How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise

TLDR: In this article, the authors show that the optimal sampling frequency is finite and derive its closed-form expression, and demonstrate that modelling the noise and using all the data is a better solution, even if one misspecifies the noise distribution.

Abstract: In theory, the sum of squares of log returns sampled at high frequency estimates their variance. When market microstructure noise is present but unaccounted for, however, we show that the optimal sampling frequency is finite and derive its closed-form expression. But even with optimal sampling, using say five minute returns when transactions are recorded every second, a vast amount of data is discarded, in contradiction to basic statistical principles. We demonstrate that modelling the noise and using all the data is a better solution, even if one misspecifies the noise distribution. So the answer is: sample as often as possible.

Figures

Figure 2 RMSE of the Estimator σ̂2 When the Presence of the Noise is Ignored

Figure 1 Various Discrete Sampling Modes — No Noise (Section 2), With Noise (Sections 3-6) and Randomly Spaced with Noise (Section 7).

Figure 4 RMSE of the Estimator σ̂2 When the Presence of Serially Correlated Noise is Ignored

Figure 3 Comparison of the Asymptotic Variances of the MLE σ̂2 Without and With Noise Taken into Account

Full text

NBER WORKING PAPER SERIES
HOW OFTEN TO SAMPLE A CONTINUOUS-TIME
PROCESS IN THE PRESENCE OF
MARKET MICROSTRUCTURE NOISE
Yacine Aït-Sahalia
Per A. Mykland
Working Paper 9611
http://www.nber.org/papers/w9611
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2003
Financial support from the NSF under grants SBR-0111140 (Aït-Sahalia) and DMS-0204639 (Mykland) is
gratefully acknowledged. The views expressed herein are those of the authors and not necessarily those of the
National Bureau of Economic Research.
©2003 by Yacine Aït-Sahalia and Per A. Mykland. All rights reserved. Short sections of text not to exceed
two paragraphs, may be quoted without explicit permission provided that full credit including ©notice, is
given to the source.

How Often to Sample a Continuous-Time Process in the Presence of
Market Microstructure Noise
Yacine Aït-Sahalia and Per A. Mykland
NBER Working Paper No. 9611
April 2003
JEL No. G12, C22
ABSTRACT
Classical statistics suggest that for inference purposes one should always use as much data as is
available. We study how the presence of market microstructure noise in high-frequency financial
data can change that result. We show that the optimal sampling frequency at which to estimate the
parameters of a discretely sampled continuous-time model can be finite when the observations are
contaminated by market microstructure effects. We then address the question of what to do about
the presence of the noise. We show that modelling the noise term explicitly restores the first order
statistical effect that sampling as often as possible is optimal. But, more surprisingly, we also
demonstrate that this is true even if one misspecifies the assumed distribution of the noise term. Not
only is it still optimal to sample as often as possible, but the estimator has the same variance as if
the noise distribution had been correctly specified, implying that attempts to incorporate the noise
into the analysis cannot do more harm than good. Finally, we study the same questions when the
observations are sampled at random time intervals, which are an essential feature of transaction-level
data.
Yacine Aït-Sahalia Per A. Mykland
Bendheim Center for Finance Department of Statistics
Princeton University University of Chicago
Princeton, NJ 08544-1021 Chicago, IL 60637-1514
and NBER Mykland@galton.uchicago.edu
yacine@princeton.edu

The notion that the observed transaction price in high frequency financial data is the unobserv-
able efficient price plus some noise component due to the imperfections of the trading process is
a well established concept in the market microstructure literature (see for instance Black (1986)).
In this paper, w e study the implications of such a data generating process for the estimation of
the parameters of the con tinuous-time efficient price process, using discretely sampled data on the
transaction price process. In particular, we focus on the effects of the presence of the noise for the
estimation of the variance of asset returns, σ
2
. In the absence of noise, it is well known that the
quadratic variation of the process (i.e., the average sum of squares of log-returns measured at high
frequency) estimates σ
2
. In theory, sampling as often as possible will produce in the limit a perfect
estimate of σ
2
. We show, however, that the situation changes radically in the presence of market
microstructure noise that is not taken into account in the analysis.
We start by asking whether it remains optimal to sample the price process as often as possible
in the presence of market microstructure noise, consistently with the basic statistical principle that,
ceteris paribus, more data is preferred to less. We show that, if noise is present but unaccounted
for, then the optimal sampling frequency is finite. The intuition for this result is as follows. The
volatility of the underlying efficient price process and the mark et microstructure noise tend to behav e
differently at different frequencies. Thinking in terms of signal-to-noise ratio, a log-return observed
from transaction prices over a tiny time in terval is mostly composed of mark et microstructure noise
and brings little information regarding the volatility of the price process since the latter is (at least in
the Brownian case) prop ortional to the time interval separating successive observations. As the time
interval separating the two prices in the log-return increases, the amount of market microstructure
noise remains constan t, since each price is measured with error, while the informational conten t of
volatility increases. Hence very high frequency data are mostly composed of mark et microstructure
noise, while the volatility of the price process is more apparent in longer horizon returns. Running
counter to this effect is the basic statistical principle mentioned above: in an idealized setting where
the data are observed without error, sampling more frequently cannot hurt. What we show is that
these two effects compensate each other and result in a finite optimal sampling frequency (in the
root mean squared error sense).
We then address the question of what to do about the presence of the noise. If, convinced by
either the empirical evidence and/or the theoretical market microstructure models, one decides to
accoun t for the presence of the noise, how should one go about doing it? We show that modelling
1

the noise term explicitly restores the first order statistical effect that sampling as often as possible
is optimal. But, more surprisingly, we also demonstrate that this is true even if one misspecifies
the assumed distribution of the noise term. If the econometrician assumes that the noise terms are
normally distributed when in fact they are not, not only is it still optimal to sample as often as
possible (unlike the result when no allowance is made for the presence of noise), but the estimator
has the same variance as if the noise distribution had been correctly specified. Put differently,
attempts to include a noise term in the econometric analysis cannot do more harm than good. This
robustness result, we think, is a major argument in favor of incorporating the presence of the noise
when estimating continuous time models with high frequency financial data, even if one is unsure
about what is the true distribution of the noise term. Finally, we study the same questions when the
observations are sampled at random time intervals, which are an essential feature of transaction-lev el
data.
Our results also have implications for the two parallel tracks that have developed in the recent
financial econometrics literature dealing with discretely observed continuous-time processes. One
strand of the literature has argued that estimation methods should be robust to the potential issues
arising in the presence of high frequency data and, consequen tly, be asymptotically valid without
requiring that the sampling interval ∆ separating successive observations tend to zero (see, e.g.,
Hansen and Scheinkman (1995), Aït-Sahalia (1996) and Aït-Sahalia (2002)). Another strand of the
literature has dispensed with that constrain t, and the asymptotic validity of these methods requires
that ∆ tend to zero instead of or in addition to, an increasing length of time T over which these
observations are recorded (see, e.g., Andersen, Bollerslev, Diebold, and Lab ys (2003), Bandi and
Phillips (2003) and Barndorff-Nielsen and Shephard (2002)).
The first strand of literature has been informally warning about the potential dangers of using
high frequency financial data without accounting for their inherent noise (see e.g., page 529 of Aït-
Sahalia (1996)), and we propose a formal modelization of that phenomenon. The implications of our
analysis are most salient for the second strand of the literature, which is predicated on the use of high
frequency data but does not account for the presence of market microstructure noise. Our results
show that the properties of estimators based on the local sample path properties of the process (such
as the quadratic variation to estimate σ
2
) change dramatically in the presence of noise, while at
the same time we suggest a robust approach to correcting for the presence of market microstructure
noise.
2

The paper is organized as follows. We start by describing in Section 1 our reduced form setup
and the underlying structural models that support it. We then review in Section 2 the base case
where no noise is present, before analyzing in Section 3 the situation where the presence of the
noise is ignored. Next, we show in Section 4 that accounting for the presence of the noise restores
the optimality of high frequency sampling. Our robustness results are presen ted in Section 5 and
in terpreted in Section 6. We incorporate random sampling intervals into the analysis in Section
7,andadrifttermin8. Sections9and10presenttwofurtherrelaxationofourassumptions,to
serially correlated and cross-correlated noise respectively. Section 11 concludes. All proofs a re in the
Appendix.
1. Setup
Our basic setup is as follows. We assume that the underlying process of interest, typ ically the
log-price of a security, is a time-homogenous diffusion on the real line
dX
t
= µ(X
t
; θ)dt + σdW
t
(1.1)
where X
0
=0,W
t
is a Brownian motion, µ(., .) is the drift function, σ
2
the diffusion coefficient and
θ the drift parameters, θ ∈ Θ and σ>0. The parameter space is an open and bounded set. As
discussed in Aït-Sahalia and Mykland (2003), the properties of parametric estimators in this model
are quite different depending upon we estimate θ alone, σ
2
alone, or both parameters together. When
the data are noisy, the main effects that we describe are already present in the simpler of these three
cases, where σ
2
alone is estimated, and so we focus on that case. Moreover, in the high frequency
contextwehaveinmind,thediffusiv e component of (1.1) is of order (dt)
1/2
while the drift component
is of order dt only, so the drift componen t is mathematically negligible at high frequencies. This is
validated empirically: including a drift actually deteriorates the performance of variance estimates
from high frequency data since the drift is estimated with a large standard error. Not cen tering
the log returns for the purpose of variance estimation produces more accurate results (see Merton
(1980)). So w e simplify the analysis one step further by setting µ =0, which w e do until Section 8,
where we then show that adding a drift term does not alter our results.
In that case,
X
t
= σW
t
. (1.2)
Un til Section 7, w e treat the case where we observations occur at equidistant time intervals ∆,in
3

Frequently asked questions

Q1. What have the authors contributed in "Nber working paper series how often to sample a continuous-time process in the presence of market microstructure noise" ?

This paper showed that the presence of market microstructure noise makes it optimal to sample less often than would otherwise be the case in the absence of noise. 

Q2. What is the likelihood function of the market microstructure noise?

Suppose that, instead of being iid with mean 0 and variance a2, the market microstructure noise follows dUt = −bUtdt+ cdZt where b > 0, c > 0 and Z is a Brownian motion independent ofW. 

Q3. What makes their situation unusual relative to quasi-likelihood?

What makes their situation unusual relative to quasi-likelihood is that the interest parameter σ2 and the nuisance parameter a2 are entangled in the same estimating equations (l̇σ2 and l̇a2 from the Gaussian likelihood) in such a way that the estimate of σ2 depends, to first order, on whether a2 is known or not. 

Q4. What is the likelihood function for the Y 0s?

The likelihood function for the Y 0s is then given byl(η, γ2) = − ln det(V )/2−N ln(2πγ2)/2− (2γ2)−1Y 0V −1Y, (4.1)where the covariance matrix for the vector Y = (Y1, ..., YN)0 is given by γ2V , whereV = [vij ]i,j=1,...,N = 1 + η2 η 0 · · · 0 η 1 + η2 η . . . ... 0 η 1 + η2 . . . 0 ... . . . . . . . . . η0 · · · 0 η 1 + η2 (4.2)Further,det(V ) = 1− η2N+2 1− η2 (4.3)and, neglecting the end effects, an approximate inverse of V is the matrix Ω = [ωij]i,j=1,...,N whereωij = ¡ 1− η2¢−1 (−η)|i−j|(see Durbin (1959)). 

Q5. What is the likelihood function of the X process?

Wτ i+1 −Wτ i ¢ are then iid N(0, σ2∆) so the likelihood function isl(σ2) = −N ln(2πσ2∆)/2− (2σ2∆)−1Y 0Y, (2.1)where Y = (Y1, ..., YN)0.. 

Q6. What is the way to solve the problem of the noise term?

The authors then addressed the issue of what to do about it, and showed that modelling the noise term explicitly restores the first order statistical effect that sampling as often as possible is optimal. 

Q7. What is the way to sample the noise term?

Their first finding in the paper is that there are situations where the presence of market microstructure noise makes it optimal to sample less often than would otherwise be the case in the absence of noise. 

Q8. What is the maximum likelihood of estimating 2?

The maximum-likelihood estimator of σ2 coincides with the discrete approximation to the quadratic variation of the processσ̂2 = 1T NX i=1 Y 2i (2.2)which has the following exact small sample moments:E £ σ̂2 ¤ = 1T NX i=1 E £ Y 2i ¤ =N ¡ σ2∆ ¢ T = σ2,V ar £ σ̂2 ¤ = 1T 2 V ar " NX i=1 Y 2i # = 1 T 2 Ã NX i=1 V ar £ Y 2i ¤! = N T 2 ¡ 2σ4∆2 ¢ = 2σ4∆ Tand the following asymptotic distributionT 1/2 ¡ σ̂2 − σ2¢ −→T−→∞ N(0, ω) (2.3)whereω = AV AR(σ̂2) = ∆E h −l̈(σ2) i−1 = 2σ4∆. (2.4)Thus selecting ∆ as small as possible is optimal for the purpose of estimating σ2. 

Q9. What is the second order correction term in the asymptotic variance?

A(2) is the base correction term present even with Gaussian noise in Theorem 2, and Cum4(U)B(2) is the further correction due to the sampling randomness. 

Q10. What is the caveat in the case of a serially correlated U?

The same caveat as in serially correlated U case applies: having modified the matrix γ2V, the artificial “normal” distribution would no longer use the correct second moment structure of the data. 

Q11. What is the usual estimate of asymptotic variance?

ThenV̂ = \\AVARnormal = µ − 1 T l̈(σ̂2, â2) ¶−1 is the usual estimate of asymptotic variance when the distribution is correctly specified as Gaussian. 

Q12. What is the asymptotic variance of the estimators?

D−1 σ2,a2 Sσ2,a2D −1 σ2,a2 . (8.3)The asymptotic variance of (σ̂2, â2) is thus the same as if µ were known, in other words, as if µ = 0, which is the case that the authors focused on in all the previous sections.