10473289%2E2011%2E637876

Journal of Business & Economic Statistics

ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20

Least Squares Inference on Integrated Volatility
and the Relationship Between Efficient Prices and
Noise
Ingmar Nolte & Valeri Voev
To cite this article: Ingmar Nolte & Valeri Voev (2012) Least Squares Inference on Integrated
Volatility and the Relationship Between Efficient Prices and Noise, Journal of Business &
Economic Statistics, 30:1, 94-108, DOI: 10.1080/10473289.2011.637876
To link to this article: http://dx.doi.org/10.1080/10473289.2011.637876

View supplementary material

Published online: 22 Feb 2012.

Submit your article to this journal

Article views: 269

View related articles

Citing articles: 1 View citing articles

Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=ubes20
Download by: [Universitas Maritim Raja Ali Haji]

Date: 11 January 2016, At: 22:29

Supplementary materials for this article are available online. Please go to http://tandfonline.com/r/JBES

Least Squares Inference on Integrated Volatility
and the Relationship Between Efficient Prices
and Noise
Ingmar NOLTE
Warwick Business School, Finance Group, Financial Econometrics Research Centre (FERC), Coventry, CV4 7AL
(Ingmar.Nolte@wbs.ac.uk)

Valeri VOEV

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:29 11 January 2016

School of Economics and Management, Aarhus University, 8000 Aarhus C, Denmark (vvoev@creates.au.dk)
The expected value of sums of squared intraday returns (realized variance) gives rise to a least squares
regression which adapts itself to the assumptions of the noise process and allows for joint inference on
integrated variance (IV), noise moments, and price-noise relations. In the iid noise case, we derive the
asymptotic variance of the IV and noise variance estimators and show that they are consistent. The joint
estimation approach is particularly attractive as it reveals important characteristics of the noise process
which can be related to liquidity and market efficiency. The analysis of dependence between the price
and noise processes provides an often missing link to market microstructure theory. We find substantial
differences in the noise characteristics of trade and quote data arising from the effect of distinct market
microstructure frictions. This article has supplementary material online.
KEY WORDS: High-frequency data; Jumps; Market microstructure; Realized volatility; Subsampling.

1.

INTRODUCTION

cle and highlight the differences (both theoretically and in the

simulations).
For the simplest case of iid noise, we derive the asymptotic
variance of the IV and noise variance estimators, and provide
explicit rules for the optimal choice of the number of frequencies. The analysis is then extended to the case of dependent,
but exogenous to the efficient price, noise process for which
the regression equation is augmented so that the autocovariance function of the noise is identified jointly with the IV. We
introduce an analytical tool related to the so-called volatility
signature plots (VSP), the Q-plot, which is a graphical representation of the estimates of IV against the lag length of the
noise autocovariance, used to guide the variable selection in the
OLS regression.
We further argue that an exogenous noise specification is not
suitable if we work with midquote prices and develop a stylized model of incomplete price adjustment which can explain
the puzzling downward bias of realized volatility (RV) found
in, for example, Hansen and Lunde (2006) and in our empirical work. The price adjustment mechanism is captured by a
parameter which reflects how much of the efficient price move
is incorporated into the observed price. Values smaller than 1
lead to a negative RV bias as the number of observations becomes large. We show how the proposed OLS regression can
be adapted to estimate the parameter in question and report empirical estimates consistent with the downward-sloping VSPs
obtained with midquote data.

Financial prices observed at high frequencies are known to
exhibit nonmartingale features often attributed to the so-called
market microstructure (MMS) noise. We provide a least squares
(LS) framework for the joint estimation of the integrated variance (IV) of the latent efficient price process and the parameters related to the characteristics of the noise process, based on
realized-type statistics computed at different frequencies. The
noise process is of interest in itself as highlighted in the works
of Hansen and Lunde (2006), Bandi and Russell (2006b), and
Oomen (2005), among others. Aı̈t-Sahalia and Yu (2009) found
that the magnitude of noise is related to an array of measures of
assets’ liquidity. By having a joint model, we can learn about
market microstructure rather than simply treating it as noise. The
observed price process is composed of signal plus noise, whose
interplay is clearly important in terms of our understanding of
financial markets.
The estimator we propose can be classified as a subsampling estimator, very much in the spirit of the two-scale realized volatility (TSRV) of Zhang, Mykland, and Aı̈t-Sahalia
(2005), the multiscale realized volatility (MSRV) of Zhang
(2006), and the multiscale discrete sine transform (MS-DST)
estimator of Corsi and Curci (2006). Other consistent approaches are the realized kernels (RK) of Barndorff-Nielsen
et al. (2008), which have been shown to be related to the
multiscale approach, and the preaveraging methodology of

Jacod et al. (2009). It should be noted that the possibility of
ordinary least squares (OLS) estimation of IV has been addressed in an independent study of Corsi and Curci (2006).
However, their main focus remained on the discrete sine transform of multiscale volatility measures and on iid noise specifications. We relate our results to theirs throughout the arti-

© 2012 American Statistical Association
Journal of Business & Economic Statistics
January 2012, Vol. 30, No. 1
DOI: 10.1080/10473289.2011.637876
94

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:29 11 January 2016

Nolte and Voev: LS Inference on IV

Another feature of financial prices that we analyze is the
presence of jumps. By using a jump-robust realized measure
such as (staggered) bipower variation as a dependent variable in
our regression, we can identify the contribution of the diffusive
component, the IV, and of the jump component, the jump variation (JV), to the total variation of the process. We indicate how a
noise-robust jump test can be performed but leave the theoretical

treatment of the problem for further research. More importantly,
our technique is the first to our knowledge that allows for joint
jump-robust estimation of the IV and the noise variance parameter. The potential of the LS methodology cannot be exhausted in
a single study. While we focus on variance estimation and noise
properties, one could possibly derive corresponding OLS regressions for other functionals of the diffusive component such
as integrated quarticity, etc. The framework can also be extended
to covariance estimation with nonsynchronous observations and
MMS noise as shown in Nolte and Voev (2008).
To evaluate the finite-sample performance of the proposed
methodology and benchmark it against other existing techniques, we conduct an extensive simulation experiment. The
results indicate that the precision of our estimators is comparable, and in many cases superior to that of the competing methods. We conduct an empirical analysis based on 27 stocks from
the DJIA index traded on New York Stock Exchange (NYSE)
and National Association of Securities Dealers Automated Quotations (NASDAQ). In order to assess the robustness of the
procedure, we compare IV estimates based on transaction and
midquote data which align closely, although the noise in both
types of data is of a very different nature.
The article is structured as follows: in Section 2 we introduce the notation, the theoretical framework, and the estimation
methodology, Sections 3 and 4 contain our simulation and empirical results, and Section 5 concludes. Proofs are collected in
an online appendix.
2.

THEORETICAL SETUP

The basic assumption is that we have irregularly spaced observations of a one-dimensional continuous time process pt ,
t ≥ 0, which is a noisy signal for an underlying process pt∗ :
pt = pt∗ + ut ,
where ut is the noise term. The process pt∗ satisfies the following
assumption:

t
Assumption 1. The process pt∗ = 0 σu dWu is a stochastic
volatility Brownian martingale process, where σt is a cádlág
stochastic process and {Wt : t ≥ 0} is a standard Brownian motion which is independent of σt for all t.
This assumption rules out leverage effects, and is needed so
that we can condition on the volatility path. We expect our results
to hold even in the presence of a leverage effect (indicated by
our simulation results), but this would make the proofs more
difficult.
integrated variation process of p∗ is given by

t The
2

IVt = 0 σu du. Our aim is to estimate the increment IV(a,b) =

b 2
a σu du = IVb − IVa for some predetermined choice of (a, b),
for example, a trading day. Henceforth, we assume that the

95

period of interest is a trading day with a = 0 and b = 1, and we
will omit a and b in the notation.
2.1

IID Noise

With respect to the noise process, we start off with the following assumption:
Assumption 2. The noise process ut satisfies the following
conditions:
(a) (a) ps∗ ⊥
⊥ ut , for all s and t; (exogeneity)
(b) (b) us ⊥
⊥ ut , for all s = t; (independence)

(c) (c) E[ut ] = 0, ω2 ≡ E[u2t ] < ∞, and µ4 ≡ E[u4t ] < ∞,
for all t.
While this assumption is unrealistic from an empirical point of
view, it is a convenient starting point for analyzing our methodology and establishing an asymptotic theory.
Consider an asset with N observations (transactions, quote
updates) within the period of interest. The grid of observation
times {tj }j =1,...,N , which we assume to be nonrandom, is divided
into subgrids {tj s+h }j =0,...,⌊ N−h
, where s = 1, . . . , S and h =
s ⌋
1, . . . , s. Here, {tj s+h }j =0,...,⌊ N−h
denotes the hth subgrid for a
s ⌋
sampling frequency of s ticks (e.g., with s = 2, we can have
two subgrids: the first one comprising the times {t1 , t3 , t5 , . . .},
and the second – the times {t2 , t4 , t6 , . . .}). In the following, we
index variables at times tj simply by j to ease the notation. For
each subgrid, we define the corresponding observed, efficient,
and noise s-tick returns for j = 1, . . . , ⌊ N−h
⌋ as

s
rj s+h ≡ pj s+h − p(j −1)s+h ,

∗
rj∗s+h ≡ pj∗s+h − p(j
−1)s+h ,

ej s+h ≡ uj s+h − u(j −1)s+h .
Since we allow for nonequidistantly spaced observations, we
need to make some assumptions regarding their regularity. Similar to Zhang (2006), we make the following assumption.
Assumption 3. The observation times {tj }j =1,...,N are nonrandom and satisfy maxj |tj − tj −1 | = O(1/N) as N → ∞.
Furthermore, the quadratic variation of time, H (t) (see Mykdiscussion
land and Zhang (2006) and Zhang (2006) for a 
of this concept), defined as H (t) = limN→∞ N tj ≤t (tj −
tj −1 )2 exists and is continuously differentiable with derivative
H ′ (t).
Assumption 3 effectively excludes the case of random and
endogenous times, whose effect on realized variance has been
studied in Li et al. (2009). Denote the number of returns on the
⌋ − 1. We define the realized

hth s-tick subgrid as Nh,s = ⌊ N−h
s
variance as a function of the number of returns on this subgrid
as
N

h,s

RV (Nh,s ) =

h,s


rj2s+h .

j =1

Under Assumptions 1–3, it holds that, conditional on the
volatility path (henceforth, all expectations are conditional on
the volatility path),


E RV h,s (Nh,s ) = IV + 2Nh,s ω2 ,
(1)

96

Journal of Business & Economic Statistics, January 2012

a result found in Hansen and Lunde (2006) and Bandi and
Russell (2006b). On the basis of the theoretical relationship in
Equation (1), we can derive an OLS regression of the form
h = 1, . . . , s,
(2)
where yh,s = RV h,s (Nh,s ), and the number of observations is
S(S + 1)/2. In the above regression, ĉ and β̂0 estimate IV and
2ω2 , respectively. Since the regressors are nonstochastic, endogeneity problems are precluded. Note that irrespective of the
degree of irregularity of the observation times, the variation in
the regressor Nh,s is mainly across s = 1, . . . , S, and that there
is only little variation for a fixed s for h = 1, . . . , s except for the
effect of rounding. The following theorem states the variance of
ĉ as an estimator of IV.

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:29 11 January 2016

yh,s = c + β0 Nh,s + εh,s ,

s = 1, . . . , S,

Theorem 1. Let N → ∞ and S = αN β for α > 0 and β ∈
[0.5, 1). Under Assumptions 1–3, the variance of ĉ is given by
Var[ĉ] =

η
β 2α2



β−1
N 1−2β (ln(N))−2 + δαN
 

discretization term
noise term







+ o N 1−2β (ln(N))−2 + o N β−1 ,
∗

. Further,
where η = 23 (π 2 a − 6(γ02 + 2γ1 )a ∗ ) and δ = 8c
15
a
= 12κω4 , a ∗ = 12κω4 − 4ω4 , c∗ = 2IQ, where IQ =
1 4 ′
0 σs H (s)ds, γ0 is the Euler–Mascheroni constant, γ1 is the
first Stieltjes constant, and κ = µ4 /3ω4 . The value of β determines which term dominates the expression asymptotically.
To gain more insight into the result mentioned above, we
discuss a corollary of Theorem 1.
Corollary 1.
(a) The highest speed of convergence of the estimator ĉ
is achieved when N 1−2β (ln(N ))−2 = N β−1 obtained for
), which converges to β = 2/3 from
βN = 32 (1 − ln(ln(N))
ln(N)
below. With S = αN βN , the asymptotic variance can be
expressed as


lim var N 1/6 (ln(N))1/3 ĉ
N→∞

= lim

η

N→∞ β 2 α 2
N

+ δα =

9η
+ δα.
4α 2

∗
Minimizing this expression with respect to α
gives α =
3 9η
for which var[N 1/6 (ln(N))1/3 ĉ] ≈ 2.48 3 δ 2 η.
2δ
(b) Assuming further a normal distribution (κ = 1) for the
noise process, we have that a = 12ω4 and a ∗ = 8ω4 .
Substituting in the expressions for η, δ results in η =
8ω4 (π 2 − 4(γ02 + 2γ1 )) and δ = 16IQ
and thus for S =
15
αN βN


lim Var N 1/6 (ln(N))1/3 ĉ
N→∞

8ω4 (π 2 − 4(γ02 + 2γ1 )) 16IQ
α
+
N→∞
15
βN2 α 2

= lim
=

18ω4 (π 2 − 4(γ02 + 2γ1 )) 16IQ
α.
+
α2
15

The optimal α, minimizing the expression above, is given
by

2
4
2
3 33.75ω (π − 4(γ0 + 2γ1 ))
.
(3)
α∗ =
IQ
Note that α ∗ is a noise-to-signal ratio, a quantity which plays
a role in the selection of subgrids and kernel length in the MSRV
and RK methods.
The OLS estimator has a slightly stronger convergence
√ compared to the TSRV, but does not achieve the optimal 4 N convergence of the MSRV and the RK. Given that we have heteroscedastic error terms, a weighed LS can achieve faster convergence. Our aim in this article, however, is not to propose the
ultimate integrated variance estimator, but rather to develop a
flexible framework for disentangling efficient price and noise
moments. Also, asymptotic results for slowly converging estimators can be a poor guide to their finite-sample performance.
Our simulations show that the OLS-RV estimator is in fact not
worse than the above-mentioned asymptotically more efficient
methods.
It is interesting to examine in more detail the relationship
between our OLS, Corsi and Curci’s (2006) multiscale least
squares (MSLS), and the MSRV estimator. All three estimators
can be written in the form
I
V =

S 
s


ws RV h,s (Nh,s ),

s=1 h=1

that is, as weighted averages of RV statistics, where the weights
ws are constant for h = 1, . . . , s. The only difference between
the approaches is 
in the 
weights that
have the properties
S 
S
s
s
S
ws
=
w
=
1
and
s
s=1 ws = 0 for all
h=1 s
h=1
s=1
s=1
three estimators, but behave differently as functions of s. For the
OLS estimator, the weights can be written as AB − ACN/s (see
the proof of Theorem 1 for the exact expressions for A, B, and
C which depend on S, but are constant for each s = 1, . . . , S).
The MSRV
weights are defined in Zhang (2006) as weights on
averages 1s sh=1 RV h,s (Nh,s ) and are quadratic in s. In the form
we have written the estimator, it follows from Equations (24)
and (25) in Zhang (2006) that the weights on the individual
RV h,s (Nh,s ) are linear in s. Finally, the MSLS weights differ
from our OLS weights since for the MSLS approach one uses
again averages over s as regressors. The sequence of weights for
the OLS, the MSRV, and the MSLS estimators is plotted in Figure 1. The lack of asymptotic efficiency of the OLS estimator is
due to the fact that ws are not linear in s and as such underweigh
RVh,s for small and large s, and overweigh RVh,s for s in the
middle range. As mentioned above, a weighted LS regression
can make the estimator more efficient. To follow up on this idea,
h,s
note that the leading
√ term in the variance of RV behaves as
1/s. Weighting by s implies linear weights ABs − ACN.
Assuming a normal distribution for the noise, the expression
for α ∗ in Corollary 1(b) depends only on IQ and ω2 . Having preliminary estimates for these quantities thus makes our approach
operational. In terms of practical implementation, we suggest
), rather than its limit β = 2/3, since
using βN = 23 (1 − ln(ln(N))
ln(N)
ln(ln(N))
converges
to
zero
extremely slowly. Taking this into
ln(N)

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:29 11 January 2016

Nolte and Voev: LS Inference on IV

97

Figure 1. Weights ws for the OLS-RV, the MSLS estimator of Corsi and Curci (2006), and the MSRV estimator for s = 1, . . . , 20. Note that
for each s, there are s values of ws (for h = 1, . . . , s) which are plotted as a single point. The online version of this figure is in color.

account in the calculation of α ∗ , we recommend in finite
samples:






15ω4 π 2 − 4 γ02 + 2γ1
3
∗
.
αN =
βN2 IQ
Regression (2) delivers another interesting statistic: ω̂2 =
β̂0 /2 as an estimator of ω2 . Since the noise variance is an important quantity, we derive the variance of ω̂2 in the following
theorem.
Theorem 2. Let N → ∞ and S = αN β for α > 0 and β ∈
[0.5, 1). The variance of ω̂2 is given by


1
c∗ α 3 3β−1
−2
a ∗ N −1 +
N
(ln(N))
var[ω̂2 ] =
4
30β 2



+ o (N −1 ) + o N 3β−1 (ln(N))−2 .

The two terms in the first bracket are of the same order, O(N −1 ),
if β = 23 (1 + ln(ln(N))
). It follows that for β < 23 (1 + ln(ln(N))
),
ln(N)
ln(N)
a∗
2
we have Var[ω̂ ] = 4N + o(N −1 ). In the case of normal noise
(κ = 1), a ∗ = 8ω4 , and the asymptotic variance of ω̂2 can be
written as limN→∞ Var[N 1/2 ω̂2 ] = 2ω4 .
The √
noise variance estimator ω̂2 converges at the parametric
speed N and has the same asymptotic variance as the ML
estimator derived in Aı̈t-Sahalia, Mykland, and Zhang (2005),
). Theorem 2 implies that choosing
whenever β < 23 (1 + ln(ln(N))
ln(N)
ln(ln(N))
2
2
) falls also in the optimal
βN = 3 (1 − ln(N) ) < 3 (1 + ln(ln(N))
ln(N)
range for β in terms of estimating ω2 .
2.2 Dependent Noise
We consider two types of dependence: in clock time, and in
tick time. We believe that the concept of clock-time dependence
is new and define it here.
2.2.1 Dependence in Clock Time. In high-frequency
datasets, observation times are always rounded to the nearest value of some minimal time increment (e.g., a second or
a millisecond). Let τ = 1, 2, . . . , τmax denote the clock time
where τmax is the maximal value of the clock for the observed

sample. For example, with NYSE TAQ data, observation times
are typically recorded as seconds elapsed since the opening
of the exchange (9:30 EST). Since the NYSE is open for 6.5
hours, a dataset containing observations of a single day has
τmax = 23,400. Until now, we worked with transaction times tj
which were standardized to take values in the interval [0, 1], so
that tj = 1/2 implies a transaction that occurred in the middle
of the trading day. We denote the clock time of the transaction
by τj = tj τmax so that continuing with the previous example
τj = 11,700. For an arbitrary t ∈ [0, 1], we define τ = ⌈tτmax ⌉,
where ⌈x⌉ denotes the nearest integer larger or equal to x. At
actual transaction times tj , we do not need the rounding, since
all transactions are recorded as a multiple of a minimal time increment. The clock-time-dependent noise satisfies the following
assumption:
Assumption 4. The noise process ut is dependent in clock
time if
(a) ps∗ ⊥
⊥ ut , for all s and t;
(b) E[ut ] = 0, for all t;
(c) ut is covariance stationary with autocovariance function
given by γ (q) = E[uτ uτ −q ], τ = ⌈tτmax ⌉, q = 1, 2, . . ..
Clock-time dependence is not the same as calendar-time dependence, since the lag q is not a fixed interval of calendar
time. Thus, q = 5 can represent 5-second or 5-millisecond dependence. Clock-time dependence has some of the features of
calendar-time dependence (since q is a measure of physical
time) and of tick-time dependence discussed below (since it
scales with the resolution of the time stamps and thus implicitly
with trading activity). The dependence structure is only defined
at the times at which the noise actually occurs (integer-valued
q), thus having an advantage over calendar-time dependence
which should be defined for any real q. The advantage over
tick-time dependence is that markets do operate in physical
time; a 1-second tick and a 1-minute tick are rather different
in terms of the amount of information processed by the market
over the particular time interval, and so it is to be expected that
the dependence structure in the noise process should take this
into account. We now consider the implications of clock-time

98

Journal of Business & Economic Statistics, January 2012

dependence for our regression framework. Under Assumptions
1, 3, and 4, it holds that
N

h,s



var[ej s+h ]
E RV h,s (Nh,s ) = IV +

∞

q=1

= IV + 2Nh,s γ (0),

Nh,s (q) (γ (0) − γ (q))

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:29 11 January 2016

≈ IV+2Nh,s γ (0)−2

Q


Nh,s (q)γ (q), (4)

q=1

where Nh,s (q) counts the number of
returns for the (h, s)-subgrid
spanning q time units, Nh,s (q) = j 1{τj s+h −τ(j −1)s+h =q} . This result is a straightforward extension of Equation (1) taking into
account that under Assumption 4, var[ej s+h ] = 2(γ (0) − γ (q)),
where q = τj s+h − τ(j −1)s+h . The approximation in Equation
(4) results from truncating the autocorrelation function at lag
Q. The equation could be made exact if one explicitly assumes
γ (q) = 0 for q > Q, for some positive Q. In terms of practical implementation, Q has to be chosen by the econometrician,
an issue we revisit later. The OLS regression corresponding to
Equation (4) takes the form
yh,s = c + β ′ xh,s + εh,s ,

s = 1, . . . , S,

N

h,s





var ej s+h
E RV h,s (Nh,s ) = IV +

j =1

j =1

= IV + 2

will have that

h = 1, . . . , s,
(5)

where yh,s = RV h,s (Nh,s ) and xh,s is the Q-dimensional vector given by xh,s = (Nh,s , Nh,s (1), . . . , Nh,s (Q))′ . As before, ĉ estimates IV, while now β̂0 , β̂1 , . . . , β̂Q estimate
2γ (0), −2γ (1), . . . , −2γ (Q), respectively. It is important to
note that clock-time dependence is applicable directly for a multivariate process. This is not the case for tick-time dependence,
which is discussed in the following section.
2.2.2 Dependence in Tick Time. The tick-time-dependent
noise satisfies the following assumption.
Assumption 5. The noise process ut is dependent in tick time

s ≥ Q̄,

which is essentially the iid noise framework (since in this case
γ (0) = ω2 ), where we have assumed that the noise in Q̄-tick returns can be considered to be iid. Given the evidence in Hansen
and Lunde (2006), Q̄ can be chosen so that there is approximately 1 minute between returns. The iid noise theory can
then be applied to the resulting Q̄-tick returns. As an alternative, if Q̄ is too large compared to an optimally selected S, we
recommend using some sparse sampling (as the approximately
1-minute sampling in Barndorff-Nielsen et al. (2008)) and applying the OLS regression for the iid noise case to the sparse
returns.
2.3 Endogenous Noise
We motivate the idea of endogenous noise from a market microstructure perspective. A feature of high-frequency data is that
prices do not always fully reflect all available information. This
phenomenon is observed, for example, in the trading model of
Campbell, Lo, and MacKinlay (1997) based on regular sampling
in the absence of trading activity. As Bandi and Russell (2006a)
argued, midquotes can be sticky at high frequencies implying
observed zero returns. One possibility for writing a model of
this kind is to assume that pj = pj −1 + φrj∗ . In this model, the
full information return rj∗ is only partially incorporated in the
current observed price pj . When φ = 0, prices are sticky and
observed returns are zero. A problem with this specification is
that it implies uj = (φ − 1)pj∗ . Since both the efficient price
and noise are Îto processes, the variation of pj∗ and uj is not
identifiable, a problem which is stressed by Aı̈t-Sahalia, Mykland, and Zhang (2006). In order to have an identifiable model
which allows for partial incorporation of information, we define
observed prices as
pj = pj∗−1 + φrj∗ + νj ,

if
(a) ps∗ ⊥
⊥ ut , for all s and t;
(b) E[ut ] = 0, for all t;
(c) ut is covariance stationary with autocovariance function
given by γ (q) = E[uj uj −q ].
The difference with respect to clock-time dependence is that
q now measures the number of ticks rather than the number of
time units between two observations. Under Assumptions 1, 3,
and 5, we have that
E[RV h,s (Nh,s )] = IV +

Nh,s


var[ej s+h ]

j =1

= IV + 2Nh,s (γ (0) − γ (s)) ,
and thus γ (0) − γ (s) cannot be identified without additional
assumptions. A possible identifying assumption is to postulate
that γ (q) = 0 for q ≥ Q̄ for some Q̄ > 0. Then for s ≥ Q̄, we

(6)

(7)

where φ plays the same role as above and νj is an additional
source of noise, which we assume to be iid with mean zero and
variance ω2 . Clearly, the case νj = 0 can be considered as a special case. The equation above implies that uj = (φ − 1)rj∗ + νj ,
a version of the specification suggested by Hansen and Lunde
(2006). In this respect, what we propose here is an intuitive reformulation of their specification, where the parameter φ represents
the degree of information processing. If φ = 1, then all information is incorporated fully and uj is an iid process. The implication of this noise specification for our OLS methodology is
the following: On the tick frequency (full grid), the return noise
process is given by ej = (φ − 1)(rj∗ − rj∗−1 ) + νj − νj −1 . It follows that E[ej2 ] = (φ − 1)2 (σj2 + σj2−1 ) + 2ω2 and E[ej rj∗ ] =
(φ − 1)σj2 , where σj2 is the variance of the one-tick return at

t
time tj , tjj−1 σs2 ds. The full-grid realized variance is given by

∗
2
∗2
RV(N) = N
j =1 (rj + 2rj ej + ej ), and thus
E [RV(N)] = IV + 2φ(φ − 1)IV + 2N ω2 ,

(8)

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:29 11 January 2016

Nolte and Voev: LS Inference on IV

99


2
where we have used as an approximation N
j =1 σj −1 = IV. If
prices adjust incompletely to the new information, φ ∈ [0, 1) so
that φ(φ − 1) ≤ 0, while φ > 1 would imply price overreaction.
Note that the polynomial φ(φ − 1) is symmetric around φ = 0.5
so that for identification purposes, we restrict the range of φ to
φ ≥ 0.5. To provide an intuition of why this arises, assume
for a moment that νj = 0, and consider the cases φ = 1 and
φ = 0, both of which lead to φ(φ − 1) = 0. These two cases are
(almost) indistinguishable in terms of their impact on the bias of
RV, since in the first case we observe the efficient price process
without noise, while in the second case, we have a one-tick
delay, that is, we observe pj∗−1 at tj . Thus, in the latter case, we
observe again all rj∗ , with the exception of the last return rN∗ . The

2
approximation N
j =1 σj −1 = IV makes both cases identical.
In order to analyze the impact of specification (7) on
our OLS regression, we need to derive E[RV h,s (Nh,s )]. We
have that by definition ej s+h = uj s+h − u(j −1)s+h , which in
this case results in ej s+h = (φ − 1)(rj∗s+h,1 − r(j∗ −1)s+h,1 ) +
νj s+h − ν(j −1)s+h , where rj∗s+h,1 = pj∗s+h − pj∗s+h−1 is a onetick return contrary to the s-tick return which we have de∗
fined earlier as rj∗s+h ≡ pj∗s+h − p(j
−1)s+h (and analogously for
2
2
2
∗
r(j −1)s+h,1 ). Thus, E[ejs+h ] = (φ − 1)2 (σjs+h
) + 2ω2
+ σ(j−1)s+h
∗
2
2
and E[ejs+h rjs+h ] = (φ − 1)σjs+h , where σj s+h (and analogously
σ(j2 −1)s+h ) is the variance of the one-tick return at time tj s+h ,

tj s+h 2
tj s+h−1 σs ds. It is important to stress that the dependence operates at the highest frequency, as a result of which the identifiability problems mentioned above do not arise. Using the definition
of the sparse-grid RV, it follows that


h,s



2

E RV (Nh,s ) = IV + (φ − 1)
+ (φ − 1)2

Nh,s

j =1

Nh,s

j =1

σj2s+h

σ(j2 −1)s+h + 2Nh,s ω2 . (9)

Given Assumptions 1 and 3, σj2s+h is of order O(1/N)
Nh,s 2
Nh,s 2
σ(j −1)s+h is
σj s+h + (φ − 1)2 j =1
and thus (φ 2 − 1) j =1
2
O(Nh,s /N). Clearly, the term 2Nh,s ω in Equation (9) dominates, so that RV h,s (Nh,s ) diverges to infinity as N → ∞. Can
we reconcile this with the downward sloping VSP obtained
for midquote data in Hansen and Lunde (2006) (and also in
our empirical analysis below)? We argue that specification (7)
can be seen as an encompassing model for which the restriction ω2 = 0 represents a model suitable for midquotes and the
restriction φ = 1 is representative for trade data. Specifying different models for both types of data can be further motivated
by the empirical studies of Dacorogna et al. (2001), BarndorffNielsen et al. (2009), and Oomen (2010), among others, documenting differences in the properties of transaction versus
midquote returns. Theoretically, one can substitute the assumption ω2 = 0 for midquote data with local-to-zero asymptotics of
the type ω2 = ω02 N −α with α ≥ 1 and a constant ω02 , discussed
by Barndorff-Nielsen et al. (2008). As they noted, however, this
is essentially equivalent to the ω2 = 0, that is, no noise, case. In
this sense, assuming ω2 = 0 is a simple way of formalizing that
the predominant source of noise in midquotes is due to incom-

plete price adjustment, consistent with downward-sloping VSP.
As far as transaction data are concerned, the assumption φ = 1
is inconsequential because as soon as there is an exogenous (not
necessarily iid) noise component of order O(1), it dominates the
effect of endogeneity of the type we have assumed. This follows
up to a first-order approximation
⎞
⎛
Nh,s
Nh,s


N ⎝ 2
σ(j2 −1)s+h ⎠ ≈ ζ (10)
σj2s+h + (φ − 1)2
(φ − 1)
Nh,s
j =1
j =1
for some constant ζ . Using this approximation, we can write


E RV h,s (Nh,s ) ≈ IV + Nh,s (2ω2 + ζ /N ).

It is clear that the slope coefficient in the OLS regression
(2) now estimates (2ω2 + ζ /N) which converges to 2ω2 as
N → ∞, while the intercept remains a consistent estimator of
IV. This analysis relates nicely to the discussion in Section 5.5
of Barndorff-Nielsen et al. (2008), who showed that the impact
of a linear endogeneity model of the type (7) is asymptotically
negligible in terms of estimation of IV. The above contemplations imply the following regression models for trade and quote
data, respectively:


with trade data, (11)
E RV h,s (Nh,s ) = IV + 2ω2 Nh,s
 h,s

E RV (Nh,s ) = IV + ζ Nh,s /N
with midquote data.
(12)

Can we learn something about the price adjustment coefficient
Nh,s 2
σj s+h ≈
φ from the estimate of ζ ? We argue that NNh,s j =1
N Nh,s 2
IV (and similarly Nh,s j =1 σ(j −1)s+h ≈ IV), considering that
1 Nh,s 2
j =1 σj s+h is an average per tick volatility which is then
Nh,s
scaled to a daily volatility by N (with constant volatility and
equidistant sampling we would have equality). Given that the
Nh,s ticks are spread over the whole day (i.e., the tick at time
tj s+h can be thought of as representative in terms of volatility for
the s ticks between t(j −1)s+h and tj s+h ) and that typically there is
not a large variation in durations, we think that this is a reasonable approximation. The implication is √
that ζ ≈ 2φ(φ − 1)IV.
IV ±

I V 2 +2I V ζ

. If ζ is negSolving for φ, we have that φ1/2 =
2I V
ative, as we would expect for midquote data, the root φ2 < 0.5,
which we assumed away for identification purposes. Thus, we
define an estimator of φ√
in terms of the coefficients in regression


2



V +2I V ζ̂
. We report estimates of φ for
model (12) as φ̂ = I V + 2II
V
midquote data in the empirical section.

2.4

Jumps

Until now we have assumed that p∗ has continuous sample
paths. We relax this assumption by introducing jumps into the
model so that now
 t
∗
pt =
σu dWu + Jt .
(13)
0

We follow Barndorff-Nielsen
 t and Shephard (2006) and assume
that Jt is given by Jt = N
j =1 cj , where Nt is a simple counting
process and cj are nonzero iid random variables. The quadratic
variation of p∗ can now be decomposed as a sum of the

100

Journal of Business & Economic Statistics, January 2012

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:29 11 January 2016


t
integrated variance IV ≡ u=0 σu2 du and the sum of squared
 Nt 2
jumps JV ≡ j =1 cj . In this section, we show how one can
adapt the OLS regression (2) to estimate the integrated variance
in the presence of jumps. The regression is based on the (staggered) bipower variation defined on the full grid as (see, e.g.,
Huang and Tauchen 2005)

 
N
N
BVi (N) = µ−2
|rj −(1+i) ||rj |,
1
N − 1 − i j =2+i

of our class of estimators against other consistent estimation
techniques, namely, the MS-DST of Corsi and Curci (2006),
the MSRV of Zhang (2006), the TSRV of Zhang, Mykland,
and Aı̈t-Sahalia (2005), and the RK of Barndorff-Nielsen et al.
(2008) using the modified Tukey-Hanning2 kernel, we resort to
Monte Carlo simulations. We employ an iid noise setup, since in
this case we have an asymptotic theory for the OLS-RV, MSRV,
TSRV, and the RK estimators, and a theoretically founded way
of choosing an optimal number of subgrids or kernel length.
The notation S is used to denote both the number of subgrids
and the number of realized autocovariances (kernel length) in
the RK framework. The simulation setup is borrowed from the
SV1FJ specification in Huang and Tauchen (2005) for which
the efficient price process is given by

√
where µ1 = 2/π is the mean of the absolute value of a standard normal random variable. The case i = 0 corresponds to
the “standard” bipower variation and i = 1, 2, . . . represents
the case when the returns are staggered by i ticks. The sparse
subgrid version of the bipower variation is given by

 
N
Nh,s
−2
(N
)
=
µ
|r(j −(1+i))s+h ||rj s+h |.
BVh,s
h,s
i
1
Nh,s − 1 − i j =2+i

p∗

dpt∗ = µdt + exp(β0 + β1 vt )dWt + dJt ,
dvt = αv vt dt + dWtv ,

v0 = 1/(−2α),

∗

where W p and W v are Brownian motion processes. We allow them to be correlated to examine whether the results are
affected by the presence of a leverage effect, which was assumed away to facilitate the proofs of the theoretical results.
Note that we also allow for drift, although in Assumption
1 we have µ = 0. The parameterization is also as in Huang
and Tauchen (2005) (with some slight modifications following
Barndorff-Nielsen et al. (2008)). We set µ = 0.03, β1 = 0.125,
αv = −0.025, and β0 = β12 /2αv , which is a normalization en
1
suring that E[ 0 σu2 du] = 1. The coefficient of correlation bep∗
tween dWt and dWtv takes values {0, −0.62}, representing the
no-leverage and leverage cases, respectively. The jump process,
Jt , is a compound Poisson process with constant jump intensity λ ∈ {0, 0.058} (representing the no-jump and jump cases,
respectively) and the jump sizes cj are N(0, 1). The noise process, u(t), is iid N(0, ω2 ), where the size of ω2 is not defined
in absolute terms,

but rather in terms of the noise-to-signal
1
ratio ξ 2 = ω2 / 0 σu4 du for which we assign values 0.0001
(low-noise regime), 0.001 (medium-noise regime), and 0.01
(high-noise regime). We also consider the case of noise due
to rounding, which is simply achieved by rounding the prices to
the nearest cent. This results in 16 scenarios that are summarized
in Table 1.
We conduct two experiments with 5,000 simulation runs,
summarized as follows:

If there is no noise and the price process is given by Equation
(13), BVi (N) is a consistent estimator of IV. The effect of MMS
noise is that rj and rj −1 are correlated so that the results in Huang
and Tauchen (2005) suggest that the bias of BV0 (N) depends,
in a complicated way, on a term measuring the expectation of
the product of the absolute value of two correlated Gaussian
variables. The consequence is that both ĉ and ω̂2 = β̂0 /2 from
regression (2) with yh,s ≡ BVh,s
0 (Nh,s ) are biased, as evidenced
in our simulation study. Given the iid noise assumption, however, rj and rj −2 are independent. With the additional assumption that ut is normally distributed, it is straightforward to show
(again, using the results in Huang and Tauchen 2005) that the
bias of the staggered (i = 1) bipower variation is simply 2N ω2 .
This has the nice implication that with yh,s ≡ BVh,s
1 (Nh,s ), we
obtain unbiased estimators of IV and ω2 , resulting in a joint
jump-robust estimation of both integrated variance and noise
variance. While jump- and noise-robust approaches for the estimation of IV do exist (see, e.g., Podolskij and Vetter 2009), we
believe ours to be the first result on jump-robust joint estimation
of IV and ω2 . As a final remark, we note that an estimator of JV
can be defined as ĉRV − ĉBV1 , that is the difference between ĉ
in regression (2), with RV and BV1 as dependent variables. The
corresponding test for jumps is beyond the scope of the article
and is left for further research.
3.

p0∗ = 0,

1. Set N = 23,400 (corresponding to 6.5 hours of second-bysecond data), vary S (the number of subgrids, or kernel
length) from 2 to 100 with a step of 2 (S = 2, 4, 6, . . . ,
100);
2. Let N vary from 1000 to 100,000 with a step of 1000, for a
total of 100 values. For each of the five estimators, choose

SIMULATION EVIDENCE

We classify our estimators depending on the variable used
as yh,s in Equation (2) as OLS-RV, OLS-BV0 and OLS-BV1 ,
BV0 and BV1 representing bipower and staggered bipower variation, respectively. To compare the finite-sample performance

Table 1. Summary of simulation scenarios. × indicates whether the particular effect is present. l, m, h, and r represent the low, medium, high,
and rounding noise scenarios, respectively
Scenario
Leverage
Jumps
Noise

1

l

2

m

3

h

4

5

6

7

8

r

×
l

×
m

×
h

×
r

9

10

11

12

13

14

15

16

×

×

×

×

l

m

h

r

×
×
l

×
×
m

×
×
h

×
×
r

Nolte and Voev: LS Inference on IV

101

Table 2. Relative RMSEs (in percentage) of the true value of IV. Smin denotes the number of subsamples (kernel length) for which the
corresponding minimum is achieved across the values of S considered in the simulation, S ∗ denotes the asymptotically optimal number of
subsamples (kernel length) using true values of IV, IQ, and ω2 , and Ŝ ∗ denotes the asymptotically optimal number of subsamples (kernel
length) based on estimated values of IV, IQ, and ω2 , following Barndorff-Nielsen et al. (2009)
V (Ŝ ∗ )

Smin

28
64
20
26
34

Scenario 2: No Lev., No Jumps, ξ 2 = 0.001
4.726
24
4.794
26
4.772
24
6.864
26
4.991
18
5.076
20
5.337
26
5.337
28
4.842
36
4.851
36

4.744
6.495
4.991
5.354
4.851

2
2
2
2
2

Scenario 4: No Lev., No Jumps, Round
1.591
2
1.591
2
1.591
2
1.591
2
1.602
4

1.591
1.591
1.591
1.591
1.791

92
32

Scenario 6: No Lev., Jumps, ξ 2 = 0.001
9.275
24
11.517
26
5.830
24
5.943
26

11.300
5.870

2
2

Scenario 8: No Lev., Jumps, Round
2.100
2
2.079
2

2.100
2.079

28
64
22
26
34

Scenario 10: Lev., No Jumps, ξ 2 = 0.001
4.714
24
4.791
26
4.791
24
6.905
26
4.963
18
5.069
18
5.305
26
5.305
26
4.815
36
4.820
36

4.733
6.536
5.069
5.305
4.820

2
2
2
2
2

Scenario 12: Lev., No Jumps, Round
1.590
2
1.589
2
1.589
2
1.590
2
1.597
4

1.590
1.589
1.589
1.590
1.765

92
32

Scenario 14: Lev., Jumps, ξ 2 = 0.001
9.334
24
11.626
26
5.882
24
5.994
26

11.403
5.921

2
4

Scenario 16: Lev., Jumps, Round
2.074
2
2.076
2

2.074
2.078

Smin
OLS-RV
MS-DST
MSRV
TSRV
RK

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:29 11 January 2016

OLS-BV0
OLS-BV1
OLS-RV
MS-DST
MSRV
TSRV
RK
OLS-BV0
OLS-BV1

V (Smin )

S∗

V (S ∗ )

Ŝ ∗

V (Smin )

S∗

V (S ∗ )

Ŝ ∗

V (Ŝ ∗ )

NOTE: Finally, V (x) is the value of the statistic (RMSE) at x. Due to the fact that in the simulations, S takes on only even values from 2 to 100, Smin is even by construction, while S ∗
and Ŝ ∗ are rounded to the nearest even number.

S in an optimal way, given the corresponding asymptotic
theory.
A summary of the results for the first simulation experiment
is presented in Table 2 for all scenarios with medium noise and
rounding. Scaling the noise parameter ξ 2 does not lead to qualitative differences and therefore, due to space limitations, the
results for the remaining scenarios are not reported here but are
available from the authors upon request. In each scenario, we
first report the value of S that minimized the root mean squared
error (RMSE) in the simulations (Smin ) and the associated value
of the RMSE. To address the applicability of the asymptotically
optimal rule for choosing S, we further report the infeasible
S ∗ , assuming that ω2 and IQ (and where necessary IV) are
known, the feasible Ŝ ∗ with estimated plug-in quantities, and
the associated RMSEs. Corsi and Curci (2006) did not provide
a rule for their estimator but, as it is a regression-type estimator,
we use the optimal S for our OLS estimator given in Corollary 1, which is also used for the OLS-BV0 and the OLS-BV1
estimators. If the source of noise is rounding, the true value of
ω2 is not available and hence the corresponding columns in the
table are missing. The results for the jump-robust estimators are
only reported in the scenarios with jumps. In these scenarios,
the other estimators measure the total quadratic variation and
are thus biased for IV. It is immediately clear from the table that
leverage does not play an important role.
Rounding can be seen as representative of a very small
nonnormal iid noise effect and, as such, is handled quite well.
As expected from the theoretical analysis in Section 2.4 the
OLS-BV0 is biased for IV, which is the reason for the fairly large
RMSE. The OLS-BV1 estimates IV very well, even though compared to the OLS-RV in the absence of jumps, there is a small

loss of efficiency. Interestingly, the asymptotically optimal rule
for the choice of S ∗ works rather well, although there is some
scope for improvement in the lines of Bandi and Russell (2008,
2009), who suggest a finite-sample RMSE correction for S ∗ .
Finally, it is evident that “borrowing” the value of S ∗ and using
it for the MS-DST as well as for the OLS-BV estimators is not a
good idea so that developing some sort of optimal rule of choice
would greatly improve their performance. Given the robustness
with respect to leverage and that ξ 2 = 0.001 is considered representative, for the second simulation experiment, we choose
scenarios 2 and 6. The results are illustrated in Figure 2. In
panel (a), we compare the OLS-RV against the four competing
estimators. The RMSEs of the RK and the OLS-RV are almost
identical (the lines practically overlap) with both estimators outperforming the TSRV, the MSRV, and the MS-DST. As we saw
in Experiment 1, the performance of the MS-DST depends on
the right choice of S, for which we do not have a selection criterion. Panel (b) compares our jump-robust estimators for which
it is again evident that the OLS-BV0 is biased while the OLSBV1 achieves a very low RMSE. Panels (c) and (d) illustrate
the RMSE of the estimators for ω2 in the absence/presence of
jumps.
√
As Theorem 2 shows, the OLS-RV estimator for ω2 is Nconsistent and, thus, much more precise than the estimator for IV
for large N, which is also the case for the MS-DST estimator. In
the presence of jumps, as discussed in Section 2.4 the OLS-BV0
is biased for ω2 and, therefore, the RMSE does not converge.
As claimed, the OLS-BV1 delivers a jump-robust estimate of
ω2 which appears to be consistent. In panel (e), we look at
the RMSE of the differences ĉRV − ĉBV0 and ĉRV − ĉBV1 as
estimators for JV. The estimator based on BV1 performs well as
expected.

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:29 11 January 2016

102

Journal of Business & Economic Statistics, January 2012

Figure 2. Summary of the results for simulation Experiment 2. Each panel illustrates the percentage RMSE of various estimators as a function
of N for (a) IV in the absence of jumps, (b) IV in the presence of jumps, (c) ω2 in the absence of jumps, (d) ω2 in the presence of jumps, and (e)
JV. The online version of this figure is in color.

4.

EMPIRICAL ANALYSIS

In this section, we apply the LS estimation framework to
high-frequency data (trades and quotes) of 27 stocks for the
period April 1, 2004, to July 31, 2008. We refer the reader to
Barndorff-Nielsen et al. (2009) for a description of the data
cleaning procedures. The ticker symbols used in the study can
be found in the first column of Table 3. The first 25 stocks trade
on NYSE and the last 2 on NASDAQ.
An empirical application of the LS estimation methodology
as proposed in this article requires a choice of Q and S. While
we provide an indication of how to choose S in Corollary 1,
the choice of Q should be data driven as it depends on the
strength of the serial dependence of the noise process. In order to analyze this dependence, we propose a graphical tool,
the Q-plot, very similar to the VSP mentioned in Section 2. A
Q-plot is a plot of the estimate of IV or ω2 against Q from the
regression in Equation (5), arising from the relation in Equation

(4). To ensure sufficient counts Nh,s (q), q = 1, . . . , Q when
Q is large (we let Q go up to 30), we use an arbitrary larger
value of S = 50, and we note that while a suboptimally chosen
S might increase the variance of the estimator, it cannot lead
to serious biases. The Q-plots are a guide for the value of Q
which should be chosen in practice so that the resulting estimate of IV is (at least approximately) unbiased. If Q is chosen
too large relative to the strength of the dependence in u, then
the estimate will be unbiased but its standard error will be increased (inclusion of irrelevant regressors). If, on the contrary,
Q is chosen too small relative to the strength of the dependence in u, then the estimate will be biased (omitted variable
bias). As an alternative, a suitable value for Q can be deduced
from autocorrelations of returns. Figures 3 and 4 are collections of Q-plots for the 27 stocks in our study for the trade and
quote data, respectively. For comparison, we also compute the
realized Parzen kernel as recommended in Barndorff-Nielsen
et al. (2009), with data sampled at ticks at approximately Q

Nolte and Voev: LS Inference on IV

103

Table 3. Results from the OLS regression (2) applied to the model in Equations (11) and (12)
Trades
ω̂2

Downloaded by [Universitas Maritim Raja Ali Haji] at 22:29 11 January 2016

ĉ
Parameter Quantile
AA
AIG
AXP
BA
BAC
C
CAT
CVX
DD
DIS
GE
GM
HD
IBM
JNJ
JPM
KO
MCD
MMM
MRK
PG
UTX
VZ
WMT
XOM
INTC
MSFT

0.05
0.96
0.40
0.31
0.57
0.28
0.36
0.67
0.45
0.51
0.54
0.31
0.69
0.63
0.41
0.24
0.41
0.32
0.48
0.40
0.52
0.33
0.51
0.43
0.48
0.45
0.86
0.47

0.5
2.04
1.14
0.75
1.21
0.69
0.90
1.50
1.32
1.20
1.25
0.68
2.47
1.36
0.91
0.59
0.95
0.70
1.15
0.95
1.23
0.70
1.15
1.11
0.94
1.16
1.88
1.01

Quotes

0.95
7.75
9.01
8.39
3.47
7.70
10.99
4.31
4.40
4.13
3.35
3.15
14.14
6.35
3.12
1.43
10.15
1.87
3.16
2.85
4.65
1.94
2.89
3.77
3.28
3.97
4.68
3.31

0.05
−0.90
−0.56
−0.38
−0.52
−0.26
−0.11
−0.98
−0.60
−0.41
0.54
0.52
−1.25
−0.09
−0.19
0.07
−0.33
0.19
−0.14
−0.64
−0.38
−0.14
−0.25
0.13
−0.04
−0.51
−0.26
0.05

0.5
0.78
0.28
0.29
0.29
0.42
0.42
0.01
0.13
0.54
1.42
0.96
0.63
0.76
0.30
0.42
0.63
0.71
0.77
0.07
0.59
0.32
0.45
0.89
0.45
0.16
1.73
1.53

ĉ
0.95
1.95
1.16
1.45
1.45
0.99
1.51
1.24
0.84
1.59
4.23
1.70
3.21
1.88
0.82
0.77
1.32
1.33
2.51
1.19
1.80
0.88
1.60
1.78
0.88
0.92
3.34
2.25

0.05
0.96
0.40
0.33
0.58
0.28
0.37
0.66
0.43
0.52
0.59
0.34
0.69
0.62
0.40
0.24
0.43
0.34
0.52
0.41
0.55
0.33
0.52
0.45
0.50
0.46
0.89
0.48

0.5
2.01
1.15
0.76
1.21
0.73
0.91
1.50
1.33
1.21
1.28
0.72
2.49
1.38
0.91
0.60
1.00
0.73
1.16
0.92
1.25
0.70
1.16
1.12
0.94
1.18
1.97
1.05

φ̂
0.95
7.66
8.66
8.53
3.50
7.92
11.16
4.27
4.35
4.17
3.39
3.19
13.81
6.39
3.20
1.42
9.88
1.90
3.14
2.83
4.45
1.92
2.89
3.78
3.21
4.01
4.80
3.40

0.05
0.69
0.68
0.68
0.67
0.69
0.67
0.63
0.62
0.67
0.80
0.75
0.67
0.71
0.67
0.73
0.71
0.77
0.70
0.63
0.70
0.67
0.71
0.73
0.71
0.63
0.65
0.65

0.5
0.94
0.91
0.94
0.92
0.98
0.92
0.88
0.90
0.95
1.07
1.00
0.91
0.94
0.93
0.9