t, t−1, t−2 . y denotes log real per capita GDP. g!", constructed as (y!" − y!"!! )/3, is the arithmetic average real GDP per capita growth rate of country i in period s. y!"!! is measured as the log real per capita GDP in the second year of the previous period (i.e., y!"!!).<sup>16</sup> We expect β! to be negative. The initial income level captures conditional convergence – as a country gets richer, it grows more slowly, _ceteris paribus_ . Aid!"!! is the log of average aid received by country i as a share of GNI in the previous period.<sup>17</sup> We use the one-period lag of aid instead of contemporaneous aid in order to allow time for aid to take effect, following Clemens et al. (2012). X!" is a vector of time-varying variables, which in our baseline specification includes the logarithm of population and it is assumed to be strictly 16 Notice that, by construction, y!"!! is not mechanically correlated with the dependent variable. Some studies in the literature use per capita real GDP in purchasing power parity terms to measure income level and to calculate growth (e.g., Boone, 1996). Real per capita GDP based on current exchange rates (in constant dollar terms, and using the Atlas method) and real per capita GDP in PPP terms are highly correlated: in our sample, the correlation is over 0.95. Growth rates constructed from the two versions are essentially the same. We use per capita GDP based on current exchange rates (in constant dollars) because there are fewer missing observations in the WDI database than for the PPP measure. Using instead the PPP measure we obtain almost identical results for our basic specifications in Table 5.

> 17 The measure of aid is slightly different from most of the literature which often use the aid to GDP or GNI ratio as the main explanatory variable. We take the log of aid since previous evidence suggests that the marginal effect of aid on growth is decreasing. The logarithmic form is a parsimonious way to introduce concavity while preserving our ability to identify aid’s causal impact with only one exogenous binary instrumental variable. However, note that instrumented aid still  takes on a large number of values on its domain, since each country’s aid is shifted by the instrument starting from different values (over time). The logarithmic specification is admittedly less flexible than a quadratic specification; in particular, it does not allow the marginal effect of aid to change its sign. Clemens et al. (2012), however, find that the effect of aid on growth does not turn negative until aid exceeds roughly 15% of GDP. In our sample, over 90% of the observations are below 11% of GDP. Thus a logarithmic specification provides a good approximation over the range of observed values on aid. Additionally, we

report as a robustness test in Appendix Table C results from using Aid!"!!∗ =(∑ !!!! !"!!"!!"!!!"!!)/3 (i.e. not logged) as the measure for aid. Aid!"!!∗ has a positive, quantitatively large, and marginally significant coefficient, despite a lower first stage F statistic. We find that a 1 percentage point increase in the aid to GNI ratio raises annual per capita GDP growth by 0.57 percentage point at the sample mean of the aid to GNI ratio. This estimate is even larger than our baseline result, reported later in Section 5.2.

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