Bayesian inference on the measurement of welfare, inequality, and deprivation
thesisposted on 02.03.2017, 23:13 by Gunawan, David
As indicators of social welfare, the degree of inequality and the incidence of deprivation are a concern to policy makers and researchers. Other things being constant, higher degrees of inequality and deprivation correspond to lower levels of social welfare. The availability of inequality and deprivation measures is essential if governments are to assess, analyse, and monitor the degrees of inequality and deprivation accurately over time and make appropriate decisions in order to reduce inequality and deprivation, and provide sustainable development to the society. This thesis examines the degree of inequality and the incidence of deprivation in Australia over the period 2001 - 2010 using income, mental health status, and education level as indicators of well-being. The study is a multidimensional one using Bayesian inference. A comparison of levels of well-being over time and for different population subgroups is done using Bayesian methodology for assessing unidimensional and multidimensional stochastic dominance. These well-being comparisons depend on the joint and marginal distributions of well-being and how they are estimated. Thus, the general objective is accompanied by several secondary objectives. Given that Bayesian inference is used, and that the available data are assigned survey sampling weights, our secondary aim is to propose and assess Bayesian methodology for including sampling weights in density estimation. We propose a new Bayesian weighted sampling algorithm to take into account the sampling weights in joint and marginal density estimation. It is applied to estimate a mixture of gamma densities to model income distributions, a mixture of beta densities to model mental health distributions, and an ordinal categorical model to model education distributions. The joint distribution of income and mental health and education level is modelled using a parametric class of copula. In Chapter 4, two new Bayesian methodologies that allow information from sampling weights to be incorporated into the estimation and inference of unknown parameters are proposed. They are a Bayesian pseudo posterior estimator (BPPE) and a Bayesian weighted estimator (BWE). The Monte Carlo simulation results show that the BPPE and BWE produce estimates whose means are close to the true values. However, the posterior variance associated with the BPPE is too small in the sense that it does not reflect the variance of the estimates in repeated samples. The posterior variance of BWE is comparable to the sandwich variance estimator used by pseudo maximum likelihood estimation (PMLE). Also, the BPPE approach does not extend easily to more general econometric models, such as mixture and latent variable models. A major advantage of BWE is its ability to be applied in a more general set of models that can be estimated by Markov chain Monte Carlo (MCMC). Chapter 5 contributes to the empirical stochastic dominance literature by developing a Bayesian approach, rather than continuing the current trend in the literature of further developing the sampling-theory approach. In this chapter, the proposed methods are applied to income distribution and welfare and inequality analysis in Australia. The results from the Bayesian analysis are reported as posterior probabilities of each of the possible outcomes, the probability that X dominates Y, the probability that Y dominates X, and the probability that neither dominates. This chapter extends the stochastic dominance analysis of Chotikapanich and Griffiths (2006) in several ways. Firstly, a mixture of gamma distributions is chosen as a more flexible model for the income distributions. A gamma mixture is specifically chosen to analyse data defined over the positive real line, which is appropriate for income distributions. Secondly, estimates of lower and upper bounds for the posterior probability of dominance are provided. Thirdly, using the Bayesian weighted estimator (BWE), the sampling weights provided are included in the estimation of gamma mixture models to ensure that conclusions drawn from the sample analysis apply to the general population as well. Lastly, the methodology developed is applied to analyse the individual disposable income distribution in Australia for the period 2001-2010. Chapter 6 considers estimation of the joint distribution of income, mental health score and education level using a Gaussian copula model. In this chapter, we show that all parameters of the marginals and copula functions can be estimated jointly by using MCMC to draw observations from the joint posterior density of all the unknown parameters. Flexible mixtures of beta and gamma densities are used as models for mental health and income distributions, respectively. A gamma mixture model is specifically chosen to analyse data defined over the positive real line and a beta mixture model is chosen to analyse data defined over the range (0,1), which is an appropriate for the index of mental health. An ordinal categorical model is used to model the education distribution. In Chapter 7, we introduce Bayesian test procedures for multidimensional stochastic dominance. In particular, we present a Bayesian framework for assessing multidimensional stochastic dominance involving two or more dominance conditions over the entire, or a subset of, the multidimensional distribution of well-being. We propose dominance conditions that belong to the classes of utility functions U1, U2, and U3. The dominance conditions that belong to the class U3 are related to the Muller and Trannoy (2011) conditions that include preference for more equal marginal distributions of income, mental health, and education, and substitutability among attributes such that priority is given to income-poor people by providing them with better access to mental health care and education to improve their level of these factors. Using income, level of mental health, and level of education as measures of well-being, we use the Bayesian test procedures to compare the social welfare and poverty for individuals 15 years of age or older in Australia over the period 2001 - 2010. Comparisons are made using the overall distribution and different population subgroups.