Asymptotics for the systematic and idiosyncratic volatility with large dimensional high-frequency data

In this paper, we decompose the volatility of a diffusion process into systematic and idiosyncratic components, which are not identified with observations discretely sampled from univariate process. Using large dimensional high-frequency data and assuming a factor structure, we obtain consistent estimates of the Laplace transforms of the systematic and idiosyncratic volatility processes. Based on the discrepancy between realized bivariate Laplace transform of the pair of systematic and idiosyncratic volatility processes and the product of the two marginal Laplace transforms, we propose a Kolmogorov–Smirnov-type independence test statistics for the two components of the volatility process. A functional central limit theorem for the discrepancy is established under the null hypothesis that the systematic and idiosyncratic volatilities are independent. The limiting Gaussian process is realized by a simulated discrete skeleton process which can be applied to define an approximate critical region for an independence test.