The income distribution with coarse data
How do you estimate poverty, inequality and earnings when the income variable consists of a combination of point-identified, interval-identified, and missing observations? This paper proposes a unifying theoretical approach to the problem of deriving point estimates when such data are present. The methodology is based on the idea of coarse data, which includes as special cases data that are censored within some predefined interval and data that are missing. A key part of the framework is to establish whether inference based on a likelihood that ignores the coarsening mechanism is equivalent to inference based on a likelihood that properly accounts for it. Results demonstrate that while the interval data are not coarsened at random (CAR), the missing data are CAR for the sample of employed economically active individuals in South Africa using the Labour Force Survey (2000 September). This requires an imputation algorithm that correctly accounts for these types of coarsening, as both univariate and multivariate parameters are affected by the choice of imputation method. It is recommended that researchers apply this framework to all analyses of the income distribution based on household survey data.
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