Supplement to “ A nondegenerate Vuong test ”

Let Y be generated from ∼N(μ υ2), where μ= √ e2·lr−1+υ − υ2, where lr ∈ {x ∈ R : e2·lr−1+υ − υ2 ≥ 0}. Under DGPs of this form, E[Λi(φ∗)] = lr . Thus, varying lr controls how far the deviation is from H0. On the other hand, when lr = 0, varying the parameter υ2 controls how large ω2 is. Setting υ2 = 1 makes ω2 = 0, and setting υ2 far from 1 makes ω2 large. First, I fix lr = 0 and study the null rejection probabilities of the different tests. The simulation results are reported in the top three subplots of Figure 4. The figure shows that my nondegenerate test has remarkable size control at all three sample sizes. On the other hand, the one-step and the two-step Vuong tests, as well as the SW tests have large size distortion at n= 100, and still some noticeable size distortion at n= 250. Second, I fix υ2 = 5 and study the power of the different tests as lr varies from 0 to 1 6 √ 250/n. The n−1/2-local power is considered because υ2 = 5 represents the nondegenerate case ωP0 > 0, and local alternatives around this null DGP should have ω 2 Pn 0. The results are reported in the middle three subplots of Figure 4. The plots show that the power figures of all four tests stay constant as the sample size increases with √ nlr kept constant. My nondegenerate test has power similar to that of the one-step and the two-step Vuong tests, and higher than that of the SW test. The power disadvantage of the SW test perhaps is due to the loss of efficiency from the sample splitting.