Estimation of Long Memory in Integrated Variance

Abstract

A stylized fact is that realized variance has long memory. We show that, when the instantaneous volatility is a long memory process of order d, the integrated variance is characterized by the same long-range dependence. We prove that the spectral density of realized variance is given by the sum of the spectral density of the integrated variance plus that of a measurement error, due to the sparse sampling and market microstructure noise. Hence, the realized volatility has the same degree of long memory as the integrated variance. The additional term in the spectral density induces a finite-sample bias in the semiparametric estimates of the long memory. A Monte Carlo simulation provides evidence that the corrected local Whittle estimator of Hurvich et al. (2005) is much less biased than the standard local Whittle estimator and the empirical application shows that it is robust to the choice of the sampling frequency used to compute the realized variance. Finally, the empirical results suggest that the volatility series are more likely to be generated by a nonstationary fractional process.

Keywords:

• Local Whittle estimator
• Long memory stochastic volatility
• Measurement error
• Realized variance

JEL Classification:

• C10
• C22
• C80

ACKNOWLEDGEMENT

The authors are grateful to Kris Boudt, Niels Haldrup, Roberto Reno, and two anonymous referees for their valuable suggestions, and to Asger Lunde for having provided the data. We also thank the participants at the ICEEE’11 conference, the Econometric Society European Meeting 2011 in Oslo, and the XII Workshop on Quantitative Finance for useful comments. The authors are grateful to CREATES (Center for Research in Econometric Analysis of Time Series) (DNRF78), funded by the Danish National Research Foundation.

Notes

The literature on long memory processes in econometrics distinguishes between type I and type II fractional Brownian motion. These processes have been carefully examined and contrasted by Marinucci and Robinson (1999) and Davidson and Hashimzade (2009).

The volatility process σ²(t) coincides almost surely with .

For example, the estimates of long memory presented in Comte and Renault (1998) are larger than 1/2.

In order to obtain non-overlapping 's the first return included in the computation of is r _{t, 2, Δ} = p _{t−1+2Δ, Δ} − p _{t−1+Δ, Δ}.

Meddahi (2002) derives a closed form expression for for the class of eigenfunction stochastic volatility models and assuming that m(t) = μ.

The double integral in (12) can only be approximated for Δ → 0.

In this Section, we will maintain the assumption that the noise term is dynamically uncorrelated with the signal and it is a white noise. As shown in Section 2.1, this is the relevant when drift in price and leverage are excluded.

Comte et al. (2010) analyze the affine fractional stochastic volatility models and characterize the autocovariance function of the expected IV _t process.

Frederiksen et al. (2008) and Nielsen (2008) suggest to approximate the log-spectrum of the short-memory component of the signal and of the perturbation by means of an even polynomial term.

http://www.stat.lsa.umich.edu/sstoev/code/ffgn.m

We avoid the possible upward bias in the semiparametric estimates of d, due to the presence of large shifts as generated by changing bull and bear markets, during the 2008–2009 financial crisis.

Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lecr