A Solution to Aggregation and an Application to Multidimensional "Well-being" Frontiers

We propose a new technique for identification and estimation of aggregation functions in multidimensional evaluations and multiple indicator settings. These functions may represent "latent" objects. They occur in many different contexts, for instance in propensity scores, multivariate measures of well-being and the related analysis of inequality and poverty, and in equivalence scales. Technical advances allow nonparametric inference on the joint distribution of continuous and discrete indicators of well-being, such as income and health, conditional on joint values of other continuous and discrete attributes, such as education and geographical groupings. In a multiattribute setting, "quantiles" are "frontiers" that define equivalent sets of covariate values. We identify these frontiers nonparametrically at first. Then we suggest "parametrically equivalent" characterizations of these frontiers that reveal likely weights for, and substitutions between, different attributes for different groups, and at different quantiles. These estimated parametric functionals are "ideal" aggregators in a certain sense, which we make clear. They correspond directly to measures of aggregate well-being popularized in the earliest multidimensional inequality measures in Maasoumi (1986). This new approach resolves a classic problem of assigning weights to multiple indicators such as dimensions of well-being, as well as empirically incorporating the key component in multidimensional analysis, the relationship between the indicators. It introduces a new way for robust estimation of "quantile frontiers," allowing "complete" assessments, such as multidimensional poverty measurements. In our substantive application, we discover extensive heterogeneity in individual evaluation functions. This leads us to perform robust, weak uniform rankings as afforded by tests for multivariate stochastic dominance. A demonstration is provided based on the Indonesian data analyzed for multidimensional poverty in Maasoumi & Lugo (2008).