example, investigate the modernization hypothesis by regressing a set of country-specific fixed effects against time-invariant country characteristics; and Brand and Davis (2011) use a similar method to assess heterogeneity in the effects of a college education on fertility. One challenge in using this two-stage approach is that _αi_ and _βi_ are estimated with error. To address this, we follow the advice of Lewis and Linzer (2005) who suggest that, in small samples, unweighted OLS regressions combined with Efron or bootstrap consistent standard errors are generally reliable when the dependent variables are themselves estimates.

For the remainder of the paper we focus on the height and BMI ratio outcomes. We do so as the ratio transformation represents the lesser of two extremes – the ratio of outcomes in the sample to that of the median in the reference distribution is not affected by (independent) changes in the variance of the reference distribution, nor does it mechanically compound initial differences. Tables 8 to 11 summarize our results for the HAR and BAR outcomes, covering each country separately. Appendix Tables A4 to A7 report the same analysis for the corresponding _z_ -scores (HAZ and BAZ). In each table, the first group of the columns (Ia-Ie) refer to the height outcomes; and the final set of columns refer to body mass outcomes. Regression results for mean child size are reported in columns Ia, Ib, IIa and IIb where the first in each set excludes community fixed effects. Columns Ic, Id, IIc and IId report the same regressions for growth velocity. Lastly, columns Ie and IIe add the estimates for child size ( ̃ _αi_ ) to the velocity regression, reflecting the potential correlation between these two variables (as captured by _ραβ_ ; see above). This extension is informative because it helps differentiate between the contributions of observed and unobserved individual determinants of growth velocity.

We highlight four main findings from these results. First, a number of the coefficient estimates are consistent with previous literature. In particular, household wealth displays a significant positive relationship with both size and velocity across most countries.<sup>22</sup> Similarly, the highest school grade attained by the mother, here transformed into standard deviation units, shows a significant positive association with predicted child size (especially height), but a less systematic relation with growth velocity. Overall, the implication is that children from more advantaged households show enhanced average development outcomes and would appear to extend their advantage over time relative to other children in the sample. However, these advantages generally do not imply that children from (relatively) wealthy families catch up toward the median of the external reference population. This is indicated in the final row of the table, which calculates the predicted size and growth velocity among children from households that are two standard deviations above the country mean wealth index level. Across all the regressions for height, the adjusted estimates of mean stature remain in the negative domain; and only in Peru and Vietnam is mean growth velocity in the positive domain for the more wealthy households. In this sense, between-group catch-up remains elusive in both Ethiopia and India, even for children from more advantaged households in the sample.

> 22 The wealth index is constructed as the standardized first principal component of three separate indexes, which are provided in the YL datasets. These pertain to ownership of consumer durables, housing quality, and access to basic amenities within the household (e.g., clean water).

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