77512 Robust Multidimensional Spatial Poverty Comparisons in Ghana, Madagascar, and Uganda Jean-Yves Duclos, David Sahn, and Stephen D. Younger Spatial poverty comparisons are investigated in three African countries using multi- dimensional indicators of well-being. The work is analogous to the univariate stochastic dominance literature in that it seeks poverty orderings that are robust to the choice of multidimensional poverty lines and indices. In addition, the study seeks to ensure that the comparisons are robust to aggregation procedures for multiple welfare variables. In contrast to earlier work, the methodology applies equally well to what can be defined as ‘‘union,’’ ‘‘intersection,’’ and ‘‘intermediate’’ approaches to dealing with multidimensional indicators of well-being. Furthermore, unlike much of the stochastic dominance literature, this work computes the sam- pling distributions of the poverty estimators to perform statistical tests of the difference in poverty measures. The methods are applied to two measures of well-being, the log of household expenditures per capita and children’s height-for- age z scores, using data from the 1988 Ghana Living Standards Study survey, the 1993 National Household Survey in Madagascar, and the 1999 National Household Survey in Uganda. Bivariate poverty comparisons are at odds with univariate compar- isons in several interesting ways. Most important, it cannot always be concluded that poverty is lower in urban areas in one region compared with that in rural areas in another, even though univariate comparisons based on household expenditures per capita almost always lead to that conclusion. It is common to assert that poverty is a multidimensional phenomenon, yet most empirical work on poverty, including spatial poverty, uses a unidimensional yardstick to judge a person’s well-being, usually household expenditures or income per capita or per adult equivalent. When studies use more than one indicator of well-being, poverty comparisons are either made independently for Jean-Yves Duclos is a professor of economics and director of the Inter-University Center on Risk, Economic Policies, and Employment (CIRPE ´ Laval; his email address is jyves@ecn.ulaval.ca. ´ E) at Universite David Sahn is a professor of economics and director of the Food and Nutrition Policy Program at Cornell University; his email address is des16@cornell.edu. Stephen D. Younger is an associate director of the Food and Nutrition Policy Program at Cornell University; his email address is sdy1@cornell.edu. The authors are grateful to three anonymous referees and the editor for comments on a previous draft. The research for this study is supported by the Strategies and Analysis for Growth and Access project, funded by a U.S. Agency for International Development cooperative agreement with Cornell University and Clark-Atlanta University and by the Poverty and Economic Policy network of the International Develop- ment Research Centre. For more information, see http://www.saga.cornell.edu and www.pep-net.org/. THE WORLD BANK ECONOMIC REVIEW, VOL. 20, NO. 1, pp. 91–113 doi:10.1093/wber/lhj005 Advance Access publication April 6, 2006 Ó The Author 2006. Published by Oxford University Press on behalf of the International Bank for Reconstruction and Development / THE WORLD BANK. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org. 91 92 THE WORLD BANK ECONOMIC REVIEW, VOL. 20, NO. 1 each indicator1 or made using an arbitrarily defined aggregation of the multiple indicators into a single index.2 In either case, aggregation across multiple welfare indicators and across the welfare statuses of individuals or households requires specific aggregation rules that are necessarily arbitrary.3 Multidimensional pov- erty comparisons also require the estimation of multidimensional poverty lines, a procedure that is problematic even in a unidimensional setting. Taking as a starting point the conviction that multidimensional poverty comparisons are ethically and theoretically attractive, the purpose here is to apply quite general methods for multidimensional poverty comparisons to the particular question of spatial poverty in three African countries—Ghana, Madagascar, and Uganda. The relevant welfare theory and accompanying statistics are developed elsewhere (Duclos, Sahn, and Younger 2003). The purpose here is to give an intuitive explanation of the methods and to show that they are both tractable and useful when applied to spatial poverty in Africa. The poverty comparisons use the dominance approach initially developed by Atkinson (1987) and Foster and Shorrocks (1988a, 1988b, 1988c) in a unidi- mensional context.4 In a review of this literature, Zheng (2000) distinguishes between poverty comparisons that are robust to the choice of a poverty line and those that are robust to the choice of a poverty measure or index. Both are attractive features of the dominance approach because they enable the analyst to avoid relying on ethically arbitrary choices of a poverty line and a poverty measure. The poverty comparisons used here are robust to the selection of both a poverty line and a poverty measure. In the multidimensional context, this includes robustness over the manner in which multiple indicators interact to generate overall individual well-being. Section I briefly presents the data and provides an intuitive discussion of multidimensional poverty comparisons. In addition to the stochastic dominance conditions that are familiar from the univariate literature, it discusses two concepts that arise only in a multivariate context. First, it distinguishes between intersection and union definitions of poverty.5 By the well-known focus axiom used in poverty measurement (see, for instance, Foster 1984), these definitions 1. This would involve, say, comparing incomes across regions and then comparing mortality rates across regions and so on. 2. The best-known example is the human development index of the United Nations Development Programme (UNDP 1990), which uses a weighted average of life expectancy, literacy, and GDP per capita across the population. 3. Such rules have been the focus of some of the recent literature. See, for instance, Tsui (2002) and Bourguignon and Chakravarty (2003). Bourguignon and Chakravarty (2002) also give several interesting examples in which poverty orderings vary with the choice of aggregation rules. 4. Atkinson and Bourguignon (1982, 1987) first used this approach in the context of multidimen- sional social welfare. See also Crawford (1999). 5. For further recent discussion, see Bourguignon and Chakravarty (2002, 2003), Atkinson (2003), and Tsui (2002). Duclos, Sahn, and Younger 93 identify the individual poverty statuses to be aggregated to obtain poverty indices. If well-being is measured in the dimensions of income and height, say, then a person whose income falls below an income poverty line or whose height falls below a height poverty line could be considered poor. This is a union definition of multidimensional poverty. By an intersection definition, however, a person would have to fall below both poverty lines to be considered poor. In contrast to earlier work, the tests used here are valid for both definitions—or for any choice of intermediate definitions for which the poverty line in one dimen- sion is a function of well-being measured in the other dimension. A second key concept that arises only in the context of multivariate poverty comparisons is that, roughly speaking, the correlation between individual measures of well-being matters. If two populations have the same univariate distributions for two measures of well-being, but one has a higher correlation between these measures, then it should not have lower poverty.6 This is because a person’s deprivation in one dimension of well- being should matter more if the person is also poorer in the other dimension. The dimensions of well-being are substitutes in the poverty measure. While this is apparently intuitive, counterexamples are also presented, although the poverty comparisons are valid only for the case in which the dimensions are substitutes. Section I concludes with examples of why the poverty comparisons here are more general than comparisons of indices such as the United Nations Develop- ment Programme’s human development index (UNDP 1990) and comparisons that consider each dimension of well-being independently. Section II applies these methods to spatial poverty comparisons in Ghana, Madagascar, and Uganda, comparing poverty across regions and areas (urban and rural) in the dimensions of household expenditures per capita and nutri- tional status for children under the age of 5. Univariate comparisons based on expenditures or nutritional status alone almost always show greater poverty in rural areas in any one region than in urban areas in any other region. Bivariate comparisons, however, are less likely to draw this conclusion for a variety of reasons. For this particular application, all of the interesting deviations from the generally accepted conclusion that poverty is higher in rural areas result from the fact that the correlation between these two dimensions of well-being is often higher in urban areas. Previous work on multidimensional poverty comparisons has ignored sam- pling variability, yet this is fundamental if the study of multidimensional poverty comparisons is to have any practical application. The poverty comparisons here are all statistical, using consistent, distribution-free estimators of the sampling distributions of the statistics of each poverty comparison. 6. Bourguignon and Chakravarty (2003, p. 31) refer to this as a ‘‘correlation increasing switch’’ and discuss it in detail. It is closely related to Tsui’s (1999) concept of correlation increasing majorization. 94 THE WORLD BANK ECONOMIC REVIEW, VOL. 20, NO. 1 I. METHODS TO COMPARE POVERTY WITH MULTIPLE INDICATORS OF WELL-BEING This section discusses the data and provides an intuitive presentation of multi- dimensional poverty comparisons. Data The data for this study come from the 1988 Ghana Living Standards Survey, the 1993 National Household Survey (Enque ˆ te Permanente aupre` s des Me ´nages) in Madagascar, and the 1999 National Household Survey in Uganda. All are nationally representative multipurpose household surveys. The first measure of well-being is household expenditures per capita, the standard variable for empirical poverty analysis in developing economies. The second is children’s height-for-age z score (HAZ) that measures how a child’s height compares with the median of the World Health Organization reference sample of healthy children (WHO 1983). In particular, the z scores standardize a child’s height by age and gender as ðxi À xmedian Þ=x , where xi is a child’s height, xmedian the median height of children in a healthy and well-nourished reference population of the same age and gender, and sx the standard deviation from the mean of the reference population. Thus, the z-score measures the number of standard deviations that a child’s height is above or below the median for a reference population of healthy children of the same age and gender. The nutrition literature includes a wealth of studies showing that in poor countries children’s height is a particularly good summary measure of children’s general health status (Cole and Parkin 1977; Mosley and Chen 1984; WHO 1995). As summarized by Beaton and others (1990, p. 2), growth failure is ‘‘the best general proxy for constraints to human welfare of the poorest, includ- ing dietary inadequacy, infectious diseases and other environmental health risks.’’ They go on to point out that the usefulness of stature is that it captures the ‘‘multiple dimensions of individual health and development and their socio- economic and environmental determinants.’’ In addition, HAZ is an interesting variable to consider with expenditures per capita because the two are, surpris- ingly, not highly correlated, so that they capture different dimensions of well- being (Haddad and others 2003).7 Univariate Poverty Dominance Methods The theoretical and statistical bases for the methods used here are developed in Duclos, Sahn, and Younger (2003). This section provides only an intuitive presentation; the formal argument is presented in the appendix. Even though the goal is to make multidimensional poverty comparisons, it is easier to grasp the intuition with a unidimensional example. 7. Pradhan, Sahn, and Younger (2003) give a more thorough defense of using children’s height as a welfare measure. Duclos, Sahn, and Younger 95 Consider the question: Is poverty greater in urban or rural areas? The dom- inance approach to poverty analysis addresses this question by making poverty comparisons that are valid for a wide range of poverty lines and a broad class of poverty measures. Figure 1 displays the cumulative density functions—or distribution functions—for real household expenditures per capita in urban and rural areas of Uganda in 1999. If the values on the x axis are thought of as potential poverty lines—the amount that a household has to spend per capita in order not to be poor—then the corresponding value on the y axis would be the headcount poverty rate—the share of people whose expenditure is below that particular poverty line. Note that this particular cumulative density function is sometimes called a poverty incidence curve. The graph makes clear that no matter which poverty line one chooses, the headcount poverty index (the share of the sample that is poor) will always be lower for urban areas than for rural. Thus, this sort of poverty comparison is robust to the choice of a poverty line. What is less obvious is that this type of comparison also permits drawing conclusions about poverty according to a very broad class of poverty measures. In particular, if the poverty incidence curve for one sample is everywhere below the poverty incidence curve for another sample over a bottom range of poverty lines, then poverty will be lower in the first sample for all those poverty lines and for all additive poverty measures that obey two conditions: they are nondecreas- ing and anonymous. Nondecreasing means that if any one person’s income increases, the poverty measure cannot increase as well. Anonymous means that F I G U R E 1 . Poverty Incidence Curves, Urban and Rural Areas of Uganda 1999 1.0 0.9 0.8 0.7 Poverty incidence 0.6 0.5 0.4 0.3 Rural 0.2 Urban 0.1 0.0 7.5 8.0 8.5 9.0 9.5 10.0 Log of household expenditures per capita Source: Authors’ analysis based on data from the Uganda 1999 National Household Survey. 96 THE WORLD BANK ECONOMIC REVIEW, VOL. 20, NO. 1 it does not matter which person occupies which position or rank in the income distribution. It is helpful to denote as Ã…1 the class of all poverty measures that have these characteristics. Ã…1 includes virtually every standard poverty measure. It should be clear that the nondecreasing and anonymous characteristics of the class Ã…1 are entirely unobjectionable. Additivity is perhaps less benign, but it is a standard feature of the poverty measures because it allows subgroup decomposi- tion (Foster, Greer, and Thorbecke 1984). Comparing cumulative density curves as in figure 1 thus enables making a very general statement about poverty in urban and rural Uganda: for any reason- able poverty line and for the class of poverty measures Ã…1, poverty is lower in urban areas than in rural areas. This is called first-order poverty dominance. The generality of such conclusions makes poverty dominance methods attractive. However, such generality comes at a cost. If the cumulative density functions cross one or more times, there is no clear ordering—it cannot be said whether poverty is lower in one group or the other. There are two ways to deal with this problem, both reasonably general. First, it is possible to conclude that poverty is lower in one sample than in another for the same large class of poverty measures, but only for poverty lines up to the first point at which the cumulative density functions cross (for a recent treatment of this, see Duclos and Makdissi 2005). If reasonable people agree that this crossing point is at a level of well-being safely beyond any sensible poverty line, this conclusion may be sufficient. Second, it is possible to make comparisons over a smaller class of poverty measures. For example, if the condition is added that the poverty measure respects the Pigou–Dalton transfer principle,8 it turns out that the areas under the crossing poverty incidence curves can be compared. If the area under one curve is less than the area under another for a bottom range of reasonable poverty lines, poverty will be lower for the first sample for all additive poverty measures that are nondecreasing, are anonymous, and obey the Pigou–Dalton transfer principle. This is called second-order poverty dom- inance, and the associated class of poverty measures is called Ã…2. While not as general as first-order dominance, it is still a quite general conclusion.9 Bivariate Poverty Dominance Methods Bivariate poverty dominance comparisons extend the univariate methods dis- cussed above. If there are two measures of well-being rather than one, figure 1 becomes a three-dimensional graph, with one measure of well-being on the x axis, a second on the y axis, and the bivariate cumulative density function 8. The Pigou–Dalton transfer principle says that a marginal transfer from a richer person to a poorer person should decrease (or not increase) the poverty measure. Again, this seems entirely sensible, but note that it does not work for the headcount whenever a richer person located initially just above the poverty line falls below the poverty line because of the transfer to the poorer person. 9. If second-order poverty dominance cannot be established, it is possible to integrate once again and check for poverty dominance for a still smaller class of poverty indices and so on. See Zheng (2000) and Davidson and Duclos (2000) for more detailed discussions. Duclos, Sahn, and Younger 97 F I G U R E 2 . Bidimensional Poverty Dominance Surface Univariate cumulative density function for HAZ 1.0 0.9 0.8 Cumulative distribution 0 .7 Univariate cumulative density 0 .6 function for household expenditure per capita 0.5 0.4 0 .3 0 .2 11.75 0.1 11.0 0 Log household 0 .0 10 .25 expenditure per 9 .5 0 capita Height-for-age z-score on the z axis (vertical), as in figure 2. The bivariate cumulative density function is now a surface rather than a line, and one cumulative density function surface is compared with another, just as in figure 1. If one such surface is everywhere below another, poverty in the first sample is lower than poverty in the second sample for a broad class of poverty measures, just as in the univariate case. It is also useful to note that univariate poverty incidence curves are the marginal cumulative densities in the picture found at the extreme edges of the bivariate surface. That class, now called Ã…1,1 to indicate that it is first order in both dimensions of well-being, has characteristics analogous to those of the univariate case— additive, nondecreasing in each dimension, and anonymous—and one more: the two dimensions of well-being must be substitutes (or more precisely, must not be complements) in the poverty measure. Roughly, this means that an increase of well-being in one dimension should have a greater effect on poverty the lower the level of well-being in the other dimension. In most cases, this restriction is sensible: if we are able to improve a child’s health, for example, it seems ethically right that this should reduce overall poverty the most when the child is very poor in the income dimension. But there are some plausible exceptions. For example, suppose that only healthy children can learn in school. Then, it might reduce poverty more to concentrate health improvements on children who are in school (better-off in the education dimension) because of the complementarity of health and education. 98 THE WORLD BANK ECONOMIC REVIEW, VOL. 20, NO. 1 Practically, it is not easy to plot two surfaces such as the one in figure 2 on the same graph and to see the differences between them, but the differences can be plotted directly. If this difference always has the same sign, one or the other of the samples has lower poverty for a large class Ã…1,1 of poverty measures. If the surfaces cross, the distributions can be compared at higher orders of dominance, just as in the univariate case. This can be done in one or both dimensions of well- being, and the restrictions on the applicable classes of poverty measures are similar to the univariate case. INTERSECTION, UNION, AND INTERMEDIATE POVERTY DEFINITIONS. In addition to the extra conditions on the class of poverty indices, multivariate dominance com- parisons require distinguishing among union, intersection, and intermediate poverty measures. This can be done with the help of figure 3 that shows the domain of dominance surfaces—the (x,y) plane. The function 1(x,y) defines an ‘‘intersection’’ poverty index: someone can be considered poor only when poor in both dimensions x and y and therefore when lying within the dashed rectangle of figure 3. The function 2(x,y) (the L-shaped dotted line) defines a union poverty index: someone can be considered poor when poor in either of the two F I G U R E 3 . Intersection, Union, and Intermediate Dominance Test Domains Duclos, Sahn, and Younger 99 dimensions and therefore when lying below or to the right of the dotted line. Finally, 3(x,y) provides an intermediate approach. Someone can be considered poor even with a y value greater than the poverty line in the y dimension if the x value is low enough to lie to the left of 3(x,y). For one sample to have less intersection poverty than another for any poverty line up to zy and zx, its dominance surface must be below the second sample’s everywhere within an area such as the one defined by 1(x,y). To have less union poverty, its surface must be below the second sample’s everywhere within an area such as the one defined by 2(x,y) and, similarly, for intermediate defini- tions and 3(x,y). The (x,y) function delimits the domain over which domi- nance tests are compared. As such, it is comparable to the maximal poverty line in a univariate comparison. MULTIVARIATE AND HUMAN DEVELOPMENT INDEX POVERTY COMPARISONS. Figure 3 is also helpful for understanding the difference between the general multivariate poverty comparisons used here and comparisons that rely on indices created with multiple indicators of well-being, the best known of which is the human development index (UNDP 1990). An individual-level index of the x and y mea- sures of well-being in figure 3 might be written as I ¼ ax x þ ay y where ax and ay are weights assigned to each variable. This index is now a univariate measure of well-being and could be used for poverty comparisons such as those in figure 1.10 The domain of this test for such an index would follow a ray starting at the origin and extending into the (x,y) plane at an angle that depends on the relative size of the weights ax and ay. Testing for dominance at these points only is clearly less general than testing over the entire area defined by a (x,y) function in figure 3. A comparison of poverty in rural Toliara and urban Mahajanga/Antsiranana in Madagascar shows why this generalization of human development index-type univariate indices is important. Table 1 summarizes the value of the t statistic for a test of the difference in the two areas’ poverty surfaces at a 10  10 grid of test points in the domain of figure 3—the (x,y) plane of that figure. The origin (the poorest people) is in the lower left corner, and the grid of test points is set at each decile of the marginal distributions.11 The significantly negative differences are highlighted in light gray and the significantly positive differences in dark gray. In 10. The human development index is actually cruder than this, as it first aggregates across individuals each dimension of well-being to generate a single scalar measure and then constructs a weighted average of those scalars to generate the index, which is also a scalar. Dutta, Pattanaik, and Xu (2003) discuss the severe restrictions needed on a social welfare function to justify such an index. 11. In theory, differences in the surfaces should be tested for everywhere, but this is computationally expensive. In practice, because the surfaces are smoothly increasing functions, it is usually sufficient to test at a grid of points, as is done here. 100 THE WORLD BANK ECONOMIC REVIEW, VOL. 20, NO. 1 T A B L E 1 . Ã…1,1 Dominance Tests for Rural and Urban Areas in Toliara, Madagascar, 1993 (Differences Between Rural and Urban Dominance Surfaces) 16.51 –8.841 –16.320 –16.580 –11.430 –8.068 –6.658 –4.174 –2.208 0.022 –0.239 13.19 –9.286 –16.780 –16.090 –11.080 –7.815 –6.221 –3.662 –0.933 2.005 2.118 12.84 –9.845 –15.690 –15.930 –10.720 –7.053 –5.253 –2.017 1.018 3.969 4.288 12.60 –3.307 –11.960 –9.174 –3.734 –0.638 1.677 5.642 8.312 11.250 11.090 12.44 1.646 –10.230 –7.667 –2.467 0.711 3.174 7.454 10.100 13.360 13.260 ln(y) 12.29 1.263 –6.159 –3.925 1.479 5.464 7.136 10.410 12.260 16.550 15.620 12.16 0.628 –3.287 –2.195 2.421 5.733 7.625 12.410 14.220 18.720 17.440 12.00 6.766 4.360 6.195 10.920 14.140 15.600 19.430 21.820 26.530 27.180 11.82 7.153 4.561 4.882 8.766 12.440 13.510 15.620 17.350 22.040 22.570 11.48 5.048 1.268 1.683 7.348 10.780 11.660 13.610 14.920 16.750 17.340 0.000 –4.01 –3.33 –2.84 –2.39 –1.98 –1.63 –1.21 –0.71 0.12 4.85 HAZ Note: The significantly negative differences are highlighted in light gray and the significantly positive differences in dark gray. Weights ax and ay are chosen, so that a human development index- type index of these two dimensions of well-being traces out the diagonal, here highlighted in bold. Source: Authors’ analysis based on data from the Madagascar 1993 National Household Survey (Enque ˆ te Permanente aupre ´nages). ` s des Me choosing the weights ax and ay so that a human development index-type index of these two dimensions of well-being traces out the diagonal of table 1, it can be concluded that poverty is higher in rural Toliara for a wide range of poverty lines—up to the 70th percentile—and all poverty measures in the Ã…1 class. However, another choice of ax and ay that gives more weight to household expenditures would yield test points on a steeper ray from the origin and thus imply a significant crossing of the index’s poverty incidence curves, yielding no dominance result. Testing over the entire two-dimensional domain rather than a single ray within that domain avoids this problem. MULTIVARIATE AND MULTIPLE UNIVARIATE POVERTY COMPARISONS. Suppose that a univariate comparison of expenditures per capita in two samples, as in figure 1, and children’s heights in two samples finds that for both variables, one sample shows lower poverty for all poverty lines and a large class of poverty measures. Is that not sufficient to conclude that poverty differs in the two samples? Unfortunately, no. The complication comes from the ‘‘hump’’ in the middle of the dominance surface shown in figure 2. How sharply the hump rises depends on the correla- tion between the two measures of well-being. If they are highly correlated, the surface rises rapidly in the center and vice versa. Thus, it is possible for one surface to be lower than another at both extremes (the edges of the surface farthest from the origin) and yet higher in the middle if the correlation between the welfare variables is higher. (The far edges of each surface integrate out one variable, and so are the univariate cumulative density functions depicted in figure 1.) Thus, in this case one surface would have lower univariate Duclos, Sahn, and Younger 101 T A B L E 2 . Ã…1,1 DOMINANCE TESTS FOR RURAL CENTRAL AND URBAN EASTERN REGIONS, UGANDA, 1999 11.660 2.637 12.510 8.720 7.938 9.993 7.941 11.170 4.484 1.109 0.000 9.276 3.458 13.930 9.712 12.030 15.540 15.410 20.020 13.550 14.130 16.400 8.996 5.519 14.940 10.590 13.920 17.110 17.110 22.360 18.330 18.410 20.250 8.803 2.559 11.910 7.156 10.320 13.760 14.730 21.160 18.730 19.030 21.460 8.664 0.610 8.643 4.224 7.651 9.988 9.820 15.270 15.010 16.430 19.950 ln(y) 8.527 0.062 8.763 5.016 8.366 9.201 12.340 17.300 15.860 17.390 19.570 8.395 –2.842 5.754 –0.025 2.692 4.249 6.958 10.650 12.260 13.580 15.240 8.249 –1.582 5.582 –0.307 2.743 2.801 5.305 8.590 11.310 13.020 13.520 8.068 –4.756 1.731 –4.960 –1.046 0.140 2.003 4.765 6.872 9.221 8.636 7.824 4.698 8.001 8.184 9.695 7.846 10.090 12.120 12.850 13.900 12.290 0.000 –3.100 –2.450 –1.970 –1.580 –1.220 –0.880 –0.500 –0.010 0.690 5.820 HAZ Note: The significantly negative differences are highlighted in light gray and the significantly positive differences in dark gray. Source: Authors’ analysis based on data from the Uganda 1999 National Household Survey. cumulative density functions, and thus lower poverty, for both measures of well-being independently, but it would not have lower bivariate poverty. Intuitively, samples with higher correlation of deprivation in multiple dimen- sions have higher poverty than samples with lower correlation because lower well-being in one dimension contributes more to poverty if well-being is also low in the other dimension.12 Consider this example. Univariate poverty is unambiguously higher in the rural Central region of Uganda than in the urban Eastern region in both dimensions—the difference between the dominance surfaces at the extreme top and right edges of table 2 is always positive—yet bivariate poverty is not unambiguously higher because of the statistically significant reversal of the dominance surfaces in the interior. Similar comparisons up to third order in each dimension also find that the dominance surfaces cross for these two areas. It is also possible that two samples with different correlations between mea- sures of well-being have univariate comparisons that are inconclusive—they cross at the extreme edges of the dominance surfaces—but have bivariate sur- faces that are different for a large part of the interior of the dominance surface. (The sample with lower correlation would have a lower dominance surface.) This would establish different intersection multivariate poverty even though either one or both of the univariate comparisons are inconclusive. It could not, 12. ‘‘Correlation’’ is actually overly strict. For instance, a recent literature has emerged on copulas, namely, functions that link two univariate distributions in ways that are more general than simple linear correlations but less flexible than the nonparametric distributions here. If these copulas differ for two groups, even if their correlations between dimensions of well-being are the same, it is still the case that one-at-a-time univariate dominance results could be reversed with a multivariate comparison. 102 THE WORLD BANK ECONOMIC REVIEW, VOL. 20, NO. 1 T A B L E 3 . Ã…2,2 Dominance Tests for Rural Central and Urban Northern Regions, Uganda, 1999 11.660 –0.824 0.263 1.863 1.217 0.048 –1.722 –2.680 –3.454 –3.200 –0.497 9.276 –6.401 –5.347 –4.431 –4.999 –5.578 –6.354 –6.607 –6.573 –5.397 –0.773 8.996 –7.860 –6.909 –6.315 –6.911 –7.340 –7.700 –7.669 –7.393 –6.083 –1.396 8.803 –9.091 –8.169 –7.775 –8.286 –8.554 –8.556 –8.240 –7.784 –6.395 –1.564 8.664 –10.090 –9.240 –8.997 –9.437 –9.571 –9.347 –8.833 –8.222 –6.765 –1.849 ln(y) 8.527 –10.750 –10.000 –9.823 –10.120 –10.080 –9.603 –8.851 –8.014 –6.456 –1.365 8.395 –11.190 –10.360 –10.100 –10.310 –10.300 –9.793 –8.981 –8.069 –6.595 –1.725 8.249 –11.820 –11.280 –10.990 –11.140 –11.190 –10.810 –10.140 –9.274 –7.970 –3.535 8.068 –12.150 –11.680 –11.270 –11.130 –11.010 –10.610 –9.910 –8.959 –7.705 –3.469 7.824 –12.240 –11.870 –11.450 –11.040 –10.650 –10.210 –9.528 –8.628 –7.559 –4.168 0.000 –3.100 –2.450 –1.970 –1.580 –1.220 –0.880 –0.500 –0.010 0.690 5.820 HAZ Note: The significantly negative differences are highlighted in light gray and the significantly positive differences in dark gray. Source: Authors’ analysis based on data from the 1999 Uganda National Household Survey. however, establish union poverty dominance, since that requires difference in the surfaces at the extremes as well as in the middle. Consider the example for rural Central and urban Northern Uganda (table 3). There is no statistically significant univariate dominance in the height-for-age dimension of well-being, and only a limited range of poverty lines for which poverty differs in the expenditure dimension, but there is a sizable domain—up to the ninth decile in each dimension—over which pov- erty is lower in the rural Central region than in the urban Northern region for all intersection poverty indices in the Ã…2,2 class. Thus, for many intersection and intermediate poverty measures, it can be concluded that the rural Central region in Uganda is less poor than the urban Northern region, even though neither univariate comparison is conclusive. II. BIVARIATE SPATIAL POVERTY COMPARISONS IN AFRICA This section applies bivariate dominance tests to spatial poverty comparisons in Ghana, Madagascar, and Uganda. Poverty, measured by household expen- ditures per capita and children’s HAZ, is compared in urban and rural areas, nationally and by region.13 The tests produce a large amount of output in the 13. The regions used in Ghana are its standard ecological zones of Coast, Forest, and Savannah. In Uganda, the four political regions are used: Central, Eastern, Western, and Northern. In Madagascar, political regions are also used, but because of small sample sizes Fianarantsoa and Toamasina are combined into one region, as are Mahajanga and Antsiranana. This choice is based on similar agro- ecological characteristics. In all countries, rural and urban areas in these regions are considered. Duclos, Sahn, and Younger 103 form of tables, such as table 1.14 Only summaries of the dominance results are reported here.15 Table 4 gives descriptive statistics for HAZ and the log of household expendi- tures per capita, ln(y). As expected, poverty measured by expenditures per capita and also stunting16 is higher in rural areas than in urban areas in each country. The same is true within each region of each country, except for the Toliara region in Madagascar, where stunting is higher in urban areas than in rural areas. In fact, with a few exceptions in Madagascar, both expenditure and height poverty are lower in urban areas in any region of each country than in rural areas in any other region in the same country. In addition to the means and poverty rates, table 4 reports the correlation between the log of expenditures per capita and HAZ. Note that in Madagascar and Uganda, expenditures and heights are more highly correlated in urban areas than in rural areas, whereas both expenditures and heights tend to be higher in urban areas. As noted, this combination can cause bivariate poverty compar- isons to differ from univariate comparisons carried out separately in each dimension of well-being.17 The dominance results for tests across urban and rural areas in Ghana, Madagascar, and Uganda show that for each country as a whole, poverty is higher in rural areas than in urban areas for each univariate poverty comparison and for both intersection and union bivariate comparisons. These results are entirely consistent with virtually every known poverty comparison based on incomes or expenditures alone—poverty is lower in urban areas. In the regional comparisons, however, a significant number of exceptions to this widely held belief emerge, especially for the bivariate comparisons. Ghana has the fewest of these exceptions, with two of nine urban–rural comparisons being statistically insignificant for both intersection and union bivariate poverty comparisons.18 In Uganda, for 4 of 16 intersection and union comparisons, the null hypothesis of nondominance cannot be rejected and two of these—rural areas in Eastern and Western regions compared with urban areas in the North- ern region—actually have somewhat limited domains over which bivariate poverty is lower in the rural area for intersection poverty measures. In 14. The results are relegated to appendixes, which are available from the authors. 15. The relevant statistics and their asymptotic standard errors can be readily computed using the software DAD (version 4.4 and higher) that is freely available at www.mimap.ecn.ulaval.ca. The authors can also provide a GAUSS program that does the same. 16. Stunting is usually defined as an HAZ of less than –2. 17. It is difficult to find universal explanations for the empirical correlations between indicators. The reasons are clearly context specific. As an example, expenditures and heights may be more highly correlated in urban than in rural areas because in urban areas the use of food markets may be prevalent. Purchasing power would then be better correlated with nutrient intake. In rural areas, nutrient intake is plausibly less correlated with purchasing power and more correlated with the proximity of food producers. 18. In each country, rural areas in each region are compared with urban areas in each region. Since there are three regions in Ghana, this yields nine comparisons. For Uganda and Madagascar, with four regions, this yields 16 comparisons. 104 THE WORLD BANK ECONOMIC REVIEW, VOL. 20, NO. 1 T A B L E 4 . Descriptive Statistics for Poverty and Stunting for Ghana, Madagascar, and Uganda Mean Percent Region HAZ ln(y) Stunted Poor N Correlation ln(y), HAZ Ghana 1988 Coast –0.98 11.90 0.22 0.41 911 0.15 Rural –1.12 11.76 0.27 0.51 488 0.10 Urban –0.82 12.06 0.16 0.30 423 0.15 Forest –1.38 11.81 0.32 0.46 1,074 0.12 Rural –1.48 11.79 0.35 0.48 793 0.11 Urban –1.10 11.88 0.24 0.39 281 0.10 Savannah –1.30 11.66 0.32 0.55 683 0.11 Rural –1.37 11.63 0.33 0.56 591 0.13 Urban –0.86 11.85 0.23 0.48 92 –0.08 National –1.22 11.80 0.28 0.47 2,668 0.14 Rural –1.35 11.73 0.32 0.51 1,872 0.11 Urban –0.92 11.97 0.19 0.35 796 0.11 Madagascar 1993 Tana –2.24 12.32 0.57 0.73 928 0.26 Rural –2.33 12.26 0.60 0.78 534 0.25 Urban –1.80 12.65 0.40 0.48 394 0.20 Fian/Toa –2.15 12.26 0.53 0.77 975 0.03 Rural –2.19 12.22 0.54 0.80 705 0.00 Urban –1.74 12.56 0.48 0.56 270 0.17 Mahajanga/ –1.35 12.62 0.34 0.55 561 –0.02 Antsiranana Rural –1.32 12.61 0.34 0.56 346 –0.04 Urban –1.44 12.71 0.34 0.50 215 0.14 Toliara –1.91 12.06 0.48 0.78 457 –0.18 Rural –1.82 11.98 0.45 0.82 302 –0.19 Urban –2.36 12.46 0.60 0.57 155 0.02 National –1.97 12.33 0.50 0.71 2,921 0.07 Rural –2.01 12.27 0.51 0.75 1,887 0.05 Urban –1.79 12.61 0.44 0.52 1,034 0.17 Uganda 1999 Central –1.00 8.80 0.25 0.19 1,806 0.07 Rural –1.08 8.65 0.27 0.23 1,390 0.04 Urban –0.77 9.22 0.18 0.08 416 0.03 Eastern –1.22 8.48 0.28 0.38 2,349 0.09 Rural –1.25 8.45 0.28 0.39 2,010 0.06 Urban –0.75 8.99 0.21 0.14 339 0.21 Western –1.42 8.63 0.34 0.28 2,096 0.12 Rural –1.46 8.60 0.35 0.29 1,860 0.07 Urban –0.59 9.35 0.15 0.06 236 0.25 Northern –1.24 8.16 0.30 0.60 1,230 0.09 Rural –1.24 8.13 0.30 0.62 1,008 0.08 Urban –1.23 8.72 0.26 0.19 222 0.36 National –1.22 8.54 0.29 0.35 7,481 0.10 Rural –1.27 8.47 0.30 0.37 6,268 0.06 Urban –0.79 9.15 0.19 0.10 1,213 0.12 Source: Authors’ analysis based on data from the 1988 Ghana Living Standards Study survey, the Madagascar 1993 National Household Survey (Enque ` s des Me ˆ te Permanente aupre ´nages), and the Uganda 1999 National Household Survey. Duclos, Sahn, and Younger 105 Madagascar, for 7 of 16 intersection comparisons and 10 of 16 union compar- isons, the null hypothesis that bivariate poverty is the same in urban and rural areas cannot be rejected, though none of these reject the null in favor of rural areas. While it is true that in only a minority of cases are urban areas not found to have significantly lower poverty, the fact that there are any such cases is surprising, given the overwhelming number of studies that find lower univariate poverty in urban areas in all developing economies. One immediate concern with these results is that the interesting cases are the ones in which the null hypothesis of nondominance is not rejected, so the results may be driven by a lack of power in the statistical tests. This concern is reinforced by the relatively few observations available in some urban areas. Review of the appendix tables shows, however, that in most cases in which bivariate dominance is not found, the dominance surfaces actually cross signifi- cantly.19 That is, there are points in the test domain where the rural surface is significantly above the urban surface and vice versa. Thus, the lack of bivariate dominance is typically not due to a lack of power. To gain a better understanding of how bivariate and univariate dominance methods can differ, we classified the results into five types. Type 1 has dom- inance (usually first order) for both univariate comparisons and for intersection and union bivariate comparisons. This is the most common result, accounting for 25 of the 41 comparisons. This is also the least interesting type of result for the methods applied here. Why bother with the more complicated bivariate comparisons if, in the end, they produce the same results as simpler univariate dominance tests or even scalar comparisons? Type 2 occurs when neither the univariate nor the bivariate method finds dominance. This is equally uninteresting for the methods used here. There is only one such case for urban and rural Mahajanga/Antsiranana region in Madagascar. Type 3 is a case in which urban areas dominate rural areas for both univariate comparisons but not for the bivariate comparisons. There are six of these cases. There is also one case, rural Mahajanga/Antsiranana compared with urban Toliara, in which the rural area dominates on both univariate comparisons but not on the bivariate comparisons. For cases in which the bivariate comparisons are inconsistent with the univariate comparisons, a type 3 result is the most common. The bivariate comparisons are more demanding than univariate com- parisons, so it makes sense that they reject the null hypothesis of nondominance less often and this happens in five of the seven cases. In two cases, both involving urban areas in the Northern region of Uganda, the dominance result is actually reversed for intersection poverty measures over a limited domain. This is surpris- ing, but understandable considering the very high correlation (0.36) between expenditures and heights in urban Northern region compared with rural Wes- tern region (0.07) and Eastern region (0.06). 19. The results are relegated to appendixes, which are available from the authors. 106 THE WORLD BANK ECONOMIC REVIEW, VOL. 20, NO. 1 Type 4 occurs when the univariate results are contradictory in the sense that univariate dominance is found in one dimension but not in the other. There are six such occurrences, and in all but one the urban area dominates in one dimension, usually expenditures, although in one case, rural Central compared with urban Northern region in Uganda, the rural area dominates, albeit only for the Ã…3 class. Of these six cases, intersection dominance is found for four bivariate tests. That is, the bivariate tests are able to ‘‘resolve’’ the conflicting univariate results for at least some classes of poverty measures20 and areas of poverty lines. Type 5 is similar to type 4 except that the contradictory univariate results are statistically significant in each univariate comparison. There are only two of these cases, rural compared with urban Toliara and rural Coast compared with urban Forest in Ghana. Unlike the type 4 results, in neither case are any of the bivariate poverty comparisons statistically significant, so the bivariate compar- isons cannot resolve the univariate conflict. Overall, sufficient evidence has not been amassed to overturn the standard presumption that poverty is lower in urban than in rural areas, but enough of the results are at odds with this idea to introduce doubt. Furthermore, the reasons that this is not found for bivariate poverty comparisons vary. For the type 4 and 5 cases, no univariate dominance is found in one dimension or another and the bivariate results follow from that. But this is relatively rare, and in about half of the cases the bivariate tests for intersection poverty measures do find that poverty is lower in urban areas despite the contradictory univariate results. Most of the differences, though, come from the fact that the two measures of well-being are often more highly correlated in urban areas than in rural areas. As noted, this correlation causes the poverty incidence surface to rise more rapidly near the origin of the distribution, raising it above the rural surface in the center even though it is below it at the extremes, where the univariate poverty incidence curves lie. In most cases, this gives results in which an urban area dominates a rural area in each dimension individually, but not jointly, because multiple deprivation is more common in urban areas. There are two cases, however, in which the dominance is reversed, so that for some intersection poverty measures the rural area dominates the urban area. III. CONCLUSIONS This article used bivariate stochastic dominance techniques to compare poverty in urban and rural areas in three African countries, measuring poverty in terms of expenditure per capita and children’s standardized heights, a good measure of children’s health status. The comparisons are shown to be more general than either a comparison of a human development index-type index or one-at-a-time 20. As noted in the methods discussion, bivariate dominance for union poverty measures requires univariate dominance in each dimension, so it is impossible for this type of result. Duclos, Sahn, and Younger 107 comparisons of multiple measures of well-being. More important, the article finds that its more general methods are at odds with simpler univariate poverty comparisons in a nontrivial number of cases. Expenditure-based urban–rural poverty comparisons almost always find that rural areas are poorer than urban areas. The results are consistent with that finding whether univariate or bivariate comparisons are used. However, differ- ences emerge when urban areas in one region of a country are compared with rural areas in another region. In several cases, univariate poverty is lower in urban areas in both dimensions, but bivariate poverty is not. This happens because the correlation between expenditures per capita and children’s heights is higher in the urban areas, so that urban residents who are expenditure poor are also more likely to be health poor. This correlation yields a higher density of observations in the poorest part of the bivariate welfare domain for urban areas, even though there are fewer observations for urban residents at the lower end of the density for each individual measure of well-being. Taking such a correlation into account is important for welfare comparisons because the social cost of poverty in one dimension, say health, is higher if the person affected is also poor in the other dimension (expenditures in this case). It is interesting to note that the share of cases in which urban areas do not dominate rural is much higher in the bivariate comparisons than in the expendi- ture- or income-based comparisons in the literature, where poverty is almost always found to be lower in urban areas. With two exceptions in Madagascar, however, the urban area in the region where the capital city is located always has lower poverty than every rural area in both univariate and bivariate compar- isons. Thus, the doubts raised here apply only to other urban areas in these countries. There are other instances in which the bivariate comparisons are at odds with univariate comparisons. Perhaps the most interesting are cases in which univari- ate results are inconclusive because one or the other univariate comparison is inconclusive, yet the bivariate results find dominance for a large domain of intersection poverty indices. This arises in about 10 percent of the examples and occurs again when the correlation between expenditures per capita and children’s heights differs significantly across areas. These results are interesting because they show that bivariate comparisons may provide statistically signifi- cant results when univariate comparisons do not. The finding that bivariate results often differ from the standard perception of greater rural poverty typically occurs not because children are taller in rural areas, but rather because the correlation between expenditures and heights is lower there than in urban areas. This, however, is based on only three countries. Pursuing similar research in other countries will yield insight into whether these results are anomalous. Why this should be is also an interesting question for future research. But a clear implication of these results for researchers and policymakers interested in multiple dimensions of poverty is that, at a minimum, one should check the correlations 108 THE WORLD BANK ECONOMIC REVIEW, VOL. 20, NO. 1 between measures of well-being in the groups of interest. Large differences in these correlations may lead to unexpected multivariate dominance comparisons. APPENDIX The following is based on the companion paper, Duclos, Sahn, and Younger (2003). Making Poverty Comparisons with Multiple Indicators of Well-Being For expositional simplicity, the focus is on the case of two dimensions of individual well-being. Let x and y be two such indicators. Assuming differentia- bility, denote by @ðx; yÞ @ðx; yÞ ðA:1Þ ðx; yÞ : < ! < 2 @ x ! 0; @ y ! 0 a summary indicator of individual well-being, analogous to but not necessarily the same as a utility function. Note that the derivative conditions in equation (A.1) simply mean that different indicators can each contribute to overall well-being. Assume that an unknown poverty frontier separates the poor from the rich, defined implicitly by a locus of the form ðx; yÞ ¼ 0 and analogous to the usual downward sloping indifference curves on the (x,y) space. The set of the poor is then obtained as: ðA:2Þ ÃðÞ ¼ fðx; yÞjððx; yÞ 0g: Letting the joint distribution of x and y be denoted by Fðx; yÞ, assume for simplicity that the multidimensional poverty indices are additive across indivi- duals and define such indices by PðÞ: ð ðA:3Þ PðÞ ¼ ðx; y; ÞdFðx; yÞ Ã ð Þ where ðx; y; Þ is the contribution to poverty of an individual with well-being indicators x and y:  ðA:4Þ ðx; y; Þ ! 0 if (x,y) 0 ¼ 0 otherwise. Here, p is the weight that the poverty measure attaches to someone inside the poverty frontier. By the focus axiom, it has to be 0 for those outside the poverty frontier. A bidimensional stochastic dominance surface can then be defined as: ð zy ð zx ðA:5Þ P x ; y ðzx ; zy Þ ¼ ðzx À xÞ x ðzy À yÞ y dFðx; yÞ: 0 0 This function looks like a two-dimensional generalization of the Foster–Greer– Thorbecke index and can also be interpreted as such. The poverty comparisons Duclos, Sahn, and Younger 109 here make use of orders of dominance, sx in the x and sy in the y dimensions, which correspond respectively to sx ¼ x þ 1 and sy ¼ y þ 1. Assume that the general poverty index in equation (A.3) is left differentiable with respect to x and y over the set ÃðÞ, up to the relevant orders of dominance, sx for derivatives with respect to x and sy for derivatives with respect to y. Denote by x the first derivative of ðx; y; Þ with respect to x, by y the first derivative of ðx; y; Þ with respect to y, and by xy the derivative of ðx; y; Þ with respect to x and to y. The following class Å€ 1;1 ðÃ Þ of bidimensional poverty indices can be defined as: 8 9 ÃðÞ & ÃðÃ Þ > > > > < = 1 € ð Þ ¼ ; 1 à  ðx ; y; Þ ¼ 0 whenever  ðx ; yÞ ¼ 0 ðA:6Þ Ã… x : > > 0 and y 0 8x; y > > : ; xy ! 0 8x; y The first line on the right of equation (A.6) defines the largest poverty set to which the poor must belong: the poverty set covered by the PðÞ indices should lie within the maximal set Ãðà Þ. The second line assumes that the poverty indices are continuous along the poverty frontier. The third line says that indices that are members of Ã… € 1;1 are weakly decreasing in x and in y. The last line assumes that the marginal poverty benefit of an increase in either x or y decreases with the value of the other variable. Denote by Ã?F ¼ FA À FB the difference between a function F for A and for B. The class of indices defined in equation (A.6) then gives rise to the following theorem: Theorem 1 Ã?PðÞ > 0; 8PðÞ 2 Å€ 1;1 ðà Þ; ðA:7Þ if Ã?P0;0 ðx; yÞ > 0; 8ðx; yÞ 2 Ãðà Þ: Proof: Denote the points on the outer poverty frontier à ðx; yÞ ¼ 0 as zx ðyÞ for a point above y and zy ðxÞ for a point above x. The derivative conditions in ð1Þ ð1Þ equation (A.1) imply that zx ðyÞ 0 and zy ðxÞ 0, where the superscript 1 indicates the first-order derivative of the function with respect to its argument. Note that the inverse of zx ðyÞ is simply zy ðxÞ: x  zx ðzy ðxÞÞ. Next, equation (A.3) is integrated by parts with respect to x, over an interval of y ranging from 0 to zy . This gives: ð zy ð zy ð zx ðyÞ z ðyÞ Pðzx ðyÞ; zy Þ ¼ ½ðx; y;  à ÞFðxjyފj0x f ðyÞdy À x ðx; y; à ÞFðxjyÞf ðyÞdxdy: 0 0 0 ðA:8Þ To integrate by parts with respect to y the second term, define a general Ã? gðyÞ function KðyÞ ¼ 0 hðx; yÞ lðx; yÞdx and note that: 110 THE WORLD BANK ECONOMIC REVIEW, VOL. 20, NO. 1 dKðyÞ ¼ gð1Þ ðyÞhðgðyÞ; yÞ lðgðyÞ; yÞ dy ð gðyÞ @ hðx; yÞ ðA:9Þ þ lðx; yÞdx 0 @y ð gðyÞ @ lðx; yÞ þ hðx; yÞ dx: 0 @y Reordering equation (A.9) and integrating it from 0 to c yields: ð c ð gðyÞ ðc @ lðx; yÞ À hðx; yÞ dxdy ¼ À KðcÞ þ Kð0Þ þ gð1Þ ðyÞhðgðyÞ; yÞ lðgðyÞ; yÞdy 0 0 @y 0 ð c ð gðyÞ @ hðx; yÞ þ lðx; yÞdxdy: 0 0 @y ðA:10Þ Now replace in equation (A.10) c by zy , gðyÞ by zx ð Ã?y Þ, hðx; yÞ by x ðx; y; à Þ, gðyÞ lðx; yÞ by Fðx; yÞ, and KðyÞ by its definition KðyÞ ¼ 0 hðx; yÞ lðx; yÞdx. This gives: ð zx ðzy Þ Pðzx ðyÞ; zy Þ ¼ À x ðx; zy ; Ã Þ P0;0 ðx; zy Þ dx ð0 zy 1Þ x 0;0 ðA:11Þ þ zð à x ðyÞ  ðzx ðyÞ; y;  Þ P ðzx ðyÞ; yÞ dy 0 ð zy ð zx ðy Þ þ xy ðx; y; Ã Þ P0;0 ðx; yÞ dx dy: 0 0 ð1Þ For the sufficiency of equation (A.7), recall that zx ðyÞ 0, x 0, and xy  ! 0, with strict inequalities for either of these inequalities over at least some inner ranges of x and y. Hence, if Ã?P0;0 ðx; yÞ > 0 for all y 2 ½0; zy Š and for all x 2 ½0; zx ðyފ (that is, for all ðx; yÞ 2 Ãðà Þ), it must be that Ã?PðÃ Þ > 0 for all of the indices that use the poverty set ÃðÃ Þ and that obey the first two lines of conditions in equation (A.6). But note that for other poverty sets ÃðÞ & Ãðà Þ, the relevant sufficient conditions are only a subset of those for Ãðà Þ. The sufficiency part of theorem 1 thus follows. For the necessity part, assume that Ã?P0;0 ðx; yÞ 0 for an area defined over x 2 ½cÀ þ À þ þ x ; cx Š and y 2 ½cy ; cy Š, with cx zy and cþ x zx ðyÞ. Then any of the poverty indices in Å€ 1;1 ðÃ Þ for which xy < 0 over that area, xy ¼ 0 outside that area, and for which x ðx; zy ; Ã Þ ¼ x ðzx ðyÞ; y; Ã Þ ¼ 0, will indicate that Ã?P < 0. Equation (A.7) is thus also a necessary condition for the ordering specified in theorem 1. Duclos, Sahn, and Younger 111 Note that similar proofs are possible for dominance comparisons at higher orders (Duclos, Sahn, and Younger 2003). Estimation and Inference Suppose a random sample of N independently and identically distributed observations drawn from the joint distribution of x and y. These observa- tions of xL and yL , drawn from a population L, can be written as ðxL L i ; yi Þ; i ¼ 1; . . . ; N : A natural estimator of the bidimensional dominance surfaces P x ; y ðzx ; zy Þ is then: ð zy ð zx ^ x ; y ðzx ; zy Þ ¼ P ðzy À yÞ y ðzx À xÞ x d F ^ L ðx; yÞ L 0 0 1X N y L x ðA:12Þ ¼ ðzy À yL L i Þ ðzx À xi Þ I ðyi zy ÞIðxL i zx Þ N i¼1 1X N y L x ¼ ðzy À yL i Þþ ðzx À xi Þþ N i¼1 where F^ denotes the empirical joint distribution function, IðÃ?Þ is an indicator function equal to 1 when its argument is true and 0 otherwise, and xþ ¼ maxð0; xÞ. This give rise to theorem 2: Theorem 2 Let the joint population moments of order 2 of ðzy À yA Þþy ðzx À xA Þ x and  ; þ ^ x y ; ðzy À yB Þþ ðzx À xB Þþx be finite. Then N1=2 P y A x y ðzx ; zy Þ À PA ðzx ; zy Þ and  ;  ^ x y ; N 1=2 P B ðzx ; zy Þ À PBx y ðzx ; zy Þ are asymptotically normal with mean 0, with asymptotic covariance structure given by ðL; M ¼ A; BÞ:  ;   lim N cov P ^ x ; y ðzx ; zy Þ ¼E ðzy À yL Þ ^ x y ðzx ; zy Þ; P y L x þ ðzx À x Þþ ðzy À y Þþ M y L M N !1  x À 1 ; x ; y ðzx À xM Þþ À PLx y ðzx ; zy Þ PM ðzx ; zy Þ: ðA:13Þ Proof: For each distribution, the existence of the appropriate population moments of order 1 permits application of the law of large numbers to equation (A.12), thus showing that P ^ x ; y ðzx ; zy Þ is a consistent estimator of K ; PKx y ðzx ; zy Þ. Given also the existence of the population moments of order 2, the central limit theorem shows that the estimator in equation (A.12) is root-N consistent and asymptotically normal with asymptotic covariance matrix given by equation (A.13). When the samples are dependent, the co- variance between the estimator for A and for B is also provided by equation (A.13). 112 THE WORLD BANK ECONOMIC REVIEW, VOL. 20, NO. 1 REFERENCES Atkinson, A. B. 1987. ‘‘On the Measurement of Poverty.’’ Econometrica 55(4):749–64. ———. 2003. ‘‘Multidimensional Deprivation: Contrasting Social Welfare and Counting Approaches.’’ Journal of Economic Inequality 1(1):51–65. Atkinson, A. B. , and F. Bourguignon. 1982. ‘‘The Comparison of Multi-Dimensional Distributions of Economic Status.’’ In A. B. Atkinson, ed., Social Justice and Public Policy. London: Harvester Wheatsheaf. ———. 1987. ‘‘Income Distribution and Differences in Needs.’’ In G. R. Feiwel, ed., Arrow and the Foundations of the Theory of Economic Policy. New York: New York Press. Beaton, G. , A. Kelly, J. Kevany, R. Martorell, and J. Mason. 1990. ‘‘Appropriate Uses of Anthropometric Indices in Children: A Report Based on an ACC/SCN Workshop.’’ ACC/SCN State-of-the-Art Series, Nutrition Policy Discussion Paper 7. United Nations, Administrative Committee on Coordination/ Subcommittee on Nutrition, New York. Bourguignon, F. , and S. R. Chakravarty. 2002. Multi-dimensional Poverty Orderings. Paris: DELTA. ———. 2003. ‘‘The Measurement of Multidimensional Poverty.’’ Journal of Economic Inequality 1(1):25–49. Cole, T. J. , and J. M. Parkin. 1977. ‘‘Infection and Its Effect on Growth of Young Children: A Comparison of the Gambia and Uganda.’’ Transactions of the Royal Society of Tropical Medicine and Hygiene 71:196–8. Crawford, I. A. 1999. ‘‘Nonparametric Tests of Stochastic Dominance in Bivariate Distributions, with an Application to UK.’’ Discussion Papers in Economics 99/07. London, UK: University College. Davidson, R. , and J.-Y. Duclos. 2000. ‘‘Statistical Inference for Stochastic Dominance and for the Measurement of Poverty and Inequality.’’ Econometrica 68(6):1435–65. Duclos, J.-Y. , D. Sahn, and S. D. Younger. 2003. ‘‘Robust Multidimensional Poverty Comparisons.’’ Working Paper 98. Ithaca, NY: Food and Nutrition Policy Program, Cornell University. Duclos, Jean-Yves , and Paul Makdissi. 2005. ‘‘Sequential Stochastic Dominance and the Robustness of Poverty Orderings.’’ Review of Income and Wealth 51(1):63–88. Dutta, I. , P. K. Pattanaik, and Y. Xu. 2003. ‘‘On Measuring Deprivation and the Standard of Living in a Multidimensional Framework on the Basis of Aggregate Data.’’ Economica 70(278):197–221. Foster, J. E. 1984. ‘‘On Economic Poverty: A Survey of Aggregate Measures.’’ In R. L. Basmann, and G. F. Rhodes, eds., Advances in Econometrics, Vol. 3. Greenwich, CT: JAI Press. Foster, J. E. , J. Greer, and E. Thorbecke. 1984. ‘‘A Class of Decomposable Poverty Measures.’’ Econo- metrica 52(3):761–76. Foster, J. E. , and A. F. Shorrocks. 1988a. ‘‘Inequality and Poverty Orderings.’’ European Economic Review 32(2/3):654–62. ———. 1988b. ‘‘Poverty Orderings.’’ Econometrica 56(1):173–7. ———. 1988c. ‘‘Poverty Orderings and Welfare Dominance.’’ Social Choice Welfare 5(2/3):179–98. Haddad, L. , H. Alderman, S. Appleton, L. Song, and Y. Yohannes. 2003. ‘‘Reducing Child Malnutrition: How Far Does Income Growth Take Us?’’ World Bank Economic Review 17(1):107–31. Mosley, W. H. , and L. C. Chen. 1984. ‘‘An Analytical Framework for the Study of Child Survival in Developing Countries.’’ Population and Development Review 10(Suppl.):25–45. Pradhan, M. , D. E. Sahn, and S. D. Younger. 2003. ‘‘Decomposing World Health Inequality.’’ Journal of Health Economics 22(2):271–93. Tsui, K. 1999. ‘‘Multidimensional Inequality and Multidimensional Generalized Entropy Measures: An Axiomatic Approach.’’ Social Choice and Welfare 16:145–48. ———. 2002. ‘‘Multidimensional Poverty Indices.’’ Social Choice and Welfare 19(1):69–93. UNDP(United Nations Development Programme). 1990. Human Development Report. New York: Oxford University Press. Duclos, Sahn, and Younger 113 WHO (World Health Organization). 1983. ‘‘Measuring Change in Nutritional Status: Guidelines for Asses- sing the Nutritional Impact of Supplementary Feeding Programmes for Vulnerable Groups.‘‘ Geneva. ———. 1995. ‘‘An Evaluation of Infant Growth: The Use and Interpretation of Anthropometry in Infants.’’ Bulletin of the World Health Organization 73:165–74. Zheng, B. 2000. ‘‘Poverty Orderings.’’ Journal of Economic Surveys 14(4):427–66.