Abstract
The chapter illustrates the main analytical properties of the \(\kappa \)-generalized net wealth distribution model, which is most able to accommodate the special features of wealth data. Negative, zero and positive data are modeled with a Weibull distribution, a point-mass at zero and a \(\kappa \)-generalized distribution, respectively. Expressions for the mean and popular tools for analyzing inequality are also derived for the assumed model of net wealth distribution. Finally, the specified model is fitted to a number of country data sets on household net wealth and its performance compared to alternative mixture models using the Dagum type I and Singh-Maddala specifications as descriptions of the positive net wealth values.
Wealth—any income that is at least one hundred dollars more a year than the income of one’s wife’s sister’s husband.
Henry Louis Mencken
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Notes
- 1.
For instance, a high income may be associated with low wealth—this is generally the case with young people starting their careers; on the other hand, a low income could accompany high wealth—this is the case with some retirees who have little income but have accumulated and paid for substantial assets that could be drawn upon in order to supplement their income and maintain the desired level of consumption.
- 2.
In the 1950s, Wold and Whittle (1957) and Sargan (1957) proposed the Pareto type I model and the lognormal distribution, respectively. Afterward, other models were proposed: in 1969 the Pareto types I and II by Stiglitz (1969) ; in 1975, the log-logistic by Atkinson (1975) and the Pearson type V by Vaughan (1975) . All of these models are restricted to describe only the positive range of wealth, since they are not defined for zero and/or negative values.
- 3.
The Gini coefficient is a relative measure of inequality, meaning that it remains invariant under equi-proportionate changes in all wealth holdings—that is, it satisfies the property of “scale invariance”. By contrast, the absolute counterpart of the aforementioned index satisfies the property of “translation invariance”, meaning that it does not alter when all the wealth values are increased or decreased by the same amount. If a relative index, when multiplied by the mean, becomes an absolute index that does not change under equal absolute translation of wealth, then it is said to possess the “compromise” property (Chakravarty 2009 , Chap. 1). The Gini index is not the only one that satisfies this property.
- 4.
See the “LWS List of Datasets” available at http://www.lisdatacenter.org/our-data/lws-database/documentation/lws-datasets-list/.
- 5.
The measure of net worth we use excludes business wealth from the definition of non-financial assets, which is otherwise included in the “net worth 2” LWS variable. Had we selected this net worth variable, however, the number of countries used in the current application would have been smaller than if business assets were excluded, since data on these assets are only available for a subset of countries (Canada, Cyprus, Finland, Italy, Norway, Sweden and the United States). We stick thus to the definition that is less inclusive but that is available for more countries.
- 6.
No comparable net wealth variable is indeed recorded in the LWS surveys of Austria, Germany and Luxembourg. For a chart showing which variables are available in each country data set, see the LWS “Variable Availability Matrix” (http://www.lisdatacenter.org/wp-content/uploads/LWS-data-availability-2012-03-08.xlsx).
- 7.
While in the case of income analysis there are good arguments for using “equivalized income” as an appropriate indicator of current individual welfare within a household, application of equivalence scales to household wealth data is more controversial (e.g. Sierminska and Smeeding 2005) . However, since we interpret wealth as the ability to finance potential consumption, and thus as a relevant measure of the living standard can be attained with, arguments for applying equivalence scales to adjust wealth for household size are strong (Cowell and Van Kerm 2015 ). The simple equivalence scale adopted here—the square root of the number of household members—is one of the most commonly used in international studies (Atkinson et al. 1995) .
- 8.
Properties of the mean excess plot are reviewed, for instance, in Beirlant et al. (2004) . Since we are interested here in the upper tail behavior of the distribution, the plots have been drawn only for the positive values of net wealth corresponding to the empirical percentiles.
- 9.
This was not the case for the UK household net worth distribution (UK00w), where the maximum likelihood estimation procedure for the Singh-Maddala mixture model failed to converge to a solution even after fiddling with different starting guess values and optimization routines. Hence, the corresponding results have not been reported in Table B.4.
- 10.
Mean net wealth estimates for the \(\kappa \)-generalized mixture model have been obtained by substituting the estimated parameters into Eq. (4.10). Mean predictions from the Singh-Maddala and Dagum type I models, in turn, were derived by replacing \(E_{3}\left( W\right) \) in formula (4.10) with \(E_{3}\left( W\right) =\frac{b\varGamma \left( 1+\frac{1}{a}\right) \varGamma \left( q-\frac{1}{a}\right) }{\varGamma \left( q\right) }\) and \(E_{3}\left( W\right) =\frac{b\varGamma \left( p+\frac{1}{a}\right) \varGamma \left( 1-\frac{1}{a}\right) }{\varGamma \left( p\right) }\) for, respectively, the Singh-Maddala and Dagum type I specifications (see e.g. Clementi et al. 2012b) . As for the Gini, we stick here to the absolute definition (4.16) of the index because of the high proportion of negative net wealth holders in Norway that causes a major inconsistency in the computation of the conventional Gini for this country (see Sect. 4.5.1). The standard Gini definition for the \(\kappa \)-generalized model of net wealth distribution is (4.15), whereas for the Singh-Maddala and Dagum type I mixtures, respectively, it is given by (Clementi et al. 2012b)
$$\begin{aligned} G = \frac{m-2\left[ \left( 1-\rho \right) ^{2}bqB\left( 2q-\frac{1}{a},1+\frac{1}{a}\right) -\lambda \theta _{1}\left( 1-\theta _{1}2^{-1-\frac{1}{s}}\right) \varGamma \left( 1+\frac{1}{s}\right) \right] }{m+\rho \lambda \theta _{1}\varGamma \left( 1+\frac{1}{s}\right) } \end{aligned}$$and
$$\begin{aligned} \begin{aligned} G =&\frac{m-2\left\{ \left( 1-\rho \right) ^{2}bp\left[ B\left( p+\frac{1}{a},1-\frac{1}{a}\right) -B\left( 2p+\frac{1}{a},1-\frac{1}{a}\right) \right] \right\} }{m+\rho \lambda \theta _{1}\varGamma \left( 1+\frac{1}{s}\right) }\\&+\frac{2\lambda \theta _{1}\left( 1-\theta _{1}2^{-1-\frac{1}{s}}\right) \varGamma \left( 1+\frac{1}{s}\right) }{m+\rho \lambda \theta _{1}\varGamma \left( 1+\frac{1}{s}\right) }. \end{aligned} \end{aligned}$$ - 11.
Corresponding all at once to the net worth distributions of Cyprus (CY02w), Finland (FI98w), Italy (IT04w), Japan (JO03w), Norway (NO02w), Sweden (SE02w), the United Kingdom (UK00w) and including the case of Norway (NO02w), where particularly large errors are found for both the mean and Absolute Gini predictions by all three models.
- 12.
The behavior around the mode of Weibull, Singh-Maddala and Dagum type I distributions is reviewed, e.g., in Kleiber and Kotz (2003) . For the \(\kappa \)-generalized distribution see Chap. 3.
- 13.
The formulas of the Lorenz curve for the Singh-Maddala and Dagum type I mixture models were derived by Clementi et al. (2012b) .
- 14.
Let \(\left( w_{i}\right) _{i=1}^{n}\) be a set of n wealth holdings for which the cumulative distribution function is \(\hat{F}_{i}=\frac{i}{n}\) and suppose that the observations are ordered from largest to smallest, so that the index i is the rank of \(w_{i}\). The Zipf plot of the sample is the graph of \(\ln \left( i\right) \) against \(\ln \left( w_{i}\right) \). Because of the ranking, \(\frac{i}{n}=1-\hat{F}_{i}\), so \(\ln \left( i\right) =\ln \left[ 1-\hat{F}_{i}\right] +\ln \left( n\right) \). Thus, the log of the rank is simply a transformation of the cumulative distribution function. For an illustration of basic properties of the Zipf plot see for instance Stanley et al. (1995) .
- 15.
The reported values of the Anderson-Darling statistic are multiplied by 100. This statistic is known to be more powerful than other measures quantifying the distance between the empirical distribution function of a uni-variate data set and the cumulative distribution function of a reference distribution, since it provides equal sensitivity at the tails as at the median of the distribution (e.g. Thode 2002) . The formula used for the Anderson-Darling distance is the one reported by Monahan (2011, p. 358) , which allows for weighted observations.
- 16.
The values of the Paretian upper tail index can be derived from parameter estimates of the Singh-Maddala, Dagum type I and \(\kappa \)-generalized distributions, respectively, as \(\gamma =aq\), \(\gamma =ap\) and \(\gamma =\frac{\alpha }{\kappa }\). For more details on the upper tail behavior of the \(\kappa \)-generalized distribution see Sect. 3.1.2. For the Singh-Maddala and Dagum type I distributions see instead Wilfling and Krämer (1993) and Kleiber (1996, 2008a).
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Clementi, F., Gallegati, M. (2016). The \(\kappa \)-Generalized Mixture Model for the Size Distribution of Wealth. In: The Distribution of Income and Wealth. New Economic Windows. Springer, Cham. https://doi.org/10.1007/978-3-319-27410-2_4
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