Table 1: Conditional mean function in each of three simulation settings within simulation scenario 1. Here, m ( a, w ) ≜ 0 . 2 ( w 2 1 + w$_{2}$ − 2 w$_{3}$w$_{4}$ ), and the third and fourth columns indi- cate, respectively, whether µ (1 , W ) and µ (0 , W ) are equal in distribution or almost surely. µ ( a, w ) d = a.s. = Simulation 1a m ( a, w ) × × Simulation 1b m ( a, w ) + 0 . 4[ aw$_{3}$ + (1 − a ) w$_{4}$ ] × Simulation 1c m ( a, w ) + 0 . 8 aw$_{3}$ 0.05 0.10 0.5 1.0 0.5 1.0 Sim 1a Sim 1b Sim 1c 125 250 500 1000 2000 Sample Size Probability of Rejecting the Null Method MMD, Variance Cutoff Racine et al. Figure 1: Empirical probability of rejecting the null when testing the null hypothesis that µ (1 , W ) − µ (0 , W ) is almost surely equal to zero (Example 1) in Simulation 1. Table 1 indicates that the null is true in Simulation 1a, and the alternative is true in Simulations 1b and 1c. Figure 1 displays the empirical coverage of our approach as well as that resulting from use of the np package. At smaller sample sizes, our method does not appear to control type I error near the nominal level. This is likely because we use an asymptotic result to compute the cutoff, even when the sample size is small. Nevertheless, as sample size grows, the type I error of our test approaches the nominal level. We note that in Racine et al. (2006), unlike in our proposal, the bootstrap was used to evaluate the significance of the proposed test. It will be interesting to see if applying a bootstrap procedure at smaller sample sizes improves our small-sample results. At larger sample sizes, it appears that the method of Racine et al. slightly outperforms our approach in terms of power in simulation scenarios 1a and 1b. In Figure 2, the empirical null rejection probability of our test is displayed for simulation scenarios 1a, 1b and 1c. In particular, we observe that our method is able to control type I error for Simulations 1a and 1b when testing the hypothesis that µ (1 , W ) is equal in distribution to µ (0 , W ). Also, the power of our test increases with sample size, as one would 28