On Dagum’s Decomposition of the Gini Coefficient

To measure the contributions to inequality from population subgroups, the Gini coefficient is often decomposed into inequality within groups, inequality between groups and a residual term arising from the overlapping of income distributions from different groups. In this paper we show that two decompositions presented separately in the literature, a traditional decomposition and a decomposition suggested by Dagum (1997), are identical, a fact not previously acknowledged in the literaturepopulation subgroups; between inequality; within inequality

A Gibbs’ Sampler for the Parameters of a Truncated Multivariate Normal Distribution

The inverse distribution function method for drawing randomly from normal and truncated normal distributions is used to set up a Gibbs’ sampler for the posterior density function of the parameters of a truncated multivariate normal distribution. The sampler is applied to shire level rainfall for five shires in Western Australia.

MULTICOLLINEARITY IN REGRESSION WITH QUADRATIC REGRESSORS

Research Methods/ Statistical Methods,

Survival on the Titantic: Illustrating Wald and LM Tests for Proportions and Logits

Students are very interested in lecture examples and class exercises involving data connected to the maiden voyage and the sinking of the liner Titanic. Information on the passengers and their fate can be used to explore relationships between various tests for differences in survival rates between different groups of passengers. Among the concepts examined are tests for differences of proportions using a normal distribution, a chi-square test for independence, a test for the equality of two logits and a test for the significance of the coefficient of a binary variable in logit model. The relationship between Wald and LM test statistics is also examined. Two related examples are given, one to be used for step by step instructional purposes and one to be given as an exercise to students.Contingency table, Difference in proportions, Logit model, Statistical tests

The determinants of research and development and intellectual property usage among Australian Companies, 1989 to 2002

This paper traces the innovation pathways of new creations from R & D activity through to intellectual property (IP) applications using enterprise panel data from 1989 to 2002. Our estimation method explicitly addresses the selection issues associated with missing R&D data which is a common problem among this type of data set. We find that R&D activity is a highly path dependent process that relies heavily on firm specific effects. These firm specific effects were subsequently found to be correlated with managerial style – more aggressive and intuitive managers have higher R&D ceteris paribus – and extensive use of incentive schemes for employees within the firm. In addition, we find that R&D is higher when the previous year’s enterprise debt ratio is lower, the speed of technological change is faster, the firm’s ability to absorb knowledge spillovers is greater and the product market is less contestable. Furthermore, these firms appear to be using the various methods of appropriation, IP and non-IP, as complementary packages to capture the quasi-rents from previous R&D expenditure rather than as substitutes.

Carnarvon Gorge: a comment on the sensitivity of consumer surplus estimation

Beal’s (1995) method of estimating the value of Carnarvon Gorge for recreational use is re‐examined. When an inconsistency in her estimation procedure is corrected, the estimated value of Carnarvon Gorge for camping is found to be six times higher. The sensitivity of the estimate to the choice of functional form is examined, and standard errors and interval estimates for consumer surplus are provided. Comments are made about functional form choice and prediction in log‐log models.Resource /Energy Economics and Policy,

Estimating Income Distributions Using a Mixture of Gamma Densities

The estimation of income distributions is important for assessing income inequality and poverty and for making comparisons of inequality and poverty over time, countries and regions, as well as before and after changes in taxation and transfer policies. Distributions have been estimated both parametrically and nonparametrically. Parametric estimation is convenient because it facilitates subsequent inferences about inequality and poverty measures and lends itself to further analysis such as the combining of regional distributions into a national distribution. Nonparametric estimation makes inferences more difficult, but it does not place what are sometimes unreasonable restrictions on the nature of the distribution. By estimating a mixture of gamma distributions, in this paper we attempt to benefit from the advantages of parametric estimation without suffering the disadvantage of inflexibility. Using a sample of Canadian income data, we use Bayesian inference to estimate gamma mixtures with two and three components. We describe how to obtain a predictive density and distribution function for income and illustrate the flexibility of the mixture. Posterior densities for Lorenz curve ordinates and the Gini coefficient are obtained

Averaging Lorenz Curves

A large number of functional forms have been suggested in the literature for estimating Lorenz curves that describe the relationship between income and population shares. One way of choosing a particular functional form is to pick the one that best fits the data in some sense. Another approach, and the one followed here, is to use Bayesian model averaging to average the alternative functional forms. In this averaging process, the different Lorenz curves are weighted by their posterior probabilities of being correct. Unlike a strategy of picking the best-fitting function, Bayesian model averaging gives posterior standard deviations that reflect the functional form uncertainty. Building on our earlier work (Chotikapanich and Griffiths 2002), we construct likelihood functions using the Dirichlet distribution and estimate a number of Lorenz functions for Australian income units. Prior information is formulated in terms of the Gini coefficient and the income shares of the poorest 10% and poorest 90% of the population. Posterior density functions for these quantities are derived for each Lorenz function and are averaged over all the Lorenz functions.Gini coefficient; Bayesian inference; Dirichlet distribution.

Estimating Lorenz Curves Using a Dirichlet Distribution

The Lorenz curve relates the cumulative proportion of income to the cumulative proportion of population. When a particular functional form of the Lorenz curve is specified it is typically estimated by linear or nonlinear least squares assuming that the error terms are independently and normally distributed. Observations on cumulative proportions are clearly neither independent nor normally distributed. This paper proposes and applies a new methodology which recognizes the cumulative proportional nature of the Lorenz curve data by assuming that the proportion of income is distributed as a Dirichlet distribution. Five Lorenz-curve specifications were used to demonstrate the technique. Once a likelihood function and the posterior probability density function for each specification are derived we can use maximum likelihood or Bayesian estimation to estimate the parameters. Maximum likelihood estimates and Bayesian posterior probability density functions for the Gini coefficient are also obtained for each Lorenz-curve specification.

Bayesian Assessment of Lorenz and Stochastic Dominance in Income Distributions

Hypothesis tests for dominance in income distributions has received considerable attention in recent literature. See, for example, Barrett and Donald (2003), Davidson and Duclos (2000) and references therein. Such tests are useful for assessing progress towards eliminating poverty and for evaluating the effectiveness of various policy initiatives directed towards welfare improvement. To date the focus in the literature has been on sampling theory tests. Such tests can be set up in various ways, with dominance as the null or alternative hypothesis, and with dominance in either direction (X dominates Y or Y dominates X). The result of a test is expressed as rejection of, or failure to reject, a null hypothesis. In this paper we develop and apply Bayesian methods of inference to problems of Lorenz and stochastic dominance. The result from a comparison of two income distributions is reported in terms of the posterior 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 uncertain outcomes in terms of probabilities has the advantage of being more informative than a simple reject / do-not-reject outcome. Whether a probability is sufficiently high or low for 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 Survey for the years 1978 and 1986. We assess the likelihood of dominance from one time period to the next. Two alternative assumptions are made about the income distributions –Dagum and Singh-Maddala – and in each case the posterior probability of dominance is given by the proportion of times a relevant parameter inequality is satisfied by the posterior observations generated by Markov chain Monte Carlo.Bayesian, Income Distributions, Lorenz