Bayesian Assessment of Lorenz andStochastic Dominance in Income Distributions

Hypothesis tests for dominance in income distributions has received considerableattention in recent literature. See, for example, Barrett and Donald (2003), Davidsonand Duclos (2000) and references therein. Such tests are useful for assessing progresstowards eliminating poverty and for evaluating the effectiveness of various policyinitiatives directed towards welfare improvement. To date the focus in the literaturehas been on sampling theory tests. Such tests can be set up in various ways, withdominance as the null or alternative hypothesis, and with dominance in eitherdirection (X dominates Y or Y dominates X). The result of a test is expressed asrejection of, or failure to reject, a null hypothesis. In this paper we develop and applyBayesian methods of inference to problems of Lorenz and stochastic dominance. Theresult from a comparison of two income distributions is reported in terms of theposterior probabilities for each of the three possible outcomes: (a) X dominates Y, (b)Y dominates X, and (c) neither X nor Y is dominant. Reporting results about uncertainoutcomes in terms of probabilities has the advantage of being more informative than asimple reject / do-not-reject outcome. Whether a probability is sufficiently high or lowfor a policy maker to take a particular action is then a decision for that policy maker.The methodology is applied to data for Canada from the Family Expenditure Surveyfor the years 1978 and 1986. We assess the likelihood of dominance from one timeperiod to the next. Two alternative assumptions are made about the incomedistributions –Dagum and Singh-Maddala – and in each case the posterior probabilityof dominance is given by the proportion of times a relevant parameter inequality issatisfied by the posterior observations generated by Markov chain Monte Carlo