Information theoretic approaches to income density estimation with an application to the U.S. income data

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Abstract

The size distribution of income is the basis of income inequality measures which in turn are needed for evaluation of social welfare. Therefore, proper specification of the income density function is of special importance. In this paper, using information theoretic approach, first, we provide a maximum entropy (ME) characterization of some well-known income distributions. Then, we suggest a class of flexible parametric densities which satisfy certain economic constraints and stylized facts of personal income data such as the weak Pareto law and a decline of the income-share elasticities. Our empirical results using the U.S. family income data show that the ME principle provides economically meaningful and a very parsimonious and, at the same time, flexible specification of the income density function.

Keywords

Income density estimation Information theoretic approach Maximum entropy Weak Pareto law 

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Notes

Acknowledgments

We are grateful to three anonymous referees, and especially the editor for many pertinent comments and helpful suggestions. We would also like to thank Duncan Foley, Essie Maasoumi, Roger Koenker, Zhongjun Qu and Ximing Wu for their comments on an earlier version of the paper. However, we retain the responsibility for any remaining errors. Financial support from the Research Board, University of Illinois at Urbana-Champaign is gratefully acknowledged.

References

  1. Aigner, D.J., Goldberger, A.S.: Estimation of Pareto’s law from grouped observation. J. Am. Stat. Assoc. 65, 712–723 (1970)CrossRefGoogle Scholar
  2. Aitchison, J., Brown, J.: The Lognormal Distribution. Cambridge University Press, Cambridge (1957)Google Scholar
  3. Berger, J.: Statistical Decision Theory and Bayesian Analysis. Springer, New York (1985)CrossRefGoogle Scholar
  4. Boccanfuso, D., Decaluwé, B., Savard, L.: Poverty, income distribution and CGE micro-simulation modeling: does the functional form of distribution matter? J. Econ. Inequal. 6, 149–184 (2008)CrossRefGoogle Scholar
  5. Bordley, R.F., McDonald, J.B., Mantrala, A.: Something new, something old: parametric models for the size distribution of income. Journal of Income Distribution 6, 91–103 (1996)Google Scholar
  6. Burkhauser, R., Feng, S., Jenkins, S.P., Larrimore, J.: Estimating trends in US income inequality using the current population survey: the importance of controlling for censoring. J. Econ. Inequal. 9, 393–415 (2011)CrossRefGoogle Scholar
  7. Burkhauser, R., Feng, S., Jenkins, S.P., Larrimore, J.: Recent trends in top income shares in the United States: reconciling estimates from March CPS and IRS tax return data. Rev. Econ. Stat. 94(2), 371–388 (2012)CrossRefGoogle Scholar
  8. Clementi, F., Gallegati, M., Kaniadakis, G.: A model of personal income distribution with application to Italian data. Empir. Econ. 39, 559–591 (2010)CrossRefGoogle Scholar
  9. Cobb, L., Koppstein, P., Chen, N.H.: Estimation and moment recursion relations for multimodal distributions of exponential family. J. Am. Stat. Assoc. 78, 124–130 (1983)CrossRefGoogle Scholar
  10. Cressie, N., Read, T.: Multinomial goodness-of-fit tests. J. R. Stat. Soc. Ser. B 46, 440–464 (1984)Google Scholar
  11. Dagum, C.: A new model of personal income distribution: specification and estimation. Econ. Appl. 30, 413–437 (1977)Google Scholar
  12. Dastrup, S.R., Hartshorn, R., McDonald, J.B.: The impact of taxes and transfer payments on the distribution of income: a parametric comparison. J. Econ. Inequal. 5, 353–369 (2007)CrossRefGoogle Scholar
  13. Dhaene, G., Hoorelbeke, D.: The information matrix test with bootstrap-based covariance matrix estimation. Econ. Lett. 82, 341–147 (2004)CrossRefGoogle Scholar
  14. Esteban, J.M.: Income-share elaticity and the size distribution of income. Int. Econ. Rev. 27, 439–444 (1986)CrossRefGoogle Scholar
  15. Feng, S., Burkhauser, R., Butler, J.: Levels and long-term trends in earnings inequality: overcoming current population survey censoring problems using the GB2 distribution. J. Bus. Econ. Stat. 24(1), 57–62 (2006)CrossRefGoogle Scholar
  16. Gastwirth, J.: The estimation of the Lorenz curve and Gini index. Rev. Econ. Stat. 54, 306–316 (1972)CrossRefGoogle Scholar
  17. Gokhale, D.: Maximum entropy characterization of some distributions. Statistical Distribution in Scientific Work 3, 299–304 (1975)CrossRefGoogle Scholar
  18. Guggenberger, P.: The impact of a Hausman pretest on the asymptotic size of a hypothesis test. Econometric Theory 26, 369–382 (2009)CrossRefGoogle Scholar
  19. Haavelmo, T.: The probability approach in Econometrics. Econometrica 12, iii–vi+ 1–115 (1944). SupplementCrossRefGoogle Scholar
  20. Jhun, M., Jeong, H.: Applications of bootstrap methods for categorical data analysis. Comput. Stat. Data Anal. 35, 83–91 (2000)CrossRefGoogle Scholar
  21. Kagan, A., Linik, Y., Rao, C.: Characterization Problems in Mathematical Statistics. Wiley, New York (1973)Google Scholar
  22. Kapur, J., Kesavan, H.: Entropy Optimization Principles with Applications. Academic Press, Boston (1992)CrossRefGoogle Scholar
  23. Kleiber, C., Kotz, S.: Statistical Size Distributions in Economics and Actuarial Sciences. Wiley, Hoboken (2003)CrossRefGoogle Scholar
  24. Kloek, T., van Dijk, H.: Further results on efficient estimation of income distribution parameters. Econ. Appl. 30, 439–459 (1977)Google Scholar
  25. Kloek, T., van Dijk, H.: Efficient estimation of income distribution parameters. J. Econ. 8, 74–81 (1978)CrossRefGoogle Scholar
  26. Kullback, S., Leibler, R.: On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951)CrossRefGoogle Scholar
  27. Leipnik, R.B.: A maximum relative entropy principle for distribution of personal income with derivations of several known income distributions. Communications in Statistics-Theory and Methods 19, 1003–1036 (1990)CrossRefGoogle Scholar
  28. Lindsay, B.: Efficiency versus robustness: the case for minimum Hellinger distance and related methods. Ann. Stat. 22, 1081–1104 (1994)CrossRefGoogle Scholar
  29. Majumder, A., Chakravarty, S.: Distribution of personal income: development of a new model and its application to U.S. income data. J. Appl. Econ. 5, 189–196 (1990)CrossRefGoogle Scholar
  30. McDonald, J.: Some generalized functions for the size distribution of income. Econometrica 52, 647–663 (1984)CrossRefGoogle Scholar
  31. McDonald, J., Mantrala, A.: The distribution of income: revisited. J. Appl. Econ. 10, 201–204 (1995)CrossRefGoogle Scholar
  32. McDonald, J., Ransom, M.: Functional forms, estimation techniques, and the distribution of income. Econometrica 47, 1513–1526 (1979)CrossRefGoogle Scholar
  33. McDonald, J., Xu, Y.J.: A generalization of the beta distribution with applications. J. Econ. 66, 133–152 (1995)CrossRefGoogle Scholar
  34. Minoiu, C., Reddy, S.G.: Kernel density estimation on grouped data: the case of poverty assessment. J. Econ. Inequal. 12(2), 163–189 (2014)CrossRefGoogle Scholar
  35. Morales, D., Pardo, L., Santamaría, L.: Bootstrap confidence regions in multinomial sampling. Appl. Math. Comput. 155, 295–315 (2004)Google Scholar
  36. Ord, J.K., Patil, G.P., Taillie, C.: Choice of a distribution to describe personal income. Statistical Distribution in Scientific Work 6, 193–201 (1981)CrossRefGoogle Scholar
  37. Pareto, V.: La legge della domanda. Giornale degli Economisti 10, 59–68 (1895). English translation in Rivista di Politica Economica 87, 691–700 (1997)Google Scholar
  38. Pareto, V.: La courbe de la répartition de la richesse. Reprinted in Busoni, G. (ed.): Eeuvres complètes de Vilfredo Pareto, Tom 3: Écrits sur la courbe de la répartition de la richesse, Geneva: Librairie Doroz. English translation in Rivista di Politica Economica, 87 (1997), 647–700 (1896)Google Scholar
  39. Pareto, V.: Aggiunta allo studio della curva della entrate. Giornale degli Economisti 14, 15–26 (1897). English translation in Rivista di Politica Economica 87, 645–700 (1997)Google Scholar
  40. Park, S.Y., Bera, A.K.: Maximum entropy autoregressive conditional heteroskedasticity model. J. Econ. 150, 219–230 (2009)CrossRefGoogle Scholar
  41. Park, S.Y., Bera, A.K.: Information Theoretic Approaches to Density Estimation with an Application to the U.S. Personal Income Data. Working Paper, University of Illinois (2013)Google Scholar
  42. Ransom, M., Cramer, J.: Income distribution with disturbances. Eur. Econ. Rev. 22, 363–372 (1983)CrossRefGoogle Scholar
  43. Renyi, A.: On measures of entropy and information. In: Proceeding of the Fourth Berkeley Symposium on Mathematics, vol. 1, pp 547–561 (1960)Google Scholar
  44. Ryu, H., Slottje, D.: Another perspective on recent changes in the U.S. income distribution: an index space representation. Adv. Econ. 12, 319–340 (1997)Google Scholar
  45. Salem, A., Mount, T.: A convenient descriptive model of income distribution: the Gamma density. Econometrica 42, 1115–1127 (1974)CrossRefGoogle Scholar
  46. Shannon, C.: The Mathematical Theory of Communication. University of Illinois Press, Urbana (1948)Google Scholar
  47. Singh, S., Maddala, G.: A function for the size distribution of income. Econometrica 44, 963–970 (1976)CrossRefGoogle Scholar
  48. U.S. Census Bureau: Money Income of Households, Families, and Persons in the United States: 1980. Current Population Reports, Series P-60, No. 132. U.S. Government Printing Office, Washington, D.C. (1982)Google Scholar
  49. U.S. Census Bureau: Money Income of Households, Families, and Persons in the United States: 1990, Current Population Reports, Series P-60, No. 174. U.S. Government Printing Office, Washington, D.C. (1991)Google Scholar
  50. Wu, X.: Calculation of maximum entropy densities with application to income distribution. J. Econ. 115, 347–354 (2003)CrossRefGoogle Scholar
  51. Wu, X., Perloff, J.M.: China’s income distribution, 1985–2001. Rev. Econ. Stat. 87, 763–775 (2005)CrossRefGoogle Scholar
  52. Wu, X., Perloff, J.M.: GMM Estimation of a maximum entropy distribution with interval data. J. Econ. 138, 532–546 (2007)CrossRefGoogle Scholar
  53. Zellner, A.: Maximal data information prior distribution. In: New Methods in the Applications of Bayesian Methods, pp. 117–132. North-Holland (1977)Google Scholar

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Authors and Affiliations

  1. 1.School of EconomicsChung-Ang UniversitySeoulKorea
  2. 2.Department of EconomicsUniversity of IllinoisUrbanaUSA

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