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Title: Integrated volatility matrix estimation with nonparametric eigenvalue regularization Authors:  Cheng Qian - London School of Economics and Political Science (United Kingdom) [presenting]
Clifford Lam - London School of Economics and Political Science (United Kingdom)
Abstract: When the number of assets $p$ is large compared with the sample size $n$, trivially extending univariate volatility matrix estimators is not advised, since they are all modifications of a sample realized covariance matrix, which suffers from bias in its extreme eigenvalues under the high dimension framework. Without implicit assumptions on the structure of the true integrated volatility matrix, we propose a nonparametric eigenvalue regularization on the multi-scale (NER-MSRVM), the kernel (NER-KRVM) and the pre-averaging (NER-PRVM) realized volatility matrix estimators. We show that our regularization can shrink nonlinearly those extreme eigenvalues on all three estimators, and are positive definite in probability. Incidentally, the bias-corrected versions of kernel and pre-averaging estimators, which have faster rate of convergence at $n^{-1/4}$, but are not guaranteed to be positive definite, are now regularized to be positive definite in probability, and we prove their rates of convergence to an ``ideal'' estimator under the spectral norm are also at $n^{-1/4}$ in high dimension scenario. Our results are extended to a jump-diffusion model for the log-price processes with jumps removed using a previous wavelet method. All methods are applied to a simulated data and real data.