Abstract
Inference on inequality indices remains challenging, even in large samples. Heavy right tails in income and wealth distributions hinder the quality and threaten the validity of asymptotic approximations to finite sample distributions. Attempts to improve on asymptotic approximations by bootstrap techniques or permutation tests are only partial successes. We evaluate a different approach to robust inference, relying on Student t statistics obtained from split samples. This relatively simple ‘t-based’ approach requires no consistent variance estimators, no random sampling of populations, and only mild distributional assumptions. We compare its performance with that of refined bootstrap and permutation techniques. We find that the more complex bootstrap methods still have the edge in one-sample tests, where the t-approach suffers from a negative skew. In two-sample comparisons though, the t-approach offers advantages: it is undersized while bootstrap tests and permutation tests are often oversized. In certain circumstances it is less powerful than permutation tests and bootstrap tests, but for large samples, this difference dissipates. It is also more generally applicable than permutation tests and easily generates confidence intervals. These differences are illustrated with an empirical application using two different sources of household data from the Russian Federation.
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Data Availibility
The raw data for the empirical application is publicly available and the few transformations undertaken are described in Section 6. The exact data used for the empirical application following these minor transformations is also directly available from the corresponding author upon request.
The simulation analysis simulates data and is fully reproducible from the R scripts built. The R scripts are directly available from the corresponding author upon request.
References
Athreya, K.B.: Bootstrap of the mean in the infinite variance case. Ann. Stat. 15(2), 724–731 (1987)
Bakirov, N.K., Székely, G.J.: Student’s t-test for Gaussian scale mixtures. J. Math. Sci. 139(3), 6497–6505 (2006)
Beran, R.: Prepivoting Test Statistics: A Bootstrap View of Asymptotic Refinements. J. Am. Stat. Assoc. 83(403), 687–697 (1988)
Bickel, P.J., Götze, F., van Zwet, W.R.: Resampling fewer than n observations: gains, losses, and remedies for losses. Statist. Sin. 7, 1–32 (1997)
Cohen, G., Ladaique, M.: Drivers of Growing Income Inequalities in OECD and European Countries. In: Carmo, R.M., Rio, C., Medgyesi, M. (eds.) Reducing Inequalities: A Challenge for the European Union?, pp. 31–43. Springer International Publishing, Cham (2018)
Cowell, F.A.: Measuring Inequality. London School of Economics Perspectives in Economic Analysis (2009)
Cowell, F.A., Flachaire, E.: Income distribution and inequality measurement: The problem of extreme values. J. Econ. 141(2), 1044–1072 (2007)
Davidson, R.: Reliable inference for the Gini index. J. Econ. 150(1), 30–40 (2009)
Davidson, R., Flachaire, E.: Asymptotic and bootstrap inference for inequality and poverty measures. J. Econ. 141(1), 141–166 (2007)
Dufour, J.M. Flachaire, E., Khalaf, L., Zalghout, A.: Identification-Robust Inequality Analysis. Working Paper, Series Cahiers de recherche (2020)
Dufour, J.M., Flachaire, E., Khalaf, L.: Permutation Tests for Comparing Inequality Measures. J. Bus. Econ. Stat. 37(3), 457–470 (2019)
Hill, B.M.: A simple general approach to the inference about the tail of a distribution. Ann. Stat. 3(5), 1163–1174 (1975)
Horowitz, J.: The bootstrap. In: Handbook of econometrics, vol. 5, chap. 52, pp. 3159–3228. Elsevier (2001)
Ibragimov, M., Ibragimov, R., Karimov, J., Yuldasheva, G.: Robust Analysis of Income Inequality Dynamics in Russia: t-Statistic Based Approaches. wiiw Balkan Observatory Working Papers No. 105 (2013)
Ibragimov, R., Kattuman, P., Skrobotov, A.: Robust Inference on Income Inequality: t-Statistic Based Approaches. Working Paper, SSRN Electronic Journal. (2021)
Ibragimov, M., Ibragimov, R.: Heavy tails and upper-tail inequality: The case of Russia. Empir. Econ. 54(2), 823–837 (2018)
Ibragimov, R., Müller, U.K.: t-Statistic Based Correlation and Heterogeneity Robust Inference. J. Bus. Econ. Stat. 28(4), 453–468 (2010)
Ibragimov, R., Müller, U.K.: Inference with few heterogeneous clusters. Rev. Econ. Stat. 98(1), 83–96 (2016)
Moran, T.P.: Statistical Inference for Measures of Inequality With a Cross-National Bootstrap Application. Sociol. Methods Res. 34(3), 296–333 (2006)
Schluter, C., van Garderen, K.J.: Edgeworth expansions and normalizing transforms for inequality measures. J. Econ. 150(1), 16–29 (2009)
Tadikamalla, P.R.: A Look at the Burr and Related Distributions. Int. Stat. Rev. Rev. Int. Stat. 48(3), 337 (1980)
Welch, B.L.: The generalisation of student’s problems when several different population variances are involved. Biometrika 34(1–2), 28–35 (1947)
Acknowledgements
Work on this paper was initiated while both authors were at Maastricht University, The Netherlands. We thank Zsolt Darvas for crucial and timely support and the referees for insightful suggestions.
Funding
This work has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 956107, “Economic Policy in Complex Environments (EPOC)”. Open Access Funding provided by Universitat Autonoma de Barcelona.
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Midões, C., de Crombrugghe, D. Assumption-light and computationally cheap inference on inequality measures by sample splitting: the Student t approach. J Econ Inequal 21, 899–924 (2023). https://doi.org/10.1007/s10888-023-09574-w
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DOI: https://doi.org/10.1007/s10888-023-09574-w
Keywords
- Inference on inequality measures
- Difference-in-inequality testing
- Bootstrap inference
- Permutation tests
- Sample splitting