MR estimates were obtained using the Wald ratio method (1). For each simulation, 2 linear regressions were performed: a regression of X on G (the first-stage regression) and a regression of Y on G (the reduced-form regression). The ratio of these estimates (the Wald estimate) and corresponding confidence intervals were obtained using the suest and nlcom commands in Stata (10). The suest (seemingly unrelated regression (SUR)) command combines the regression estimates into 1 parameter vector and a simultaneous sandwich (robust) variance-covariance matrix. The nlcom command computes standard errors and confidence intervals for nonlinear combinations of parameter estimates using the delta method. We did not use the traditional 2-stage least-squares procedure (11), because this method discards persons with missing data on X, whereas the Wald method can include such persons in the reduced-form regression. Power was defined as the proportion of the 10,000 data sets in which a statistically significant effect of X on Y was observed (2-sided P < 0.05).
