Are countries becoming equally unequal?

Abstract

Literature on convergence in inequality is sparse and has almost entirely focused on the notion of testing beta convergence in the Gini indices. In this paper, for the first time, we test for sigma convergence in decile income shares across countries. We compile panel data on decile income shares for more than 60 countries over the last 25 years. Regardless of the level of development, within-country inequality increased; income shares of the poorest deciles declined and those of the top decile increased significantly. Importantly, the decile income shares exhibited a statistically significant decline in dispersion between 1985 and 2011, providing strong evidence of sigma convergence in inequality. Convergence was more prominent among developing countries and less so among developed countries. The findings are robust to an array of sensitivity tests. Our analysis suggests that cross-country income distributions became more unequal but noticeably similar over time.

Appendices

Appendix

See Table 10.

Table 10 Construction of a balanced panel

Testing normality of the distribution of decile income shares

To formally test normality, we employ a nonparametric bootstrap test, developed by Fan (1994). It states that the unknown empirical distribution of the data $(f\left( x \right) )$ equals a given parametric distribution $(g\left( {x,\theta } \right) )$, against the alternative that these distributions are unequal:

$$\begin{aligned}&H_0 :f\left( x \right) =g\left( {x,\theta } \right) \nonumber \\&H_1 :\,f\left( x \right) \ne g\left( {x,\theta } \right) \end{aligned}$$

(9)

The test compares the two distributions by way of their integrated squared errors (ISE):

$$\begin{aligned} {\text {ISE}}\left( {f,g} \right) =\mathop \smallint \nolimits _x \left[ {f\left( x \right) -g\left( {x,\theta } \right) } \right] ^{2}\mathrm{d}x \end{aligned}$$

(10)

Equation (10) is estimated by replacing $f\left( x \right)$ with the kernel density of x and substituting $g\left( {x,\theta } \right)$ with the kernel-smoothed maximum likelihood estimate of g. For the case of a Gaussian distribution under the null, Eq. (10) can be estimated by the following closed-form representation:

$$\begin{aligned} \widehat{\mathrm{ISE}}&=\frac{1}{N\left( {N-1} \right) h}\sum \limits _{i=1}^N\sum \limits _{j=1}^N \left( {\frac{x_i -x_j }{h}} \right) +\frac{1}{2\sqrt{\pi }\left( {h^{2}+\widehat{\sigma } ^{2}} \right) ^{\frac{1}{2}}}\nonumber \\&\quad -\frac{2}{N\sqrt{2\pi }\left( {h^{2}+\widehat{\sigma }^{2}} \right) ^{1/2}}\sum \limits _{i=1}^N {\exp }\,\left( {-\frac{\left( {x_i -\widehat{\mu }}\right) ^{2}}{2\left( {h^{2}+ \widehat{\sigma }^{2}} \right) }} \right) \end{aligned}$$

(11)

where N is the sample size, h is the nonparametric window width, and $\hat{\mu }$ and $\hat{\sigma }^{2}$ are the maximum likelihood estimates of the first and second moments of g.¹⁶ Using Eq. (11), the resulting test statistic (for univariate x) equals

$$\begin{aligned} \hat{T}_N =\left( {Nh} \right) ^{\frac{1}{2}}\frac{ \widehat{\mathrm{ISE}}}{\widehat{\sigma }}\mathop \rightarrow \limits ^d \mathrm{Normal}\left( {0,1} \right) \end{aligned}$$

(12)

Although Eq. (12) has a standard normal limiting distribution, we follow Henderson and Parmeter (2015) and use bootstrap-based critical values when conducting the test.¹⁷ Table 11 shows results of the Fan (1994) tests performed on both time period 1 and period 5 for each of the ten deciles. In every instance, we fail to reject the null hypothesis that the distribution of a given decile share of income for a given time period is normally distributed.

Table 11 Tests for normality of income shares