Errors and disturbances and the performance of some common income distribution functions

Recent papers by Kloek & van Dijk and by McDonald & Ransom report maximum likelihood estimates cE various income distribution functions and provide test statistics of goodness of fit. According to the latter virtually all models should be rejected, yet this conclusion is avoided, and rightly so for the test is too strict. It allows for sampling variation only, while in fact income distribution functions like any other econometric model are not expected to hold exactly. If this is so one should allow for the presence of errors and disturbances in analysing income distribution data. The simplest approach is to treat observed income as the sum of a systematic component which follows some specific distribution and an independent normal error. We apply this model to three traditional two-parameter functions, viz. the Pareto, Gamma and Lognormal distribution. Since maximum likelihood estimation is impracticable we proceed by the method of moments. The evidence of a few data sets suggests that the Pareto nor the Gamma distribution are redeemed by the introduction of an error term. The verdict on the Lognormal must be postponed; as a by-product of the analysis we find that it does rather better than was suggested some years ago by Salem & Mount. A major difficulty is that estimation by moments turns out to be a delicate operation when higher moments are involved, especially if income class frequency data must be used.