Abstract
Our target in this chapter is to illustrate one of the major advantages of the GMD regressions: they offer a complete framework for checking and dealing with some of the assumptions imposed on the data in a multiple regression problem. There are two approaches that are related to the Gini—the semi-parametric approach and the minimization approach. The interaction between the two gives tools for assessing the adequacy of the model. In addition, there are two tools that enable the researcher to investigate the curvature of the regression curve: the extended Gini regression and the NLMA curve. The basic idea is the following: there is an unknown regression curve that relates the dependent variable Y and (all or some out of) a set of explanatory variables X1,…,Xn. The shape of the curve is not known. The curve is approximated by a linear model (which is then estimated from the data). However, each approach mentioned above leads to a (possibly different) linear model. The interaction between the two approaches can help to decide whether the original curve is linear (in each individual explanatory variable) or not. The suggested stages are the following: first one estimates the regression coefficients according to the semi-parametric approach without specifying a linear model. This means that at this stage the researcher decides only on the set of explanatory variables to be included in the regression model but not on the functional form. Then one uses the residuals from the fitted curve and tests whether they fulfill the necessary conditions for the minimization approach (which were obtained assuming linearity) for each explanatory variable separately. If for any given explanatory variable the above conditions are fulfilled; that is, if the hypothesis that the two regression coefficients are equal is not rejected, then one concludes that the regression curve is linear in this variable. Otherwise it is not (see Chap. 7 for details or below for a brief review). This property is especially important in regressions with several explanatory variables. It enables the investigator to find a set of variables that allows linear predictions without having to commit to the linearity of the model as a whole. Provided that the linearity hypothesis is not rejected for all explanatory variables one can examine the properties of the residuals such as their distribution, whether it is symmetric around the regression line or not, the serial correlation between them, etc., using the methodologies that will keep the analysis under the Gini framework. Although each stage could be performed by alternative methods, we are not aware of any methodology that can offer a complete set of tests that is governed by a unified framework and therefore offers a method to test the assumptions behind the regression with an internal consistency. We note in passing that the suggested test for linearity does not require replications of observations, as is the case in the common tests for linearity.
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- 1.
To be accurate, the investigator has also to decide whether the model is multiplicative or additive.
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- 3.
If one is interested in overcoming the restriction, then one should use EG regression. See Chap. 21.
- 4.
In the empirical application we use GR* = 1 − cov(e,r(e))/cov(y,r(y)).
- 5.
It is worth emphasizing that the connection between R-regression and GMD was not recognized in the literature mentioned above. Many of the properties of those regressions can be traced to the properties of GMD. Bowie and Bradfield (1998) compare the robustness of several alternative estimation methods in the simple regression case and find the minimization of the GMD of the residuals among the most robust methods.
- 6.
Because (20.8), the GMD of the residuals, is a piecewise linear function, its partial derivative with respect to bM may not exist because the derivative is a step function. In this case the solutions bM to (20.9) form a segment on the real line and bM is determined up to a range. The larger the sample the lower the probability that such an event occurs.
- 7.
The semi-parametric estimators can be viewed as OLS instrumental variable (IV) estimators, with the rank of each variable being used as an IV. However, note that the assumptions that are assumed here are entirely different (see Yitzhaki and Schechtman (2004)). Therefore the inference cannot be drawn from there.
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- 9.
The French tax system resembles this structure.
- 10.
There are two problems with these results. The first problem is that household’s size is a discrete variable. In this case there is a mismatch between the LMA curve and the definition of cumulative distribution, because the empirical cumulative distribution is defined as a step function, while in an LMA (and Lorenz) curve one connects different points of the curve by straight lines, which implies continuity (see Chap. 5). The other problem is the issue of rounding errors because of small numbers involved. Therefore one should be careful in interpreting this result. Further research is required to resolve this issue.
- 11.
Standard errors were calculated using Jackknife fast method.
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Yitzhaki, S., Schechtman, E. (2013). Gini’s Multiple Regressions: Two Approaches and Their Interaction. In: The Gini Methodology. Springer Series in Statistics, vol 272. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4720-7_20
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