Using the Dagum model to explain changes in personal income distribution

Abstract

This article studies the evolution of personal income distribution in Spain from 1995 to 2005 using the Dagum model. Dagum's three parameter model (type I) provides a good fitting to empirical income distributions in Spain. Moreover, its parameters can be interpreted as economic indicators of income distribution changes. After studying the economic interpretation of the Dagum model parameters, we analyse the impact of parameter changes on different income percentiles and also on the evolution of inequality in Spain using data from the European Community Household Panel (ECHP) and the European Union Statistics on Income and Living Conditions (EU-SILC).

Notes

¹ Dagum (1990), Majumder and Chakravarty (1990) and Callealta et al. (1996) present different lists of other desirable properties for the parametric income models.

² Kleiber and Kotz (2003) present a complete study about the most often used parametric income models.

³ Recently, McDonald and Ransom (2009) have expressed inequality indexes in terms of the parameters of GB and have also explored the use of numerical integration techniques for calculating these indexes.

⁴ McDonald (1984) gives a detailed list of the models nested by GB2.

⁵ This author studies the relationship between the Dagum and Singh–Maddala models. He also notes that the upper tail of the Singh–Maddala model is governed by two parameters and the region near the origin (where there are more individuals) is controlled only by one, while the opposite happens with the Dagum model.

⁶ Other techniques used to estimate the Dagum parameters are nonlinear least square (Dagum, 1980) and robust methods based on bounding the influence function (Victoria-Feser, 2000).

⁷ The goodness-of-fit could be also analysed by means of statistical tests such as the Anderson–Darling, the Cramer–von Mises, and the Kolmogorov–Smirnov tests. A complete review of these tests could be found in Stephens (1986). Unfortunately, assuming that the Dagum model is the true model with unknown parameters, the distribution of these statistics tests are not defined.

⁸ García et al. (2006) calculated the percentiles and income mean elasticities with respect to the different parameters of the model.

⁹ It can be demonstrated that, in some cases, the partial derivative of mean income with respect to δ is negative.

¹⁰ The Wolfson index of polarization has been also reformulated in terms of the between-groups and within-groups Gini coefficient (Rodríguez and Salas, 2003).

¹¹ Our conclusions regarding the interpretation of the Dagum model parameters are in accordance with the results of Gertel et al. (2001), obtained by means of numerical simulations.

¹² The EU-SILC substitutes to the ECHP that concluded in 2000.

¹³ The modified OECD equivalence scale assigns a value of 1 to the head of the household, 0.5 to each additional adult member and 0.3 to each child under age 14.

¹⁴ Prieto-Alaiz and Victoria-Feser (1996), Callealta et al. (1996), Prieto and García (2007) also demonstrated the good fit of the Dagum distribution using micro-data from Spain for all the years considered in these studies.

¹⁵ Dagum (2004) points out that an increment of parameter β and a decrease of δ provokes a sharp polarisation in distribution and establishes, as characteristics of the distributions of the developing countries, estimations of β which tend to be greater than 1; likewise, he points out that the industrialised countries systematically present greater values of δ than those which are obtained in the distributions of the developing countries. To offer a comparative idea with other countries of the EU, in Bandourian et al. (2003) in 1994, values were obtained for France of 3.55 for the parameter δ and 0.38 for β. In the case of Germany, these values are 4.41 and 0.34, respectively.