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2005 | 52 | 1 | 5-35

Article title

Measuring income mobility


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The paper contains a proposal of the mixed approach to the multivariate distribution of incomes of a member of a population over multiple time periods. Research works in this area are based on panel data.Traditional parametric approach based on econometric models assumes usually that logincomes of an individual (or incomes transformed by some other simple function) follow the multivariate normal distribution, which means that in fact all information on dependency between incomes in subsequent years is contained in the respective correlation matrix. Consequently, any conclusions about trends in short term mobility, possible linear time-series model linking logincomes in subsequent years (with possible interpretation in terms of permanent positions and transitory migrations of individuals in the income hierarchy) etc. are derivable from this matrix. The proposal of the mixed approach comes from the simple remark, that essential properties of the above model (and its various generalizations) enabling interesting interpretations are in fact properties of the Gaussian copula, whereas assumptions on marginal distributions (as lognormality for instance) are irrelevant in this respect. The standard operation of 'normalizing' the original income figure D(t) by taking logarithm can be treated as a special case of the general transformation (resp. equation given). Three, the most important aspects of this approach are analysed: (1) Provided the marginal cumulative distribution functions of income are known, we can compare the Van der Waerden rank correlation sample coefficient with the standard sample correlation coefficient, treating both of them as alternative estimators of the true coefficient of the parameter of continuous version of the copula; (2) Properties of the Van der Waerden and related Fisher- Yates coefficients applied to groupped data are analysed. Grouping of individuals into quantiles (deciles, quintiles) of marginal distributions are of special attention, as it produces empirical 'transition matrices' that are a usual basis for a traditional fully non-parametric approach to income mobility. Interesting new interpretation of the second-largest eigenvalue of the transition matrix (a well known 'ad hoc' mobility measure) is derived. The interpretation is based on treating the normalized 'transition matrix' as the discretised version of the Gaussian copula density. The second largest eigenvalue of this matrix appears to be just the approximation of the parameter of continuous version of the copula; (3) Practical result of the proposal (being due to flexibility of rank correlation coefficients in respect of treatment of 'tied observations'), is a possibility to integrate two aspects of mobility (separately treated until now) in one mobility measure: migrations inside the sub-group of individuals reporting positive incomes and migrations in and out of zero-income group. Being a by-product of more sophisticated technicalities of the proposal, this result is of major importance for practical applications, as a typical motivation for research on income mobility is poverty.








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  • W. Otto, Uniwersytet Warszawski, Wydzial Nauk Ekonomicznych, Katedra Statystyki i Ekonometrii, ul. Dluga 44/50, 00-241 Warszawa, Poland


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