Source: Ann. Statist.
Volume 37, Number 4
We consider a bivariate process Xt=(Xt1, Xt2), which is observed on a finite time interval [0, T] at discrete times 0, Δn, 2Δn, …. Assuming that its two components X1 and X2 have jumps on [0, T], we derive tests to decide whether they have at least one jump occurring at the same time (“common jumps”) or not (“disjoint jumps”). There are two different tests for the two possible null hypotheses (common jumps or disjoint jumps). Those tests have a prescribed asymptotic level, as the mesh Δn goes to 0. We show on some simulations that these tests perform reasonably well even in the finite sample case, and we also put them in use for some exchange rates data.
 Aït Sahalia, Y. and Jacod, J. (2008). Testing for jumps in a discretely observed process. Ann. Statist. To appear.
 Andersen, T., Bollerslev, T. and Diebold, F. (2007). Roughing it up: Disentagling continuous and jump components in measuring, modeling and forecasting asset return volatility. Review of Economics and Statistics 89 701–720.
 Barndorff-Nielsen, O. E. and Shephard, N. (2004). Power and bipower variation with stochastic volatility and jumps. J. Financial Econometrics 2 1–37.
 Barndorff-Nielsen, O. E. and Shephard, N. (2006). Econometrics of testing for jumps in financial economics using bipower variation. J. Financial Econometrics 4 1–30.
 Barndorff-Nielsen, O. E. and Shephard, N. (2007). Variation, jumps and high frequency data in financial econometrics. In Advances in Economics and Econometrics. Theory and Applications, Ninth World Congress (R. Blundell, T. Persson and W. Newey, eds.). Cambridge Univ. Press.
 Bollerslev, T., Law, T. and Tauchen, G. (2008). Risk, jumps and diversification. J. Econometrics 144 234–256.
 Huang, X. and Tauchen, G. (2006). The relative contribution of jumps to total price variance. J. Financial Econometrics 4 456–499.
 Jacod, J. (2008). Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Process. Appl. 118 517–559.
 Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Springer, Berlin.
 Jacod, J., Kurtz, T. G., Méléard, S. and Protter, P. (2005). The approximate Euler method for Lévy driven stochastic differential equations. Ann. Inst. H. Poincaré Probab. Statist. 41 523–558.
 Jacod, J. (2008). Statistics and high-frequency data. SEMSTAT Course in La Manga. To appear.
 Jiang, G. and Oomen, R. (2008). A new test for jumps in asset prices. J. Econometrics. To appear.
 Lee, S. and Mykland, P. A. (2008). Jumps in financial markets: A new nonparametric test and jump clustering. Review of Financial Studies 21 2535–2563.
 Mancini, C. (2001). Disentangling the jumps of the diffusion in a geometric jumping Brownian motion. Giornale dell’Instituto Italiano degli Attuari LXIV 19–47.
 Mancini, C. (2006). Estimating the integrated volatility in stochastic volatility models with Lévy type jumps. Technical report, Univ. di Firenze.
 Merton, R. (1976). Otion pricing when underlying stock returns are discontinuous. J. Financial Economics 3 125–144.
 Woerner, J. (2006). Power and multipower variation: Inference for high frequency data. In Stochastic Finance (A. N. Shiryaev, M. do Rosário Grosshino, P. Oliviera, M. Esquivel, eds.) 343–354. Springer, Berlin.