10 Under this linearity assumption, {w�} - where w�= Cψ<sup>�</sup> - is a parametric equivalent to the smoothed distribution �μik� previously described. It is a distribution where individual values of the wealth index have been replaced by the mean conditional on circumstances, much as before. Whereas a nonparametric approach, using the cell means, is clearly preferable when data permits it, the parametric approach based on estimating the reduced-form equation (4) may be preferable when _K_ is large relative to _N_ , so that many cells are sparsely populated, and their means imprecisely estimated. Given the properties of the distribution of w, we follow Ferreira et al. (2011) in measuring its inequality simply by the variance: V({w�}) .

An important caveat about these measures is that, in practice, not all relevant circumstance variables may be observed in the data. If the vector of _observed_ circumstances has dimension less than J, then both the non-parametric index I��μik�� and the parametric measure V({w�}) are lower-bound estimates of true inequality of opportunity. See Ferreira and Gignoux (2011) for a formal proof. In addition, in the presence of omitted circumstances, clearly neither the non-parametric decomposition nor the reducedform regression (4) can be used to identify the effect of individual circumstance variables. We do know the direction of bias – downward – for the overall measures of inequality of opportunity, however, which is why they are lower-bound estimators.

# **4. Econometric specification and data sources**

Our aim is to investigate whether decomposing inequality into inequality of opportunity and a residual term (comprising inequality arising from efforts, as well as omitted circumstances) helps resolve the inconclusiveness about the effects of inequality on subsequent growth in the empirical cross-country literature. We first estimate the following equation, which is identical to the specification employed in Forbes (2000): git = β1yi,t−5 + β2I(y)i,t−5 + β3MEi,t−5 + β4FEi,t−5 + β5PPPIi,t−5 + αi + ηt + uit (5) We estimate equation (5) (and equation (6), which replaces overall inequality with inequality of opportunity and a residual component, described below) in two panel data sets: one consisting of income and expenditure surveys (IES), and another comprised of DHS surveys. These data sets are described in detail below. In both data sets, the dependent variable, git , is the average annual growth rate of per capita gross national income in a five-year interval. The data comes from the World Bank’s World Development Indicators data set, from which we also obtain the (five-year) lagged national income per capita, yi,t−5 , expressed in constant 2005 US dollars.<sup>16</sup> I(y)i,t−5 – our measure of overall inequality – is the key variable that varies between the two samples<sup>17</sup> : in the IES sample, it denotes the mean logarithmic deviation of incomes (or expenditures) at the beginning of the five-year interval. In the DHS sample, it denotes the (overall) variance of the asset index ( V(w) ), also at the beginning of the five-

> 16 With the exception of the Czech Republic, Estonia and Ireland in the case of the IES sample and of Haiti for the DHS sample, where GDP is used instead of GNI.

> 17 To be precise, we divide the survey years into five-year bins. For example, the value of inequality of opportunity in 2005 may come from any year between 2001 and 2005. In a small number of cases, we have stretched the boundaries slightly: in Romania, e.g., we use the 2002 survey for 1996-2000 and the 2006 survey for 2001-2005. We only extend the boundaries forward and not backward (e.g. we do not use a 2000 survey for the 2001-2005 bin). Please see Tables A1 and A2 for details.