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We find the asymptotic distribution of the multi-dimensional multi-scale andkernel estimators for high-frequency financial data with microstructure.Sampling times are allowed to be asynchronous and endogenous. In the process,we show that the classes of multi-scale and kernel estimators for smoothingnoise perturbation are asymptotically equivalent in the sense of having thesame asymptotic distribution for corresponding kernel and weight functions. Thetheory leads to multi-dimensional stable central limit theorems and feasibleversions. Hence they allow to draw statistical inference for a broad class ofmultivariate models which paves the way to tests and confidence intervals inrisk measurement for arbitrary portfolios composed of high-frequently observedassets. As an application, we enhance the approach to construct a test forinvestigating hypotheses that correlated assets are independent conditional ona common factor.

Inference for Multi-Dimensional High-Frequency Data: Equivalence of  Methods, Central Limit Theorems, and an Application to Conditional  Independence Testing