توزیع درآمد دو متغیره برای ارزیابی نابرابری و فقر تحت نمونه های وابسته

As indicators of social welfare, the incidence of inequality and poverty is of ongoing concern to policy makers and researchers alike. Of particular interest are the changes in inequality and poverty over time, which are typically assessed through the estimation of income distributions. From this, income inequality and poverty measures, along with their differences and standard errors, can be derived and compared. With panel data becoming more frequently used to make such comparisons, traditional methods which treat income distributions from different years independently and estimate them on a univariate basis, fail to capture the dependence inherent in a sample taken from a panel study. Consequently, parameter estimates are likely to be less efficient, and the standard errors for between-year differences in various inequality and poverty measures will be incorrect. This paper addresses the issue of sample dependence by suggesting a number of bivariate distributions, with Singh–Maddala or Dagum marginals, for a partially dependent sample of household income for two years. Specifically, the distributions considered are the bivariate Singh–Maddala distribution, proposed by Takahasi (1965), and bivariate distributions belonging to the copula class of multivariate distributions, which are an increasingly popular approach to modelling joint distributions. Each bivariate income distribution is estimated via full information maximum likelihood using data from the Household, Income and Labour Dynamics in Australia (HILDA) Survey for 2001 and 2005. Parameter estimates for each bivariate income distribution are used to obtain values for mean income and modal income, the Gini inequality coefficient and the headcount ratio poverty measure, along with their differences, enabling the assessment of changes in such measures over time. In addition, the standard errors of each summary measure and their differences, which are of particular interest in this analysis, are calculated using the delta method.

The incidence of inequality and poverty is of ongoing concern not only in developing regions but also amongst some of the world's economic leaders. As indicators of social welfare, much of the literature has been drawn towards the accurate specification and estimation of various inequality and poverty measures. In particular, policy makers and researchers alike are often interested in assessing the changes in inequality and poverty over time. That is, one would ideally expect to observe a reduction in inequality and poverty from one year to another in order to determine whether policies implemented for that purpose have been effective. Such assessments are typically performed through the estimation of income distributions, from which income inequality and poverty measures, along with their differences and standard errors, can be derived and compared. Traditionally, comparisons of inequality and poverty over time have been made with income distributions for different years being treated as independent. Various univariate functional forms have been suggested in the literature and income distributions have been estimated accordingly using conventional inference techniques. Initially, the gamma, lognormal and Pareto distributions were commonly used, with the gamma distribution generally found to fit better than the lognormal distribution (Salem and Mount, 1974, McDonald and Ransom, 1979 and McDonald, 1984). Other two-parameter distributions that have been considered include the beta, Fisk and Weibull distributions. In addition, a number of three-parameter distributions have been proposed in the literature, including the Singh–Maddala distribution, which contains the Pareto, Fisk and Weibull distributions as special cases, and the Dagum distribution, which, although not as widely applied as the Singh–Maddala distribution, has shown to provide a better fit (Kleiber, 1996 and Kleiber and Kotz, 2003). A major issue in taking the univariate approach, however, is that panel data are becoming more frequently used to make comparisons of inequality and poverty over time. Consequently, as some members of the panel will be common between years, recorded incomes are likely to be correlated, resulting in a dependent sample. Therefore, treating one income distribution for any given year independently of another does not take into account that those who earned a high income in one year are also likely to earn a high income in a subsequent year and vice versa. This is of concern particularly in regions which exhibit low income mobility, as it coincides with a high degree of correlation. There are two main consequences of estimating separate univariate distributions for different years of a panel. The first is that the parameter estimates are likely to be less efficient than a bivariate or multivariate approach that recognises correlation in incomes from year to year. The second is that the standard errors for between-year differences in various inequality and poverty measures will be incorrect. This is due to the estimated differences being functions of the parameter estimates from both marginal distributions. These parameter estimates will be correlated, and separate estimation does not provide a covariance term for computing the standard error of the difference. These features provide motivation for the use of multivariate techniques for the distribution of income when using panel data, in order to account for possible correlation between years. The presence of dependence in a sample of income taken from a panel study has been left largely unaddressed in the literature. In a paper by Kmietowicz (1984), a bivariate lognormal distribution is suggested for the joint distribution of household size and income, rather than income over time, which is then used to derive estimates of the Gini inequality measure. Sarabia et al. (2005) adapt this model by deriving extensions of the bivariate lognormal distribution and applying each to data from the European Community Household Panel. In both papers, the proposed models have marginal income distributions which follow the univariate lognormal distribution. However, it has been historically found at the univariate case that although the lognormal distribution performs well at lower income levels, it fits poorly at higher income levels (Singh and Maddala, 1976). In addition, the Singh–Maddala and Dagum distributions have been subsequently shown to provide a better fit than the lognormal distribution (Singh and Maddala, 1976, McDonald and Ransom, 1979 and McDonald, 1984). Therefore, a multivariate distribution which has either Singh–Maddala or Dagum marginals would be better suited to approximating the joint distribution of income rather than one with lognormal marginal distributions. Other studies which have recognised the issue of dependent samples often ignore the problem by selecting a subsample of the data to create either an independent sample or a completely dependent sample, with both Kmietowicz, 1984 and Sarabia et al., 2005 guilty of the latter. This is of concern as the disregard of large proportions of available data creates the potential for the marginal distributions of income to be estimated inaccurately. Intuitively, a solution would be the use of a partially dependent sample, which contains both the dependent observations within a panel as well as the independent observations. This paper seeks to address the issue of sample dependence by applying various bivariate distributions, with Singh–Maddala or Dagum marginals, to a partially dependent sample of household income for two (non-consecutive) years, with a view to assessing the changes in inequality and poverty over that period. For ease of analysis only the bivariate case is being considered in this paper. One of the distributions suggested is the bivariate Singh–Maddala distribution proposed by Takahasi (1965). The appeal of this distribution is that both the marginal and conditional distributions follow a univariate Singh–Maddala specification (Kleiber and Kotz, 2003). Other bivariate distributions being considered belong to the copula class of multivariate distributions. Using copulas to model multivariate distributions is extremely popular in the finance and actuarial context, particularly for capturing dependence amongst stocks. This approach is appealing as copulas are easily estimated using maximum likelihood techniques, and there are many variants available in the literature which capture a wide range of dependence structures beyond simple correlation. In addition, copulas are flexible in that they can be applied to any specification of the marginal distribution, including allowing for the marginal distributions to have different specifications. This provides an attractive method for capturing the dependence structure contained in the joint distribution of income under partially dependent samples. Each of the above bivariate income distributions is estimated via full information maximum likelihood using income data from the Household, Income and Labour Dynamics in Australia (HILDA) Survey for 2001 and 2005. Once the parameters for each bivariate income distribution have been estimated, values for various measures of inequality and poverty are obtained for each marginal distribution along with their differences, enabling the assessment of changes in such measures over time. More specifically, the summary measures to be considered in this paper include mean income and modal income, the Gini inequality coefficient and the headcount ratio poverty measure. In addition, the standard errors of each of the differences, which are of particular interest in this analysis, are calculated using the delta method. For comparative purposes, estimates of each measure will also be obtained for the Singh–Maddala and Dagum marginal distributions which are estimated as univariate distributions under independence. The remainder of this paper is organised as follows. Section 2 discusses and defines the concept of a partially dependent sample. Specifications for each of the bivariate distributions proposed for the joint distribution of income, along with the inequality and poverty measures considered in this analysis are provided in Section 3. Section 4 defines a likelihood function for partially dependent samples with particular emphasis on the likelihood function for a copula. Section 5 summarises the characteristics of the HILDA panel data used in the analysis. Empirical results of the analysis as applied to the data, including parameter estimates, tests for independence, and estimates for the inequality and poverty measures are presented and discussed in Section 6; conclusions appear in Section 7.

When analysing the incidence of inequality and poverty through the estimation of income distributions, the presence of dependence in a sample of income taken from a panel study has been left largely unaddressed in the literature on estimation of parametric income distributions. The main objective of this paper was to suggest various bivariate distributions for modelling a partially dependent sample of household income, from which inequality and poverty measures could be produced to examine changes in such measures over time. Of particular interest were the standard errors for the differences of each welfare measure between two years of a panel study, as they are incorrectly estimated under the univariate approach. Maximum likelihood was used to fit the Gaussian, Clayton and Gumbel copulas, each with either two Singh–Maddala marginals or two Dagum marginals, along with a bivariate Singh–Maddala distribution, to real equivalised household income in Australia. In comparing the parameter estimates of the bivariate income distributions to those obtained for the univariate income distributions, it was found that the former performed better overall as they produced higher log-likelihood values. This was consolidated by tests for independence conducted on the dependence parameter for each copula. On a comparison of the bivariate models, it was found that the Gumbel copula with two Singh–Maddala marginals best approximated the distribution of income in Australia. In addition, the parameter estimates were used to obtain estimates for the mean income, modal income, the Gini inequality measure, and the headcount ratio poverty measure for each marginal distribution. Each measure was shown to be close to its respective nonparametric estimate, with small standard errors also indicating that each was estimated with a high degree of precision. In terms of the differences for each summary measure, all bivariate distributions considered produced more efficient estimates than those obtained from the univariate distributions. It was shown that the Gumbel copula with two Singh–Maddala marginals produced estimates for the differences in mean income, modal income and the Gini coefficient with the smallest standard errors. However, for the headcount ratio, the Clayton copula with two Dagum marginals produced estimates of the difference with the smallest standard errors. When using the differences of each summary measure to analyse trends in income inequality and poverty, it was found that estimates of mean and modal income indicated that on average, Australian households were better off in 2005 than in 2001. In analysing the trend in inequality, the sample Gini coefficients suggested that there had been no improvement over the five-year period. Conversely, the estimates obtained from the bivariate distributions suggested that there had been a slight reduction in inequality. With respect to trends in poverty, both the sample headcount ratio and associated estimates from the bivariate distributions indicated a slight decrease in the proportion of the population considered to be living in poverty between 2001 and 2005. The methodology proposed in this paper can be easily extended to a multivariate case, which would allow the use of data from each available year of a panel study, rather than only two years. Various multivariate copulas are available in the literature, including multivariate extensions of the Gaussian, Clayton and Gumbel copulas. This would enable the researcher to gain a larger picture of the trends in inequality and poverty in each year. Further extensions of the proposed methodology also include the Bayesian analysis of such multivariate income distributions. The application of Bayesian inference methods is gaining interest in the field of income inequality and poverty analysis at the univariate level. The methodology described in this paper will facilitate the production of posterior densities for not only the parameters, but also inequality and poverty measures of interest, and their differences, when using data from a panel study.