!!!!!!!!!!!!!!!!!!!!!!!!Optimal combinationofrealizedvolatilityestimators!!!!!!!!!!!!

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!!!!!!!!!!!!!!!!!!!!!!!!Optimal combinationofrealizedvolatilityestimators!!!!!!!!!!!!

  1. 1. 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. 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 NonlinearEconomics 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:andrew.patton@economics.ox.ac.uk and kevin.sheppard@economics.ox.ac.uk.
  2. 2. 1 IntroductionThe development of new estimators of asset price variability has been an active area of econometricresearch in the past decade. These estimators, known as “realised volatility estimators”or “realisedmeasures” 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 papersin 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, andrecent work1 has built on this to propose estimators that are more e¢ cient, robust to marketmicrostructure e¤ects, and that can estimate the variation due to the continuous part of the priceprocess 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-evolvingliterature. 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 inthe forecasting literature that combinations of individual forecasts often out-perform even the bestindividual forecast, see Bates and Granger (1969), Newbold and Granger (1974) and Stock andWatson (2004), for example, and see Clemen (1989) and Timmermann (2006) for a more reviewsof this …eld2 . Timmermann (2006) summarises three explanations for why combination forecastswork well in practice: they combine the information contained in each individual forecast; theyaverage across di¤erences in the way individual forecasts are a¤ected by structural breaks; theyare less sensitive to possible mis-speci…cation of individual forecasting models (see also Clementsand Hendry, 1998, on forecast model mis-speci…cation). Each of these three points applies equallyto the problem of estimating price variability: individual realised measures use di¤erent pieces ofinformation from high frequency data, they may be di¤erently a¤ected by structural breaks (causedby, for example, changes in the market microstructure), and they may be a¤ected to various de- 1 See 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 andRussell (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. 2 See Halperin (1961) and Reid (1968) for interesting early work on combining di¤erent estimates of a mean, anddi¤erent noisy estimates of GDP, as opposed to combining forecasts. 1
  3. 3. grees by mis-speci…cation. Thus there is reason to believe that a combination realised measure mayout-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 ofcombination estimators for asset price variability (measured by its quadratic variation, QV) di¤ersin an important way from the usual forecast combination problem: the task is complicated by thefact that QV is not observed, even ex post. This means that measuring the accuracy of a givenestimator of QV, or constructing a combination estimator that is as accurate as possible, has tobe done using proxies (or, in our case, instruments) for the true latent QV. Our theoretical workextends the data-based method for estimating the relative accuracy of realised measures suggestedby 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 andmoment 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 30di¤erent realised measures, across 6 distinct classes of estimators, estimated on high frequency dataon IBM over the period 1996-2007. We present results on the ranking of the individual estimatorsand “simple” combination estimators such as the arithmetic mean, the geometric mean and themedian, 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 measuredescribed 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 squaredreturns, the standard 5-minute realised volatility estimator, or a simple equally-weighted averageof 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 bothcases includes a simple combination estimator. Furthermore, using the Romano-Wolf test, we …ndthat no realised measure signi…cantly out-performs the simple average in the full sample, or in anysub-sample, while many signi…cantly under -perform the simple average estimator. We also estimate optimal linear and multiplicative combination estimators, under both MSE 2
  4. 4. and QLIKE, and examine which individual realised measures enter signi…cantly into the optimalcombination forecast, or enter with non-zero weight in an optimal constrained forecast. We …ndthat weight is given to a variety of realised measures, including both simple and more sophisticatedestimators. Importantly, we …nd that none of the individual estimators encompasses the informationin 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 thearray of di¤erent volatility estimators proposed in the literature to date.2 Combining realised measures2.1 NotationThe latent target variable, generally the quadratic variation (QV) or integrated variance (IV) ofan 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. Inaddition to being estimators with di¤erent functional forms, these may include the same type ofestimator 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) = log 1 (2) X XThe QLIKE distance measure is a simple modi…cation of the familiar Gaussian log-likelihood, withthe modi…cation being such that the minimum distance of zero is obtained when X = . Ourresult below will be shown to hold for a more general class of distance measures, namely the classof “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 ofboth ~ and X, and where C is the anti-derivative of C. In this class each pseudo-distance measure 3
  5. 5. L is completely determined by the choice of C, and MSE and QLIKE are obtained (up to locationand scale constants) when C (z) = z and C (z) = 1=z respectively. Finally, it is convenient tointroduce the following quantities: L (w) E [L ( t ; g (Xt ; w))] (4) ~ L (w) E [L (Yt ; g (Xt ; w))] (5) T X 1 LT (w) L (Yt ; g (Xt ; w)) (6) T t=1 ~where the dependence of LT ; L and L on the function g is suppressed for simplicity. Yt is anobservable proxy for the latent t; and is further discussed in the following section.2.2 Estimating optimal combinations of realised measuresIn 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 genericcombination estimator as g (Xt ; w). A concrete example of a combination estimator is the linearcombination: n X gL (Xt ; w) = ! 0 + ! i Xit (7) i=1In volatility applications multiplicative forecast combinations are also used: n Y ! gM (Xt ; w) = ! 0 Xit i (8) i=1We will de…ne the optimal combination parameter, w ; the feasible optimal combination parameter,w ; and the estimated combination parameter, wT ; as follows:~ ^ w arg min L (w) ; w ~ ~ arg min L (w) ; wT ^ arg min LT (w) (9) w2W w2W w2W ~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 thatquadratic 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 targetvariable 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
  6. 6. (2008) discuss this problem in volatility forecasting. Unfortunately, the methods developed forvolatility 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 informationset 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 inPatton (2008), this subtle change in information sets forces a substantial change in methods tocompare and combine realised measures. Ignoring this change leads to combination estimators thatare biased and inconsistent. These problems arise because the error in the proxy for t is correlatedwith the error in the estimator, a problem that does not arise, under basic assumptions, in volatilityforecasting applications. Following Patton (2008), we will consider the case that an unbiased proxy for t is known to beavailable. An example of this is the daily squared return, which can plausibly assumed to be freefrom microstructure and other biases, and so is an unbiased, albeit noisy, estimator of QV. Assumption P1: ~t = t + t, with E [ t jFt 1; t] = 0. In order the break the problem of correlated measurement errors in ~t and in Xit identi…ed inPatton (2008), we will use a lead, or combination of leads, of ~t in the estimation of the optimalcombination weights. Denoting this as Yt ; we make the following assumption: P PJ Assumption P2: Yt = J ~ j=1 j t+j , where 1 J < 1; j 0 8 j and j=1 j = 1: 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 willfocus on that case; the extension to the AR(p) approximation is straightforward. Assumption T1: t = t 1 + t, with E [ t jFt 1] = 0: To obtain the asymptotic distribution of the estimated optimal combination weights we requireassumptions su¢ cient for a central limit theorem to hold. Several di¤erent sets of assumptionsmay be employed here; we use high-level assumptions from Gallant and White (1988), and referthe 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 conciseand very readable overview of asymptotic normality for m-estimators. p Assumption A1(a): LT (w) ~ L (w) ! 0 uniformly on W. 5
  7. 7. ~ Assumption A1(b): L (w) has a unique minimiser w . ~ Assumption A1(c): g is twice continuously di¤erentiable with respect to w. p Assumption A1(d): Let AT (w) rww LT (w), then AT (w) A (w) ! 0 uniformly on W;where A (w) is a …nite positive de…nite matrix of constants for all w 2W. XT D Assumption A1(e): T 1=2 rw L (Yt ; g (Xt ; w)) ! N (0; B (w)), where B (w) is …nite t=1and 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: 1=2 p D ^ VT T (wT ^ w ) ! N (0; I) T " T # ^ ^ ^ ^ 1 ^ 1 1X 1 Xwhere VT AT BT AT ; AT rww L (Yt ; g (Xt ; wT )) ; BT ^ V p rw L (Yt ; g (Xt ; wT )) ^ T T t=1 t=1 p ^ ^and BT is some symmetric and positive de…nite estimator of BT such that BT BT ! 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: 2 • @L ( t ; g (Xt ; w)) 1 @ L t ; g (Xt ; w) 2L (Yt ; g (Xt ; w)) = L ( t ; g (Xt ; w)) + (Yt t )+ (Yt t) @ 2 @ 2where •t = t Yt + (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: 2 3 1 @ 2 L •t ; g (Xt ; w) E [L (Yt ; g (Xt ; w))] = E [L ( t ; g (Xt ; w))] + E 4 (Yt t) 5 2 2 @ 2 2Distance measures in the class in equation (3) yield @ 2 L ( ; X) =@ = C 0 ( ) ; and so 1 h 0 • 2 i E [L (Yt ; g (Xt ; w))] = E [L ( t ; g (Xt ; w))] E C t (Yt t) 2Notice that the second term above does not depend on w; thus the parameter that minimisesE [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 wT : This part of the proof uses standard results ^from m-estimation theory, see Gallant and White (1988) for example. Under assumptions A1(a) 6
  8. 8. pand A1(b), Theorem 3.3 of Gallant and White (1988) yields wT ^ w ! 0: Combining this with ~ pthe fact that w = w yields consistency of wT for the parameter of interest: wT ~ ^ ^ w ! 0: UnderAssumption A1, Theorem 5.1 of Gallant and White (1988) yields the asymptotic normality of wT ; ^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 combinationweights from the data, by employing a “robust”loss function of the form in equation (3) and usinga lead (or combination of leads) of a conditionally unbiased proxy for t: This proposition furthershows 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 thanthose that would be obtained if t was observable, but nevertheless these standard errors can beestimated using the expressions above.3 Empirical application3.1 Data descriptionIn 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 estimatorsand sampling frequencies. We use data on NYSE trade prices from the TAQ database over theperiod from January 1996 to June 2007, yielding a total of 2893 daily observations3 . Figure 1reveals that the sample includes periods of rising prices and moderate-to-high volatility (roughly1996-1999), of slightly falling prices and relatively high volatility (roughly 2000-2003), and of stableprices and relatively low volatility (2004-2007). In addition to considering the full sample estimatesof optimal combination estimators, we will consider the results in each of these three sub-samples. [ INSERT FIGURE 1 HERE ] 3 We use trade prices from the NYSE only, between 9:45am and 4:00pm, with a g127 code of 0 or 40, a corr code of0 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-tradechange of more than 10% relative to the opening price. Finally, if more than one price was observed with the sametime stamp then we used the median of these prices. See Barndor¤-Nielsen, et al. (2008b) for a discussion of cleaninghigh frequency data. 7
  9. 9. 3.2 Description of the individual estimatorsThe motivation for our study of realised measures is that the varied forms of realised measuresthat have been proposed in the literature to date, and the di¤erent pieces of information capturedby each, may allow for the construction of combination estimators that out-perform any givenindividual 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 ofthe original paper as closely as possible (and in most cases, exactly). We omit detailed de…nitionsand descriptions of each estimator in the interests of space and instead refer the interested readerto the original papers. We …rstly consider standard realised variance, de…ned as: Xm (m) 2 RVt = rt;j (10) j=1where m is the number of intra-daily returns used, and rt;j is the j th return on day t: The numberof intra-daily returns used can vary from 22,500 (if we sample prices every second between 9:45amand 4pm) to 1, if we sample just the open and the close price. To keep the number of estimatorstractable, and with the high degree of correlation between RV estimators with similar samplingfrequencies 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 ofminutes 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 spacedin “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 approximatelyhomoskedastic. Theory suggests that this should improve the accuracy of RV estimation, see Hansenand Lunde (2006a) and Oomen (2006). We consider average sampling frequencies in tick-time thatcorrespond to those used in calendar time: 1 second, 5 seconds, 1 minute, 5 minutes, 62.5 minutesand 375 minutes. The highest and the lowest of these frequencies lead to estimators that arenumerically identical to calendar-time RV, and so we drop these from the analysis. 8
  10. 10. We then draw on the work of Bandi and Russell (2006, 2008), who provide a method to estimatethe optimal (calendar-time) sampling frequency, for each day, for realised variance in the presenceof market microstructure noise. This formula relies on estimates of the variance and kurtosis of themicrostructure noise, as well as preliminary estimates of the integrated variance (IV) and integratedquarticity (IQ) of the price process: we follow Bandi and Russell (2008), who also study IBM stockreturns, and estimate the moments of the microstructure noise using 1-second returns, and use15-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 ofthe microstructure noise: we consider the Bandi-Russell RV estimator both with RV BR;bc andwithout 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’ implementation of this estimator and use 5-minute blocks. Rather than estimate the number sof prices to use within each block from the number of non-zero return changes, as in Christensenand 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 RRVusing 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.’ application sto 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 bothcalendar-time and tick-time sampling. The fourth set of realised measures are the “bi-power variation” (BPV) of Barndor¤-Nielsenand Shephard (2006). These authors implemented their estimator using 5-minute calendar-timereturns, however this was presumably partially dictated by their data (indicative quotes for theUS dollar/German Deutsche mark and US dollar/Japanese yen exchange rate), for which this wasthe highest frequency. We thus implement BPV at both 5-minute and 1-minute frequencies, usingboth 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 partof quadratic variation due to the continuous semimartingale (that is, the integrated variance, IV) and so ignore thecontribution to QV from jumps. If jumps are unpredictable (or less predictable than IV), as found by Andersen, 9
  11. 11. 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 equivalentto the “multi-scale”estimator (MSRV) of Zhang (2006), (and which in turn was shown by Zhang tobe more e¢ cient than 2SRV). For both RK bart and RK cubic we consider both 5-tick sampling and5-second sampling, and in all cases we use the optimal bandwidth for a given kernel provided inBNHLS5;6 . Next we consider the “modi…ed Tukey-Hanning2 ”(TH) kernel, following their empiricalapplication to General Electric stock returns. They suggest using 1-minute tick-time sampling withoptimal bandwidth, which we implement here. We also consider the TH kernel with bandwidth…xed to equal 5 RK T H 5bw , and the TH kernel with optimal bandwidth applied to 5-tick returns RK T H;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 guaranteedfor the other RK estimators considered above). We implement this with their optimal bandwidthformula, 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, Largeproposes a novel estimator of the QV of the price series. Our application to IBM stock returnsdoes not satisfy this assumption, and so formally ALTRV is not applicable here, but we implementa modi…cation of this estimator to see whether it nevertheless provides useful information in theoptimal combination estimator. Following Large (2005), we implement this on quote prices ratherthan trade prices. To handle the fact that non-zero absolute IBM price changes are not always ofBollerslev and Diebold (2006), then there may be bene…ts to using estimators that focus on IV rather than QV inforecasting applications. 5 These optimal bandwidths, like the RVBR sampling frequencies, are estimated for each day in the sample and socan change with market conditions, the level of market microstructure noise, etc. 6 These optimal bandwidth formulas require estimates of moments of the noise and preliminary estimates of IVand 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 RKestimate used to bias-correct the noise estimate. These are slightly di¤erent to the choices in the web appendix toBarndor¤-Nielsen, et al. (2008a); see http://www.hha.dk/~alunde/BNHLS/BNHLS.htm for further discussion. 10
  12. 12. the same size (i.e. 1 tick), on days when there are more than one size for non-zero absolute pricechanges 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 otherestimators considered here) rather than the price, we scale ALTRV by the squared opening pricefor 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 tothe mid-quote (ALTRVmq ), and the average of the ALTRVs applied separately to the bid and theask prices (ALTRVcombo ). In total we have 30 di¤erent realised measures, from 6 di¤erent classes of estimators, with avariety of sampling frequencies and sampling schemes. To the best of our knowledge, this is thelargest collection of realised measures considered in a single empirical study. Table 1 presents some summary statistics for these 30 estimators. ALT RV mq has the smallestaverage 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% annualisedstandard deviation. Whilst RV 1day has a reasonable average value, it performs poorly on theother summary statistics: it has the highest standard deviation, skewness, and kurtosis of all 30estimators. BP V 1 min is the estimator with the lowest standard deviation, while QRV has thelowest skewness and kurtosis. RV 1day is also the only estimator to generate estimates of QV thatare not strictly positive: its minimum value is zero, which it attains on 42 days (around 1.5% ofthe sample). None of the other estimators have a minimum value that is non-positive, includingthe 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 twostandard estimators in the literature (RV1day and RV5 min ), a naïve choice given high frequencydata (RV1 sec ), and three of the estimators that ultimately perform well in our empirical analysis(RVtick;1 min ; BPVtick;1 min and RKcubic ). This table shows that these estimators are generally highlycorrelated (as one would expect); the average correlation across all elements of their correlationmatrix is 0.861. This should be kept in mind when interpreting the estimated optimal combinationweights in Section 3.4. The highest correlation between any two estimators is between RKtick;cubicand RKT H;5tick ; which is 0.9992. The lowest correlation between any two estimators is betweenRV1day and ALTRVcombo , at 0.425. 11
  13. 13. [ INSERT TABLES 1 AND 2 HERE ]3.3 Results using simple combination estimatorsIn Tables 3 and 4 we present the …rst set of empirical results of the paper. These tables present theestimated accuracy of each of the estimators using the ranking methodology in Patton (2008). Ourpreferred measure of accuracy is the QLIKE distance measure, and we also present results for thefamiliar MSE distance measure. We use the random walk approximation (assumption T1 above),with a one-period lead of the RV 5 min as the instrument for the latent quadratic variation to obtainthese estimates. The ranking method of Patton (2008) can only estimate the accuracy of an estimator relativeto some other estimator, and in Tables 3 and 4 we use RV 5 min as the base estimator; this choiceis purely a normalisation and has no e¤ect on the conclusions. Negative values in the …rst columnsof Tables 3 and 4 indicate that a given estimator has lower average MSE or QLIKE distance to thelatent QV (i.e., greater accuracy) than RV 5 min , while positive values indicate higher average MSEor QLIKE distance than RV 5 min : We consider the 30 individual realised measures discussed in the previous section, as well as threesimple combination estimators: the equally-weighted arithmetic mean, equally-weighted geometricmean, and the median, leading to a total of 33 estimators. Under QLIKE the most accurateestimator of QV is the simple RV tick;1 min ; 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 ranked20th in the last sub-period. The top 5 estimators, averaging across the full-sample ranks under bothMSE and QLIKE are QRV tick (…rst), RK cubic (second), RV BR and QRV (equal third), RV M eanand RV M edian (equal …fth). The discussion of rankings of average accuracy is a useful initial look at the results, but amore formal analysis is desirable. We use two approaches: The …rst is the “model con…denceset” (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 ofmodels 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
  14. 14. 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;1 min , QRV tick , RK cubic , and RV M ean : Under MSE distance the MCS contains 10 estimators:RV 1 min ; RV tick;1 min , RV BR , QRV , QRV tick , BP V 1 min , BP V tick;5 min , RV tick;bart , RV cubic , RV M edian :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, whichis 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 Pattonand 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 estimatorsuses the stepwise multiple testing method of Romano and Wolf (2005). This method identi…es theestimators 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 considerthree choices of benchmark estimator: RV 1day ; which is the standard estimator in the absence ofhigh frequency data; RV 5 min ; which is based on a rule-of-thumb from earlier papers in the RVliterature (see Andersen, et al. (2001b) and Barndor¤-Nielsen and Shephard (2002) for example);and RV M ean ; which is the standard simple combination estimator. The results of these tests arepresented 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 thefull sample and in all three sub-samples. This clearly a strong signal that high frequency data, whenused in any one of the 32 other estimators in this study, yields more precise estimates of QV thandaily data can. When the RV 5 min is taken as a benchmark we see more variation in the results: someestimators are signi…cantly better, others signi…cantly worse and some not signi…cantly di¤erent.Broadly stated, the estimators that out-perform RV 5 min include RV sampled at the 1-minute 7 The MCS is implemented via a bootstrap re-sampling scheme. We use Politis and Romano’ (1994) stationary sbootstrap with an average block length of 10 days and 1000 bootstrap replications for each test. 8 The Romano-Wolf testing method is also implemented using Politis and Romano’ (1994) stationary bootstrap swith an average block length of 10 days, and we again use 1000 bootstrap replications for each test. 13
  15. 15. frequency (either in tick time or calendar time), RRV and QRV, RK with the Bartlett, cubicor Tukey-Hanning kernel (when sampled every 5 ticks or 5 seconds) and the three combinationestimators. The estimators that under -perform RV 5 min include RV sampled at the 1 hour or 1 dayfrequency, RKT H sampled at the 1 minute frequency, and ALTRV (though recall that the latterestimator was not designed for a stock like IBM). Finally, when RV M ean is taken as the benchmark estimator in the Romano-Wolf testing methoda very clear conclusion emerges: no estimator signi…cantly out-performs RV M ean . Several estima-tors are not signi…cantly di¤erent, but none are signi…cantly better in the full sample or in anyof the sub-samples. This is a strong endorsement of using this simple combination estimator inpractice9 . 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 5 min or RV M ean as the benchmarkthere are very few rejections of the null hypothesis. [ INSERT TABLES 5 AND 6 HERE ]3.4 Results using optimal combination estimatorsIn this section we present our estimated optimal combination estimators. We consider two standardparametric combination estimators, a linear combination and a multiplicative combination: n X ^ lin = w0 + Xt ^ wi Xit ; and ^ (11) i=1 n Y ^ mult = ! 0 !^ Xt ^ Xit i ; (12) i=1estimated 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 spaceand so permit us to compute standard errors using Proposition 1, and also constrained combinationestimators, 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 RV M edian as thebenchmarks. Under MSE neither of these was ever signi…cantly out-performed, while under QLIKE RV Geo meanwas signi…cantly beaten (amongst others, by RV M ean ) while RV M edian was never signi…cantly beaten. These resultsare available upon request. 14
  16. 16. estimation makes obtaining standard errors di¢ cult (and we do not pursue this here) but providesadditional 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 optimalconstrained combination. The largest weights are assigned to QRV (in calendar time and ticktime) with some weight also given to RV tick;5 sec ; RK T H ; RK T H;5tick : Small weight (0.0039) isalso given to RV 1day . In the unconstrained combination the results are broadly similar: somesigni…cant weight is attached to QRV (in calendar time and tick time), and to RK T H ; RK T H;5tickand RK tick;cubic : Interestingly, although the ALTRV estimators do not get signi…cant weight in thefull sample, they do have signi…cant coe¢ cients in the 1996-1999 sub-sample. This correspondswith the period where the assumption of single-tick price changes is most plausible for IBM: theminimum 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 theoptimal constrained combination estimator we again see substantial weight on QRV, and alsoon RV N F P arzen ; and we also see non-zero weights on a collection of simple RVs, with samplingfrequencies from 5 seconds up to and including 1 day. As in the optimal MSE combination, theunconstrained combination has few individually signi…cant coe¢ cients, though there are signi…cantcoe¢ cients on BPV sampled at the 1-minute frequency, RK T H ; and simple RV with samplingfrequency ranging from 1 second to 1day. Also similar to the optimal MSE combination, we againsee that the ALTRV estimators receive signi…cant weight in the …rst sub-sample but not in thelatter 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, theresults are broadly in line with the optimal linear combination results: the optimal combinationsfor MSE put substantial weights on QRV, RK T H ; RK T H;5tick and RV tick;5 sec ; while the optimalcombinations for QLIKE again put weight on QRV, RV N F P arzen and a collection of simple RVswith sampling frequencies between 5 seconds and 1 day. 10 Source: New York Stock Exchange web site, http://www.nyse.com/about/history/timeline_chronology_index.html. 15
  17. 17. Often of interest in the forecasting literature is whether the estimated optimal combinationis signi…cantly di¤erent from a simple equally-weighted average. If we let wi denote the optimallinear combination weights, and ! i denote the optimal multiplicative combination weights, thehypotheses of interest are: L H0 : w0 = 0 w1 = w2 = ::: = wn = 1=n (13) L vs. Ha : w0 6= 0 [ wi 6= 1=n for some i = 1; 2; :::; n M and H0 : ! 0 = 1 ! 1 = ! 2 = ::: = ! n = 1=n (14) M vs. Ha : ! 0 6= 1 [ ! i 6= 1=n for some i = 1; 2; :::; nUsing Proposition 1 these hypotheses can be tested using Wald tests, and we …nd on the fullsample of data that the null that the optimal combination is an equally-weighted combination canbe 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 wasnot signi…cantly beaten by any individual estimator, it can still be improved: an optimally formedlinear 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: L;i H0 : wi = 1 wj = 0 8 j 6= i (15) L;i vs. Ha : wi 6= 1 [ wj 6= 0 for some j 6= i M;i and H0 : ! 0 = ! i = 1 ! j = 0 8 j 6= 0; i (16) M;i vs. Ha : ! i 6= 1 [ ! 0 6= 1 [ wj 6= 0 for some j 6= 0; ifor i = 1; 2; :::; n: We …nd that the null hypothesis is rejected for every single estimator, under bothMSE and QLIKE distance, for both linear and multiplicative combinations, with all p-values lessthan 0:01. This is strong evidence that there are gains to considering combination estimators ofquadratic variation: no single estimator dominates all others. This result is new to the realisedvolatility literature, but is probably not surprising to those familiar with with forecasting in practice. 16
  18. 18. 4 Summary and conclusionRecent advances in …nancial econometrics have led to the development of new estimators of assetprice variability using high frequency price data. These estimators are based on a variety of di¤erentassumptions and take many di¤erent functional forms. Motivated by the success of combinationforecasts over individual forecasts in a range of forecasting applications, see Clemen (1989) andTimmermann (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 isa resounding “yes”. This paper presents a novel method for combining individual realised measures to form newestimators 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 6distinct 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…cantlyout-performs, in terms of average accuracy, a simple equally-weighted average. Further, we …nd thatnone of the individual estimators encompasses the information in all other estimators, providingfurther support for the use of combination realised measures. Overall our results suggest that thereare indeed bene…ts from combining the information contained in the array of di¤erent volatilityestimators proposed in the literature to date. Several extensions of our analysis are possible. Firstly, it may be interesting, both economicallyand statistically, to try to understand why certain estimators out-perform, or enter with greaterweight 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 estimatorrelative 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 thelines of Elliott and Timmermann (2005), rather than the static combination estimators consideredin this paper. 17
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  22. 22. [48] Zhang, L., 2006, E¢ cient Estimation of Stochastic Volatility using Noisy Observations: A Multi-Scale Approach, Bernoulli , 12, 1019-1043.[49] Zhang, L., P.A. Mykland, and Y. Aït-Sahalia, 2005, A Tale of Two Time Scales: Determining Integrated Volatility With Noisy High-Frequency Data, Journal of the American Statistical Association, 100, 1394-1411.[50] Zhou, B., 1996, High-Frequency Data and Volatility in Foreign-Exchange Rates, Journal of Business and Economic Statistics, 14, 45-52. Price of IBM (adjusted f or splits) 150 100 50 0 Jan96 Jan97 Jan98 Jan99 Jan00 Jan01 Jan02 Jan03 Jan04 Jan05 Jan06 Jan07 Annualised volatility of IBM, using RV5min 100 80 60 40 20 0 Jan96 Jan97 Jan98 Jan99 Jan00 Jan01 Jan02 Jan03 Jan04 Jan05 Jan06 Jan07Figure 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 based pon 5-minute calendar-time trade prices, annualised using the formula t = 252 RVt : . 21
  23. 23. Table 1: Summary statistics of the realised measures Standard Mean Deviation Skewness Kurtosis Minimum RV1 sec 3.128 2.952 2.710 18.255 0.149 RV5 sec 2.985 2.804 2.736 18.866 0.130 RV1 min 2.388 2.377 3.731 34.841 0.092 RV5 min 2.296 2.937 7.339 120.425 0.081 RV1hr 2.223 3.414 5.010 42.997 0.010 RV1day 2.343 5.864 9.732 163.764 0.000 RVtick;5 sec 3.314 3.113 2.583 16.387 0.191 RVtick;1 min 2.517 2.592 4.204 41.515 0.114 RVtick;5 min 2.461 3.060 6.896 111.351 0.076 RVtick;1hr 2.423 3.972 7.456 104.018 0.012 RVBR 2.385 2.274 3.315 27.760 0.103 RVBR;bc 2.664 2.532 2.832 18.914 0.107 RRV 2.382 2.651 4.562 48.207 0.120 RRVtick 2.378 2.713 5.208 65.155 0.119 QRV 2.509 2.327 2.352 11.026 0.130 QRVtick 2.483 2.295 2.503 12.629 0.108 BPV1 min 2.173 2.170 2.651 13.156 0.110 BPV5 min 2.278 2.730 4.116 31.550 0.098 BPVtick;1 min 2.162 2.268 3.856 35.470 0.095 BPVtick;5 min 2.296 2.658 3.929 29.194 0.056 RKbart 2.638 2.447 2.968 21.904 0.138 RKtick;bart 2.480 2.509 4.127 45.574 0.107 RKcubic 2.468 2.296 3.106 23.317 0.111 RKtick;cubic 2.533 3.006 6.796 114.502 0.099 RKT H 2.323 2.899 4.535 39.456 0.026 RKT H bw5 2.322 3.406 6.824 86.401 0.031 RKT H 5tick 2.535 3.012 6.761 113.514 0.099 RKN F P arzen 2.373 2.930 5.013 49.124 0.062 ALTRVmq 2.158 2.562 5.893 65.928 0.113 ALTRVcombo 2.765 3.372 7.014 95.550 0.139 RVM ean 2.495 2.663 4.072 36.688 0.112 RVGeo mean 2.270 2.446 3.914 33.699 0.094 RVM edian 2.404 2.473 3.753 33.029 0.114 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
  24. 24. Table 2: Correlation between the realised measures RV1 sec RV5 min RV1day RVtick;1 min BPVtick;1 min RKcubic RV1 sec 1 0.836 0.445 0.922 0.925 0.944 RV5 sec 0.999 0.848 0.455 0.934 0.935 0.954 RV1 min 0.934 0.941 0.527 0.977 0.978 0.986 RV5 min 0.836 1 0.568 0.942 0.924 0.915 RV1hr 0.683 0.806 0.668 0.775 0.776 0.767 RV1day 0.445 0.568 1 0.526 0.524 0.512 RVtick;5 sec 0.998 0.832 0.447 0.922 0.926 0.945 RVtick;1 min 0.922 0.942 0.526 1 0.978 0.971 RVtick;5 min 0.843 0.967 0.588 0.939 0.935 0.925 RVtick;1hr 0.641 0.791 0.695 0.746 0.736 0.730 RVBR 0.948 0.919 0.508 0.972 0.976 0.991 RVBR;bc 0.976 0.889 0.486 0.953 0.956 0.978 RRV 0.892 0.977 0.563 0.974 0.971 0.966 RRVtick 0.888 0.969 0.570 0.976 0.974 0.961 QRV 0.905 0.845 0.484 0.923 0.936 0.954 QRVtick 0.905 0.852 0.507 0.931 0.944 0.953 BPV1 min 0.912 0.867 0.493 0.943 0.955 0.960 BPV5 min 0.835 0.943 0.545 0.923 0.916 0.918 BPVtick;1 min 0.925 0.924 0.524 0.978 1 0.977 BPVtick;5 min 0.829 0.895 0.571 0.908 0.913 0.911 RKbart 0.967 0.896 0.497 0.966 0.973 0.992 RKtick;bart 0.925 0.946 0.533 0.981 0.981 0.987 RKcubic 0.944 0.915 0.512 0.971 0.977 1 RKtick;cubic 0.855 0.973 0.571 0.965 0.951 0.941 RKT H 0.801 0.912 0.599 0.896 0.907 0.896 RKT H bw5 0.726 0.871 0.653 0.827 0.828 0.817 RKT H 5tick 0.853 0.975 0.574 0.965 0.951 0.940 RKN F P arzen 0.820 0.931 0.608 0.909 0.913 0.908 ALTRVmq 0.749 0.811 0.472 0.844 0.856 0.865 ALTRVcombo 0.702 0.751 0.425 0.793 0.816 0.818 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 RV5 min ), a naïve choice given high frequency data (RV1 sec ), and threeof the estimators that turn out to perform well in our empirical analysis (RVtick;1 min ; BPVtick;1 minand RKcubic ). The complete correlation matrix is available from the authors on request. 23
  25. 25. Notes to Table 3: The …rst column of this table presents the average di¤erence in MSEdistance of each realised measure, relative to RV 5 min ; with negative (positive) values indicatingthat the estimator was closer (further) on average to the target variable than RV 5 min : Columns2– present the rank of each estimator using MSE distance, for the full sample period and for 5three 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– present an indicator for whether the 9estimator 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 5 min ; with negative (positive) values indicatingthat the estimator was closer (further) on average to the target variable than RV 5 min : Columns2– present the rank of each estimator using QLIKE distance, for the full sample period and for 5three 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– present an indicator for whether the 9estimator 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
  26. 26. 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 04-07RV1 sec 0.117 26 28 22 30 – – – –RV5 sec -0.504 19 25 19 26 – – – –RV1 min -1.783 7 6 7 9 X X X –RV5 min 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;1 min -1.292 14 15 14 16 X X X –RVtick;5 min 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 –BPV1 min -2.033 3 5 3 12 X X X –BPV5 min -0.762 18 24 17 13 – – – –BPVtick;1 min -1.878 6 9 5 5 X X X XBPVtick;5 min -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 – – – –RKT H -0.398 20 27 20 4 – – – XRKT H bw5 2.515 29 31 29 24 – – – –RKT H 5tick -0.024 23 20 23 17 – – – –RKN F P arzen -0.243 21 26 21 3 – – – XALTRVmq -0.038 22 18 25 27 – – – –ALTRVcombo 3.019 31 17 32 32 – X – –RVM ean -1.300 13 16 13 8 – X – –RVGeo mean -1.528 12 7 12 1 – X – XRVM edian -1.715 8 10 9 6 X X X –Notes: See page 24 for a description of this table. 25
  27. 27. 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 04-07RV1 sec -0.008 19 13 26 23 – – – –RV5 sec -0.018 18 12 24 21 – – – –RV1 min -0.041 4 5 6 5 – – X XRV5 min 0.000 22 21 22 27 – – – –RV1hr 0.731 32 32 32 32 – – – –RV1day 33.671 33 33 33 33 – – – –RVtick;5 sec 0.002 24 15 29 25 – – – –RVtick;1 min -0.046 1 1 2 4 X X X XRVtick;5 min -0.022 14 14 18 19 – – – –RVtick;1hr 0.484 31 31 31 31 – – – –RVBR -0.041 6 4 7 10 – – – –RVBR;bc -0.034 11 6 17 17 – – – –RRV -0.029 13 16 9 12 – – X –RRVtick -0.031 12 17 8 3 – – X XQRV -0.038 9 9 10 9 – – – XQRVtick -0.041 5 8 1 11 X – X –BPV1 min 0.010 25 28 13 6 – – – XBPV5 min 0.018 27 26 23 26 – – – –BPVtick;1 min -0.003 20 27 12 8 – – – XBPVtick;5 min 0.001 23 24 20 22 – – – –RKbart -0.038 8 3 14 16 – X – –RKtick;bart -0.036 10 10 11 7 – – – XRKcubic -0.042 3 2 5 13 X X X –RKtick;cubic -0.020 17 20 15 14 – – – –RKT H 0.071 29 30 28 28 – – – –RKT H bw5 0.148 30 29 30 30 – – – –RKT H 5tick -0.021 16 19 16 15 – – – –RKN F P arzen -0.003 21 22 21 24 – – – –ALTRVmq 0.015 26 25 25 20 – – – –ALTRVcombo 0.031 28 23 27 29 – – – –RVM ean -0.043 2 7 4 1 X – X XRVGeo mean -0.021 15 18 19 18 – – – –RVM edian -0.039 7 11 3 2 – – X X Notes: See page 24 for a description of this table. 26
  28. 28. Notes to Table 5: This table presents the results of the Romano-Wolf “stepwise”test for threedi¤erent choices of benchmark estimator. Columns 1– present indicators for when the benchmark 4is 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’ accuracy is not signi…cantly di¤erent from RV1day : The four columns refer to the full ssample period and three sub-samples, 1996-1999, 2000-2003 and 2004-2007. Columns 5– present 8corresponding results when the benchmark estimator is set to RV 5 min ; and columns 9–12 presentresults when the benchmark estimator is set to RVM ean : 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– present indicators for when the benchmark 4is set to RV 1day : 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’ accuracy is not signi…cantly di¤erent from RV1day : The four columns refer to sthe 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 RV5 min ; and columns 9– 12present results when the benchmark estimator is set to RVM ean : The benchmark estimator in eachcolumn is denoted with a F: 27
  29. 29. Table 5: Romano-Wolf tests on the realised measures, under MSE distance Romano-Wolf: RV1day Romano-Wolf: RV5min Romano-Wolf: RVM ean Full 96-99 00-03 04-07 Full 96-99 00-03 04-07 Full 96-99 00-03 04-07RV1 sec X X X X – – – – –RV5 sec X X X X – – – – – –RV1 min X X X X – – – – – – – –RV5 min 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;1 min X X X X – – – – – – – –RVtick;5 min X X X X – – – – – – – –RVtick;1hr X X X X – – – – – –RVBR X X X X – – – – – – – –RVBR;bc X X X X – – – – – – – –RRV X X X X – – – – – – – –RRVtick X X X X – – – – – – – –QRV X X X X – – – – – – – –QRVtick X X X X – – – – – – – –BPV1 min X X X X – – – – – – – –BPV5 min X X X X – – – – – – –BPVtick;1 min X X X X – – – – – – – –BPVtick;5 min X X X X – – – – – – – –RKbart X X X X – – – – – – – –RKtick;bart X X X X – – – – – – – –RKcubic X X X X – – – – – – – –RKtick;cubic X X X X – – – – – – – –RKT H X X X X – – – – – – – –RKT H bw5 X X X X – – – – – –RKT H 5tick X X X X – – – – – – – –RKN F P arzen X X X X – – – – – – – –ALTRVmq X X X X – – – – – – – –ALTRVcombo X X X – – – – – – –RVM ean X X X X – – – – F F F FRVGeo mean X X X X – – – – – – – –RVM edian X X X X – – – – – – – – Notes: See page 27 for a description of this table. 28
  30. 30. Table 6: Romano-Wolf tests on the realised measures, under QLIKE distance Romano-Wolf: RV1day Romano-Wolf: RV5min Romano-Wolf: RVmean Full 96-99 00-03 04-07 Full 96-99 00-03 04-07 Full 96-99 00-03 04-07RV1 sec X X X X – – – –RV5 sec X X X X – – – – –RV1 min X X X X X X X X – – – –RV5 min X X X X F F F FRV1hr X X X XRV1day F F F FRVtick;5 sec X X X X – – –RVtick;1 min X X X X X X X X – – – –RVtick;5 min X X X X X – – XRVtick;1hr X X X XRVBR 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 – – –BPV1 min X X X X – X XBPV5 min X X X X – –BPVtick;1 min X X X X – X X –BPVtick;5 min 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 XRKT H X X X X –RKT H bw5 X X X XRKT H 5tick X X X X X – X XRKN F P arzen X X X X – – – –ALTRVmq X X X X –ALTRVcombo X X X X –RVM ean X X X X X X X X F F F FRVGeo mean X X X X X – – XRVM edian X X X X X X X X – – Notes: See page 27 for a description of this table. 29
  31. 31. Notes to Table 7: This table presents the MSE-optimal linear combination weights. Columns1– present these weights for the unconstrained case, with estimates that are signi…cantly di¤erent 4from 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– present the optimal linear combination weights, 8using 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 are ,reported to two decimal places. Notes to Table 8: This table presents the QLIKE-optimal linear combination weights.Columns 1– present these weights for the unconstrained case, with estimates that are signi…- 4cantly 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– present the optimal linear combi- 8nation 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 from ,the boundary are reported to two decimal places. Notes to Table 9: This table presents the MSE-optimal multiplicative combination weights.Columns 1– present these weights for the unconstrained case, with estimates that are signi…cantly 4di¤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– present the optimal multiplicative 8combination 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 from ,the boundary are reported to two decimal places. Notes to Table 10: This table presents the QLIKE-optimal multiplicative combinationweights. Columns 1– present these weights for the unconstrained case, with estimates that are sig- 4ni…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– present the optimal multiplicative 8combination 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 away ,from the boundary are reported to two decimal places. 30
  32. 32. Table 7: Optimal linear combinations under MSE distance Unconstrained Constrained Full 96-99 00-03 04-07 Full 96-99 00-03 04-07 Constant 0.21 0.87 0.24 0.28 0.27 1.16 0.28 0.32 RV1 sec -0.70 -0.25 0.58 -0.39 0 0 0 0 RV5 sec 0.18 -0.40 -0.20 0.19 0 0 0 0 RV1 min -0.57 -0.58 -1.18 -0.20 0 0 0 0 RV5 min 0.31 0.37 0.50 -0.15 0.01 0 0 0 RV1hr 0.00 0.09 -0.07 0.02 0 0.08 0 0 RV1day 0.00 0.03 -0.03 -0.01 0.00 0.03 0 0 RVtick;5 sec 0.84 0.61 0.03 0.64 0.11 0 0.17 0.11 RVtick;1 min -0.33 0.23 -0.94 -0.22 0 0 0 0 RVtick;5 min -0.25 -0.11 -0.17 -0.19 0 0 0 0 RVtick;1hr 0.04 0.04 0.07 0.04 0 0.02 0 0.02 RVBR 0.07 0.70 -0.71 0.04 0 0 0 0 RVBR;bc 0.01 -0.24 0.45 -0.13 0 0 0 0 RRV -0.21 0.33 -0.71 0.07 0 0 0 0 RRVtick 0.62 0.02 0.74 0.12 0 0 0 0 QRV 0.38 0.39 0.43 0.17 0.20 0.35 0 0.07 QRVtick 0.47 -0.08 0.92 0.11 0.40 0 0.62 0 BPV1 min 0.03 0.15 0.16 0.07 0 0 0 0 BPV5 min -0.16 -0.51 0.09 -0.02 0 0 0 0 BPVtick;1 min -0.23 -0.54 0.09 -0.06 0 0 0 0 BPVtick;5 min -0.03 0.15 -0.25 0.15 0 0 0 0 RKbart -0.40 -0.04 -1.19 0.14 0 0 0 0 RKtick;bart -0.21 -0.21 -0.23 -0.40 0 0 0 0 RKcubic 0.70 0.43 1.89 -0.24 0 0 0 0 RKtick;cubic -3.31 -3.18 0.28 0.24 0 0 0 0.03 RKT H 0.26 0.05 0.17 0.50 0.03 0 0 0.17 RKT H bw5 -0.09 -0.08 -0.09 -0.01 0 0 0 0 RKT H 5tick 3.60 3.38 0.47 0.12 0.02 0.04 0 0 RKN F P arzen -0.22 -0.12 -0.34 -0.25 0 0 0 0 ALTRVmq -0.24 -0.77 0.00 0.13 0 0 0 0 ALTRVcombo 0.14 0.61 -0.01 -0.02 0 0 0 0.03Notes: See page 30 for a description of this table. 31

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