is evident.<sup>15</sup> There are two main off-diagonal elements. The first is the expanding middle-class amongst those countries that started with middle-class shares less than 40%. Secondly there is a marked fall in the size of the middle-class in some countries that started with a large share (over 80%) of the population in the $2-$13 interval; some of this is undoubtedly measurement error.

Figure 5 plots the size of the middle class for the latest survey against the earliest (i.e., _Ait_ against _Ait_ −τ ). The regression line has a slope significantly less than unity; the regression coefficient is 0.704 with a robust standard error of 0.050, and the intercept is 18.776% (s.e.=3.149).16 (The fixed point solution is at a middle-class share of 63.5%.) The share of the population living between $2 and $13 a day rose in 64 out of the 99 countries. The mean _d i_ (<sup>_A_</sup> _it_ ) was 0.22 percentage points per year and the median was 0.45 percentage points per year. The mean _gi_ (<sup>_R_</sup> _it_ ) is 0.024 (and the median is 0.011) or a growth rate of about 2% per year.

There is a markedly bimodal distribution across countries in the population shares living between $2 and $13 a day; this is evident in the clustering of data points in Figure 5 but is easier to see in the kernel density functions for _Ait_ and _Ait_ −τ in Figure 6.17 Taking a share of 40% as the cut-off point, 30 countries are in the lower mode and 69 are in the upper one for the most recent survey; the corresponding counts for the earliest surveys are 42 and 57 (Table 4). Over time, there has been a shift of density toward the upper mode, away from the lower one.

There is a notable regional concentration of Sub-Saharan African (SSA) countries in the lower mode. Two-thirds (20 out of 30) of the lower-mode countries in Figure 6 for the most recent survey rounds in SSA (versus 29% for the sample as a whole) and over two-thirds (20 out of 29) of the SSA countries in the sample are in the lower mode. In the latest survey rounds, the mean population share in the $2-$13 interval for SSA was 27.6% (s.e.=3.5%) as compared to 63.5% (s.e.=2.3%) in the set of all other countries. In the earliest survey rounds, the corresponding means were 22.9% (4.3%) and 58.8% (2.9%).<sup>18</sup> (In both cases, the difference between SSA and non-SSA countries is statistically significant at the 1% level.)

> 15  Cramer’s V statistic is 0.453 and the contingency coefficient is 0.671. Pearson’s test gives a Chi-square of 81.35, which is significant at better than the 1% level.

> 16  All standard errors reported in this paper are corrected for heteroskedasticity using White’s method. 17  The fact that Figure 4 suggests “convergence” to a common signal middle-class share yet there is marked bimodality in the densities echoes Quah’s (1996) critique of past interpretations of “growth regressions.” 18  The population share in the $2-$13 interval is also correlated with whether the survey used consumption or income, with lower share for consumption surveys. However, this difference vanishes if one controls for SSA countries; the correlation stems from the fact that SSA countries have mostly used consumption.

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