The persistence term (ρ+1), which is expected to be below but not too far from 1, controls the dynamics of the steady-state (Solow) income and growth path.<sup>12</sup> Taking first differences of equation (2) leads to: which highlights that innovations (Δ) in X have an effect on the subsequent growth spell, but that the effect on income _growth_ is not permanent but transitory (and fades out via the persistence parameter [ρ+1]), while it is permanent on income _levels_ , again consistent with the Solow model.

Given the nature of income data at several parts of the distribution, estimating equation (1) is preferred over estimating equation (2). Direct reverse causality can be limited in this case (see below for a more detailed discussion): the growth rate after _t_ should not influence other variables at time _t_ .<sup>13</sup> Estimating equation (1) is also preferred over estimating equation (3), as the latter would lead to a greater loss of observations because of the first differences. Furthermore, equation (1) makes it clearer that the explanatory variables in _X_ can generally be seen as enabling factors for subsequent income generation, which makes it appealing for the underlying question analyzed here.<sup>14</sup> Overall, the estimated parameter _b_ from (1) will give us an indication on how important that variable is for generating subsequent income increases.

# **_Econometric Issues_**

As all panel studies, our estimations could face endogeneity issues. The endogeneity bias may arise from measurement errors, omitted variables or potential reverse causality between the dependent variable, income growth, and our variables of interest. It is possible, for instance, that as income grows, a larger share of the population becomes better educated or lives longer. The use of country fixed effects and the consideration of initial income would try to capture relevant initial conditions and variables that could have been omitted.

Furthermore, studies on poverty or on income typically address reverse causality by lagging the explanatory variables so that income growth in a given year is explained by the values on the explanatory variables in the previous year. In this spirit and similar to Dollar and Kraay (2002) and Dollar et al. (2016), we have used the value of the variable at the beginning of the spell. For example, it seems reasonable to assume that income-growth over the period 2000-2005 would be unlikely to influence life expectancy in 2000.

> 12 Accordingly, ρ itself is expected to be negative but not too far from 0. Since the model includes mean income growth, the term reflects specifically steady-state convergence for the respective income group _j_ under investigation. 13 In macro panels with a sufficiently large time dimension, this endogeneity issue can be addressed using internal instruments. However, the short time dimension of household-survey-based income data makes this approach infeasible or at least highly unreliable.

> 14 Of course, this positive effect will be balanced by the negative effect of the higher income level (via _ρY-t_ ) that corresponds to the higher _X_ . Nevertheless, we find the formulation as in equation (1) instructive.

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