Nevertheless, there is a simple albeit unconventional way to repair this example. Let A be a Bernoulli random variable, independent of all other variables, with fixed probability of success p ∈ (0 , 1). Replace S$_{P}$ with o ↦→ E$_{P}$ ( Y | A = 1 ,W ( − k ) = w ( − k )) from Example 2, yielding then It then follows that D R 0 ̸≡ D S 0 and in particular Γ$_{0}$ is no longer constant. In this case, the limit distribution given in Theorem 4 is non-degenerate. Consistent estimation of q$_{1}$$_{−}$$_{α}$ thus yields a test that asymptotically controls type I error. Given that the proposed estimator ψ$_{n}$ converges to zero faster than n − $^{1}$, the probability of rejecting the null approaches zero as sample size grows. In principle, we could have chosen any positive cutoff given that nψ$_{n}$ → 0 in probability, but choosing a more principled cutoff seems judicious. Because p is known, the remainder term Rem S P is equal to zero. Furthermore, in view of the independence between A and all other variables, one can estimate E$_{P}$$_{0}$ ( Y | A = 0 ,W ( − k )) by regressing Y on W ( − k ) using all of the data without including the covariate A . In future work, it may also be worth checking to see if the parameter is third-order differentiable under the null, and if so whether or not this allows us to construct an α -level test without resorting to an artificial source of randomness. In simulation studies, we have explored the performance of our proposed test in the context of Examples 1, 2 and 3, and have also compared our method to the approach of Racine et al. (2006) for which software is readily available – see, e.g., the R package np (Hayfield and Racine, 2008). We report the results of our simulation studies in this section. 6 Simulation studies D S $_{P}$( o ) = a p [ y − E$_{P}$ ( Y | A,W ( − k ) = w ( − k ))] . 26