based on Bossert et al. (2013, Table 2).

  24. Austria, Belgium, Croatia, the Czech Republic, Germany, Denmark, Estonia, Spain, Finland, France, Greece, Hungary, Ireland, Iceland, Italy, Lithuania, Luxembourg, the Netherlands, Norway, Poland, Portugal, Sweden, Slovenia, Slovakia, and the United Kingdom.

  25. The first stage does not lead to an advance research qualification, whereas the second stage does. See EUROSTAT (2007).

  26. This question is about the ability to keep the house adequately warm, including the ability to pay the respective bills, independently of whether the house needs to be heated. See EUROSTAT (2007).

  27. As Atkinson and Bourguignon (1982) explain, comparing the joint, cumulative or survival, distributions suffices to ascertain the fulfillment of the dominance conditions. These comparisons of joint, cumulative or survival, distributions are also necessary when the individual poverty (or wellbeing) function has non-zero cross-partial derivatives or differences.

  28. All detailed results for this section are available upon request.

  29. I am indebted to Gordon Anderson for a conversation on this issue.

  30. Although not increasing the sample size when more variables are added may worsen the problem by generating comparisons of empty cells, or with very few observations, in the case of discrete variables.

  31. I am indebted to Carlos Gradin for having suggested this course of action at the 2012 IARIW Conference.

  32. The derivation for three or more variables follows the same rationale. In the union approach the formula for three or more variables resembles (16), with \(p\) being replaced by \(g\). In the intersection approach, the formula for three or more variables is: \(P=\sum _{i=1}^{t_{1}}\ldots \sum _{m=1}^{t_{D}}\Pr [x_{1}=i,\ldots ,x_{D}=m]p( i,\ldots ,m;W,Z,k)\).

  33. However the maximum is not always realized. For instance, if all weights are equal then \(c_n\) can only take \(D+1\) values: \(0,\frac{1}{D}, \frac{2}{D}, ..., 1.\)

  34. These are the discrete counterparts of the second-order dominance conditions derived by Atkinson and Bourguignon (1982, pp. 188–189).

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Acknowledgments

I would like to thank two anonymous referees for very helpful comments, and Gordon Anderson and Casilda Lasso de la Vega for substantial comments on earlier drafts. I would also like to thank Carlos Gradin, Nicole Rippin, Suman Seth, Jose Manuel Roche, Sabina Alkire, Julie Lichtfield, Paul Segal and participants at the 32nd IARIW Conference, University of the Basque Country, Maastricth School of Governance, OPHI, and the XVI IEA World Congress for helpful comments. I would like to thank the European Commission, Eurostat, for permission to use the EU-SILC 2007 user database, release 2 August 2009, under contract EU-SILC/2009/33 between Eurostat and University of Oxford and its Colleges. Eurostat has no responsibility for the results or the conclusions of this paper.

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  1. Leeds University Business School, Maurice Keyworth Building, Leeds , LS2 9JT, UK

    Gaston Yalonetzky

  2. OPHI, Oxford University, Oxford, UK

    Gaston Yalonetzky

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Appendices

Appendix 1: Cross-partial difference of \(\psi (c_{n}; k)\)

$$\begin{aligned}&\varphi _{12}(i,j) =\mathbb {I}(i-1<z_{1}<i) \mathbb {I}(j-1<z_{2}<j) \nonumber \\&\quad [\mathbb {I}(c_{n}-w_{1}<k) \mathbb {I}( c_{n}-w_{2}< k)\mathbb {I}(c_{n}\ge k) -\mathbb {I}( c_{n}-w_{1}-w_{2}<k) \mathbb {I}( c_{n}-w_{1}\ge k) \nonumber \\&\qquad -\mathbb {I}(c_{n}-w_{1}-w_{2}<k) \mathbb {I}( c_{n}-w_{2}\ge k) ] \end{aligned}$$
(23)

Appendix 2: Basic cross-partial differences of \(p\) in the intersection approach

$$\begin{aligned} p_{12}&= g\varphi _{12}\left( c_{n};1\right) +g_{1}\varphi _{2}\left( c_{n};1\right) +g_{2}\varphi _{1}\left( c_{n};1\right) +\varphi \left( c_{n};1\right) g_{12} \end{aligned}$$
(24)
$$\begin{aligned} p_{123}&= g_{1}\varphi _{23}\left( c_{n};1\right) +g_{2}\varphi _{13}\left( c_{n};1\right) +g_{3}\varphi _{12}\left( c_{n};1\right) \\&+\,\varphi _{3}\left( c_{n};1\right) g_{12}+\varphi _{2}\left( c_{n};1\right) g_{13}+\varphi _{1}\left( c_{n};1\right) g_{23} \nonumber \\&+\,g\varphi _{123}\left( c_{n};1\right) +g_{123}\varphi \left( c_{n};1\right) \nonumber \end{aligned}$$
(25)

Appendix 3: Sample sizes and descriptive statistics

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Yalonetzky, G. Conditions for the most robust multidimensional poverty comparisons using counting measures and ordinal variables. Soc Choice Welf 43, 773–807 (2014). https://doi.org/10.1007/s00355-014-0810-2

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