Expenditure modeling with a mixture of lognormal distributions

Motivation
The adequate evaluation of success of market reforms in transition economies necessarily includes the assessment of the reform social cost, including welfare redistribution. The main source of information on the distribution of income, expenditures and wealth are population surveys
[1,2]. Various distortions and deficiencies of the available survey micro data complicate this assessment. Because of wage arrears, as well as high shares of informal economic activities, including home production, the welfare of a
household is better represented by (per capita) expenditures than by the officially reported income. Besides, survey participation rates tend to differ in different welfare groups. One of the manifestations of those deficiencies is a huge discrepancy between the mean income as found from the macroeconomic statistics, and one found from survey data. For the time period analyzed in this paper, the macroeconomic mean income for the Q4 1998 as reported in [3] is 1211 rub., while the sample mean from the raw data [2] is 913 rub. The distributional model currently used by the Russian statistical authority, Goskomstat (The State Committee on Statistics) is the lognormal distribution [4], for which the location parameter (mean or mode) is estimated from macroeconomic trade statistics, and the variance parameter is estimated from sample income data. We propose several refinements to this model. The first one is to use expenditure information that seems to represent the household financial situation better than income. The second is to approximate the shape of expenditure distribution by a univariate mixture of lognormal components. Such a model can be estimated by the maximum likelihood method from survey data, with special attention paid to the choice of thappropriate number of the mixture components. Third, we introduce weights to account for propensity to avoid disclosing income information. Finally, having estimated the above model, we use a parametric bootstrap to reconstruct the observations from the range of very high expenditures not touched upon by the sample. The estimates of the expenditure distribution thus obtained are used to construct popular inequality and poverty indices. The results suggest that unadjusted estimates of income inequality and poverty (including the officially reported poverty rates and the values of Gini index) might be seriously biased downwards.

• http://www.komkon.org/~tacik/science/skolenik-jsm2001.pdf