Gini’s multiple regressions: two approaches and their interaction. (English) Zbl 1329.62026
Summary: Two regressions can be interpreted as based on Gini’s Mean Difference (GMD): a semiparametric approach, which relies on weighted average of slopes defined between adjacent observations and a minimization approach, which is based on minimization of the GMD of the residuals. The estimators obtained by the semiparametric approach have representations that resemble the OLS estimators. In addition they are robust with respect to extreme observations and monotonic transformations. The estimators obtained by the minimization approach do not have a closed form. The relationship between the estimators obtained by the two methods is studied in this paper. Combination of the methods provides tools for challenging the specification of the model. In particular it provides tools for assessing the linearity of the model. It can be applied to each explanatory variable individually and to several explanatory variables simultaneously without requiring replications. The semiparametric method and its relationship with the minimization approach are illustrated using consumption data. It is shown that the linearity of the Engel curve, and therefore the ‘linear expenditures system’ is not supported by the data.
MSC:
| 62-03 | History of statistics |
| 62J05 | Linear regression; mixed models |
| 62G08 | Nonparametric regression and quantile regression |
| 62P20 | Applications of statistics to economics |
| 01A70 | Biographies, obituaries, personalia, bibliographies |
Keywords:
Gini’s mean difference; average derivative; linear expenditure system; monotonicity; child benefitsBiographic References:
Gini, CorradoReferences:
| [1] | Banks, J., Blundell, R. and Lewbel, A. (1997) Quadratic Engel curves and consumer demand, Review of Economics and Statistics, 79, 4, (November), 527–539. · doi:10.1162/003465397557015 |
| [2] | Bassett, G., Jr. and Koenker R. (1978) Asymptotic theory of least absolute error regression, Journal of the American Statistical Association, 73, 618–622. · Zbl 0391.62046 · doi:10.1080/01621459.1978.10480065 |
| [3] | Bowie, C. D. and Bradfield, D. J. (1998) Robust estimation of beta coefficients: Evidence from small stock market, Journal of Business Finance & Accounting, 25(3)&(4), April/May, 439–454. · doi:10.1111/1468-5957.00196 |
| [4] | Bruno, M. and J. Habib (1976) Taxes, family grants and redistribution, Journal of Public Economics, vol. 5, 57–79. · doi:10.1016/0047-2727(76)90060-8 |
| [5] | Central Bureau of Statistics (2009) Family expenditure survey 2008, Special Series No. 1363, Jerusalem, Israel. |
| [6] | D’Agostino, R. B. (1971) An omnibus test of normality for moderate and large size samples, Biometrika, 58, 341–48. · Zbl 0227.62060 · doi:10.1093/biomet/58.2.341 |
| [7] | D’Agostino, R. B. (1972) Small sample probability points for the D test of normality, Biometrika, 59, 219–221. · doi:10.2307/2334638 |
| [8] | Ebert, U. (2005) Optimal anti poverty programmes: horizontal equity and the paradox of targeting, Economica, 72, 453–468. · doi:10.1111/j.0013-0427.2005.00425.x |
| [9] | Eubank, R. L. and Spiegelman, C. H. (1990) Testing the goodness of fit of a linear model via nonparametric regression techniques, Journal of the American Statistical Association, 85, 387–392. · Zbl 0702.62037 · doi:10.1080/01621459.1990.10476211 |
| [10] | Frick, R. J., Goebel, J., Schechtman, E., Wagner, G., and Yitzhaki, S. (2006) Using analysis of Gini (ANOGI) for detecting whether two sub-samples represent the same universe: the German Socio-Economic Panel Study (SOEP) experience, Sociological Methods and Research, 34, 4 (May), 427–468. · doi:10.1177/0049124105283109 |
| [11] | Feldstein, M. (1976) On the theory of tax reforms, Journal of Public Economics, 6, 77–104. · doi:10.1016/0047-2727(76)90042-6 |
| [12] | Goldberger, A. S. (1984) Reverse regression and salary discrimination, The Journal of Human Resources, 19, 293–319. · doi:10.2307/145875 |
| [13] | Grether, D. M. (1974) Correlation with ordinal data, Journal of Econometrics, 2, 241–246. · Zbl 0288.62025 · doi:10.1016/0304-4076(74)90003-7 |
| [14] | Härdle, W. and Stoker, T. M. (1989) Investigating smooth multiple regression by the method of average derivative, Journal of the American Statistical Association, 84, 408, Theory and Methods, 986–995. · Zbl 0703.62052 |
| [15] | Heckman, J. J. (2001) Micro data, heterogeneity, and the evaluation of public policy: Nobel lecture, Journal of Political Economy, 109(4), 673–748. · doi:10.1086/322086 |
| [16] | Heckman, J. J., Urzua, S. and Vytlacil, E. (2004) Understanding instrumental variables in models with essential heterogeneity, mimeo, http://jenni.ucicago.edu/underiv/ |
| [17] | Hettmansperger, T. P. (1984) Statistical Inference Based on Ranks, New York: John Wiley & Sons. · Zbl 0592.62031 |
| [18] | Jaeckel, L. A. (1972) Estimating regression coefficients by minimizing the dispersion of the residuals, Annals of Mathematical Statistics, 43, 1449–1458. · Zbl 0277.62049 · doi:10.1214/aoms/1177692377 |
| [19] | Jurečková, J. (1969) Asymptotic linearity of a rank statistic in regression parameter, Annals of Mathematical Statistics, 40, 1889–1900. · Zbl 0188.51003 · doi:10.1214/aoms/1177697273 |
| [20] | Jurečková, J. (1971) Nonparametric estimates of regression coefficients, Annals of Mathematical Statistics, 42, 1328–1338. · Zbl 0225.62052 · doi:10.1214/aoms/1177693245 |
| [21] | Kakwani, N. C. (1980) Income Inequality and Poverty, Oxford: Oxford University Press. · Zbl 0436.90036 |
| [22] | Koenker, R. and Bassett, G. Jr. (1978) Regression quantiles, Econometrica, 46, 33–50. · Zbl 0373.62038 · doi:10.2307/1913643 |
| [23] | Kozek, A. S. (1990) A nonparametric test of fit of a linear model, Communications in Statistics – Theory and Methods, 19(1), 169–179. · Zbl 0900.62222 · doi:10.1080/03610929008830195 |
| [24] | Lewbel, A. (1995) Consistent nonparametric hypothesis tests with an application to Slutsky symmetry, Journal of Econometrics, 67, 379–401. · Zbl 0820.62042 · doi:10.1016/0304-4076(94)01637-F |
| [25] | McKean, J. W. and Hettmansperger, T. P. (1978) A robust analysis of the general linear model based on one step Restimates, Biometrika, 65, 571–579. · Zbl 0391.62026 |
| [26] | Neill, J. W. and Johnson, D. E. (1984) Testing for lack of fit in regression – a review, Communications in Statistics – Theory and Methods, 13(4), 485–511. · Zbl 0552.62057 · doi:10.1080/03610928408828696 |
| [27] | Olkin I. and Yitzhaki, S. (1992) Gini regression analysis, International Statistical Review, 60, 2, (August), 185–196. · Zbl 0755.62053 · doi:10.2307/1403649 |
| [28] | Rilstone, P. (1991) Nonparametric hypothesis testing with parametric rates of convergence, International Economic Review, 32, 1 (February), 209–227. · Zbl 0714.62034 · doi:10.2307/2526941 |
| [29] | Schechtman, E. and Yitzhaki, S. (1987) Ameasureof association based on Gini’smean difference, Communications in Statistics – Theory and Methods, A16, 1, 207–231. · Zbl 0617.62061 · doi:10.1080/03610928708829359 |
| [30] | Schechtman, E. and Yitzhaki, S. (1999) On the proper bounds of the Gini correlation, Economics Letters, 63, 133–138. · Zbl 0924.90043 · doi:10.1016/S0165-1765(99)00033-6 |
| [31] | Schechtman, E. and Yitzhaki, S. (2008) Calculating the extended Gini coefficient from grouped data: a covariance presentation approach, Bulletin of Statistics & Economics, 2, S08, (Spring), 64–69. |
| [32] | Schechtman, E., Shelef, A., Yitzhaki, S. and Zitikis, R. (2008a) Testing hypotheses about absolute concentration curves and marginal conditional stochastic dominance, Econometric Theory, 24, 1044–1062. · Zbl 1284.62284 · doi:10.1017/S0266466608080407 |
| [33] | Schechtman, E., Yitzhaki, S. and Artzev, Y. (2008b) Who does not respond in the household expenditure survey: An exercise in extended Gini regressions, Journal of Business & Economics Statistics, 26, Number 3, 329–344. · doi:10.1198/00 |
| [34] | Yitzhaki, S. (1990) On the sensitivity of a regression coefficient to monotonic transformations, Econometric Theory, 6, No. 2, 165–169. · doi:10.1017/S0266466600005107 |
| [35] | Yitzhaki, S. (1991) Calculating jackknife variance estimators for parameters of the Gini method, Journal of Business & Economic Statistics, 9, No. 2 (April), 235–9. |
| [36] | Yitzhaki, S. (1996) On using linear regressions in welfare economics, Journal of Business & Economic Statistics, 14, 4, 478–486. |
| [37] | Yitzhaki, S. (1998) More than a dozen alternative ways of spelling Gini, Research on Economic Inequality, 8, 13–30. |
| [38] | Yitzhaki, S. (2003) Gini’s mean difference: A superior measure of variability for non-normal distributions, Metron, LXI, 2, 285–316. |
| [39] | Yitzhaki, S. and Olkin, I. (1991) Concentration curves and concentration indices, In: Stochastic Orders and Decisions under Risk (eds. Karl Mosler and Marco Scarsini), Institute of Mathematical Statistics, Lecture-Notes Monograph Series, Vol. 19, 380–392. · Zbl 0755.90016 |
| [40] | Yitzhaki, S. and Schechtman, E. (2004) The Gini instrumental variable, or the ”double IV” estimator, Metron, LXII, 3, 287–313. |
| [41] | Yitzhaki, S. and Schechtman, E. (2009) Identifying monotonic and non-monotonic relationships. Paper presented in ISI 57-th Meeting Proceedings: Development of Progress in Applied Statistical Modelling for the Economic Sciences Using Non-paramtric Regression Methods, Statistics South Africa, http://www.statssa.gov.za/isi2009/ScientificProgramme/IPMS/1665.pdf |
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