parameters must be estimated. As such, the gain in fit of the FEIS model, relative to alternatives, merits examination. Second, under the FEIS estimator, all time-invariant elements of _x_ are absorbed by the latent variables. This naturally focuses attention on the predicted effects of age-specific changes in the elements of _x_ (within-unit variation), holding each individual’s mean size (intercept) and growth rate (slope) fixed. As Bell and Jones (2015) explain, there is no necessary reason why the effect of within-unit variation in _x_ on the outcome will always be the same as the effect of between-unit variation, one reason being that the latter reflects conditions at a higher level of aggregation. In longitudinal individual data, the distinction of within- versus between-unit variation effectively captures the difference between transitory (short-run) and permanent (long-run) effects. So, applying an FEIS estimator to equation (6a) would effectively address bias from unobserved static heterogeneity. But the downside is these estimates do not yield direct insights into the effects of long-run elements of _x_ on the outcome, including time-invariant characteristics. This drawback is not crucial for testing the basic hypotheses about catch-up. Even so, in Section 4.4 we show how further insight can be gained by examining the correlates of the estimated fixed effects in a second stage analysis.

The remaining issue is that hypotheses about catch-up (Table 2) are composite in nature. In order to evaluate claims about catch-up, we must combine results from various individual hypothesis tests, while also minimizing type I and II errors. Following Wilkinson (1951), the maximum probability (maxP) associated with each individual null hypothesis represents a straightforward omnibus test of whether they can be rejected in all cases simultaneously (i.e., it is a test of their disjunction). We use this approach here.<sup>12</sup> To address type II errors from multiple testing, we further adjust the confidence level using the conservative Bonferonni procedure, which is robust to arbitrary dependence between the individual tests (Clarke and Hall 2009). So, to test for between-group convergence, we take the highest probability of the separate one-sided nulls ( _α_ 0 _≥_ 0; _β_ 0 _≤_ 0) and adjust the chosen confidence level by a factor of two. Similarly, to test for both within- and between-convergence, we take the highest probability of the three one-sided nulls and adjust the chosen confidence level by a factor of three.

# **4 Application to Young Lives**

The rest of this paper applies the proposed approach to the rich longitudinal data on child development collected under the Young Lives (YL) initiative, which covers four low-income countries.<sup>13</sup> Section 4.1 briefly introduces the data; Section 4.2 reports the baseline results for tests of the

> 12 Various alternative omnibus tests have been proposed. However, many of these are designed for cases where the number of hypotheses to be tested (jointly) is very large (e.g., Roback and Askins 2005). Also, various other omnibus tests, such as Fisher’s combined probability test, essentially provide a kind of average probability and therefore can lead to a rejection of the null even if some individual probabilities (hypotheses) are insignificant.

> 13 The YL data have been used extensively before, therefore detailed introduction is not necessary. For further details and examples see ( _inter alia_ ) Barnett et al. (2012), Crookston et al. (2013), Lundeen et al. (2014a).

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