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In this paper we study the effects of noise on the bipower variation (BPV),realized volatility (RV) and testing for co-jumps in high-frequency data underthe small noise framework. We first establish asymptotic properties of the BPVin this framework. In the presence of the small noise, the RV is asymptoticallybiased and the additional asymptotic conditional variance term appears in itslimit distribution. We also give feasible estimation methods of the asymptoticconditional variances of the RV. Second, we derive the asymptotic distributionof the test statistic proposed in Jacod and Todorov(2009) under the presence ofsmall noise for testing the presence of co-jumps in two dimensional It\^osemimartingale. In contrast to the setting in Jacod and Todorov(2009), we showthat the additional conditional asymptotic variance terms appear, and giveconsistent estimation procedures for the asymptotic conditional variances inorder to make the test feasible. Simulation experiments show that ourasymptotic results give reasonable approximations in the finite sample cases.

Power variations and testing for co-jumps: the small noise approach