Bernstein’s inequality; LASSO estimation; Low rank estimation; Quadratic variation; Rank recovery; Realized variance; Shrinkage estimator; Statistics and Probability
Abstract :
[en] In this paper, we develop a penalized realized variance (PRV) estimator of the quadratic variation (QV) of a high-dimensional continuous Itô semimartingale. We adapt the principle idea of regularization from linear regression to covariance estimation in a continuous-time high-frequency setting. We show that under a nuclear norm penalization, the PRV is computed by soft-thresholding the eigenvalues of realized variance (RV). It therefore encourages sparsity of singular values or, equivalently, low rank of the solution. We prove our estimator is minimax optimal up to a logarithmic factor. We derive a concentration inequality, which reveals that the rank of PRV is—with a high probability—the number of non-negligible eigenvalues of the QV. Moreover, we also provide the associated non-asymptotic analysis for the spot variance. We suggest an intuitive data-driven subsampling procedure to select the shrinkage parameter. Our theory is supplemented by a simulation study and an empirical application. The PRV detects about three–five factors in the equity market, with a notable rank decrease during times of distress in financial markets. This is consistent with most standard asset pricing models, where a limited amount of systematic factors driving the cross-section of stock returns are perturbed by idiosyncratic errors, rendering the QV—and also RV—of full rank.
Disciplines :
Mathematics
Author, co-author :
Christensen, Kim; Department of Economics and Business Economics, Aarhus University, Aarhus, Denmark
Nielsen, Mikkel Slot; Department of Mathematics, Aarhus University, Aarhus, Denmark
PODOLSKIJ, Mark ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
External co-authors :
yes
Language :
English
Title :
High-dimensional estimation of quadratic variation based on penalized realized variance
H2020 - 815703 - STAMFORD - Statistical Methods For High Dimensional Diffusions
Name of the research project :
Statistical Methods For High Dimensional Diffusions
Funders :
FP7 Ideas: European Research Council Danmarks Frie Forskningsfond Union Européenne
Funding number :
815703
Funding text :
Christensen and Nielsen were supported by the Independent Research Fund Denmark under grant 1028–00030B and 9056–00011B. Podolskij acknowledges funding from the ERC Consolidator Grant 815703 “STAMFORD: Statistical Methods for High Dimensional Diffusions”.
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