GMM Estimation of Mixtures from Grouped Data:
We show how the generalized method of moments (GMM) framework developed in Hajargasht et al. (2012) for estimating income distributions from grouped data can be adapted for estimating mixtures. This approach can be used to estimate a mixture of any distributions where the moments and moment distribution functions of the mixture components can be expressed in terms of the parameters of those components. The required expressions for mixtures of lognormal and gamma densities are provided; in our empirical work we focus on estimation of mixtures of lognormal distributions. Two- and three-component lognormal mixtures are estimated for the income distributions of China rural, China urban, India rural, India urban, Pakistan, Russia, South Africa, Brazil and Indonesia. Their performance, in terms of goodness-of-fit and validity of moment conditions, is compared with that of a generalized beta (GB2) distribution. We find that the three-component lognormal mixture always outperforms the GB2 distribution, but the two-component mixture does not. For Brazil and Indonesia we have single observations, making it possible to compare maximum likelihood estimation of the mixtures from a complete set of single observations with GMM estimates obtained after grouping the data. Estimates from both procedures are found to be comparable, lending support to the usefulness of the GMM approach.
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