Measurement error and the distribution of income

A model for measurement error is developed, based on the assumption that measurement error is random, multiplicative, and independent of the level of actual income. Thus, measured income is defined as the product of actual income and measurement error. Flexible parametric forms are utilized to model the distributions of actual income (generalized gamma) and measurement error (inverse generalized gamma). The probability density of measured income is then derived as a generalized beta of the second kind (GB2). Estimation of the parameters of the GB2 (measured income), then allows an estimate to be made of the pdf of actual income, from which corresponding estimated means, variances, Gini coefficients, and Lorenz curves are obtained. An identification problem in the parameters of actual and measured income is solved with additional information as to the average fraction of actual income reported. The implied characteristics of measurement error are also obtained. The procedure is applied to income data from several Latin American economies, and estimates of actual income distribution characteristics are derived from measured income. It is found, in some cases, that when measured income inequality moved in one direction over time, actual income inequality moved in the opposite direction. This finding has important implications for the evaluation of policies designed to affect relative equality in the distribution of income.