× V k1
V(lnyM,P). (18) Table 1 presents the summary of theoretical estimates that would be observed in case Cov(µ, σ) = 0 holds, on average, for the lognormal distribution and in case Cov(lnym, k) = 0 holds, on average, for the Pareto distribution. Transmission of changes in median incomes is close to be proportionally distributed across the pop-ulation while changes in mean incomes fail to be homogeneously distributed.
[TABLE 1 HERE]
[SUMMARY OF BIAS IN ˆα1]
4 Simulations and Empirical Illustration 4.1 Simulations
In this section, we present the results of simulations carried out separately for both lognormal and Pareto distributions. For the lognormal distribution, a random sample of 200 pairs (µ, σ) uniformly distributed on [4; 9]×[0.45; 2] was created. The range of values for σ is derived from the relationship between the Gini index and the variance under lognormality13 as described in Appendix A.2. For each pair, a distribution of income for a population of 1000 observations was generated. For each simulated population (distribution), we extracted the empirical values of yLN,yLNp , yM,LN andyM,LNp and ran the regression specified by Equation 10 with both the mean and median as measures of income. The results are reported in the first panel of Table 2 below.
[TABLE 2 HERE]
[REGRESSION OF THE MEASURE OF INCOME OF POOR ON THE MEASURE OF INCOME OF THE POPULATION]
This Table shows that when mean incomes are employed as measures, the coefficient estimates of α1 are not ‘close’ to 1, the true value. The same conclusions can be drawn when the changes on this measure are used in the regressions. Estimates in regres-sions using mean incomes are different from 1 at usual levels of statistical significance.
13There is no formal justification for the range of values forµ.
Chapter I Simulations and Empirical Illustration
Further, looking at the R2 values for these regressions, one can see that both levels or changes of the mean income of the population have a lower explanatory power in explaining changes or levels of the mean income of the poor14. When median incomes are used, the estimate is not significantly different from 1, the true value, and this assumption cannot be statistically rejected.
Results in the second panel of Table 2 correspond to simulations done with the Pareto distribution with values randomly generated for k and ym over the space [1.125; 2.5]× [10000; 50000]. Here again, the range of values for k is determined by the relationship between the Gini index and the dispersion parameter as described in Appendix A.215. The estimates for the Pareto distribution show similar characteristics to those obtained for the lognormal distribution. We therefore conclude that increases in the median income of the population seem to induce proportional increases of the level of incomes for the poor. Increases in the mean income are not necessarily proportional for all individuals given that estimates of the elasticity are lower than 1.
Working under a particular income distribution as done above can be seen as restric-tive since any observed income distribution does not need to necessarily be close to one single distribution. In order to simulate distributions that could better resemble empirically observed distributions, we decided to generate samples from a mix of the above two distributions and explore the relationship (Equation 10) using samples from the ‘mixed’ distributions. Our results confirm our findings with samples derived from single parametric distributions. They are presented and discussed in Appendix A.5.
They clearly show that the elasticity in median incomes is larger than the elasticity in mean incomes and the former elasticity is not statistically different from 1.
4.2 Bootstrap
Next, we perform a small bootstrap analysis (recreating the sample of 200 populations 40 times) in order to study the finite sample properties of our estimates. Figure 2a presents the resulting boxplot for the estimate of α1 in the each of the four versions of the regression for the lognormal distribution and Figure 2b for the Pareto distribution.
The empirical distributions confirm our theoretical findings. Estimates using median measures of income are higher than estimates with mean incomes and nearer to 1.
Regressions on mean incomes are less likely to capture the true (theoretical) value of the elasticity under the assumptions (i) that the above distributions can describe incomes and (ii) that Cov(µ, σ) = Cov(lnym, k) = 0 hold on average even though not
14Sensitivity to extreme values would be a potential explanation for a lower explanatory in regres-sions on mean values. Nonetheless, all measures considered here will be affected by extreme values except the overall median income of the lognormal distribution.
15Values for ym represent the lowest income observed in each economy and we consider that the proposed range would be in line with empirical observations of these minimum values.
Simulations and Empirical Illustration
necessarily in every sample due to its random generation.
[FIGURE 2 HERE]
[ELASTICITY ESTIMATES FOR DIFFERENT DISTRIBUTIONS]
Figure 2c and 2d show the results obtained with two other distributions frequently used to describe incomes, the Weibull and the Singh-Maddala distributions, for which we implement the same bootstrap analysis. The ranges of values for parameters follow the same logic as in the two previous cases and they aim to replicate observed values for Gini indices. Given the expressions of mean/median incomes for the Weibull and Singh-Maddala distributions, it is not possible to clearly derive the expression of estimates as shown in Equations 16 and 17. Nonetheless, the empirical behavior of estimates for the several versions of Equation 10 allows us to confirm the results obtained with the lognormal and the Pareto distributions and we conclude that increases in median incomes are more homogenously distributed across the population than increases in mean incomes.
The assumption of no covariance between central tendency and dispersion might be seen as a restrictive and therefore, not interesting for the simulation exercise. Appendix A.6 aims at addressing doubts concerning the efficiency of the regression in mean incomes to capture ‘pro-poor’ spells of growth when this covariance is positive. For these simu-lations, increases in median incomes are negatively correlated with increases in income inequality and a doubling of median incomes implies a 5 percent reduction in income inequality, measured by the Gini index. Despite this clearly ‘pro-poor’ scenario, the regression in mean incomes fails to capture this characteristic for the lognormal and the Pareto distributions. On the other hand, the regression in median incomes captures is able to capture this characteristic in both cases.
4.3 Illustration with US income data and in a panel of countries (1978-2012)
Using annual data for income quintiles from the US Census Bureau for the period 1966-2009, we empirically verify the results we obtain in simulations. Mean incomes per quintile are provided and thus we can calculate the mean income of the population by addition. The results in the first column of Table 3 correspond to Equation 10 run on mean incomes. Median incomes by quintile are not available and hence we take the mean incomes of the first and third quintile as proxies for the median incomes of the poor and of the population respectively. Results in the second column are from the regression using these proxies. These results confirm our findings from simulations that changes in median income tend to be more evenly distributed within the population.
Since some methodological changes were introduced in 1993 affecting the calculations of
Chapter I Effect of growth on poverty taking account of inequality change
the quintile mean incomes16, we also performed the same regressions separately for the periods before and after 1993 and the results do not change in any significant manner.
[TABLE 3 HERE]
[REGRESSION WITH US INCOME DATA]
Table 4 uses all available data on per capita income and income shares for the first and third quintiles of the income distribution from the World Development Indicators.
The resulting sample includes 153 countries over the period 1978-2012 and we use this dataset to estimate the regressions on mean/median incomes following the specification in Equation 10. Here again, we approximate the median income of the population and the median income of the poor by the mean income of the third and first quintile, respectively. Three versions of each regression are run: (i) a first one with all available data, (ii) a second one with data since 2005 and (iii) a last one with the most recent observation since 2005 for all countries.
Two important results need to be highlighted here. First, the estimate of α1 based on median incomes is consistently higher than the one based on mean incomes which implies that changes in median incomes are more homogeneously distributed than changes in mean incomes and this comparison corroborates results obtained through simulations.
Second, the estimate of α1 based on median incomes is not statistically different from one in any of the three versions of this specification. On the other hand, the estimate based on mean incomes is always lower than one at 10% significance level. These results allow us to confirm that the assumption of no covariance between central tendency and dispersion is not arbitrary, otherwise α1 would not be close to one for regressions in median incomes.
[TABLE 4 HERE]
[REGRESSIONS IN A PANEL OF COUNTRIES WITH INCOME SHARES 1978-2012]