country.<sup>18</sup> We compute OLS and bias-corrected OLS estimates of � based on the evolution over time of the mean of log income, and based on the evolution over time of the variance of log income, for each country. We also compute the MSE-minimizing weighted average of the two for each equation separately, and the MSE-minimizing weighted average of these across the two equations to arrive at our preferred estimate of � for each country.

Our second application of this approach draws on the PovcalNet database maintained by the World Bank. This database is a large compendium of household survey data for developing countries, and is the basis for the World Bank’s global poverty estimates. The database, as accessed in September 2016, contains records corresponding to 1,411 household surveys covering virtually the entire developing world, and for some countries extends back to the 1980s. For each survey, the PovcalNet database reports the mean of either per capita income or per capita consumption, depending on the welfare measure used in the survey, as well a number of summary measures of poverty and inequality. We extract time series data on mean income and the Gini coefficient for 28 countries with at least 10 household surveys. Two countries in our sample, China and Indonesia, have separate surveys for rural and urban populations, resulting in a total of 30 time-series of household surveys to which to apply our methodology.<sup>19</sup> Again relying on the assumption of lognormality, we convert these series into series for the mean and variance of log income, and then we implement the procedure described in Section 3 to obtain estimates of � and upper and lower bounds on mobility. Because the available time series in PovcalNet is much shorter than in the WID, we do not allow for trend breaks when estimating Equations (8) and (9). A visual inspection of the time-series of the mean and variance of log income for these countries shows no obvious signs of structural breaks in the data. As with the WID dataset, we generate OLS and bias-corrected OLS estimates of � , as well as an MSE-minimizing combination of the two, for each of the 30 surveys.

We report the results of these two empirical applications of our methodology in Tables 2 and 3. In the first column of both tables we present our preferred estimate of � for each country. The underlying OLS and bias-corrected OLS estimates from the equations for the evolution of the mean and variance of log income, and the corresponding MSE-minimizing weighted averages, are reported in Appendix Tables

> 18 To minimize the influence of a small number of observations corresponding to large swings in the mean and variance of log income, we eliminate from our estimation sample observations for which Cook’s Distance exceeds 0.2 and/or the Studentized residuals exceed 3.5.

> 19 Most of the countries we selected from the PovcalNet database have annual household surveys, but a few have regularly surveys once every two or three years.  We annualize our estimates of � and ���� for these countries to make them comparable to those based on annual data.  Expressions for the irregularly-spaced versions of our main results are detailed in Appendix A.

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