This supplemental appendix provides the following supporting materials. Sections SA – SC provide the proofs of Lemmas A1 – A7 in the appendix to the main text Cheng, Dou, and Liao (2021). Section SA provides the proofs of several lemmas on the asymptotic convergence of the random components in the test statistic T and the conditional critical value cα( ˆd). Section SB verifies the bounded Lipschitz properties of the test statistic and the conditional critical value, which are used to show their weak convergence in large samples. Section SC includes some auxiliary lemmas. Section SD provides additional theoretical results on the power of the proposed conditional test. Section SE provides comparison with some power envelopes through simulations. Section SF collects details and additional results of the empirical application.