Identifying the Brownian Covariation from the Co-Jumps Given Discrete Observations

In this paper we consider two semimartingales driven by Wiener processes and (possibly infinite activity) jumps. Given discrete observations we separately estimate the integrated covariation IC from the sum of the co-jumps. The Realized Covariation (RC) approaches the sum of IC with the co-jumps as the number of observations increases to infinity. Our threshold (or truncated) estimator \hat{IC}_n excludes from RC all the terms containing jumps in the finite activity case and the terms containing jumps over the threshold in the infinite activity case, and is consistent. To further measure the dependence between the two processes also the betas, \beta^{(1,2)} and \beta^{(2,1)}, and the correlation coefficient \rho^{(1,2)} among the Brownian semimartingale parts are consistently estimated. In presence of only finite activity jumps: 1) we reach CLTs for \hat{IC}_n, \hat\beta^{(i,j)} and \hat \rho^{(1,2)}; 2) combining thresholding with the observations selection proposed in Hayashi and Yoshida (2005) we reach an estimate of IC which is robust to asynchronous data. We report the results of an illustrative application, made in a web appendix (on www.dmd.unifi.it/upload/sub/persone/mancini/WebAppendix3.pdf), to two very different simulated realistic asset price models and we see that the finite sample performances of \hat{IC}_n and of the sum of the co-jumps estimator are good for values of the observation step large enough to avoid the typical problems arising in presence of microstructure noises in the data. However we find that the co-jumps estimators are more sensible than \hat{IC}_n to the choice of the threshold. Finding criteria for optimal threshold selection is object of further research.