We can also provide a number of other statistical tests that confirm this choice. There is a high correlation between the poverty lines implied by (5) for our sample and the nonparametric regression function in Figure 1 (r=0.994) as well as with the data on national poverty lines (r=0.836). Equation (5) also outperforms a wide range of smooth parametric functional forms. Indeed, remarkably, the standard error in predicting the national lines is actually lower using equation (5) than the nonparametric regression in Figure 1; the standard deviation of the error is $1.19 for our poverty lines versus $1.20 for the fitted values using the smoothness parameter for the regression in Figure 1. (Of course, a sufficiently less smooth nonparametric regression would do better than our piece-wise linear model.) There is no correlation between the errors in predicting the national poverty lines using (5) and the fitted values of the nonparametric regression in Figure 1 (the correlation coefficient is 0.02). As a further test, neither the fitted values from the nonparametric regression in Figure 1 nor a cubic polynomial in _M_ were significant when added to a regression of the national poverty lines on _Z_ given by (5).<sup>30</sup> The bold unbroken line in Figure 2 gives our weakly-relative schedule in (5). In our data set of national poverty lines,<sup>_Z_</sup> _i_<sup>varies from $1.25 a day to $8 a day.31</sup> The fact that the rising portion of our poverty lines in (5) is not homogeneous immediately implies that the elasticity of the poverty line to mean consumption is below unity throughout (the elasticity goes to unity in the limit, as consumption goes to infinity). The elasticity is zero at _M_ < $.195 and then rises from 0.5 to close to 1.0 over the sample range. The consumption level at which the kink (above which the poverty line rises with the mean) occurred in the Chen Ravallion (2001) version of the AB schedule of relative poverty lines is appreciably higher than for our new schedule. For the Chen and Ravallion (2001) relative lines the kink was at a consumption level of $3.24 per day at 1993 PPP, while the new schedule of relative poverty lines in equation (5) has a kink at $1.95 a day at 2005 PPP. If we had instead chosen max($.125, _M i_ / 3) as the relative poverty line at 2005 PPP, the kink would be at a consumption level of $3.75 a day instead of $1.95. This reflects

> 30 The joint F test of the null that the three parameters in the cubic function of _M_ are all zero in the regression of national poverty lines on _Z_ given by equation (5) gave F(3,69)=0.14 (prob.=0.93) while the t-test on the coefficient on the fitted values when added to the same regression was t=0.44.

> 31 There are three special cases: China, India and Indonesia. For these countries, we have separate rural and urban distribution data from 1981 to 2005. In addition, for China and India we have separate rural and urban CPI over time. We treat the relative poverty line based on (5) as the national line for India and Indonesia, and then back out the rural and urban poverty lines using the urban-rural differentials in national lines. For China, the 2005 PPP is an urban PPP, so we set the urban relative poverty line as the national line, and adjust the rural relative poverty line down according to the ratio of urban to rural poverty lines (following Chen and Ravallion 2008b).

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