Mixture models and poverty measurement

The long-established practice of poverty analysis is to identify the poor by specifying a poverty threshold in terms of a variable that reflects individual wellbeing (typically income or consumption) so that individuals below that boundary are classified as poor. More generally, a society is partitioned into groups or classes according to an observed outcome, and groups are defined by fixing lower and upper thresholds for that outcome. The identification of these boundaries has long been a matter of considerable debate. Determining the poor has probably been the most contentious dispute (Sen, 1983; Foster, 1998), but things are not different when the focus of the analysis is on the middle or rich class (Atkinson and Brandolini, 2013; Saez and Veall, 2005). A less conventional perspective speculates that the observed outcome distribution, say income, is the aggregate result of a small number of homogeneous classes, and attention is paid to the extent to which group members are similar not only based on income but also based on other social circumstances or exogenous factors people face during their lives. These circumstances may in fact limit or bound individual income opportunities in different ways. Individuals belonging to a certain class share comparable income opportunities that can be described by a distribution function. Within each class, income is concentrated around a pole (the expected mean income) with some degree of dispersion. A direct consequence of the income dispersion within each class is the presence of some degree of overlap between the distributions of the income classes. Therefore, agents with certain observed outcome levels cannot be assigned exclusively to one class; rather, only the probability of category membership can be determined for each agent. Following this line of reasoning, the entire population is viewed as a collection of a small number of classes k=1,…, K determined by inherent circumstances, such that an individual belonging to class k faces income opportunities described by a distribution fk. The circumstances of an individual’s class are both observable (such as educational attainment, parental backgrounds, ethnicity, location) and unobservable (such as childhood circumstances, genetic endowment, the freedoms they enjoy, the capabilities they possess, the security they experience in their actions). The overall income distribution will be a mixture of income classes, with mixing weights equal to the proportions of society that are members of the respective classes. If these sub-distributions and their respective weights can be estimated, much can be said about the behaviour and state of wellbeing of classes without resorting to debates about defining boundaries. Poverty measurement then becomes a matter of identifying and measuring aspects of these class distributions driven by different income opportunities. Since these classes cannot be defined a priori, finite mixture models (FMM), also known as latent class models (LCM), is the natural technique for recovering hidden groups from observed data – ‘the art of unscrambling eggs’ according to the picturesque definition of Oberski (2016) – and for modelling unobserved population heterogeneity. In what follows, we briefly present a mixture model that simultaneously estimates class membership probabilities together with the influence of possible correlates of the membership. Some prospect in conducting poverty analysis and, more generally, interclass comparisons using the estimated model are discussed, followed by reports on an illustrative application to Kyrgyzstan household incomes. The final section offers some conclusions.