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

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 under the small noise framework. We first establish asymptotic properties of the BPV in this framework. In the presence of the small noise, the RV is asymptotically biased and the additional asymptotic conditional variance term appears in its limit distribution. We also give feasible estimation methods of the asymptotic conditional variances of the RV. Second, we derive the asymptotic distribution of the test statistic proposed in Jacod and Todorov(2009) under the presence of small noise for testing the presence of co-jumps in two dimensional It\^o semimartingale. In contrast to the setting in Jacod and Todorov(2009), we show that the additional conditional asymptotic variance terms appear, and give consistent estimation procedures for the asymptotic conditional variances in order to make the test feasible. Simulation experiments show that our asymptotic results give reasonable approximations in the finite sample cases.