Optimal Combinations of Realised Volatility Estimators Andrew J. Patton and Kevin Sheppard University of Oxford 12 September 2008 Abstract Recent advances in nancial econometrics have led to the development of new estimators of asset price variability using frequently-sampled price data, known as realised volatility estima- torsor simply realised measures. These estimators rely on a variety of di/erent assumptions and take many di/erent functional forms. Motivated by the success of combination forecasts over individual forecasts in many forecasting applications, this paper presents a novel approach for combining individual realised measures to form new estimators of price variability, overcom- ing the obstacle that the quantity of interest is not observable, even ex post. In an application to high frequency IBM price data over the period 1996-2007, we consider 30 di/erent realised measures from 6 distinct classes of estimators. We nd that a simple equally-weighted average of these estimators is not signicantly out-performed, in terms of accuracy, by any individual es- timator. Moreover, we nd that none of the individual estimators encompasses the information in all other estimators, providing further support for the use of combination realised measures. Keywords: realised variance, volatility forecasting, forecast comparison, forecast combination. J.E.L. codes: C52, C22, C53. We thank Asger Lunde, Neil Shephard, and seminar participants at the third ESRC seminar on Nonlinear Economics and Finance at Keele University. Contact address: Department of Economics, and Oxford-Man Insti- tute of Quantitative Finance, University of Oxford, Manor Road, Oxford OX1 3UQ, United Kingdom. Email: [email protected] and [email protected].
Transcript
Optimal Combinations of Realised Volatility Estimators�
Andrew J. Patton and Kevin Sheppard
University of Oxford
12 September 2008
Abstract
Recent advances in �nancial econometrics have led to the development of new estimators of
asset price variability using frequently-sampled price data, known as �realised volatility estima-
tors�or simply �realised measures�. These estimators rely on a variety of di¤erent assumptions
and take many di¤erent functional forms. Motivated by the success of combination forecasts
over individual forecasts in many forecasting applications, this paper presents a novel approach
for combining individual realised measures to form new estimators of price variability, overcom-
ing the obstacle that the quantity of interest is not observable, even ex post. In an application
to high frequency IBM price data over the period 1996-2007, we consider 30 di¤erent realised
measures from 6 distinct classes of estimators. We �nd that a simple equally-weighted average
of these estimators is not signi�cantly out-performed, in terms of accuracy, by any individual es-
timator. Moreover, we �nd that none of the individual estimators encompasses the information
in all other estimators, providing further support for the use of combination realised measures.
The development of new estimators of asset price variability has been an active area of econometric
research in the past decade. These estimators, known as �realised volatility estimators�or �realised
measures�, exploit the information in high frequency data on asset prices (e.g., 5-minute prices) to
estimate the variability of the price process over a longer period, commonly one day. Older papers
in this �realised volatility� literature, such as Merton (1980), French, Schwert and Stambaugh
(1987) and Zhou (1996) recognised the bene�ts in increased accuracy from such an approach, and
recent work1 has built on this to propose estimators that are more e¢ cient, robust to market
microstructure e¤ects, and that can estimate the variation due to the continuous part of the price
process separately from the variation due to the �jump�part of the price process. See Andersen,
et al. (2006) and Barndor¤-Nielsen and Shephard (2007) for recent reviews of this rapidly-evolving
literature.
This paper seeks to answer the following simple question: do combinations of the above esti-
mators o¤er gains in average accuracy relative to individual estimators? It has long been known in
the forecasting literature that combinations of individual forecasts often out-perform even the best
individual forecast, see Bates and Granger (1969), Newbold and Granger (1974) and Stock and
Watson (2004), for example, and see Clemen (1989) and Timmermann (2006) for a more reviews
of this �eld2. Timmermann (2006) summarises three explanations for why combination forecasts
work well in practice: they combine the information contained in each individual forecast; they
average across di¤erences in the way individual forecasts are a¤ected by structural breaks; they
are less sensitive to possible mis-speci�cation of individual forecasting models (see also Clements
and Hendry, 1998, on forecast model mis-speci�cation). Each of these three points applies equally
to the problem of estimating price variability: individual realised measures use di¤erent pieces of
information from high frequency data, they may be di¤erently a¤ected by structural breaks (caused
by, for example, changes in the market microstructure), and they may be a¤ected to various de-
1See Andersen and Bollerslev (1998), Andersen, et al. (2001, 2003), Barndor¤-Nielsen and Shephard (2002, 2004,
2006), Aït-Sahalia, Mykland and Zhang (2005), Large (2005), Zhang, Mykland and Aït-Sahalia (2005), Bandi and
Russell (2006, 2008), Hansen and Lunde (2006a), Oomen (2006), Christensen and Podolskij (2007), Christensen,
Oomen and Podolskij (2008), Barndor¤-Nielsen, et al. (2008a,b) amongst others.2See Halperin (1961) and Reid (1968) for interesting early work on combining di¤erent estimates of a mean, and
di¤erent noisy estimates of GDP, as opposed to combining forecasts.
1
grees by mis-speci�cation. Thus there is reason to believe that a combination realised measure may
out-perform individual realised measures.
The theoretical contribution of this paper is to propose methods for constructing optimal com-
binations of realised measures, where optimality is de�ned formally below. The construction of
combination estimators for asset price variability (measured by its quadratic variation, QV) di¤ers
in an important way from the usual forecast combination problem: the task is complicated by the
fact that QV is not observed, even ex post. This means that measuring the accuracy of a given
estimator of QV, or constructing a combination estimator that is as accurate as possible, has to
be done using proxies (or, in our case, instruments) for the true latent QV. Our theoretical work
extends the data-based method for estimating the relative accuracy of realised measures suggested
by Patton (2008) to allow the estimation of optimal combination weights, or optimal combina-
tion functional forms more generally. Our methods use the time series aspect of the data (i.e.,
they are �large T�), which enables us to avoid making strong assumptions about the underlying
price process, but at the cost of having to employ some assumptions (such as standard mixing and
moment conditions) to ensure that a central limit theorem can be invoked.
The main contribution of this paper is to apply our combination methods to a collection of 30
di¤erent realised measures, across 6 distinct classes of estimators, estimated on high frequency data
on IBM over the period 1996-2007. We present results on the ranking of the individual estimators
and �simple� combination estimators such as the arithmetic mean, the geometric mean and the
median, over the full sample period and over three sub-samples (1996-1999, 2000-2003, 2004-2007),
using two distance measures, the mean squared error (MSE) and the QLIKE distance measure
described below. We use the step-wise hypothesis testing method of Romano and Wolf (2005) to
�nd the estimators that are signi�cantly better (and signi�cantly worse) than simple daily squared
returns, the standard 5-minute realised volatility estimator, or a simple equally-weighted average
of all estimators. We also use the �model con�dence set� (MCS) of Hansen, Lunde and Nason
(2005) to �nd the set of estimators that is not signi�cantly di¤erent from the best estimator. We
�nd that the MCS contains 4 to 10 estimators (for QLIKE and MSE respectively), and in both
cases includes a simple combination estimator. Furthermore, using the Romano-Wolf test, we �nd
that no realised measure signi�cantly out-performs the simple average in the full sample, or in any
sub-sample, while many signi�cantly under -perform the simple average estimator.
We also estimate optimal linear and multiplicative combination estimators, under both MSE
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and QLIKE, and examine which individual realised measures enter signi�cantly into the optimal
combination forecast, or enter with non-zero weight in an optimal constrained forecast. We �nd
that weight is given to a variety of realised measures, including both simple and more sophisticated
estimators. Importantly, we �nd that none of the individual estimators encompasses the information
in all other estimators, providing further support for the use of combination realised measures.
Overall our results suggest that there are bene�ts from combining the information contained in the
array of di¤erent volatility estimators proposed in the literature to date.
2 Combining realised measures
2.1 Notation
The latent target variable, generally the quadratic variation (QV) or integrated variance (IV) of
an asset price process, is denoted �t. We assume that �t is a Ft-measurable scalar, where Ftis the information set generated by the complete path of the log-price process. The estimators
(�realised measures� or �realised volatility estimators�) of �t are denoted Xi;t, i = 1; 2; :::; n. In
addition to being estimators with di¤erent functional forms, these may include the same type of
estimator applied to data sampling at di¤erent frequencies (e.g., standard RV estimated on 1-
minute or 5-minute data). Let g (Xt;w) denote a parametric combination estimator, where w is a
�nite-dimensional vector of parameters to be estimated from the data.
De�ning an �optimal� combination estimator requires a measure of accuracy for a given esti-
mator. Two popular measures in the volatility literature are the MSE and QLIKE measures:
MSE L (�;X) = (� �X)2 (1)
QLIKE L (�;X) =�
X� log
��
X
�� 1 (2)
The QLIKE distance measure is a simple modi�cation of the familiar Gaussian log-likelihood, with
the modi�cation being such that the minimum distance of zero is obtained when X = �. Our
result below will be shown to hold for a more general class of distance measures, namely the class
of �robust�pseudo-distance measures proposed in Patton (2006):
L (�;X) = ~C (X)� ~C (�) + C (X) (� �X) (3)
with C being some function that is decreasing and twice-di¤erentiable function on the supports of
both � and X, and where ~C is the anti-derivative of C. In this class each pseudo-distance measure
3
L is completely determined by the choice of C, and MSE and QLIKE are obtained (up to location
and scale constants) when C (z) = �z and C (z) = 1=z respectively. Finally, it is convenient to
introduce the following quantities:
L� (w) � E [L (�t; g (Xt;w))] (4)
~L� (w) � E [L (Yt; g (Xt;w))] (5)
�LT (w) � 1
T
TXt=1
L (Yt; g (Xt;w)) (6)
where the dependence of �LT ; ~L� and L� on the function g is suppressed for simplicity. Yt is an
observable proxy for the latent �t; and is further discussed in the following section.
2.2 Estimating optimal combinations of realised measures
In this section we provide the theory underlying the estimation of optimal combination estimators,
building on Patton (2008) who considered rankings of realised measures. We will denote a generic
combination estimator as g (Xt;w). A concrete example of a combination estimator is the linear
combination:
gL (Xt;w) = !0 +
nXi=1
!iXit (7)
In volatility applications multiplicative forecast combinations are also used:
gM (Xt;w) = !0 �nYi=1
X!iit (8)
We will de�ne the optimal combination parameter, w�; the feasible optimal combination parameter,
~w�; and the estimated combination parameter, w�T ; as follows:
w� � arg minw2W
L� (w) ; ~w� � arg minw2W
~L� (w) ; w�T � arg minw2W
�LT (w) (9)
where W is the parameter space and L�; ~L�, and �LT are de�ned in equations (4) to (6).
The di¢ culty in estimating optimal combinations of realised measures lies in the fact that
quadratic variation is not observable, even ex post. Thus measuring the accuracy of a given estima-
tor, or constructing a combination estimator that is as accurate as possible, is not straightforward.
This is related to the problem of volatility forecast evaluation and comparison, where the target
variable is also unobservable. Andersen and Bollerslev (1998), Meddahi (2001), Andersen, Boller-
slev and Meddahi (2005), Hansen and Lunde (2006b), Patton (2006), and Patton and Sheppard
4
(2008) discuss this problem in volatility forecasting. Unfortunately, the methods developed for
volatility forecasting are not directly applicable to the problem of evaluating realised measure ac-
curacy, or the construction of combinations of realised measures, due to a change in the information
set that is used: in forecasting applications, the estimate of �t will be based on Ft�1, while in re-
alised volatility estimation applications, the estimate of �t will be based on Ft: As discussed in
Patton (2008), this subtle change in information sets forces a substantial change in methods to
compare and combine realised measures. Ignoring this change leads to combination estimators that
are biased and inconsistent. These problems arise because the error in the proxy for �t is correlated
with the error in the estimator, a problem that does not arise, under basic assumptions, in volatility
forecasting applications.
Following Patton (2008), we will consider the case that an unbiased proxy for �t is known to be
available. An example of this is the daily squared return, which can plausibly assumed to be free
from microstructure and other biases, and so is an unbiased, albeit noisy, estimator of QV.
Assumption P1: ~�t = �t + �t, with E [�tjFt�1; �t] = 0.
In order the break the problem of correlated measurement errors in ~�t and in Xit identi�ed in
Patton (2008), we will use a lead, or combination of leads, of ~�t in the estimation of the optimal
combination weights. Denoting this as Yt; we make the following assumption:
Finally, we need an assumption about the dynamics of the target variable �t: Patton (2008)
considers two assumptions here, that �t follows a (possibly heteroskedastic) random walk, or that
�t follows a stationary AR(p) process. For the high frequency IBM data studied below, Patton
(2008) found that the random walk approximation was satisfactory, and so for simplicity we will
focus on that case; the extension to the AR(p) approximation is straightforward.
Assumption T1: �t = �t�1 + �t, with E [�tjFt�1] = 0:
To obtain the asymptotic distribution of the estimated optimal combination weights we require
assumptions su¢ cient for a central limit theorem to hold. Several di¤erent sets of assumptions
may be employed here; we use high-level assumptions from Gallant and White (1988), and refer
the interested reader to that book for more primitive assumptions that may be used for non-linear,
dynamic m-estimation problems such as ours. See Davidson and MacKinnon (1993) for a concise
and very readable overview of asymptotic normality for m-estimators.
Assumption A1(a): �LT (w)� ~L� (w)p! 0 uniformly on W.
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Assumption A1(b): ~L� (w) has a unique minimiser ~w�.
Assumption A1(c): g is twice continuously di¤erentiable with respect to w.
Assumption A1(d): Let AT (w) � rww �LT (w), then AT (w)� A (w)p! 0 uniformly on W;
where A (w) is a �nite positive de�nite matrix of constants for all w 2W.
Assumption A1(e): T�1=2XT
t=1rwL (Yt; g (Xt;w))
D! N (0; B (w)), where B (w) is �nite
and positive de�nite for all w 2W.
With the above assumptions in hand, we now present our main theoretical result.
Proposition 1 If the distance measure L is a member of the class in equation (3) and if ~w�
interior to W, then under assumptions P1, P2, T1 and A1, we have:
V�1=2T
pT (w�T �w�)
D! N (0; I)
where VT � A�1T BT A�1T ; AT �
1
T
TXt=1
rwwL (Yt; g (Xt; w�T )) ; BT � V"1pT
TXt=1
rwL (Yt; g (Xt; w�T ))#
and BT is some symmetric and positive de�nite estimator of BT such that BT �BTp! 0:
Proof. We �rst show that w� = ~w�:This part of the proof is a corollary to Proposition 2(a) of
Patton (2008). Consider a second-order mean-value expansion of L (Yt; g (Xt;w)) around �t:
L (Yt; g (Xt;w)) = L (�t; g (Xt;w)) +@L (�t; g (Xt;w))
@�(Yt � �t) +
1
2
@2L���t; g (Xt;w)
�@�2
(Yt � �t)2
where ��t = �tYt+(1� �t) �t for some �t 2 [0; 1] : Under assumptions P1, P2 and T1, Patton (2008)
shows that the second term in this expansion has mean zero, and so we obtain:
E [L (Yt; g (Xt;w))] = E [L (�t; g (Xt;w))] +1
2E
24@2L���t; g (Xt;w)
�@�2
(Yt � �t)235
Distance measures in the class in equation (3) yield @2L (�;X) =@�2 = �C 0 (�) ; and so
E [L (Yt; g (Xt;w))] = E [L (�t; g (Xt;w))]�1
2EhC 0���t
�(Yt � �t)2
iNotice that the second term above does not depend on w; thus the parameter that minimises
E [L (Yt; g (Xt;w))] is the same as that which minimises E [L (�t; g (Xt;w))] : Thus ~w� = w�:
Next we obtain the asymptotic distribution of w�T : This part of the proof uses standard results
from m-estimation theory, see Gallant and White (1988) for example. Under assumptions A1(a)
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and A1(b), Theorem 3.3 of Gallant and White (1988) yields w�T � ~w�p! 0: Combining this with
the fact that ~w� = w� yields consistency of w�T for the parameter of interest: w�T �w�
p! 0: Under
Assumption A1, Theorem 5.1 of Gallant and White (1988) yields the asymptotic normality of w�T ;
centered around ~w�: Combining this with ~w� = w� from above yields the desired result.
The above proposition shows that it is possible to consistently estimate the optimal combination
weights from the data, by employing a �robust�loss function of the form in equation (3) and using
a lead (or combination of leads) of a conditionally unbiased proxy for �t: This proposition further
shows how to compute standard errors on these estimated optimal weights. The use of a proxy, Yt;
for the true quadratic variation, �t; means that these standard errors will generally be larger than
those that would be obtained if �t was observable, but nevertheless these standard errors can be
estimated using the expressions above.
3 Empirical application
3.1 Data description
In this section we consider the problem of estimating the quadratic variation of the open-to-close
(9:45am to 4pm) continuously-compounded return on IBM, using a variety of di¤erent estimators
and sampling frequencies. We use data on NYSE trade prices from the TAQ database over the
period from January 1996 to June 2007, yielding a total of 2893 daily observations3. Figure 1
reveals that the sample includes periods of rising prices and moderate-to-high volatility (roughly
1996-1999), of slightly falling prices and relatively high volatility (roughly 2000-2003), and of stable
prices and relatively low volatility (2004-2007). In addition to considering the full sample estimates
of optimal combination estimators, we will consider the results in each of these three sub-samples.
[ INSERT FIGURE 1 HERE ]
3We use trade prices from the NYSE only, between 9:45am and 4:00pm, with a g127 code of 0 or 40, a corr code of
0 or 1, positive size, and cond not equal to �O�, �Z�, �B�, �T�, �L�, �G�, �W�, �J�, or �K�. Further, the data were
cleaned for outliers and related problems: prices of zero were dropped, as were prices that suggested a single-trade
change of more than 10% relative to the opening price. Finally, if more than one price was observed with the same
time stamp then we used the median of these prices. See Barndor¤-Nielsen, et al. (2008b) for a discussion of cleaning
high frequency data.
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3.2 Description of the individual estimators
The motivation for our study of realised measures is that the varied forms of realised measures
that have been proposed in the literature to date, and the di¤erent pieces of information captured
by each, may allow for the construction of combination estimators that out-perform any given
individual estimator. With this in mind, we consider a large collection of di¤erent realised measures.
In the implementation of each of these estimators, we follow the implementation of the authors of
the original paper as closely as possible (and in most cases, exactly). We omit detailed de�nitions
and descriptions of each estimator in the interests of space and instead refer the interested reader
to the original papers.
We �rstly consider standard realised variance, de�ned as:
RV(m)t =
mXj=1
r2t;j (10)
where m is the number of intra-daily returns used, and rt;j is the jth return on day t: The number
of intra-daily returns used can vary from 22,500 (if we sample prices every second between 9:45am
and 4pm) to 1, if we sample just the open and the close price. To keep the number of estimators
tractable, and with the high degree of correlation between RV estimators with similar sampling
frequencies in mind, we select six sampling frequencies: 1 second, 5 seconds, 1 minute, 5 minutes,
62.5 minutes (which we will abbreviate as �1 hour�), and 1 trade day (375 minutes). The �rst
set of RV estimators is based on prices sampled in �calendar time�using last-price interpolation,
meaning that we construct a grid of times between 9:45am and 4pm with the speci�ed number of
minutes between each point, and use the most recent price as the one for a given grid point.
Next we consider RV estimators computed using the same formula, but with prices sampled in
�tick time�(also known as �event time�or �trade time�). In this sampling scheme a price series
for each day is constructed by skipping every x trades: this leads to prices that are evenly spaced
in �event time�, but generally not in calendar time. If the trade arrival rate is correlated with the
level of volatility then tick-time sampling produces high-frequency returns which are approximately
homoskedastic. Theory suggests that this should improve the accuracy of RV estimation, see Hansen
and Lunde (2006a) and Oomen (2006). We consider average sampling frequencies in tick-time that
correspond to those used in calendar time: 1 second, 5 seconds, 1 minute, 5 minutes, 62.5 minutes
and 375 minutes. The highest and the lowest of these frequencies lead to estimators that are
numerically identical to calendar-time RV, and so we drop these from the analysis.
8
We then draw on the work of Bandi and Russell (2006, 2008), who provide a method to estimate
the optimal (calendar-time) sampling frequency, for each day, for realised variance in the presence
of market microstructure noise. This formula relies on estimates of the variance and kurtosis of the
microstructure noise, as well as preliminary estimates of the integrated variance (IV) and integrated
quarticity (IQ) of the price process: we follow Bandi and Russell (2008), who also study IBM stock
returns, and estimate the moments of the microstructure noise using 1-second returns, and use
15-minute returns to obtain preliminary estimates of the IV and IQ. Bandi and Russell (2008)
also propose a bias-corrected realised variance estimator, which removes the estimated impact of
the microstructure noise: we consider the Bandi-Russell RV estimator both with�RV BR;bc
�and
without�RV BR
�this bias correction.
Our second set of realised measures are the �realised range-based variance� (RRV) of Chris-
tensen and Podolskij (2007) and Martens and van Dijk (2007). We follow Christensen and Podol-
skij�s implementation of this estimator and use 5-minute blocks. Rather than estimate the number
of prices to use within each block from the number of non-zero return changes, as in Christensen
and Podolskij (2007), we simply use 1-minute prices within each block, giving us 5 prices per block,
compared with around 7 in their application to General Motors stock returns. We implement RRV
using both calendar-time and tick-time sampling.
The third set of realised measures are the �quantile-based realised variance� (QRV) of Chris-
tensen, Oomen and Podolskij (2008). For implementation we follow Christensen, et al.�s application
to Apple stock returns and use quantiles of 0:85; 0:90; and 0:96; and prices sampled every 1 minute,
with the number of subintervals (�n�in their notation) set to one. We implement QRV using both
calendar-time and tick-time sampling.
The fourth set of realised measures are the �bi-power variation� (BPV) of Barndor¤-Nielsen
and Shephard (2006). These authors implemented their estimator using 5-minute calendar-time
returns, however this was presumably partially dictated by their data (indicative quotes for the
US dollar/German Deutsche mark and US dollar/Japanese yen exchange rate), for which this was
the highest frequency. We thus implement BPV at both 5-minute and 1-minute frequencies, using
both calendar-time and tick-time sampling4.
4 It is worth noting that, unlike the other estimators considered in this paper, BPV and QRV estimate the part
of quadratic variation due to the continuous semimartingale (that is, the integrated variance, IV) and so ignore the
contribution to QV from jumps. If jumps are unpredictable (or less predictable than IV), as found by Andersen,
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The �fth set of estimators are the �realised kernels�(RK) of Barndor¤-Nielsen, et al. (2008a),
BNHLS henceforth. This is a broad class of estimators and we consider several variations. Firstly,
we consider RK with the Bartlett kernel, as this estimator was shown by BNHLS to be asymptot-
ically equivalent to the �two-scale�estimator (2SRV) of Zhang, Mykland and Aït-Sahalia (2005).
Second, we consider RK with the �cubic�kernel, which was shown to be asymptotically equivalent
to the �multi-scale�estimator (MSRV) of Zhang (2006), (and which in turn was shown by Zhang to
be more e¢ cient than 2SRV). For both RKbart and RKcubic we consider both 5-tick sampling and
5-second sampling, and in all cases we use the optimal bandwidth for a given kernel provided in
BNHLS5 ;6. Next we consider the �modi�ed Tukey-Hanning2�(TH) kernel, following their empirical
application to General Electric stock returns. They suggest using 1-minute tick-time sampling with
optimal bandwidth, which we implement here. We also consider the TH kernel with bandwidth
�xed to equal 5�RKTH�5bw�, and the TH kernel with optimal bandwidth applied to 5-tick returns�
RKTH;5tick�. Finally, we consider the �non-�at-top Parzen� kernel of Barndor¤-Nielsen, et al.
(2008b), which is designed to guarantee non-negativity of the estimator (which is not guaranteed
for the other RK estimators considered above). We implement this with their optimal bandwidth
formula, and following their application to Alcoa stock returns, we use 1-minute tick-time sampling.
Our sixth and �nal realised measure is the �alternation� estimator of Large (2005), ALTRV.
This estimator was designed for use on assets whose prices can only change by one tick size (e.g.,
one cent, one penny, one-sixteenth of a dollar, etc.), and with this restriction in mind, Large
proposes a novel estimator of the QV of the price series. Our application to IBM stock returns
does not satisfy this assumption, and so formally ALTRV is not applicable here, but we implement
a modi�cation of this estimator to see whether it nevertheless provides useful information in the
optimal combination estimator. Following Large (2005), we implement this on quote prices rather
than trade prices. To handle the fact that non-zero absolute IBM price changes are not always of
Bollerslev and Diebold (2006), then there may be bene�ts to using estimators that focus on IV rather than QV in
forecasting applications.5These optimal bandwidths, like the RVBR sampling frequencies, are estimated for each day in the sample and so
can change with market conditions, the level of market microstructure noise, etc.6These optimal bandwidth formulas require estimates of moments of the noise and preliminary estimates of IV
and QV. We use 1-second returns to obtain estimates of the moments of the noise as in Bandi and Russell (2008),
10-minute returns to obtain a preliminary estimate of IV and QV, and 1-minute returns to obtain a preliminary RK
estimate used to bias-correct the noise estimate. These are slightly di¤erent to the choices in the web appendix to
Barndor¤-Nielsen, et al. (2008a); see http://www.hha.dk/~alunde/BNHLS/BNHLS.htm for further discussion.
10
the same size (i.e. 1 tick), on days when there are more than one size for non-zero absolute price
changes we use the mode non-zero price change as the �tick�, and implement ALTRV as though
this was the tick size. To make ALTRV an estimator of the QV of the log-price (as for the other
estimators considered here) rather than the price, we scale ALTRV by the squared opening price
for each day (this is a form of �rst-order Taylor series adjustment term). ALTRV uses every quote,
and so we do not have to choose a sampling scheme or frequency. We consider ALTRV applied to
the mid-quote (ALTRVmq), and the average of the ALTRVs applied separately to the bid and the
ask prices (ALTRVcombo).
In total we have 30 di¤erent realised measures, from 6 di¤erent classes of estimators, with a
variety of sampling frequencies and sampling schemes. To the best of our knowledge, this is the
largest collection of realised measures considered in a single empirical study.
Table 1 presents some summary statistics for these 30 estimators. ALTRV mq has the smallest
average value, 2.158, and RV tick;5 sec has the largest average at 3.314. More familiar estimators,
such as RV 1day and RV 5min have average values of around 2.3, corresponding to 24.1% annualised
standard deviation. Whilst RV 1day has a reasonable average value, it performs poorly on the
other summary statistics: it has the highest standard deviation, skewness, and kurtosis of all 30
estimators. BPV 1min is the estimator with the lowest standard deviation, while QRV has the
lowest skewness and kurtosis. RV 1day is also the only estimator to generate estimates of QV that
are not strictly positive: its minimum value is zero, which it attains on 42 days (around 1.5% of
the sample). None of the other estimators have a minimum value that is non-positive, including
the RK estimators which do not ensure non-negativity of the estimates.
Table 2 presents a subset of the correlation matrix of these estimators (the complete corre-
lation matrix is available upon request). We present the correlation of each estimator with two
standard estimators in the literature (RV1day and RV5min), a naïve choice given high frequency
data (RV1 sec), and three of the estimators that ultimately perform well in our empirical analysis
(RVtick;1min; BPVtick;1min and RKcubic). This table shows that these estimators are generally highly
correlated (as one would expect); the average correlation across all elements of their correlation
matrix is 0.861. This should be kept in mind when interpreting the estimated optimal combination
weights in Section 3.4. The highest correlation between any two estimators is between RKtick;cubic
and RKTH;5tick; which is 0.9992. The lowest correlation between any two estimators is between
RV1day and ALTRVcombo, at 0.425.
11
[ INSERT TABLES 1 AND 2 HERE ]
3.3 Results using simple combination estimators
In Tables 3 and 4 we present the �rst set of empirical results of the paper. These tables present the
estimated accuracy of each of the estimators using the ranking methodology in Patton (2008). Our
preferred measure of accuracy is the QLIKE distance measure, and we also present results for the
familiar MSE distance measure. We use the random walk approximation (assumption T1 above),
with a one-period lead of the RV 5min as the instrument for the latent quadratic variation to obtain
these estimates.
The ranking method of Patton (2008) can only estimate the accuracy of an estimator relative
to some other estimator, and in Tables 3 and 4 we use RV 5min as the base estimator; this choice
is purely a normalisation and has no e¤ect on the conclusions. Negative values in the �rst columns
of Tables 3 and 4 indicate that a given estimator has lower average MSE or QLIKE distance to the
latent QV (i.e., greater accuracy) than RV 5min, while positive values indicate higher average MSE
or QLIKE distance than RV 5min:
We consider the 30 individual realised measures discussed in the previous section, as well as three
simple combination estimators: the equally-weighted arithmetic mean, equally-weighted geometric
mean, and the median, leading to a total of 33 estimators. Under QLIKE the most accurate
estimator of QV is the simple RV tick;1min; which is ranked in the top 4 in all three sub-periods.
Under MSE, QRVtick is ranked the highest and does well in the �rst two sub-periods but is ranked
20th in the last sub-period. The top 5 estimators, averaging across the full-sample ranks under both
MSE and QLIKE are QRV tick (�rst), RKcubic (second), RV BR and QRV (equal third), RVMean
and RVMedian (equal �fth).
The discussion of rankings of average accuracy is a useful initial look at the results, but a
more formal analysis is desirable. We use two approaches: The �rst is the �model con�dence
set� (MCS) of Hansen, Lunde and Nason (2005), which was developed to obtain a set of fore-
casting models that contains the true �best� model out of the entire set of forecasting mod-
els with some speci�ed level of con�dence. It allows the researcher to identify the sub-set of
models that are �not signi�cantly di¤erent� from the unknown true best model. Patton (2008)
shows that this methodology may be adapted to the problem of identifying the most accurate re-
alised measures, under the assumptions discussed in Section 2. The last four columns of Tables
12
3 and 4 show the results of the MCS procedure on the full sample and on three sub-samples7.
Under QLIKE the full-sample MCS, at the 90% con�dence level, contains just 4 estimators:
RV tick;1min, QRV tick, RKcubic, and RVMean: Under MSE distance the MCS contains 10 estimators:
Thus under both distance measures a simple combination estimator appears in the MCS. Inter-
estingly, it is a di¤erent combination estimator depending on the distance measure: under MSE,
which is more sensitive to outliers, the preferred simple combination estimator is the median, which
is a more robust combination estimator, see Stock and Watson (2001) for example. Under QLIKE,
which is less sensitive to large over-predictions and is invariant to the level of volatility, see Patton
and Sheppard (2008), the preferred combination estimator is the mean.
[ INSERT TABLES 3 AND 4 HERE ]
The second formal analysis of the individual estimators and simple combination estimators
uses the stepwise multiple testing method of Romano and Wolf (2005). This method identi�es the
estimators that are signi�cantly better, or signi�cantly worse, than a given benchmark estimator,
while controlling the family-wise error rate of the complete set of hypothesis tests. We consider
three choices of benchmark estimator: RV 1day; which is the standard estimator in the absence of
high frequency data; RV 5min; which is based on a rule-of-thumb from earlier papers in the RV
literature (see Andersen, et al. (2001b) and Barndor¤-Nielsen and Shephard (2002) for example);
and RVMean; which is the standard simple combination estimator. The results of these tests are
presented in Tables 5 and 6 for MSE and QLIKE distance respectively8.
The results of the Romano-Wolf method under QLIKE distance reveal some interesting patterns.
Firstly, at the 10% level of signi�cance, every estimator signi�cantly outperforms RV 1day, in the
full sample and in all three sub-samples. This clearly a strong signal that high frequency data, when
used in any one of the 32 other estimators in this study, yields more precise estimates of QV than
daily data can. When theRV 5min is taken as a benchmark we see more variation in the results: some
estimators are signi�cantly better, others signi�cantly worse and some not signi�cantly di¤erent.
Broadly stated, the estimators that out-perform RV 5min include RV sampled at the 1-minute7The MCS is implemented via a bootstrap re-sampling scheme. We use Politis and Romano�s (1994) stationary
bootstrap with an average block length of 10 days and 1000 bootstrap replications for each test.8The Romano-Wolf testing method is also implemented using Politis and Romano�s (1994) stationary bootstrap
with an average block length of 10 days, and we again use 1000 bootstrap replications for each test.
13
frequency (either in tick time or calendar time), RRV and QRV, RK with the Bartlett, cubic
or Tukey-Hanning kernel (when sampled every 5 ticks or 5 seconds) and the three combination
estimators. The estimators that under -perform RV 5min include RV sampled at the 1 hour or 1 day
frequency, RKTH sampled at the 1 minute frequency, and ALTRV (though recall that the latter
estimator was not designed for a stock like IBM).
Finally, when RVMean is taken as the benchmark estimator in the Romano-Wolf testing method
a very clear conclusion emerges: no estimator signi�cantly out-performs RVMean. Several estima-
tors are not signi�cantly di¤erent, but none are signi�cantly better in the full sample or in any
of the sub-samples. This is a strong endorsement of using this simple combination estimator in
practice9.
When using MSE distance there are fewer signi�cant results: all estimators continue to signi�-
cantly out-perform RV 1day; whereas in the comparisons with RV 5min or RVMean as the benchmark
there are very few rejections of the null hypothesis.
[ INSERT TABLES 5 AND 6 HERE ]
3.4 Results using optimal combination estimators
In this section we present our estimated optimal combination estimators. We consider two standard
parametric combination estimators, a linear combination and a multiplicative combination:
X lint = w0 +
nXi=1
wiXit; and (11)
Xmultt = !0 �
nYi=1
X !iit ; (12)
estimated using Proposition 1 above. We consider both unconstrained combination estimators,
which satisfy the condition that the unknown weights all lie in the interior of the parameter space
and so permit us to compute standard errors using Proposition 1, and also constrained combination
estimators, with the constraint being that all weights must be non-negative. The constrained
9 In unreported results, we also implemented the Romano-Wolf method with RV Geo�mean and RVMedian as the
benchmarks. Under MSE neither of these was ever signi�cantly out-performed, while under QLIKE RV Geo�mean
was signi�cantly beaten (amongst others, by RVMean) while RVMedian was never signi�cantly beaten. These results
are available upon request.
14
estimation makes obtaining standard errors di¢ cult (and we do not pursue this here) but provides
additional information on the individual estimators that are most useful in a combination estimator.
Table 7 presents the optimal linear combination estimators under MSE distance. Looking
�rst at the constrained combination, we see many estimators get a weight of zero in the optimal
constrained combination. The largest weights are assigned to QRV (in calendar time and tick
time) with some weight also given to RV tick;5 sec; RKTH ; RKTH;5tick: Small weight (0.0039) is
also given to RV 1day. In the unconstrained combination the results are broadly similar: some
signi�cant weight is attached to QRV (in calendar time and tick time), and to RKTH ; RKTH;5tick
and RKtick;cubic: Interestingly, although the ALTRV estimators do not get signi�cant weight in the
full sample, they do have signi�cant coe¢ cients in the 1996-1999 sub-sample. This corresponds
with the period where the assumption of single-tick price changes is most plausible for IBM: the
minimum tick size on the New York stock exchange was one-eighth of a dollar until June 24, 1997,
and was one-sixteenth of a dollar until January 29, 2001, when decimal pricing began10.
[ INSERT TABLES 7 AND 8 HERE ]
Table 8 presents the results for optimal linear combinations under QLIKE distance. In the
optimal constrained combination estimator we again see substantial weight on QRV, and also
on RV NFParzen; and we also see non-zero weights on a collection of simple RVs, with sampling
frequencies from 5 seconds up to and including 1 day. As in the optimal MSE combination, the
unconstrained combination has few individually signi�cant coe¢ cients, though there are signi�cant
coe¢ cients on BPV sampled at the 1-minute frequency, RKTH ; and simple RV with sampling
frequency ranging from 1 second to 1day. Also similar to the optimal MSE combination, we again
see that the ALTRV estimators receive signi�cant weight in the �rst sub-sample but not in the
latter sub-samples.
Tables 9 and 10 present the estimated combination weights for multiplicative combinations.
Perhaps surprisingly, given the change in the functional form of these combination estimators, the
results are broadly in line with the optimal linear combination results: the optimal combinations
for MSE put substantial weights on QRV, RKTH ; RKTH;5tick and RV tick;5 sec; while the optimal
combinations for QLIKE again put weight on QRV, RV NFParzen and a collection of simple RVs
with sampling frequencies between 5 seconds and 1 day.
10Source: New York Stock Exchange web site, http://www.nyse.com/about/history/timeline_chronology_index.html.
15
Often of interest in the forecasting literature is whether the estimated optimal combination
is signi�cantly di¤erent from a simple equally-weighted average. If we let w�i denote the optimal
linear combination weights, and !�i denote the optimal multiplicative combination weights, the
hypotheses of interest are:
HL0 : w0 = 0 \ w1 = w2 = ::: = wn = 1=n (13)
vs. HLa : w0 6= 0 [ wi 6= 1=n for some i = 1; 2; :::; n
and HM0 : !0 = 1 \ !1 = !2 = ::: = !n = 1=n (14)
vs. HMa : !0 6= 1 [ !i 6= 1=n for some i = 1; 2; :::; n
Using Proposition 1 these hypotheses can be tested using Wald tests, and we �nd on the full
sample of data that the null that the optimal combination is an equally-weighted combination can
be rejected with a p-value of less than 0.01 in all four cases (linear and multiplicative combinations,
under MSE and QLIKE distance). Thus, while Section 3.3 revealed that the simple mean was
not signi�cantly beaten by any individual estimator, it can still be improved: an optimally formed
linear combination is signi�cantly more accurate than an equally-weighted average.
Finally, we conduct a set of tests related to idea of forecast encompassing, see Chong and Hendry
(1986) and Fair and Shiller (1990). We test the null hypothesis that a single realised measure (i)
encompasses the information in all other estimators:
HL;i0 : wi = 1 \ wj = 0 8 j 6= i (15)
vs. HL;ia : wi 6= 1 [ wj 6= 0 for some j 6= i
and HM;i0 : !0 = !i = 1 \ !j = 0 8 j 6= 0; i (16)
vs. HM;ia : !i 6= 1 [ !0 6= 1 [ wj 6= 0 for some j 6= 0; i
for i = 1; 2; :::; n:We �nd that the null hypothesis is rejected for every single estimator, under both
MSE and QLIKE distance, for both linear and multiplicative combinations, with all p-values less
than 0:01. This is strong evidence that there are gains to considering combination estimators of
quadratic variation: no single estimator dominates all others. This result is new to the realised
volatility literature, but is probably not surprising to those familiar with with forecasting in practice.
16
4 Summary and conclusion
Recent advances in �nancial econometrics have led to the development of new estimators of asset
price variability using high frequency price data. These estimators are based on a variety of di¤erent
assumptions and take many di¤erent functional forms. Motivated by the success of combination
forecasts over individual forecasts in a range of forecasting applications, see Clemen (1989) and
Timmermann (2006) for example, this paper sought to answer the following question: do combina-
tions of individual estimators o¤er accuracy gains relative to individual estimators? The answer is
a resounding �yes�.
This paper presents a novel method for combining individual realised measures to form new
estimators of price variability, overcoming the obstacle that the quantity of interest is not observable,
even ex post. We applied this method to a collection of 30 di¤erent realised measures, across 6
distinct classes of estimators, estimated on high frequency IBM price data over the period 1996-
2007. Using the Romano-Wolf (2005) test, we �nd that no individual realised measure signi�cantly
out-performs, in terms of average accuracy, a simple equally-weighted average. Further, we �nd that
none of the individual estimators encompasses the information in all other estimators, providing
further support for the use of combination realised measures. Overall our results suggest that there
are indeed bene�ts from combining the information contained in the array of di¤erent volatility
estimators proposed in the literature to date.
Several extensions of our analysis are possible. Firstly, it may be interesting, both economically
and statistically, to try to understand why certain estimators out-perform, or enter with greater
weight into the optimal combination estimator than others. It may be that certain market condi-
tions (e.g., high vs. low volatility or small vs. large bid-ask spreads) favour one class of estimator
relative to another, and identifying these may aid in the construction of improved estimators. Sec-
ondly, it may be bene�cial to consider time-varying combinations of realised measures, along the
lines of Elliott and Timmermann (2005), rather than the static combination estimators considered
in this paper.
17
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Figure 1: This �gure plots IBM price and volatility over the period January 1996 to June 2007.The price is adjusted for stock splits, and the volatility is computed using realised volatility basedon 5-minute calendar-time trade prices, annualised using the formula �t =
p252�RVt:
.
21
Table 1: Summary statistics of the realised measures
Notes: This table presents basic summary statistics on the 30 di¤erent realised measures con-sidered in this paper, and 3 simple combination estimators.
22
Table 2: Correlation between the realised measures
Notes: This table presents the a sub-set of the correlation matrix of the 30 di¤erent realised mea-sures considered in this paper. The estimators in the columns correspond to standard choices in theextant literature (RV1day and RV5min), a naïve choice given high frequency data (RV1 sec), and threeof the estimators that turn out to perform well in our empirical analysis (RVtick;1min; BPVtick;1min
and RKcubic). The complete correlation matrix is available from the authors on request.
23
Notes to Table 3: The �rst column of this table presents the average di¤erence in MSEdistance of each realised measure, relative to RV 5min; with negative (positive) values indicatingthat the estimator was closer (further) on average to the target variable than RV 5min: Columns2�5 present the rank of each estimator using MSE distance, for the full sample period and forthree sub-samples, 1996-1999, 2000-2003 and 2004-2007. The most accurate estimator is ranked 1,and the least accurate estimator is ranked 33. Columns 6�9 present an indicator for whether theestimator was in the �model con�dence set�(equal to X if in, �if not) in the full sample and eachof the three sub-samples.
Notes to Table 4: The �rst column of this table presents the average di¤erence in QLIKEdistance of each realised measure, relative to RV 5min; with negative (positive) values indicatingthat the estimator was closer (further) on average to the target variable than RV 5min: Columns2�5 present the rank of each estimator using QLIKE distance, for the full sample period and forthree sub-samples, 1996-1999, 2000-2003 and 2004-2007. The most accurate estimator is ranked 1,and the least accurate estimator is ranked 33. Columns 6�9 present an indicator for whether theestimator was in the �model con�dence set�(equal to X if in, �if not) in the full sample and eachof the three sub-samples.
24
Table 3: Performance of the realised measures under MSE distance
Avg �MSE MSE rank In MCS?Full Full 96-99 00-03 04-07 Full 96-99 00-03 04-07
RV1 sec 0.117 26 28 22 30 � � � �RV5 sec -0.504 19 25 19 26 � � � �RV1min -1.783 7 6 7 9 X X X �RV5min 0.000 24 19 26 11 � X � �RV1hr 3.014 30 30 30 28 � � � �RV1day 24.005 33 33 33 33 � � � �RVtick;5 sec 0.806 28 29 28 29 � � � �RVtick;1min -1.292 14 15 14 16 X X X �RVtick;5min 0.198 27 23 27 15 � � � �RVtick;1hr 6.096 32 32 31 31 � � � �RVBR -1.957 5 1 6 14 X X X �RVBR;bc -1.541 11 13 10 22 � X � �RRV -1.286 15 12 16 7 � X � �RRVtick -1.084 16 14 18 2 � X � XQRV -2.100 2 2 2 21 X X X �QRVtick -2.186 1 4 1 20 X X X �BPV1min -2.033 3 5 3 12 X X X �BPV5min -0.762 18 24 17 13 � � � �BPVtick;1min -1.878 6 9 5 5 X X X XBPVtick;5min -1.023 17 21 15 10 � � � �RKbart -1.660 9 11 8 25 � X � �RKtick;bart -1.564 10 8 11 19 X X X �RKcubic -1.967 4 3 4 23 X X X �RKtick;cubic 0.045 25 22 24 18 � � � �RKTH -0.398 20 27 20 4 � � � XRKTH�bw5 2.515 29 31 29 24 � � � �RKTH�5tick -0.024 23 20 23 17 � � � �RKNFParzen -0.243 21 26 21 3 � � � XALTRVmq -0.038 22 18 25 27 � � � �ALTRVcombo 3.019 31 17 32 32 � X � �RVMean -1.300 13 16 13 8 � X � �RVGeo�mean -1.528 12 7 12 1 � X � XRVMedian -1.715 8 10 9 6 X X X �
Notes: See page 24 for a description of this table.
25
Table 4: Performance of the realised measures under QLIKE distance
Avg �QLIKE QLIKE rank In MCS?Full Full 96-99 00-03 04-07 Full 96-99 00-03 04-07
Notes: See page 24 for a description of this table.
26
Notes to Table 5: This table presents the results of the Romano-Wolf �stepwise�test for threedi¤erent choices of benchmark estimator. Columns 1�4 present indicators for when the benchmarkis set to RV1day : using MSE distance, the indicator is set to X if the estimator is signi�cantly moreaccurate than RV1day; to � if the estimator is signi�cantly less accurate than RV1day; and to �if theestimator�s accuracy is not signi�cantly di¤erent from RV1day: The four columns refer to the fullsample period and three sub-samples, 1996-1999, 2000-2003 and 2004-2007. Columns 5�8 presentcorresponding results when the benchmark estimator is set to RV5min; and columns 9�12 presentresults when the benchmark estimator is set to RVMean: The benchmark estimator in each columnis denoted with a F:
Notes to Table 6: This table presents the results of the Romano-Wolf �stepwise�test for threedi¤erent choices of benchmark estimator. Columns 1�4 present indicators for when the benchmarkis set to RV1day : using QLIKE distance, the indicator is set to X if the estimator is signi�cantlymore accurate than RV1day, to � if the estimator is signi�cantly less accurate than RV1day; and to� if the estimator�s accuracy is not signi�cantly di¤erent from RV1day: The four columns refer tothe full sample period and three sub-samples, 1996-1999, 2000-2003 and 2004-2007. Columns 5�8present corresponding results when the benchmark estimator is set to RV5min; and columns 9�12present results when the benchmark estimator is set to RVMean: The benchmark estimator in eachcolumn is denoted with a F:
27
Table 5: Romano-Wolf tests on the realised measures, under MSE distance
Full 96-99 00-03 04-07 Full 96-99 00-03 04-07 Full 96-99 00-03 04-07
RV1 sec X X X X � � � � � � � �RV5 sec X X X X � � � � � � � �RV1min X X X X X X X X � � � �RV5min X X X X F F F F � � � �RV1hr X X X X � � � � � � � �RV1day F F F F � � � � � � � �RVtick;5 sec X X X X � � � � � � � �RVtick;1min X X X X X X X X � � � �RVtick;5min X X X X X � � X � � � �RVtick;1hr X X X X � � � � � � � �RVBR X X X X X X X X � � � �RVBR;bc X X X X X X � X � � � �RRV X X X X X � X X � � � �RRVtick X X X X X � X X � � � �QRV X X X X X X X X � � � �QRVtick X X X X X X X X � � � �BPV1min X X X X � � X X � � � �BPV5min X X X X � � � � � � � �BPVtick;1min X X X X � � X X � � � �BPVtick;5min X X X X � � � � � � � �RKbart X X X X X X X � � � � �RKtick;bart X X X X X X X X � � � �RKcubic X X X X X X X X � � � �RKtick;cubic X X X X X � X X � � � �RKTH X X X X � � � � � � � �RKTH�bw5 X X X X � � � � � � � �RKTH�5tick X X X X X � X X � � � �RKNFParzen X X X X � � � � � � � �ALTRVmq X X X X � � � � � � � �ALTRVcombo X X X X � � � � � � � �RVMean X X X X X X X X F F F FRVGeo�mean X X X X X � � X � � � �RVMedian X X X X X X X X � � � �
Notes: See page 27 for a description of this table.
29
Notes to Table 7: This table presents the MSE-optimal linear combination weights. Columns1�4 present these weights for the unconstrained case, with estimates that are signi�cantly di¤erentfrom zero at the 10% level highlighted in bold. (Standard errors were computed using Proposition1, with Newey-West (1987) estimates of the covariance matrix, BT ; and are not reported here inthe interests of space.) The four columns refer to the full sample period and three sub-samples,1996-1999, 2000-2003 and 2004-2007. Columns 5�8 present the optimal linear combination weights,using MSE distance, imposing the constraint that each weight must be weakly positive. Weightsthat were on the boundary at zero are reported as �0�, while those away from the boundary arereported to two decimal places.
Notes to Table 8: This table presents the QLIKE-optimal linear combination weights.Columns 1�4 present these weights for the unconstrained case, with estimates that are signi�-cantly di¤erent from zero at the 10% level highlighted in bold. (Standard errors were computedusing Proposition 1, with Newey-West (1987) estimates of the covariance matrix, BT ; and are notreported here in the interests of space.) The four columns refer to the full sample period and threesub-samples, 1996-1999, 2000-2003 and 2004-2007. Columns 5�8 present the optimal linear combi-nation weights, using QLIKE distance, imposing the constraint that each weight must be weaklypositive. Weights that were on the boundary at zero are reported as �0�, while those away fromthe boundary are reported to two decimal places.
Notes to Table 9: This table presents the MSE-optimal multiplicative combination weights.Columns 1�4 present these weights for the unconstrained case, with estimates that are signi�cantlydi¤erent from zero at the 10% level highlighted in bold. (Standard errors were computed usingProposition 1, with Newey-West (1987) estimates of the covariance matrix, BT ; and are not reportedhere in the interests of space.) The four columns refer to the full sample period and three sub-samples, 1996-1999, 2000-2003 and 2004-2007. Columns 5�8 present the optimal multiplicativecombination weights, using MSE distance, imposing the constraint that each weight must be weaklypositive. Weights that were on the boundary at zero are reported as �0�, while those away fromthe boundary are reported to two decimal places.
Notes to Table 10: This table presents the QLIKE-optimal multiplicative combinationweights. Columns 1�4 present these weights for the unconstrained case, with estimates that are sig-ni�cantly di¤erent from zero at the 10% level highlighted in bold. (Standard errors were computedusing Proposition 1, with Newey-West (1987) estimates of the covariance matrix, BT ; and are notreported here in the interests of space.) The four columns refer to the full sample period and threesub-samples, 1996-1999, 2000-2003 and 2004-2007. Columns 5�8 present the optimal multiplicativecombination weights, using QLIKE distance, imposing the constraint that each weight must beweakly positive. Weights that were on the boundary at zero are reported as �0�, while those awayfrom the boundary are reported to two decimal places.
30
Table 7: Optimal linear combinations under MSE distance
