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Inference for Income Distributions Using Grouped Data

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  • Gholamreza Hajargsht, William E. Griffiths, Joseph Brice, D.S. Prasada Rao, Duangkamon Chotikapanich
Abstract
We develop a general approach to estimation and inference for income distributions using grouped or aggregate data that are typically available in the form of population shares and class mean incomes, with unknown group bounds. Generic moment conditions and an optimal weight matrix that can be used for GMM estimation of any parametric income distribution are derived. Our derivation of the weight matrix and its inverse allows us to express the seemingly complex GMM objective function in a relatively simple form that facilitates estimation. We show that our proposed approach, that incorporates information on class means as well as population proportions, is more efficient than maximum likelihood estimation of the multinomial distribution that uses only population proportions. In contrast to the earlier work of Chotikapanich et al. (2007, 2012), that did not specify a formal GMM framework, did not provide methodology for obtaining standard errors, and restricted the analysis to the beta-2 distribution, we provide standard errors for estimated parameters and relevant functions of them, such as inequality and poverty measures, and we provide methodology for all distributions. A test statistic for testing the adequacy of a distribution is proposed. Using eight countries/regions for the year 2005, we show how the methodology can be applied to estimate the parameters of the generalized beta distribution of the second kind, and its special-case distributions, the beta-2, Singh-Maddala, Dagum, generalized gamma and lognormal distributions. We test the adequacy of each distribution and compare predicted and actual income shares, where the number of groups used for prediction can differ from the number used in estimation. Estimates and standard errors for inequality and poverty measures are provided.

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  • Gholamreza Hajargsht, William E. Griffiths, Joseph Brice, D.S. Prasada Rao, Duangkamon Chotikapanich, 2012. "Inference for Income Distributions Using Grouped Data," Department of Economics - Working Papers Series 1140, The University of Melbourne.
    • Handle: RePEc:mlb:wpaper:1140
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    1. Branko Milanovic, 2002. "True World Income Distribution, 1988 and 1993: First Calculation Based on Household Surveys Alone," Economic Journal, Royal Economic Society, vol. 112(476), pages 51-92, January.
    2. James B. McDonald, 2008. "Some Generalized Functions for the Size Distribution of Income," Economic Studies in Inequality, Social Exclusion, and Well-Being, in: Duangkamon Chotikapanich (ed.), Modeling Income Distributions and Lorenz Curves, chapter 3, pages 37-55, Springer.
    3. Chotikapanich, Duangkamon & Griffiths, William E. & Rao, D. S. Prasada, 2007. "Estimating and Combining National Income Distributions Using Limited Data," Journal of Business & Economic Statistics, American Statistical Association, vol. 25, pages 97-109, January.
    4. Wu, Ximing & Perloff, Jeffrey M., 2007. "GMM estimation of a maximum entropy distribution with interval data," Journal of Econometrics, Elsevier, vol. 138(2), pages 532-546, June.
    5. Cameron,A. Colin & Trivedi,Pravin K., 2005. "Microeconometrics," Cambridge Books, Cambridge University Press, number 9780521848053, September.
    6. Butler, Richard J. & McDonald, James B., 1989. "Using incomplete moments to measure inequality," Journal of Econometrics, Elsevier, vol. 42(1), pages 109-119, September.
    7. Ximing Wu & Jeffrey M. Perloff, 2005. "China's Income Distribution, 1985-2001," The Review of Economics and Statistics, MIT Press, vol. 87(4), pages 763-775, November.
    8. McDonald, James B. & Xu, Yexiao J., 1995. "A generalization of the beta distribution with applications," Journal of Econometrics, Elsevier, vol. 69(2), pages 427-428, October.
    9. Duangkamon Chotikapanich & William E Griffiths & D.S. Prasada Rao & Vicar Valencia, 2009. "Global Income Distribution and Inequality: 1993 and 2000," Department of Economics - Working Papers Series 1062, The University of Melbourne.
    10. Duangkamon Chotikapanich & William E. Griffiths & D. S. Prasada Rao & Vicar Valencia, 2012. "Global Income Distributions and Inequality, 1993 and 2000: Incorporating Country-Level Inequality Modeled with Beta Distributions," The Review of Economics and Statistics, MIT Press, vol. 94(1), pages 52-73, February.
    11. Duangkamon Chotikapanich (ed.), 2008. "Modeling Income Distributions and Lorenz Curves," Economic Studies in Inequality, Social Exclusion, and Well-Being, Springer, number 978-0-387-72796-7, July.
    12. McDonald, James B & Ransom, Michael R, 1979. "Functional Forms, Estimation Techniques and the Distribution of Income," Econometrica, Econometric Society, vol. 47(6), pages 1513-1525, November.
    13. James B. McDonald & Michael Ransom, 2008. "The Generalized Beta Distribution as a Model for the Distribution of Income: Estimation of Related Measures of Inequality," Economic Studies in Inequality, Social Exclusion, and Well-Being, in: Duangkamon Chotikapanich (ed.), Modeling Income Distributions and Lorenz Curves, chapter 8, pages 147-166, Springer.
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    Cited by:

    1. Duangkamon Chotikapanich & William Griffiths & Wasana Karunarathne & D.S. Prasada Rao, 2013. "Calculating Poverty Measures from the Generalised Beta Income Distribution," The Economic Record, The Economic Society of Australia, vol. 89, pages 48-66, June.
    2. Gholamreza Hajargasht & William E. Griffiths, 2016. "Inference for Lorenz Curves," Department of Economics - Working Papers Series 2022, The University of Melbourne.
    3. Griffiths, William & Hajargasht, Gholamreza, 2015. "On GMM estimation of distributions from grouped data," Economics Letters, Elsevier, vol. 126(C), pages 122-126.
    4. Mathias Silva, 2023. "Parametric estimation of income distributions using grouped data: an Approximate Bayesian Computation approach [Working Papers / Documents de travail]," Working Papers hal-04066544, HAL.
    5. Vanesa Jorda & José María Sarabia & Markus Jäntti, 2021. "Inequality measurement with grouped data: Parametric and non‐parametric methods," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 184(3), pages 964-984, July.
    6. William E. Griffiths and Gholamreza Hajargasht, 2012. "GMM Estimation of Mixtures from Grouped Data:," Department of Economics - Working Papers Series 1148, The University of Melbourne.
    7. Chotikapanich, Duangkamon & Griffiths, William E. & Rao, D.S. Prasada & Karunarathne, Wasana, 2014. "Income Distributions, Inequality, and Poverty in Asia, 1992–2010," ADBI Working Papers 468, Asian Development Bank Institute.
    8. Kazuhiko Kakamu & Haruhisa Nishino, 2019. "Bayesian Estimation of Beta-type Distribution Parameters Based on Grouped Data," Computational Economics, Springer;Society for Computational Economics, vol. 54(2), pages 625-645, August.
    9. Lee, Ji Hyung & Sasaki, Yuya & Toda, Alexis Akira & Wang, Yulong, 2024. "Tuning parameter-free nonparametric density estimation from tabulated summary data," Journal of Econometrics, Elsevier, vol. 238(1).
    10. Hajargasht, Gholamreza & Griffiths, William E., 2013. "Pareto–lognormal distributions: Inequality, poverty, and estimation from grouped income data," Economic Modelling, Elsevier, vol. 33(C), pages 593-604.
    11. Alexis Akira Toda & Yulong Wang, 2021. "Efficient minimum distance estimation of Pareto exponent from top income shares," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 36(2), pages 228-243, March.
    12. Gholamreza Hajargasht and William E. Griffiths, 2012. "Pareto-Lognormal Income Distributions:Inequality and Poverty Measures, Estimation and Performance," Department of Economics - Working Papers Series 1149, The University of Melbourne.
    13. Tsvetana Spasova, 2019. "Regional Income Distribution in the European Union: A Parametric Approach," Research on Economic Inequality, in: What Drives Inequality?, volume 27, pages 1-18, Emerald Group Publishing Limited.
    14. Michał Brzeziński, 2013. "Parametric Modelling of Income Distribution in Central and Eastern Europe," Central European Journal of Economic Modelling and Econometrics, Central European Journal of Economic Modelling and Econometrics, vol. 5(3), pages 207-230, September.
    15. Kazuhiko Kakamu, 2016. "Simulation Studies Comparing Dagum and Singh–Maddala Income Distributions," Computational Economics, Springer;Society for Computational Economics, vol. 48(4), pages 593-605, December.
    16. Kazuhiko Kakamu & Haruhisa Nishino, 2016. "Bayesian Estimation Of Beta-Type Distribution Parameters Based On Grouped Data," Discussion Papers 2016-08, Kobe University, Graduate School of Business Administration.
    17. Jordá, Vanesa & Niño-Zarazúa, Miguel, 2019. "Global inequality: How large is the effect of top incomes?," World Development, Elsevier, vol. 123(C), pages 1-1.
    18. Tobias Eckernkemper & Bastian Gribisch, 2021. "Classical and Bayesian Inference for Income Distributions using Grouped Data," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 83(1), pages 32-65, February.
    19. Duangkamon Chotikapanich & William E. Griffiths & Gholamreza Hajargasht & Wasana Karunarathne & D.S. Prasada Rao, 2018. "Using the GB2 Income Distribution: A Review," Department of Economics - Working Papers Series 2036, The University of Melbourne.
    20. Fernández-Morales, Antonio, 2016. "Measuring poverty with the Foster, Greer and Thorbecke indexes based on the Gamma distribution," MPRA Paper 69648, University Library of Munich, Germany.
    21. David Gunawan & William E. Griffiths & Duangkamon Chotikapanich, 2021. "Posterior Probabilities for Lorenz and Stochastic Dominance of Australian Income Distributions," The Economic Record, The Economic Society of Australia, vol. 97(319), pages 504-524, December.
    22. Duangkamon Chotikapanich & William E. Griffiths & Gholamreza Hajargasht & Wasana Karunarathne & D. S. Prasada Rao, 2018. "Using the GB2 Income Distribution," Econometrics, MDPI, vol. 6(2), pages 1-24, April.
    23. Sugasawa, Shonosuke & Kobayashi, Genya & Kawakubo, Yuki, 2020. "Estimation and inference for area-wise spatial income distributions from grouped data," Computational Statistics & Data Analysis, Elsevier, vol. 145(C).
    24. Helton Saulo & Roberto Vila & Giovanna V. Borges & Marcelo Bourguignon, 2022. "Parametric quantile regression for income data," Papers 2207.06558, arXiv.org.

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    Keywords

    GMM; Generalized beta distribution; Inequality and poverty.;
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