lagged outcome variable) using a single baseline and single follow-up survey. The power will also depends on the correlation between baseline profits (revenues) and follow-up profits (revenues). McKenzie (2011, 2012) examines several firm surveys and finds autocorrelations in the 0.3 to 0.5 range, reflecting the fact that firms experience a large amount of idiosyncratic variation in their profits and revenues from one period to the next (Fafchamps et al, 2012). This autocorrelation is likely to be stronger for larger firms (see Bloom et al, 2012), and weakest for the smallest microenterprises. We assume an autocorrelation of 0.5, and ignore any survey attrition, which would further weaken power. However, we also ignore any reduction in heterogeneity that comes from stratified randomization or other methods to improve power (Bruhn and McKenzie, 2009).<sup>5</sup> Nevertheless, these calculations provide a useful approximation of the approximate power of different studies. Group-randomized studies add a further complication for power comparisons, since their power is reduced relative to individual level randomization by a factor which depends on the intra-group correlation. Our solution is to provide a range of power, where the higher number would occur if the intra-group correlation is zero (equivalent to individual randomization) and the lower number would occur if the intra-group correlation is one (equivalent to having only one observation per group). Given the heterogeneity in firms, it is likely the intra-group correlation is relatively low, making actual power closer to the higher number.

Many funding agencies consider 80% to 90% power an appropriate target (Duflo et al, 2008) and power of 80% or more is the standard in medical trials (Schultz and Grimes, 2005). Table 5 reveals that many, if not most, business training experiments fall well below these levels in terms of power to detect a 25 percent or 50 percent increase in profits or revenues. For a microenterprise earning $25 per month (about $1 per day), a 25 percent increase in profits would be $75 per year, or around 75 percent of the direct costs of a typical microenterprise training program. So a reasonable assessment of impact should have the power to measure returns at least of this level. But in fact _none_ of the studies achieve 80% power to detect a 25 percent increase in revenues, and only de Mel et al. (2012) and possibly Berge et al. (2011) exceed 80% power for a 25% increase in profits.<sup>6</sup> The industrial cluster studies (Mano et al, 2012; Sonobe et al., 2011) and Bruhn and Zia (2012) are particularly low-powered. However, Valdivia (2012) demonstrates the importance of heterogeneity and take-up. Although the study has the largest sample size of any individual experiment, high heterogeneity among firms and a low 51 percent take-up rate (requiring four times the sample size to achieve a given power compared with 100 percent take-up) yield very low power.

In practice some of these studies attempt to further increase power by reducing the standard deviation of revenues or profits through truncating (capping the outcome at a specified level) or trimming (dropping observations with outcomes above a certain level) the top 1 or 5 percent of observations. This

> 5 Power can be further increased by taking multiple rounds of follow-up surveys and pooling them together to look at an average impact over the pooled survey rounds (McKenzie, 2012). However, de Mel et al. (2012) and Calderon et al. (2012) are the only business training studies to do this, so we ignore this element in our power calculations. 6 Note that Berge et al. (2012) take existing loan groups who meet on a given day in a given branch and randomly assign training to one out of two days in each of the two branches. True randomization thus only involves choosing one of four possible allocations, and thus has zero power according to permutation analysis. The authors claim that since loan groups are offered time on the basis of availability, this is as good as random, and so proceed with analysis as if randomization was at the group-level. Our table does the same, but this caveat should be noted.

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