- , (S.58) where cT (u0, k) (resp. cPU ,T (u0, k)) is equal to e cT (u0, k) (resp. cPU ,T (u0, k)) but with |K2 () | in place of K2 (). Since K2 () ⥠0 by definition, Proposition 3.1 in Casini (2021) implies VarPU aâ² e cT (u0, k) a = 1 Tb2,T Ë 1 0 K2 2 (x) dx â X l=ââ aâ² cPU (u0, l) [cPU (u0, l) + cPU (u0, l + 2k)]â² a + 1 Tb2,T Ë 1 0 K2 2 (x) dx â X h1=ââ â X h2=ââ κPU ,aV,Tu0 (h1, 0, h2) + o b4 2,T + O (1/ (b2,T T)) = VarPU aâ² cT (u0, k) a . (S.59) S-22 long-run variance estimation under nonstationarity Next, we discuss the bias. We have, sup PâP U,2 EP aâ² e cT (u0, k) a â aâ² cP (u0, k) a = lim Tââ sup PâP U,2 (Tb2,T )â1 T X s=k+1 K2 ((r + 1) nT â (s + k/2)) /T b2,T ! aâ² EP VsV â² sâk a â aâ² cP (u0, k) a ⤠1 2 b2 2,T Ë 1 0 x2 K2 (x) dx Ë 1 0 aâ² â2 â2u cPU (u0, k) a du + o b2 2,T + O 1 Tb2,T !
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