Supplementary Materials: The Dependent Wild Bootstrap

0.1 Proofs of Theorems 4.1, 4.2 & 5.2 Proof of Theorem 4.1: Let bn = pn+ln, where pn = b √ nc. Define a block of observations as Bn(i) = i + bn(0, 1], i ∈ Z. We first divide Rn into non-overlapped blocks of observations. Let Kn = {k ∈ Z : Bn(bnk) ⊂ Rn} represents the index set of all complete blocks Bn(bnk) = bn(k + (0, 1]) lying inside Rn. For each k ∈ Kn, we further divide each block into large and small blocks, i.e., Bn(k) = B n (k) ∪ B n (k), where B n (k) = k + pn(0, 1] and B n (k) = Bn(k)−B n (k). Denote by ΣL (ΣS) the sum over all time points in the big (small) blocks and ΣNB the sum over all time points that are not in the complete blocks, i.e., {Rn − ∪k∈KnBn(k)} ∩ Z. Write ΣB = ΣL + ΣS. For ‖v‖1 = 1, denote by f̂n,v(0) the lag window estimator of the univariate time series {Xt }. Denote by cv = DH(μ)/v! and ĉv = D H(X̄n)/v!. We apply a third-order Taylor expansion to θ̂∗ n = H(X̄ ∗ n) around X̄n and write θ̂ ∗ n − H(X̄n) = J1n + J2n + J3n, where