Bootstrapping high-frequency jump tests : Supplementary Appendix ∗

This supplementary appendix is organized as follows. In Section S1, we first provide an auxiliary lemma and then provide proofs of the general bootstrap results appearing in Section 3 of the main paper. In Section S2, we establish the results appearing in Section 4 of the main paper. In particular, this section contains the asymptotic expansion of the cumulants of the asymptotic test statistic Tn and its bootstrap versions T ∗ n and T̄ ∗ n . The limits of these cumulants are derived by relying on some auxiliary lemmas that are introduced and proved in this section of the appendix. Detailed formulas useful for the implementation of the log version of our tests are provided in Appendix S3. Finally, Section S4 presents the theoretical justification for the local Gaussian bootstrap when applied to two alternative jump tests: the test of Podolskij and Ziggel (2010) and the big jumps test of Lee and Hannig (2010).