“…Note that, by Lemma 2, for each j 2 J 0 , y jt is integrated at frequency y j and, for each jeJ, y jt is stationary. Therefore, combining the argument of Lemma 4 of Shin and Lee (2001) and proof of Theorem 6 of Caner and Hansen (2001), we have that n À2 P y Jt y 0 Jt I kt and n À1 P ðy À Jt 0 I kt , z à kt 0 Þ 0 ðy À Jt 0 I kt , z à kt 0 Þ converge in distribution to positive definite matrices, n À3=2 P y Jt y À Jt 0 I kt ¼ o p ð1Þ for jeJ 0 and n À3=2 P y Jt z à Lemma A.1. If p ðiÞ ¼ 0 and p ðjÞ ¼ 0, iaj, then (i) P t sgn i 1 ðy i 1 ;t Þy j 1 ;t ¼ o p ðn 3=2 Þ, (ii) P t sgn i 1 ðy i 1 ;t Þsgn j 1 ðy j 1 ;t Þ ¼ o p ðnÞ,…”