Estimating Covariation: Epps Effect, Microstructure Noise

This paper is about how to estimate the integrated covariance T of two assets over a fixed time horizon [0,T], when the observations of X and Y are "contaminated" and when such noisy observations are at discrete, but not synchronized, times. We show that the usual previous-tick covariance estimator is biased, and the size of the bias is more pronounced for less liquid assets. This is an analytic characterization of the Epps effect. We also provide the optimal sampling frequency which balances the tradeoff between the bias and various sources of stochastic error terms, including nonsynchronous trading, microstructure noise, and time discretization. Finally, a two scales covariance estimator is provided which simultaneously cancels (to first order) the Epps effect and the effect of microstructure noise. The gain is demonstrated in data.

[1]  Myron S. Scholes,et al.  Estimating betas from nonsynchronous data , 1977 .

[2]  David Aldous,et al.  On Mixing and Stability of Limit Theorems , 1978 .

[3]  W. Fuller,et al.  Distribution of the Estimators for Autoregressive Time Series with a Unit Root , 1979 .

[4]  T. W. Epps.  Comovements in Stock Prices in the Very Short Run , 1979 .

[5]  P. Hall,et al.  Martingale Limit Theory and Its Application , 1980 .

[6]  H. Rootzén.  Limit Distributions for the Error in Approximations of Stochastic Integrals , 1980 .

[7]  L. Devroye.  Laws of the Iterated Logarithm for Order Statistics of Uniform Spacings , 1981 .

[8]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[9]  Luc Devroye,et al.  A Log Log Law for Maximal Uniform Spacings , 1982 .

[10]  P. Hall,et al.  Martingale Limit Theory and its Application. , 1984 .

[11]  P. Phillips.  Testing for a Unit Root in Time Series Regression , 1988 .

[12]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[13]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[14]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[15]  P. McCullagh.  Tensor Methods in Statistics , 1987 .

[16]  C. Z. Wei,et al.  Asymptotic Inference for Nearly Nonstationary AR(1) Processes , 1987 .

[17]  Alan G. White,et al.  The Pricing of Options on Assets with Stochastic Volatilities , 1987 .

[18]  D. Aldous.  Probability Approximations via the Poisson Clumping Heuristic , 1988 .

[19]  A. Lo,et al.  An Econometric Analysis of Nonsynchronous Trading , 1989 .

[20]  Per A. Mykland,et al.  Bartlett Type Identities for Martingales , 1994 .

[21]  Robert F. Engle,et al.  The Econometrics of Ultra-High Frequency Data , 1996 .

[22]  S. Delattre,et al.  A central limit theorem for normalized functions of the increments of a diffusion process, in the presence of round-off errors , 1997 .

[23]  T. Bollerslev,et al.  ANSWERING THE SKEPTICS: YES, STANDARD VOLATILITY MODELS DO PROVIDE ACCURATE FORECASTS* , 1998 .

[24]  R. Gencay,et al.  Real-Time Trading Models and the Statistical Properties of Foreign Exchange Rates , 1998 .

[25]  P. Protter,et al.  Asymptotic error distributions for the Euler method for stochastic differential equations , 1998 .

[26]  George Tauchen,et al.  Using Daily Range Data to Calibrate Volatility Diffusions and Extract the Forward Integrated Variance , 1999 .

[27]  E. Ghysels,et al.  A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation , 2000 .

[28]  R. Gencay,et al.  An Introduc-tion to High-Frequency Finance , 2001 .

[29]  Lan Zhang,et al.  From martingales to ANOVA : implied and realized volatility , 2001 .

[30]  F. Diebold,et al.  The Distribution of Realized Exchange Rate Volatility , 2000 .

[31]  F. Diebold,et al.  The distribution of realized stock return volatility , 2001 .

[32]  GenÇay Ramazan,et al.  Real-Time Trading Models and the Statistical Properties of Foreign Exchange Rates*: REAL-TIME TRADING MODELS , 2002 .

[33]  Dudley,et al.  Real Analysis and Probability: Measurability: Borel Isomorphism and Analytic Sets , 2002 .

[34]  N. Shephard,et al.  Econometric analysis of realized volatility and its use in estimating stochastic volatility models , 2002 .

[35]  N. Shephard,et al.  Estimating quadratic variation using realized variance , 2002 .

[36]  Yacine Ait-Sahalia,et al.  How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise , 2003 .

[37]  Lan Zhang,et al.  A Tale of Two Time Scales , 2003 .

[38]  R. Oomen.  Modelling Realized Variance when Returns are Serially Correlated , 2004 .

[39]  Lan Zhang.  Efficient Estimation of Stochastic Volatility Using Noisy Observations: A Multi-Scale Approach , 2004, math/0411397.

[40]  P. Mykland,et al.  How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise , 2003 .

[41]  P. Mykland,et al.  Edgeworth Expansions for Realized Volatility and Related Estimators , 2005 .

[42]  N. Yoshida,et al.  On covariance estimation of non-synchronously observed diffusion processes , 2005 .

[43]  A. Lunde,et al.  Integrated Covariance Estimation using High-frequency Data in the Presence of Noise , 2006 .

[44]  N. Shephard,et al.  Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise , 2006 .

[45]  P. Mykland,et al.  ANOVA for diffusions and Itô processes , 2006, math/0611274.

[46]  Mark Podolskij,et al.  Estimation of Volatility Functionals in the Simultaneous Presence of Microstructure Noise and Jumps , 2006 .

[47]  N. Shephard,et al.  Subsampling Realised Kernels , 2007 .

[48]  Jean Jacod,et al.  Microstructure Noise in the Continuous Case: The Pre-Averaging Approach - JLMPV-9 , 2007 .

[49]  Mark Podolskij,et al.  Estimation of Volatility Functionals in the Simultaneous Presence of Microstructure Noise and Jumps , 2007, 0909.0827.

[50]  Neil Shephard,et al.  Designing Realised Kernels to Measure the Ex-Post Variation of Equity Prices in the Presence of Noise , 2008 .

[51]  N. Shephard,et al.  Multivariate Realised Kernels: Consistent Positive Semi-Definite Estimators of the Covariation of Equity Prices with Noise and Non-Synchronous Trading , 2010 .