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Journal of Business & Economic Statistics
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Comment
Ole E Barndorff-Nielsen & Neil Shephard
To cite this article: Ole E Barndorff-Nielsen & Neil Shephard (2006) Comment, Journal of Business & Economic Statistics, 24:2, 179-181, DOI: 10.1198/073500106000000099 To link to this article: http://dx.doi.org/10.1198/073500106000000099
Published online: 01 Jan 2012.
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Barndorff-Nielsen and Shephard: Comment 179
Andersen, T. G., Bollerslev, T., and Huang, X. (2005), “A Semiparametric Framework for Modeling and Forecasting Jumps and Volatility in Specu-lative Prices,” unpublished manuscript, Northwestern University and Duke University.
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Eraker, B. (2004), “Do Stock Prices and Volatility Jump? Reconciling Evidence From Spot and Option Prices,”Journal of Finance, 59, 1367–1403. Eraker, B., Johannes, M. S., and Polson, N. G. (2003), “The Impact of Jumps
in Volatility,”Journal of Finance, 58, 1269–1300.
Garcia, R., Ghysels, E., and Renault, E. (2004), “The Econometrics of Op-tion Pricing,” inHandbook of Financial Econometrics, eds. L. P. Hansen and Y. Aït-Sahalia, New York: Elsevier Science, forthcoming.
Johannes, M. (2004), “The Statistical and Economic Role of Jumps in Continuous-Time Interest Rate Models,”Journal of Finance, 59, 227–260. Pan, J. (2002), “The Jump-Risk Premia Implicit in Options: Evidence From an
Integrated Time Series Study,”Journal of Financial Economics, 63, 3–50. Schaumburg, E. (2004), “Estimation of Markov Processes With Levy-Type
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Todorov, V. (2005), “Econometric Analysis of Jump-Driven Stochastic Volatil-ity Models,” unpublished manuscript, Duke UniversVolatil-ity, Dept. of Economics.
Comment
Ole E. B
ARNDORFF-N
IELSENThe T.N. Thiele Centre for Mathematics in Natural Science, Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark (oebn@imf.au.dk)
Neil S
HEPHARDNuffield College, University of Oxford, Oxford OX1 1NF, U.K. (neil.shephard@nuffield.ox.ac.uk)
1. INTRODUCTION
The article by Hansen and Lunde provides an excellent win-dow through which researchers and new students can view how econometricians have been trying to nonparametrically estimate quadratic variation (QV) in the presence of market frictions. The story that the authors tell us demonstrates an enormous leap forward in recent years, underpinned by the availability of high-frequency data. The matching of continu-ous-time arbitrage-free price processes with the econometrics has driven this subject and provides a solid basis for fur-ther work. Our recent survey (Barndorff-Nielsen and Shephard 2007) provided an overview of recent work that has a rather different emphasis than that provided by Hansen and Lunde.
Fascinatingly, Hansen and Lunde have shown that as the-oretical research has progressed, the properties of friction on U.S. equity markets have changed due to the advent of new technology, which has reduced the tick size on many markets, dramatically reducing the variance of frictions and moderately increasing the temporal dependence. The result is that exist-ing no-noise analysis, such as the feasible central limit theorem (CLT) introduced by Barndorff-Nielsen and Shephard (2002), appears to give pretty good predictions of finite-sample be-havior when returns are measured through midpoints recorded every 10 or 20 minutes. Given that this was the goal of that research, it is rather pleasing to have it confirmed.
The current research agenda focuses on two issues:
1. Can we exploit even higher-frequency data than 10 mi-nutes to improve the efficiency of our estimator of QV? 2. Do these methods extend to the multivariate case?
We briefly discuss both of these issues. Finally, we make some general observations on the model used for the efficient price, a Brownian stochastic volatility model, and the role of jumps.
2. HIGHER–FREQUENCY DATA
Hansen and Lunde suggest that two steps are involved in tackling issue 1. For returns down to the level of 1 minute, a useful approximation is that frictions are uncorrelated with the underlying price. This is a helpful simplifying assumption that appears in much recent econometric theory on this subject, in-cluding the subsamplers of Zhou (1996), Zhang, Mykland, and Aït-Sahalia (2005), Zhang (2004), and Aït-Sahalia, Mykland, and Zhang (2005b) and the general kernel approach studied by Barndorff-Nielsen, Hansen, Lunde, and Shephard (2004). The finite-sample behavior of these estimators using 1-minute re-turn data is unclear, however, although we expect this issue to be clarified soon.
The gains from using 1-minute data rather than 10-minute data are less than might be expected. Under the 10-minute re-turn data, we know that the no-noise CLT provides a reasonable approximation, which says that
process updating the price every δ units of time. Hence if the no-noise assumption were true, then moving from 10 minutes to 1 minute would reduce the variance of the estimator by a fac-tor of 10. However, as Hansen and Lunde argue, when there is noise, one should change the estimator to make it more robust. A simple estimator for this is the kernel estimator. Barndorff-Nielsen et al. (2004) extended (1) to general kernels. In partic-ular, in the best-case scenario for the larger data approach, if
© 2006 American Statistical Association Journal of Business & Economic Statistics April 2006, Vol. 24, No. 2 DOI 10.1198/073500106000000099
180 Journal of Business & Economic Statistics, April 2006
there were no noise and if one were to use even weights and use five lags, then the following result would hold:
δ−1/2
Hence if one were to use 1-minute return data in (2) and 10-minute return data in (1), then the 1-minute–based kernel estimator would have approximately the same variance as the one based on 10-minute data. Of course, by using more sophis-ticated weights, one could make the kernel more efficient, but the general point would still hold. Using more data can have payoffs, but the econometrics are subtle.
A deeper issue is whether subsampling or kernels will con-tinue to work when the uncorrelatedness assumption between the efficient price and the frictions is removed. This will be needed when we examine returns recorded over finer time inter-vals than 1 minute. As far as we can see, this remains an open problem.
3. MULTIVARIATE PROBLEMS
The foregoing suggests that we may be rather skeptical of the research being carried out on bringing the effect of frictions to the forefront of this area of research. But we are not. First, as Hansen and Lunde have done, it is intellectually important to clarify the impact of frictions and to understand the possible robust estimators that are less influenced by market frictions. However, there is a second reason why we should be interested in this area of econometrics that is not mentioned by Hansen and Lunde. In our view, the main importance of this type of research comes, from the multivariate case. A rare econometrics article that discusses some of the multivariate issues is that of Bandi and Russell (2005), whereas the case of no noise was dealt with by Barndorff-Nielsen and Shephard (2004a).
It has been known since the work of Epps (1979) that when we compute realized covariances, the resulting realized cor-relation tends to dramatically fall as δ↓0, which should not occur from a continuous sample path viewpoint. This phenom-enon, which has been documented in great detail by Sheppard (2005), is probably caused by the fact that asset prices tend to not move simultaneously due to nonsynchronous trading, and the differential rate at which information of different types is absorbed into individual stock prices. The first of these effects has been studied extensively in empirical finance by, for ex-ample, Scholes and Williams (1977) and Lo and MacKinlay (1990). Martens (2003) provided a review of some of this work and more modern works, and made some contributions of his own as well. These new effects simply do not appear in the uni-variate case.
In the statistics and mathematical finance literature, much attention has been given to this type of problem. Interest-ing work on this topic includes that of Hayashi and Yoshida (2005) and Malliavin and Mancino (2002). The latter article has been influential, and applied work based on it includes that of Barucci and Reno (2002a,b), Reno (2003), Kanatani (2004a,b),
Precup and Iori (2005), Nielsen and Frederiksen (2005), and Mancino and Reno (2005). It seems an open issue as to whether subsampler or kernels will provide good estimators in this chal-lenging case.
4. NON–BROWNIAN SV MODELS
Finally, it seems to us that some discussion is needed on what should be called volatility and how should it be modeled. Sup-pose that we take the viewpoint of Carr and Wu (2004) that time-changed (non-Gaussian) Lévy processesYare sufficient to model the efficient log price process. LetLbe the Lévy process and letTbe the time change, and for concreteness letLbe the symmetric NIG Lévy process (Barndorff-Nielsen 1997), so that
Lt=(B◦S)t,
that is,Lt=BSt, whereBis Brownian motion andSis the
in-verse Gaussian subordinator. Then
Yt=B◦(S◦T)t.
What is the volatility process here—T orS◦Tor[Y]? Now suppose that we add an independent noise processZ
and let this be stationary and of the form
Zt=Z0+σ
t
0
gt−sdWs,
whereW is another Brownian motion and g is deterministic withg0=1. Then, letting
Using bipower variation (Barndorff-Nielsen and Shephard 2004b, 2006a), we can determineσ2t
0g 2
udu and hence[Y]t.
In other words, we can separate the variation in Y from the variation in Z, contrary to what is the case when Y follows a Brownian stochastic volatility model, for instance. Hence bipower variation joins subsampling and kernels as a potential way to deal with the effect of market frictions.
5. CONCLUSION
The article by Hansen and Lunde discusses one of the most interesting active areas in econometrics. Their detailed empir-ical work suggests various modeling approaches for the new theory remaining to be developed. We encourage econometri-cians to think about the multivariate case as well as being able to tackle the univariate examples.
ADDITIONAL REFERENCES
Bandi, F. M., and Russell, J. R. (2005), “Realized Covariation, Realized Beta and Microstructure Noise,” unpublished paper, Graduate School of Business, University of Chicago.
Barndorff-Nielsen, O. E. (1997), “Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling,”Scandinavian Journal of Statistics, 24, 1–14.
Diebold: Comment 181
Barndorff-Nielsen, O. E., and Shephard, N. (2004a), “Econometric Analysis of Realised Covariation: High-Frequency Covariance, Regression and Correla-tion in Financial Economics,”Econometrica, 72, 885–925.
(2004b), “Power and Bipower Variation With Stochastic Volatility and Jumps” (with discussion),Journal of Financial Econometrics, 2, 1–48. Barucci, E., and Reno, R. (2002a), “On Measuring Volatility and the GARCH
Forecasting Performance,”Journal of International Financial Markets, Insti-tutions and Money, 12, 182–200.
(2002b), “On Measuring Volatility of Diffusion Processes With High-Frequency Data,”Economic Letters, 74, 371–378.
Carr, P., and Wu, L. (2004), “Time-Changed Lévy Processes and Option Pric-ing,”Journal of Financial Economics, 71, 113–141.
Epps, T. W. (1979), “Comovements in Stock Prices in the Very Short Run,”
Journal of the American Statistical Association, 74, 291–296.
Hayashi, T., and Yoshida, N. (2005), “On Covariance Estimation of Non-Synchronously Observed Diffusion Processes,”Bernoulli, 11, 359–379. Kanatani, T. (2004a), “High-Frequency Data and Realized Volatility,”
unpub-lished doctoral thesis, Kyoto University, Graduate School of Economics. (2004b), “Integrated Volatility Measuring From Unevenly Sampled Observations,”Economics Bulletin, 3, 1–8.
Lo, A., and MacKinlay, C. (1990), “The Econometric Analysis of Nonsynchro-nous Trading,”Journal of Econometrics, 45, 181–211.
Malliavin, P., and Mancino, M. E. (2002), “Fourier Series Method for Measure-ment of Multivariate Volatilities,”Finance and Stochastics, 6, 49–61. Mancino, M., and Reno, R. (2005), “Dynamic Principal Component Analysis
of Multivariate Volatilites via Fourier Analysis,”Applied Mathematical Fi-nance, 12, 187–199.
Martens, M. (2003), “Estimating Unbiased and Precise Realized Covariances,” unpublished manuscript (EFA 2004 Maastricht Meetings Paper No. 4299). Nielsen, M. O., and Frederiksen, P. H. (2005), “Finite-Sample Accuracy of
In-tegrated Volatility Estimators,” unpublished manuscript, Cornell University, Dept. of Economics.
Precup, O. V., and Iori, G. (2005), “Cross-Correlation Measures in High-Frequency Domain,” unpublished manuscript, City University, Dept. of Eco-nomics.
Reno, R. (2003), “A Closer Look at the Epps Effect,”International Journal of Theoretical and Applied Finance, 6, 87–102.
Scholes, M., and Williams, J. (1977), “Estimating Betas From Nonsynchronous Trading,”Journal of Financial Economics, 5, 309–327.
Sheppard, K. (2005), “Measuring Realized Covariance,” unpublished manu-script, University of Oxford, Dept. of Economics.
Comment
Francis X. D
IEBOLDDepartment of Economics, University of Pennsylvania, PA 19104, and National Bureau of Economic Research (fdiebold@sas.upenn.edu)
The research program of Hansen and Lunde (HL) is gen-erally first rate, displaying a rare blend of theoretical prowess and applied sense. The present article is no exception. In a ma-jor theoretical advance, HL allow for correlation between mi-crostructure (MS) noise and latent price. (I prefer “latent price” to such terms as “efficient price” or “true price,” which carry a lot of excess baggage.) In a parallel major substantive advance, HL provide a pioneering empirical investigation of thenatureof the correlation between MS noise and latent price, document-ing a negative correlation at high frequencies. My admiration of the article hinges on the aforementioned contributions and is indeed most genuine. Nevertheless, much of what follows is rather critical of the extant literature, including certain key ele-ments of HL’s approach. My intention is for my criticism to be constructive, promoting and hastening additional progress.
1. ON THE DYNAMICS OF LATENT PRICE
HL work in the framework
p(t)=p∗(t)+ν(t) (1) and
dp∗(t)=µdt+σ (t)dW(t), (2) where p(t) is observed (log) price, p∗(t) is the latent (log) price,ν(t)is MS noise,µis a fixed expected return (actually HL go even farther and restrictµ=0), and dWt is an
incre-ment of standard Brownian motion. Without additional assump-tions, (1) is tautological, defining MS noise simply asν(t)=
p(t)−p∗(t). Hence everything hinges on the assumed specifi-cations ofp∗(t)andν(t)—neither of which is observable—and assumptions regarding their interaction.
Before HL’s work, the literature effectively focused on spec-ifications with latent price assumed to be uncorrelated with the MS noise,
corr(ν(t),dW(t))=0. (3) HL progress by allowing instead
corr(ν(t),dW(t))=ρ. (4) Importantly, their allowance for corr(ν(t),dW(t))=0 is in ac-cordance with both MS theory (more on this later) and with em-pirical fact (as HL emphasize). Nonetheless, HL’s specifications (1), (2), and (4) remain quite limited relative to one allowing for
time-varying expected returnsandjumps, as in
dp∗(t)=µ(t)dt+σ (t)dW(t)+κ(t)dq(t), (5) whereµ(t)is the time-varying expected return,κ(t)=p(t)−
p(t−)is jump size, andq(t)is a counting process with possibly time-varying intensityλ(t)such thatP[dq(t)=1] =λ(t)dt.
First, consider the possibility of time-varying expected re-turns. Note that p∗(t)is a real-world price, not a risk-neutral price, so there is no reason for p∗(t) to follow a martingale. Hence allowance for time-varying expected returns is impor-tant in principle. In practice, one might argue that, at least in high-frequency environments [e.g., hourly returns used to con-struct daily realized volatility (RV)], time variation in expected returns is likely to be negligible and thus can be safely ignored. Fair enough, but at least three caveats are in order. First, and
© 2006 American Statistical Association Journal of Business & Economic Statistics April 2006, Vol. 24, No. 2 DOI 10.1198/073500106000000143
