For a study of 10,000 persons with data on Y, Figure 1 (left panel) shows how varying the size of the subsample (nX) affects power to detect a significant effect of X on Y. For all effect sizes considered, the power of the subsample IV estimator has an upper bound approximately equal to the power of the reduced-form estimator (shown as horizontal dashed lines), which is approximately equal to the power for a traditional IV approach for these scenarios, where complete data are available for all nY individuals. As nX increases, power approaches this upper bound, and gains in power diminish. For these scenarios, our results indicate that more than 90% of the maximum power can be achieved by obtaining exposure data on only 20% of the sample. Figure 1 (right panel) shows the standard errors for these scenarios. In general, the standard errors for subsample IV estimates are larger when the effect size is larger, and standard errors decrease as nX increases. Standard errors converge to the full analysis standard error of 0.063 for all effect sizes as nX
