Kim Christensen, Mikkel Slot Nielsen, Mark Podolskij
arXiv 4 Mar 2021 · Econometrics · publishedStatistical Inference for Stochastic Processes (2022) · 5 citations (OpenAlex)
arXiv:2103.03237 · PDF · DOI · OpenAlex · Extracted main text
In this paper, we develop a penalized realized variance (PRV) estimator of the quadratic variation (QV) of a high-dimensional continuous It\^{o} 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 bootstrap 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.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Aït-Sahalia and Xiu (2019) Principal component analysis of high-frequency data | 1.000 | 5 | 3 | 100% |
| 2 | Aït-Sahalia and Xiu (2017) Using principal component analysis to estimate a high dimensional factor model with high-frequency data | 0.874 | 5 | 2 | 100% |
| 3 | Koltchinskii, Lounici, and Tsybakov (2011) Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion | 0.843 | 4 | 3 | 75% |
| 4 | Christensen, Podolskij, Thamrongrat, and Veliyev (2017) Inference from high-frequency data: A subsampling approach | 0.737 | 3 | 2 | 100% |
| 5 | Jacod and Protter (2012) Discretization of Processes | 0.737 | 3 | 2 | 100% |
| 6 | Barndorff-Nielsen and Shephard (2004) Econometric analysis of realized covariation: High frequency based covariance, regression, and correlation in financial economics | 0.644 | 2 | 2 | 100% |
| 7 | Lounici (2014) High-dimensional covariance matrix estimation with missing observations | 0.511 | 2 | 2 | 50% |
| 8 | Tropp (2011) Freedman's inequality for matrix martingales | 0.511 | 2 | 2 | 50% |
| 9 | Jacod (1994) Limit of random measures associated with the increments of a Brownian semimartingale | 0.511 | 2 | 1 | 100% |
| 10 | Tropp (2015) An introduction to matrix concentration inequalities | 0.511 | 2 | 1 | 100% |
Showing the top 10 of 50 scored citations.
arXiv econ.EM papers that cite this one, ranked by how heavily they lean on it.
| Citing paper | Intensity | Mentions | Sections | |
|---|---|---|---|---|
| 1 | Realized Regularized Regressions | 0.405 | 1 | 1 |