Testing for stochastic dominance among additive, multivariate welfare functions with discrete variables1

A‡ourishing literature on robustness in multidimensional welfare and poverty comparisons has aroused the interest on multidimensional stochastic dominance. By generalizing the dominance conditions of Atkinson and Bourguignon (1982) this paper o¤ers complete conditions, alternative to those proposed by Duclos et al. (2006a,b). We also show how to test these conditions for discrete variables extending the non-parametric test by Anderson (1996) to multiple dimensions. An empirical application illustrates these tests. Key Words: Multidimensional welfare comparisons, stochastic dominance, nonparametric tests JEL Classi…cation: C14, D30 This version: June 2009. 1. Introduction A recent literature on multidimensional wellbeing emphasizes comparing individuals or groups with welfare functions which aggregate separate dimensions of well-being into a single indicator. Such dimensions can include consumption items, health attributes and, in general, any other aspect of wellbeing that "people have reason to value" (Sen, 2001). Since several 1I am very grateful to Ian Crawford for his comments and support. I would also like to thank Jean-Yves Duclos, Sabina Alkire and participants at the OPHI workshop on Robustness in Multdimensional Welfare Analysis for their valuable comments. I am very grateful to Lorenzo Oimas for helping me with securing the dataset. such evaluative functions can be proposed, the choices of functional forms, weights attached to dimensions and other details may a¤ect the consistency of the rankings of societies or individuals. In the twin literature of multidimensional poverty an additional source of ranking inconsistency is the choice of multidimensional poverty line and criterion for the identi…cation of the poor (Bourguignon and Chakravarty, 2002; Duclos et al., 2006). Two approaches have been proposed to provide robust welfare comparisons in the presence of such range of choices: one is to conduct sensitivity analysis of a given indicator to gauge the impact of a change in an aspect of the indicator (e.g. the weights) on its ranking performance. Another approach is to provide multidimensional stochastic dominance conditions which state the conditions under which a broad class of welfare functions consistently rank multivariate distributions of individuals or societies. Such exercise has two components: the dominance conditions themselves and the corresponding statistical tests to probe them empirically. To date there is a broad literature both on stochastic dominance conditions for comparisons of welfare indicators2, poverty indicators3 , Lorenz curves4 , even economic mobility (Fields et al., 2002). Similarly statistical tests abound (e.g. Anderson, 1996; Xu and Osberg, 1998; Davidson and Duclos, 2000; Barrett and Donald, 2003; Thuysbaert and Zitikis, 2005; Linton et al., 2005; Bennett, 2008). They all work with univariate distributions. For multidimensional comparisons the …rst package of dominance conditions and corresponding statistical test was provided by Duclos et al. (2006,2007), in turn based on the theory laid out by Davidson and Duclos (2000). Notwithstanding the soundness of their technique, the conditions of Duclos et al. (2006,2007) do not cover all the potential signs of the crossderivatives of the welfare functions. By contrast the older conditions laid out by Atkinson and Bourguignon (1982) cover di¤erent signs of the cross-derivatives. In this paper we show with a bivariate illustration that the conditions by Atkinson and Bourguignon (1982) include all the conditions of Duclos et al. (2006) and are more complete in that they also consider all possibilities in terms of signs of cross-derivatives.5 Thereby this paper provides justi…cation to …nding alternative statistical techniques to test the conditions of Atkinson and Bourguignon which are generalized, in this paper, to n dimensions and up to third-order 2For instance, Atkinson and Bourguignon (1982), Muller and Trannoy (2003). 3For instance, Foster and Shorrocks (1988), Davidson and Duclos (2000), Duclos and Makdissi (2005), Duclos et al. (2006). 4For instance, Dardanoni and Forcina (1999) 5On the other hand, the robustness work of Davidson and Duclos (2000) as well as Duclos et al. (2006) is not restricted to dominance conditions and tests. They also show how to estimate consistently critical frontiers, i.e. the points at which cumulative distributions or dominance surfaces cross.