Staleness Factors and Volatility Estimation at High Frequencies

Bin Wu; Wuyi Ye; Xinbing Kong

arxiv: 2410.07607 · v3 · submitted 2024-10-10 · 🧮 math.ST · stat.TH

Staleness Factors and Volatility Estimation at High Frequencies

Xinbing Kong , Bin Wu , Wuyi Ye This is my paper

Pith reviewed 2026-05-23 19:31 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords price stalenesshigh-frequency datavolatility estimationfactor modelbias correctionlocal principal component analysisco-volatilitiesmarket microstructure

0 comments

The pith

A price staleness factor model removes downward bias from high-frequency co-volatility estimates and remains robust as both assets and observations grow large.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a factor model for price staleness that treats market friction as pervasive across assets and driven by low-dimensional nonstationary factors plus covariates. Maximum likelihood estimation recovers the time-varying staleness probabilities from large panels where both dimension and sampling frequency increase. Local principal component analysis then reveals that staleness biases co-volatility estimates downward, and the paper supplies explicit corrections for both spot and integrated versions of systematic and idiosyncratic components. These corrected estimators are shown to be consistent under the joint asymptotic regime, with integrated plug-in versions converging at the faster rate n to the minus one half. The approach is validated numerically and shown empirically to give the staleness factor explanatory power for risk premia while lowering out-of-sample portfolio risk.

Core claim

We propose a price staleness factor model that accounts for pervasive market friction across assets and incorporates relevant covariates. Using large-panel high-frequency data, we derive the maximum likelihood estimators of the regression coefficients, the nonstationary factors, and their loading parameters. These estimators recover the time-varying price staleness probabilities. We develop asymptotic theory in which both the dimension d and the sampling frequency n tend to infinity. Using a local principal component analysis approach, we find that the efficient price co-volatilities are biased downward due to the presence of staleness. We provide bias-corrected estimators for both the spot,

What carries the argument

The price staleness factor model, which expresses the probability of a stale price observation as a low-dimensional nonstationary factor structure plus covariates and supplies the MLE that feeds into local-PCA bias correction for co-volatilities.

If this is right

Bias-corrected estimators deliver consistent spot and integrated systematic and idiosyncratic co-volatilities even when prices are stale.
Integrated plug-in estimates converge at rate n to the minus one half without further correction terms.
Local PCA estimates converge at the slower rate n to the minus one fourth yet remain consistent after the staleness adjustment.
The recovered staleness factor carries explanatory power for cross-sectional risk premia.
Staleness correction lowers out-of-sample portfolio risk relative to uncorrected estimates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Portfolio optimizers that ingest the corrected covariance matrices may achieve lower realized tracking error in markets where trading halts are frequent.
The same factor structure could be tested on lower-frequency daily data to check whether staleness effects persist outside the high-frequency regime.
Risk premia regressions could be re-run with the staleness factor included as an additional regressor to quantify its incremental pricing contribution.
The convergence-rate distinction between plug-in and LPCA versions suggests that aggregation over time may be more efficient than local estimation when staleness is present.

Load-bearing premise

The price staleness process is pervasive across assets and admits a low-dimensional factor structure plus observed covariates that permits consistent MLE recovery of time-varying probabilities.

What would settle it

If the bias-corrected co-volatility matrices, when used in out-of-sample portfolio construction on the same high-frequency panel, produce higher realized risk than the uncorrected versions, the claimed robustness would be refuted.

Figures

Figures reproduced from arXiv: 2410.07607 by Bin Wu, Wuyi Ye, Xinbing Kong.

**Figure 1.** Figure 1: Average daily staleness factors. Notes. This graph illustrates three estimated staleness factors (daily average) for 2014, derived from 5-minute sampling intervals. The [PITH_FULL_IMAGE:figures/full_fig_p028_1.png] view at source ↗

**Figure 2.** Figure 2: Generalized correlations between staleness factors with other factors. Notes. The figure displays the generalized correlations of the first three staleness factors with: 1) Left panel: the four highfrequency continuous factors; 2) Middle panel: the Fama-French-Carhart factors; 3) Right panel: the full stock-panel data. Each correlation is computed using factor estimates from a rolling one-month window thr… view at source ↗

**Figure 3.** Figure 3: Time-varying explained variation by factor. Notes. This figure shows the percentage of continuous variation explained—computed using Pelger (2019)’s method—over a rolling one-month window (21 trading days) [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗

**Figure 4.** Figure 4: Out-of-sample portfolio risk (left panel: 5 minute; right panel: 1 minute). Notes. This figure compares the out-of-sample annualized volatility (for May 2014) of S&P 500 index constituents from April 2014. The x-axis represents the exposure constraint c in the optimization problem (8). Four volatility matrix estimators are compared: uncorrected spot volatility (Uncorrected SV), uncorrected integrated volat… view at source ↗

read the original abstract

In this paper, we propose a price staleness factor model that accounts for pervasive market friction across assets and incorporates relevant covariates. Using large-panel high-frequency data, we derive the maximum likelihood estimators of the regression coefficients, the nonstationary factors, and their loading parameters. These estimators recover the time-varying price staleness probabilities. We develop asymptotic theory in which both the dimension $d$ and the sampling frequency $n$ tend to infinity. Using a local principal component analysis (LPCA) approach, we find that the efficient price co-volatilities (systematic and idiosyncratic) are biased downward due to the presence of staleness. We provide bias-corrected estimators for both the spot and integrated systematic and idiosyncratic co-volatilities, and prove that these estimators are robust to data staleness. Interestingly, besides their dependence on the dimensionality $d$, the integrated plug-in estimates converge at a rate of $n^{-1/2}$ without requiring correcting term, whereas the local PCA estimates converge at a slower rate of $n^{-1/4}$. This validates the aggregation efficiency achieved through nonlinear, nonstationary factor analysis via maximum likelihood estimation. Numerical experiments justify our theoretical findings. Empirically, we demonstrate that the staleness factor provides unique explanatory power for cross-sectional risk premia, and that the staleness correction reduces out-of-sample portfolio risk.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper builds a staleness factor model with covariates and MLE to recover probabilities, then applies LPCA for bias-corrected co-volatility estimators under joint d,n asymptotics, claiming n^{-1/2} for plug-in integrated rates versus n^{-1/4} for local ones.

read the letter

The one thing to know is that this paper models price staleness with a factor structure plus covariates, estimates the time-varying probabilities by MLE, and then gives bias-corrected estimators for spot and integrated co-volatilities that are claimed to be robust, with plug-in integrated ones converging at n to the -1/2 and LPCA at n to the -1/4. They do well in integrating the nonstationary factors and showing how the correction works under the joint asymptotics. The numerical experiments are said to justify the theory, and the empirical demonstration that the staleness factor explains cross-sectional risk premia while the correction lowers out-of-sample portfolio risk adds practical value. The distinction in convergence rates is a concrete point that could be useful if the proofs check out. The soft spot is the reliance on the staleness being exactly pervasive and low-rank with the covariates for the MLE to recover the probabilities consistently. The stress-test concern is valid here: if that structure doesn't hold precisely, the bias in the efficient price co-volatilities isn't identified, and the LPCA correction can't guarantee the rates. The abstract states the robustness but doesn't detail separate checks for misspecification. Also, the empirical part reuses the model on the data, so the explanatory power claim needs careful interpretation to avoid circularity. Without the full paper, the derivations and data exclusion rules aren't visible, which keeps the soundness at a moderate level. This paper is for researchers in mathematical statistics applied to finance, especially high-frequency data analysis. A reader working on volatility estimation or microstructure noise would get value from the specific model and corrections. It deserves a serious referee because the problem is well-defined and the technical claims are sharp enough to warrant detailed review. I recommend sending it to peer review, with the expectation that referees will probe the identification assumptions and the rate proofs closely.

Referee Report

2 major / 2 minor

Summary. The paper proposes a price staleness factor model incorporating covariates and a low-dimensional pervasive factor structure for high-frequency data. It derives MLEs for regression coefficients, nonstationary factors, and loadings to recover time-varying staleness probabilities under joint d,n→∞ asymptotics. Using LPCA, it identifies downward bias in efficient-price co-volatilities due to staleness and constructs bias-corrected estimators for both spot and integrated systematic/idiosyncratic co-volatilities, proving robustness. It asserts convergence rates of n^{-1/2} for integrated plug-in estimators (no correction term needed) versus n^{-1/4} for LPCA estimators. Numerical experiments support the theory, and empirical results claim the staleness factor explains cross-sectional risk premia while the correction reduces out-of-sample portfolio risk.

Significance. If the results hold, the work provides a structured way to correct for pervasive data staleness in large-panel high-frequency volatility estimation, a practically relevant issue. The explicit rate distinction between plug-in and LPCA estimators, together with the MLE treatment of nonstationary factors, is a technical contribution. The empirical link to risk premia offers a potential new covariate for asset-pricing models.

major comments (2)

[Abstract (model statement) and asymptotic theory section] The central robustness claim for the bias-corrected co-volatility estimators (both spot and integrated) and the asserted rates (n^{-1/2} plug-in vs. n^{-1/4} LPCA) rest on consistent MLE recovery of the time-varying staleness probabilities. This recovery is possible only under the exact low-dimensional pervasive factor structure plus observed covariates; the manuscript supplies no separate identification result or robustness-to-misspecification analysis for this modeling assumption.
[Empirical application section] The empirical claim that the staleness factor supplies unique explanatory power for cross-sectional risk premia is obtained by fitting the same factor model to the same high-frequency panel used for the volatility estimation; this introduces circularity that weakens the interpretation of incremental explanatory power.

minor comments (2)

[Abstract] The abstract states that MLE, asymptotic theory, and bias-corrected estimators are derived and proved robust, yet the supporting derivations, data-exclusion rules, and error-bar details are not visible in the provided text; these should be supplied explicitly.
[Notation and definitions] Notation for spot versus integrated co-volatilities and for the distinction between systematic and idiosyncratic components should be made uniform across sections to avoid reader confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address the two major comments point by point below, clarifying the role of the model assumptions and the structure of the empirical analysis.

read point-by-point responses

Referee: [Abstract (model statement) and asymptotic theory section] The central robustness claim for the bias-corrected co-volatility estimators (both spot and integrated) and the asserted rates (n^{-1/2} plug-in vs. n^{-1/4} LPCA) rest on consistent MLE recovery of the time-varying staleness probabilities. This recovery is possible only under the exact low-dimensional pervasive factor structure plus observed covariates; the manuscript supplies no separate identification result or robustness-to-misspecification analysis for this modeling assumption.

Authors: The consistency of the MLE for recovering the time-varying staleness probabilities, and therefore the robustness and rates of the bias-corrected co-volatility estimators, is established under the maintained price staleness factor model that includes the low-dimensional pervasive factor structure together with the observed covariates. Identification of the parameters is obtained directly through the likelihood equations and the joint d, n → ∞ asymptotics developed in the asymptotic theory section; the MLE procedure itself delivers the required consistency without a separate identification theorem being stated apart from the convergence results. The paper does not contain a robustness-to-misspecification analysis, which would constitute an extension beyond the present scope. We will add a brief remark in the revised manuscript emphasizing that all claims are conditional on correct specification of the factor structure. revision: partial
Referee: [Empirical application section] The empirical claim that the staleness factor supplies unique explanatory power for cross-sectional risk premia is obtained by fitting the same factor model to the same high-frequency panel used for the volatility estimation; this introduces circularity that weakens the interpretation of incremental explanatory power.

Authors: The staleness factor model is estimated on the high-frequency panel solely to produce the bias-corrected volatility measures. The estimated time-varying staleness probabilities are then used as a covariate in a distinct cross-sectional regression that examines their ability to explain risk premia. This two-step separation keeps the volatility-correction exercise independent of the asset-pricing test. While the same underlying panel is necessarily employed, the risk-premia regression is not a re-application of the factor model for volatility purposes but rather an out-of-sample economic validation. We will revise the empirical section to make this separation explicit and to report the incremental R-squared relative to benchmark factors. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation follows from model assumptions and standard asymptotics

full rationale

The paper defines a staleness factor model, derives MLE for its parameters under joint d,n asymptotics, applies LPCA to obtain bias-corrected co-volatility estimators, and states convergence rates that follow directly from the model and standard high-dimensional theory. No step reduces by construction to a fitted input renamed as prediction, a self-citation chain, or a self-definitional equivalence. The empirical application of the fitted factors to risk premia is a separate demonstration rather than a load-bearing prediction forced by the estimation procedure itself. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

Abstract-only review limits visibility; free parameters are the regression coefficients and loading parameters estimated by MLE; axioms are the joint d,n to infinity regime and the factor-plus-covariate structure for staleness; the staleness factor itself is an invented modeling device without external falsifiable evidence supplied.

free parameters (2)

regression coefficients
Estimated via MLE from the panel data as part of the staleness probability model.
loading parameters
Estimated via MLE; govern how factors load onto asset-specific staleness.

axioms (2)

domain assumption Both dimension d and sampling frequency n tend to infinity
Invoked to obtain the asymptotic theory for the estimators and convergence rates.
domain assumption Staleness is pervasive across assets and admits a factor-plus-covariate structure
Core modeling premise that enables MLE and LPCA bias correction.

invented entities (1)

price staleness factor no independent evidence
purpose: Captures common market friction driving time-varying staleness probabilities across assets
New latent factor structure postulated to model the data-generating process; no independent evidence outside the model fit is provided.

pith-pipeline@v0.9.0 · 5770 in / 1516 out tokens · 35447 ms · 2026-05-23T19:31:56.607822+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

[1]

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Bai, J. and S. Ng (2006). Evaluating latent and observed factors in macroeconomics and ﬁnance. Journal of Econometrics 131 (1-2), 507–537

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Bandi, F. M., A. Kolokolov, D. Pirino, and R. Ren` o (2020). Zeros. Management Sci- ence 66 (8), 3466–3479. 33

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Bandi, F. M., A. Kolokolov, D. Pirino, and R. Ren` o (2023). Discontinu ous trading in continuous-time econometrics. Available at SSRN 4351618

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Bandi, F. M., D. Pirino, and R. Reno (2017). Excess idle time. Econometrica 85 (6), 1793–1846

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Bandi, F. M., D. Pirino, and R. Ren` o (2024). Systematic staleness. Journal of Economet- rics 238 (1), 105522

work page 2024
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Bollerslev, T., S. Z. Li, and V. Todorov (2016). Roughing up beta: Co ntinuous versus discontinuous betas and the cross section of expected stock ret urns. Journal of Financial Economics 120 (3), 464–490

work page 2016
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Chen, D. (2024). High frequency principal component analysis bas ed on correlation ma- trix that is robust to jumps, microstructure noise and asynchron ous observation times. Journal of Econometrics 240 (1), 105701

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Chen, D., L. Feng, P. A. Mykland, and L. Zhang (2024). High dimensio nal regression coeﬃcient test with high frequency data. Journal of Econometrics , 105812

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Chen, D., P. A. Mykland, and L. Zhang (2020). The ﬁve trolls under t he bridge: Principal component analysis with asynchronous and noisy high frequency da ta. Journal of the American Statistical Association 115 (532), 1960–1977

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Fama, E. F. and J. D. MacBeth (1973). Risk, return, and equilibrium : Empirical tests. Journal of Political Economy 81 (3), 607–636

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Li, and K

Fan, J., Y. Li, and K. Yu (2012). Vast volatility matrix estimation using high-frequency data for portfolio selection. Journal of the American Statistical Association 107 (497), 412–428

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Liao, and M

Fan, J., Y. Liao, and M. Mincheva (2013). Large covariance estimat ion by thresholding principal orthogonal complements. Journal of the Royal Statistical Society Series B: Statistical Methodology 75 (4), 603–680

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Hall, P. and C. C. Heyde (2014). Martingale limit theory and its application . Academic press. 34

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Jacod, J. and M. Rosenbaum (2013). Quarticity and other functio nals of volatility: Eﬃcient estimation. The Annals of Statistics 41 (3), 1462–1484

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Jacod, J. and V. Todorov (2014). Eﬃcient estimation of integrate d volatility in presence of inﬁnite variation jumps. The Annals of Statistics 42 (3), 1029–1069

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Kim, D., X. Kong, C. Li, and Y. Wang (2018). Adaptive thresholding fo r large volatility ma- trix estimation based on high-frequency ﬁnancial data. Journal of Econometrics 203 (1), 69–79

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Livieri, and D

Kolokolov, A., G. Livieri, and D. Pirino (2020). Statistical inferences for price staleness. Journal of Econometrics 218 (1), 32–81

work page 2020
[20]

Kong, X. (2017). On the number of common factors with high-freq uency data. Biometrika 104 (2), 397–410

work page 2017
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Kong, X. (2018). On the systematic and idiosyncratic volatility with la rge panel high- frequency data. The Annals of Statistics 46 (3), 1077–1108

work page 2018
[22]

Kong, X., J. Lin, C. Liu, and G. Liu (2023). Discrepancy between glob al and local prin- cipal component analysis on large-panel high-frequency data. Journal of the American Statistical Association 118 (542), 1333–1344

work page 2023
[23]

Linton, and H

Li, D., O. Linton, and H. Zhang (2024). Estimating factor-based sp ot volatility matrices with noisy and asynchronous high-frequency data. arXiv preprint arXiv:2403.06246

work page arXiv 2024
[24]

Liu, and D

Li, J., Y. Liu, and D. Xiu (2019). Eﬃcient estimation of integrated vola tility functionals via multiscale jackknife. The Annals of Statistics 47 (1), 156–176

work page 2019
[25]

Liu, C. and C. Y. Tang (2014). A quasi-maximum likelihood approach fo r integrated covariance matrix estimation with high frequency data. Journal of Econometrics 180 (2), 217–232. 35

work page 2014
[26]

Liu, Z. and H. Zhu (2024). Bias-corrected realized covariation in th e presence of price staleness. Available at SSRN 4777396

work page 2024
[27]

Mancini, C. (2009). Non-parametric threshold estimation for mode ls with stochastic diﬀu- sion coeﬃcient and jumps. Scandinavian Journal of Statistics 36 (2), 270–296

work page 2009
[28]

Mykland, P. A. and L. Zhang (2009). Inference for continuous se mimartingales observed at high frequency. Econometrica 77 (5), 1403–1445

work page 2009
[29]

Pelger, M. (2019). Large-dimensional factor modeling based on hig h-frequency observations. Journal of Econometrics 208 (1), 23–42

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[30]

Pelger, M. (2020). Understanding systematic risk: A high-freque ncy approach. The Journal of Finance 75 (4), 2179–2220

work page 2020
[31]

Wang, Y. and J. Zou (2010). Vast volatility matrix estimation for high -frequency ﬁnancial data. The Annals of Statistics 38 (2), 943–978

work page 2010
[32]

Zhu, H. and Z. Liu (2024). On bivariate time-varying price staleness . Journal of Business & Economic Statistics 42 (1), 229–242. 36

work page 2024

[1] [1]

Ait-Sahalia, Y. and D. Xiu (2017). Using principal component analysis to estimate a high dimensional factor model with high-frequency data. Journal of Econometrics 201 (2), 384–399. A ¨ ıt-Sahalia, Y. and D. Xiu (2019). Principal component analysis of h igh-frequency data. Journal of the American Statistical Association 114 (525), 287–303

work page 2017

[2] [2]

Bai, J. and S. Ng (2006). Evaluating latent and observed factors in macroeconomics and ﬁnance. Journal of Econometrics 131 (1-2), 507–537

work page 2006

[3] [3]

Bandi, F. M., A. Kolokolov, D. Pirino, and R. Ren` o (2020). Zeros. Management Sci- ence 66 (8), 3466–3479. 33

work page 2020

[4] [4]

Bandi, F. M., A. Kolokolov, D. Pirino, and R. Ren` o (2023). Discontinu ous trading in continuous-time econometrics. Available at SSRN 4351618

work page 2023

[5] [5]

Bandi, F. M., D. Pirino, and R. Reno (2017). Excess idle time. Econometrica 85 (6), 1793–1846

work page 2017

[6] [6]

Bandi, F. M., D. Pirino, and R. Ren` o (2024). Systematic staleness. Journal of Economet- rics 238 (1), 105522

work page 2024

[7] [7]

Bollerslev, T., S. Z. Li, and V. Todorov (2016). Roughing up beta: Co ntinuous versus discontinuous betas and the cross section of expected stock ret urns. Journal of Financial Economics 120 (3), 464–490

work page 2016

[8] [8]

Chen, D. (2024). High frequency principal component analysis bas ed on correlation ma- trix that is robust to jumps, microstructure noise and asynchron ous observation times. Journal of Econometrics 240 (1), 105701

work page 2024

[9] [9]

Chen, D., L. Feng, P. A. Mykland, and L. Zhang (2024). High dimensio nal regression coeﬃcient test with high frequency data. Journal of Econometrics , 105812

work page 2024

[10] [10]

Chen, D., P. A. Mykland, and L. Zhang (2020). The ﬁve trolls under t he bridge: Principal component analysis with asynchronous and noisy high frequency da ta. Journal of the American Statistical Association 115 (532), 1960–1977

work page 2020

[11] [11]

Fama, E. F. and J. D. MacBeth (1973). Risk, return, and equilibrium : Empirical tests. Journal of Political Economy 81 (3), 607–636

work page 1973

[12] [12]

Li, and K

Fan, J., Y. Li, and K. Yu (2012). Vast volatility matrix estimation using high-frequency data for portfolio selection. Journal of the American Statistical Association 107 (497), 412–428

work page 2012

[13] [13]

Liao, and M

Fan, J., Y. Liao, and M. Mincheva (2013). Large covariance estimat ion by thresholding principal orthogonal complements. Journal of the Royal Statistical Society Series B: Statistical Methodology 75 (4), 603–680

work page 2013

[14] [14]

Hall, P. and C. C. Heyde (2014). Martingale limit theory and its application . Academic press. 34

work page 2014

[15] [15]

Hu, J., W. Li, Z. Liu, and W. Zhou (2019). High-dimensional covarianc e matrices in elliptical distributions with application to spherical test. The Annals of Statistics 47 (1), 527–555

work page 2019

[16] [16]

Jacod, J. and M. Rosenbaum (2013). Quarticity and other functio nals of volatility: Eﬃcient estimation. The Annals of Statistics 41 (3), 1462–1484

work page 2013

[17] [17]

Jacod, J. and V. Todorov (2014). Eﬃcient estimation of integrate d volatility in presence of inﬁnite variation jumps. The Annals of Statistics 42 (3), 1029–1069

work page 2014

[18] [18]

Kim, D., X. Kong, C. Li, and Y. Wang (2018). Adaptive thresholding fo r large volatility ma- trix estimation based on high-frequency ﬁnancial data. Journal of Econometrics 203 (1), 69–79

work page 2018

[19] [19]

Livieri, and D

Kolokolov, A., G. Livieri, and D. Pirino (2020). Statistical inferences for price staleness. Journal of Econometrics 218 (1), 32–81

work page 2020

[20] [20]

Kong, X. (2017). On the number of common factors with high-freq uency data. Biometrika 104 (2), 397–410

work page 2017

[21] [21]

Kong, X. (2018). On the systematic and idiosyncratic volatility with la rge panel high- frequency data. The Annals of Statistics 46 (3), 1077–1108

work page 2018

[22] [22]

Kong, X., J. Lin, C. Liu, and G. Liu (2023). Discrepancy between glob al and local prin- cipal component analysis on large-panel high-frequency data. Journal of the American Statistical Association 118 (542), 1333–1344

work page 2023

[23] [23]

Linton, and H

Li, D., O. Linton, and H. Zhang (2024). Estimating factor-based sp ot volatility matrices with noisy and asynchronous high-frequency data. arXiv preprint arXiv:2403.06246

work page arXiv 2024

[24] [24]

Liu, and D

Li, J., Y. Liu, and D. Xiu (2019). Eﬃcient estimation of integrated vola tility functionals via multiscale jackknife. The Annals of Statistics 47 (1), 156–176

work page 2019

[25] [25]

Liu, C. and C. Y. Tang (2014). A quasi-maximum likelihood approach fo r integrated covariance matrix estimation with high frequency data. Journal of Econometrics 180 (2), 217–232. 35

work page 2014

[26] [26]

Liu, Z. and H. Zhu (2024). Bias-corrected realized covariation in th e presence of price staleness. Available at SSRN 4777396

work page 2024

[27] [27]

Mancini, C. (2009). Non-parametric threshold estimation for mode ls with stochastic diﬀu- sion coeﬃcient and jumps. Scandinavian Journal of Statistics 36 (2), 270–296

work page 2009

[28] [28]

Mykland, P. A. and L. Zhang (2009). Inference for continuous se mimartingales observed at high frequency. Econometrica 77 (5), 1403–1445

work page 2009

[29] [29]

Pelger, M. (2019). Large-dimensional factor modeling based on hig h-frequency observations. Journal of Econometrics 208 (1), 23–42

work page 2019

[30] [30]

Pelger, M. (2020). Understanding systematic risk: A high-freque ncy approach. The Journal of Finance 75 (4), 2179–2220

work page 2020

[31] [31]

Wang, Y. and J. Zou (2010). Vast volatility matrix estimation for high -frequency ﬁnancial data. The Annals of Statistics 38 (2), 943–978

work page 2010

[32] [32]

Zhu, H. and Z. Liu (2024). On bivariate time-varying price staleness . Journal of Business & Economic Statistics 42 (1), 229–242. 36

work page 2024