- (2014). There, the asymptotic theory is pursued for continuous observations, but, once we have the illustration above for I−1 k , the analysis is the same. Using block-wise transformations which diagonalize Σkhn and transfer the noise level (5) to the identity matrix, i.e., Λkhn = OkHkhn Σkhn Hkhn O> k , with Ok being orthogonal matrices and Λkhn being diagonal, we can infer the asymptotic form via COV vec ˆ Σor s = bsh−1 n c+Kn X k=bsh−1 n c−Kn (2Kn + 1)−2 OkH−1 k −⊗2 ˜ I−1 k H−1 k O> k −⊗2 Z
Paper not yet in RePEc: Add citation now
- + O(K−1 n ), with a diagonalized version ˜ Ik of Ik. Along the same lines as in the proof of Corollary 4.3 in Bibinger et al. (2014), we derive that COV vec ˆ Σor s = (2 + O(1)) bsh−1 n c+Kn X k=bsh−1 n c−Kn (2Kn + 1)−2 × Σkhn ⊗ Σkhn H 1/2 + Σkhn H 1/2 ⊗ Σkhn
Paper not yet in RePEc: Add citation now
- ——— (2011): “Multivariate realised kernels: consistent positive semi-definite estimators of the covariation of equity prices with noise and non-synchronous trading,†Journal of Econometrics, 162, 149–169.
Paper not yet in RePEc: Add citation now
- ABADIR, K. M. AND J. R. MAGNUS (2005): Matrix algebra, vol. 1 of Econometric Exercises, Cambridge: Cambridge University Press.
Paper not yet in RePEc: Add citation now
- AIT-SAHALIA, Y. AND J. JACOD (2009): “Testing for jumps in a discretely observed process,†Annals of Statistics, 37, 184–222.
Paper not yet in RePEc: Add citation now
AIT-SAHALIA, Y., J. FAN, AND D. XIU (2010): “High-Frequency Estimates With Noisy and Asynchronous Financial Data,†Journal of the American Statistical Association, 105, 1504–1516.
AIT-SAHALIA, Y., L. ZHANG, AND P. A. MYKLAND (2011): “Ultra High Frequency Volatility Estimation with Dependent Microstructure Noise,†Journal of Econometrics, 160, 160–165.
ALTMEYER, R. AND M. BIBINGER (2014): “Functional Stable Limit Theorems for Efficient Spectral Covolatility Estimators,†SFB 649 Discussion Papers SFB649DP2014-005, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
ANDERSEN, T. G. AND T. BOLLERSLEV (1997): “Intraday Periodicity and Volatility Persistence in Financial Markets,†Journal of Empirical Finance, 4, 115–158.
ANDERSEN, T. G., D. DOBREV, AND E. SCHAUMBURG (2009): “Duration-Based Volatility Estimation,†Global COE Hi-Stat Discussion Paper Series gd08-034, Institute of Economic Research, Hitotsubashi University.
ANDERSEN, T. G., T. BOLLERSLEV, F. X. DIEBOLD, AND P. LABYS (2003): “Modeling and Forecasting Realized Volatility,†Econometrica, 71, 579–625.
- Φjk t (p) i −t (p) i−1 ! 1≤p≤d . (26) On the compact interval [0, 1], it can be assumed that kbsk, kÃsk, k˜ bsk, k˜ Ãsk are uniformly bounded. This is based on Jacod (2012), Lemma 6. 6 in Section 6. 3.
Paper not yet in RePEc: Add citation now
- B Stochastic Volatility Specifications for Simulation Study Following Huang and Tauchen (2005), we consider one- and two-factor stochastic volatility models for the efficient log-price process in (24a). The one-factor specification reads ˜ Ãt = exp(β0 + β1vt), dvt = αvtdt + dWt, (40) where Wt is a standard Brownian motion with Corr(dWt, dBt) = ÃÂ. The parameter values are chosen as β0 = 0, β1 = 0.125, α = −0.025 and à= −0.62.
Paper not yet in RePEc: Add citation now
BANDI, F. M. AND R. RENO (2009): “Nonparametric Stochastic Volatility,†Global COE Hi-Stat Discussion Paper Series gd08-035, Institute of Economic Research, Hitotsubashi University.
BARNDORFF-NIELSEN, O. E. AND N. SHEPHARD (2004): “Econometric Analysis of Realized Covariation: High Frequency Based Covariance, Regression, and Correlation in Financial Economics,†Econometrica, 72, 885–925.
BARNDORFF-NIELSEN, O. E., P. R. HANSEN, A. LUNDE, AND N. SHEPHARD (2009): “Realized Kernels in Practice: Trades and Quotes,†Econometrics Journal, 12, C1–C32.
BIBINGER, M. AND L. WINKELMANN (2013): “Econometrics of Co-Jumps in High-Frequency Data with Noise,†SFB 649 Discussion Papers SFB649DP2013-021, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
BIBINGER, M., N. HAUTSCH, P. MALEC, AND M. REISS (2014): “Estimating the Quadratic Covariation Matrix from Noisy Observations: Local Method of Moments and Efficiency,†Annals of Statistics, 42, 1312–1346.
- BILLINGSLEY, P. (1991): Probability and Measure, Springer, New York, 2 ed.
Paper not yet in RePEc: Add citation now
BOS, C. S., P. JANUS, AND S. J. KOOPMAN (2012): “Spot Variance Path Estimation and Its Application to High-Frequency Jump Testing,†Journal of Financial Econometrics, 10, 354–389.
- CFTC AND SEC (2010): “Findings Regarding the Market Events of May 6, 2010,†Tech. rep., U.S. Commodity Futures Trading Commission and U.S. Securities & Exchange Commission.
Paper not yet in RePEc: Add citation now
CHERNOV, M., A. R. GALLANT, E. GHYSELS, AND G. TAUCHEN (2003): “Alternative Models for Stock Price Dynamics,†Journal of Econometrics, 116, 225 – 257.
CHRISTENSEN, K., M. PODOLSKIJ, AND M. VETTER (2013): “On covariation estimation for multivariate continuous Itˆ o semimartingales with noise in non-synchronous observation schemes.†Journal of Multivariate Analysis, 120, 59–84.
CHRISTENSEN, K., R. OOMEN, AND M. PODOLSKIJ (2014): “Fact or Friction: Jumps at Ultra High Frequency,†Journal of Financial Economics, forthcoming.
- CURCI, G. AND F. CORSI (2012): “Discrete Sine Transform for Multi-Scales Realized Volatility Measures,†Quantitative Finance, 12, 263–279.
Paper not yet in RePEc: Add citation now
DE SANTIS, G. AND B. GERARD (1997): “International Asset Pricing and Portfolio Diversification with Time-Varying Risk,†The Journal of Finance, 52, 1881–1912.
- FAN, J. AND Y. WANG (2008): “Spot Volatility Estimation for High-Frequency Data,†Statistics and Its Interface, 1, 279–288.
Paper not yet in RePEc: Add citation now
- For the order of the weights we have by Lemma C.1 of Bibinger et al. (2014) uniformly over all k that kWj Hn k , Σkhn k . (log (n))−1 1 + j2 (nh2 n)−1 −2 . (27) We introduce the short notation t (p) i = (1/2) t (p) i + t (p) i−1
Paper not yet in RePEc: Add citation now
FOSTER, D. P. AND D. B. NELSON (1996): “Continuous Record Asymptotics for Rolling Sample Variance Estimators,†Econometrica, 64, 139–174.
GALLANT, A. R. (1981): “On the Bias in Flexible Functional Forms and an Essentially Unbiased Form : The Fourier Flexible Form,†Journal of Econometrics, 15, 211–245.
HANSEN, P. R. AND A. LUNDE (2006): “Realized Variance and Market Microstructure Noise,†Journal of Business & Economic Statistics, 24, 127–161.
HANSEN, P. R., J. LARGE, AND A. LUNDE (2008): “Moving Average-Based Estimators of Integrated Variance,†Econometric Reviews, 27, 79–111.
HAUTSCH, N. AND M. PODOLSKIJ (2013): “Preaveraging-Based Estimation of Quadratic Variation in the Presence of Noise and Jumps: Theory, Implementation, and Empirical Evidence,†Journal of Business & Economic Statistics, 31, 165–183.
HAYASHI, T. AND N. YOSHIDA (2011): “Nonsynchronous covariation process and limit theorems,†Stochastic Processes and their Applications, 121, 2416–2454.
- HUANG, R. AND T. POLAK (2011): “LOBSTER: Limit Order Book Reconstruction System,†Technical report, Humboldt-Universit at zu Berlin.
Paper not yet in RePEc: Add citation now
HUANG, X. AND G. TAUCHEN (2005): “The Relative Contribution of Jumps to Total Price Variance,†Journal of Financial Econometrics, 3, 456–499.
- Inserting the observed returns ∆iY as estimators of the noise increments ∆i gives consistent estimators of the variances. Sufficient conditions for a central limit theorem can easily be shown here by applying, for example, Theorem 27.4 from Billingsley (1991).
Paper not yet in RePEc: Add citation now
- JACOD, J. (2012): “Statistics and high frequency data.†Proceedings of the 7th S eminaire Europ een de Statistique, La Manga, 2007: Statistical methods for stochastic differential equations, edited by M. Kessler, A. Lindner and M. Sørensen.
Paper not yet in RePEc: Add citation now
- JACOD, J., Y. LI, P. A. MYKLAND, M. PODOLSKIJ, AND M. VETTER (2009): “Microstructure Noise in the Continous Case: the Pre-Averaging Approach,†Stochastic Processes and their Applications, 119, 2803–2831.
Paper not yet in RePEc: Add citation now
KALNINA, I. AND O. LINTON (2008): “Estimating Quadratic Variation Consistently in the Presence of Endogenous and Diurnal Measurement Error,†Journal of Econometrics, 147, 47–59.
- KOIKE, Y. (2014): “Quadratic covariation estimation of an irregularly observed semimartingale with jumps and noise,†preprint, arXiv:1408.0938.
Paper not yet in RePEc: Add citation now
KRISTENSEN, D. (2010): “Nonparametric filtering of the realized spot volatility: a kernel-based approach,†Econometric Theory, 26, 60–93.
LI, Y., P. A. MYKLAND, E. RENAULT, L. ZHANG, AND X. ZHENG (2014): “Realized Volatility when Sampling Times are Possibly Endogenous,†Econometric Theory, 30, 580– 605.
LONGIN, F. AND B. SOLNIK (2001): “Extreme Correlation of International Equity Markets,†The Journal of Finance, 56, 649–676.
- MANCINI, C., V. MATTIUSSI, AND R. RENO’ (2012): “Spot Volatility Estimation Using Delta Sequences,†Working Papers - Mathematical Economics 2012-10, Universita’ degli Studi di Firenze, Dipartimento di Scienze per l’Economia e l’Impresa.
Paper not yet in RePEc: Add citation now
- MYKLAND, P. A. AND L. ZHANG (2008): “Inference for Volatility-Type Objects and Implications for Hedging,†Statistics and Its Interface, 1, 255–278.
Paper not yet in RePEc: Add citation now
- OOMEN, R. C. (2006): “Comment on Hansen, P. R., and Lunde, A. (2006), “Realized Variance and Market Microstructure Noiseâ€Â,†Journal of Business & Economic Statistics, 24, 195–202.
Paper not yet in RePEc: Add citation now
- The two-factor model introduced by Chernov et al. (2003) allows for more pronounced movements in the instantaneous volatility by a feedback mechanism. The corresponding parameterization is ˜ Ãt = s–exp(β0 + β1v1,t + β2v2,t), (41) dv1,t = α1v1,tdt+dW1,t, dv2,t = α2v2,tdt + (1 + βvv2,t) dW2,t, s–exp(u) = ( exp(u) if u ≤ u0 exp(u0) p 1 − u0 + u2/u0 else, where W1,t and W2,t are standard Brownian motions with Corr(dW1,t, dBt) = ÃÂ1 and Corr(dW2,t, dBt) = ÃÂ2. We consider the configuration β0 = −1.2, β1 = 0.04, β2 = 1.5, α1 = −0.137e−2, α2 = −1.386, βv = 0.25, ÃÂ1 = ÃÂ2 = −0.3 and u0 = ln (1.5). C Tables and Figures
Paper not yet in RePEc: Add citation now
ZHANG, L. (2011): “Estimating Covariation: Epps Effect and Microstructure Noise,†Journal of Econometrics, 160, 33–47.
ZHANG, L., P. A. MYKLAND, AND Y. AIT-SAHALIA (2005): “A Tale of Two Time Scales: Determining Integrated Volatility With Noisy High-Frequency Data,†Journal of the American Statistical Association, 100, 1394–1411.
ZU, Y. AND P. H. BOSWIJK (2014): “Estimating Spot Volatility with High-Frequency Financial Data,†Journal of Econometrics, forthcoming. A Proofs A.1 Preliminaries Consider the process ˜ Xt = Z t Ãbsh−1 n chn dBs , (25) without drift and with block-wise constant volatility as a simplified approximation of X. In the following, we distinguish between the estimator of the spot covariance matrix (12) based on oracle optimal weights (13), ˆ Σor s , and the adaptive estimator ˆ Σs. Furthermore, we write ˆ Σs( ˜ X + ) for the estimator built from observations in the simplified model in which ˜ X is observed in noise and denote the associated spectral statistics by: ˜ Sjk = Àjh−1 n np X i=1 ˜ X (p) t (p) i + (p) i − ˜ X (p) t (p) i−1 − (p) i−1 !