| Financial practices often need to estimate an integrated volatility matrix of a large number of assets using noisy high-frequency financial data. This estimation problem is a challenging one for four reasons: (1) high-frequency financial data are discrete observations of the underlying assets' price processes; (2) due to market micro-structure noise, high-frequency data are observed with measurement errors; (3) different assets are traded at different time points, which is the so-called non-synchronization phenomenon in high-frequency financial data; (4) the number of assets may be comparable to or even exceed the observations, and hence many existing estimators of small size volatility matrices become inconsistent when the size of the matrix is close to or larger than the sample size.;In this dissertation, we focus on large volatility matrix inference for high-frequency financial data, which can be summarized in three aspects. On the methodological aspect, we propose a new threshold MSRVM estimator of large volatility matrix. This estimator can deal with all the four challenges, and is consistent when both sample size and matrix size go to infinity.;On the theoretical aspect, we study the optimal convergence rate for the volatility matrix estimation, by building the asymptotic theory for the proposed estimator and deriving a minimax lower bound for this estimation problem. The proposed threshold MSRVM estimator has a risk matching with the lower bound up to a constant factor, and hence it achieves an optimal convergence rate.;As for the applications, we develop a novel approach to predict the volatility matrix. The approach extends the applicability of classical low-frequency models such as matrix factor models and vector autoregressive models to the high-frequency data. With this approach, we pool together the strengths of both classical low-frequency models and new high-frequency estimation methodologies.;Furthermore, numerical studies are conducted to test the finite sample performance of the proposed estimators, to support the established asymptotic theories. |
