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the fact that A is binary. We also scaled the function µ (1 , w ) − µ (0 , w ) by a factor of 5 to ensure most of the probability mass of R$_{0}$ falls between − 1 and 1 (around 99% when β = 0) – this is equivalent to selecting a bandwidth of 1 / 5 for the Gaussian kernel in the definition of the MMD. We also considered a bandwidth of 5 / 2: the results were essentially identical and therefore omitted here. We note that even with scaling the variable Y is not bounded as our regularity conditions require. Nonetheless, an evaluation of our method under violations of our assumptions can itself be very informative. 

Figure 3 displays the empirical null rejection probability of our test as well as that of Racine et al. (2006). In this setup, used by the authors themselves to showcase their test procedure, our method outperforms their proposal, both in terms of type I error control and power. 

## 6.3 Simulation scenario 3: higher dimensions 

We also explored the performance of our method as extended to tackle higher-dimensional hypotheses, as discussed in Section 7. To do this, we used the same distribution as for Sim- ulation 1 but with Y now a 20-dimensional random variable. Our objective here was to test µ (1 , W ) − µ (0 , W ) is equal to (0 , 0 , . . . , 0) in probability, where µ ( a, w ) ≜ ( µ$_{1}$ ( a, w ) , µ$_{2}$ ( a, w ) , . . . , µ$_{20}$ ( a, w )) with µ$_{j}$ ( a, w ) ≜ E$_{0}$ ( Y$_{j}$ | A = a, W = w ). Conditional on A and W , the coordinates of Y are independent. We varied the number of coordinates that represent signal and noise. For 

