Realized Variance and IID Market Microstructure Noise

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1 Realized Variance and IID Market Microstructure Noise Peter R. Hansen a, Asger Lunde b a Brown University, Departent of Econoics, Box B,Providence, RI 02912, USA b Aarhus School of Business, Departent of Inforation Science, Denark Version: February 20, 2004 Abstract We analyze the properties of a bias-corrected realized variance (RV) in the presence of iid arket icrostructure noise. The bias correction is based on the first-order autocorrelation of intraday returns and we derive the optial sapling frequency as defined by the ean squared error (MSE) criterion. The biascorrected RV is bencharked to the standard easure of RV and an epirical analysis shows that the forer can reduce the MSE by 50%-90%. Our epirical analysis also shows that the iid noise assuption does not hold in practice. While this need not affect the RVs that are based on low-frequency intraday returns, it has iportant iplications for those based on high-frequency returns. Keywords: Realized Variance; High-Frequency Data; Integrated Variance. JEL Classification: C10; C22; C Introduction The realized variance (RV) has becoe a popular epirical easure of volatility, and the RV yields a perfect estiate of volatility in the hypothetical situation where prices are observed in continuous tie and without easureent error. This result suggests that the RV, which is a su-of-squared returns, should be based on returns that are sapled at the highest possible frequency (tick-by-tick data). However, in practice this leads to a well-known bias proble due to arket icrostructure noise, see e.g. Andreou & Ghysels (2002) and Ooen (2002a). 1 So there is a trade-off between bias and variance when choosing the sapling frequency, and this is the reason that returns are typically sapled at a oderate frequency, such as 5-inute sapling. An alternative way to handle the bias proble is to use bias correction techniques. In this paper, we analyze an estiator that utilize the first-order autocorrelation to bias-correct the RV. This estiator is denoted by RV AC1 and has previously been used by French, Schwert & Stabaugh (1987) and Zhou (1996), who applied it to Corresponding author, eail: Peter Hansen@brown.edu 1 The bias is particularly evident fro the so-called volatility signature plots that were introduced by Andersen, Bollerslev, Diebold & Labys (2000). 1

2 daily returns and intraday returns, respectively. 2 The subscript refers to the fact that we use one (the first) autocorrelation of intraday returns to correct for the bias. We ake three contributions in this paper. First, we derive the bias and variance properties of the RV AC1 and the optial sapling frequency as defined by the ean squared error (MSE) criterion. Second, we derive the asyptotic distribution of RV AC1 and show that its asyptotic variance is saller than that of the standard RV. Third, the analysis is based on a particular type of arket icrostructure noise, which has previously been analyzed by Corsi, Zubach, Müller & Dacorogna (2001), Zhang, Mykland & Aït-Sahalia (2003), and Bandi & Russell (2003). Here it is assued that the noise is independent and identically distributed (across tie) and that the noise is independent of the true price process. We label this type of noise as iid noise. An iportant result of our epirical analysis is that the iid noise assuption does not hold in practice. Under the iid noise assuption the RV AC1 is unbiased at any sapling frequency, however the RV AC1 is clearly biased when returns are sapled at high frequencies. While the RV AC1 should reduce the MSE by 80% 90% copared to the standard RV, when based on its optial saple frequency (about five-second sapling), we conclude that the iplications of the iid noise assuption are only valid when we saple every 30 seconds (or slower). At this sapling frequency the unbiased RV AC1 leads to a reduction of the MSE by a little ore than 50% in our epirical analysis. The paper is organized as follows. In Section 2, we define the RV AC1 and derives its properties. Section 3 contains an epirical analysis that quantifies the relative MSE of RV AC1 to that of the standard RV, and Section 4 contains concluding rearks. All proofs are given in the appendix. 2. Definitions and Theoretical Results Let {p (t)} be a latent log-price process in continuous tie and let {p(t)} be the observable logprices process, such that the easureent error process is given by u(t) p(t) p (t). The noise process, u, ay be due arket icrostructure effects such as bit-ask bounces, but the discrepancy between p and p can also be a result of the technique that is used to construct p(t). For exaple, p is often constructed artificially fro observed trades and quotes using the previous-tick ethod or the linear interpolation ethod. 3 We assue that the specification for p is a siple stochastic volatility odel and our assup- 2 Other approached to bias correcting the RV include the filtering techniques by Andersen, Bollerslev, Diebold & Ebens (2001) (oving average) and Bollen & Inder (2002) (autoregressive). 3 The forer was proposed by Wasserfallen & Zierann (1985) and the latter was used by Andersen & Bollerslev (1997). For a discussion of the two, see Dacorogna, Gencay, Müller, Olsen & Pictet (2001, sec ). Soe additional approaches to calculate a easure for the realized variance are discussed in Andersen, Bollerslev & Diebold (2003). 2

3 tions about the (continuous-tie) noise process, are analogous to standard (discrete-tie) assuptions in the literature. We need the following definition. Definition 1 (Gaussian iid process) We call u(t) a Gaussian iid process with ean µ and variance ω 2 if u(t) and u(s) are independent for all t = s and u(t) N(µ,ω 2 ) for all t R. Lea 1 The Gaussian iid process exists and (u(t 1 ),...,u(t k )) N k (µ,ω 2 I k ) for any k-tuple (t 1,...,t k ) of distinct points, where µ = (µ,...,µ) and I k is the k k identity atrix. Assuption 1 (i) The true price process is given fro dp (t) = σ(t)dw(t), where w(t) is a standard Brownian otion, σ(t) is a tie-varying (rando) function that is independent of w, and σ 2 (t) is Lipschitz (alost surely). (ii) The noise process, u, is a Gaussian iid process with ean zero and variance ω 2 that is independent of p. Although we allow the volatility function, σ(t), to be rando we shall condition on σ(t) in our analysis, because our object of interest is the integrated variance, IV b a σ 2 (t)dt. The Lipschitz condition is a soothness condition that requires σ 2 (t) σ 2 (t + δ) < ɛδ for soe ɛ and all t and δ (with probability one). This specification for the noise process is siilar (or identical) to those in Corsi et al. (2001), Zhang et al. (2003), and Bandi & Russell (2003). Assuing a Gaussian distribution is not crucial but akes the analysis ore tractable. We partition the interval [a, b] into intervals of equal length, (b a)/, and obtain the returns, y i, p (a + i ) p (a + i ), i = 1,...,, that will be referred to as intraday returns. Siilarly we define y i, and e i, to be the increents in p and u, respectively, and note that e i, = y i, y i,. The realized variance for p is defined by RV () y 2 i,, and it follows that RV() consistent for the IV, as, see e.g. Meddahi (2002). An asyptotic distribution theory of realized variance (in relation to integrated variance) is established in Barndorff-Nielsen & Shephard (2002). While RV () is the ideal estiator it is not a feasible estiator, because p is latent. The realized variance of p, which is given by RV () y2 i,, is observable but suffers fro a well-known bias proble and is inconsistent for the IV. The bias-variance properties of the RV () have been established by Zhang et al. (2003) and Bandi & Russell (2003) under an iid noise assuption. The following lea suarizes soe of their results in our fraework, where our Gaussian assuptions lead to ore detailed (and sipler) expressions. First we define σ 2 i, a+i a+i σ 2 (t)dt and we note that var(y i, ) = E(y 2 i, ) = σ 2 i,. is 3

4 Lea 2 Given Assuption 1 it holds that E(RV () ) = IV + 2ω 2, var(rv () ) = 12ω 4 + 2ω 2 σ i, 2 4ω4 + 2 σ i, 4, and the asyptotic distribution is given by RV () 2ω 2 12ω4 = /3( RV() 2ω 2 1) d N(0, 1), as. Next, we consider the alternative easure of the realized variance, that is given by RV () yi, 2 + y i, y i 1, + y i, y i+1,. This quantity incorporates the epirical first-order autocorrelation which explains the subscript. This odification aounts to a bias reduction that works the sae way that robust covariance estiators, such as that of Newey & West (1987), achieve their consistency. Lea 3 Given Assuption 1 it holds that E(RV () ) = IV, var(rv () ) = 8ω 4 + 6ω 2 ( σ i, 2 ω2 ) + 6 and the asyptotic distribution is given by RV () IV 8ω4 d N(0, 1), as. An iportant result of Lea 3 is that RV () σ i, 4 + ω2 (σ 2 0, + σ 2 +1, ) + O( 2 ), is unbiased for the IV (conditionally on {σ(s), a s b}), such that an unbiased easure is available in the presence of arket icrostructure noise. A rather rearkable result of Lea 3 is that the bias corrected estiator, RV (), has a saller asyptotic variance than the unadjusted estiator, RV (). Usually a bias correction leads to a larger asyptotic variance. Also note that the asyptotic results of Lea 3 is ore useful than that of Lea 2, because the result of Lea 2 does not involve the object of interest, IV, but only shed light on aspects of the RV s bias. Note, however, that the asyptotic result of Lea 3 does not suggest that RV () should be sapled at the highest possible frequency, since the asyptotic variance is increasing in. Our expression for the variance is approxiately given by var[rv () ] 8ω 4 + 6ω 2 [ b a σ 2 (s) ω 2 ]+6 b a σ 4 (s)ds 1, where the last ter involves the integrated quarticity that was introduced by Barndorff-Nielsen & Shephard (2002). Next we copare RV () to RV () in ters of their ean square error (MSE) and their respective optial sapling frequencies for a special case. Corollary 4 Suppose that the volatility is constant such that σ 2 i, = σ 2 /, where σ 2 = IV and define the noise-to-signal ratio, λ ω 2 /σ 2. The ean squared errors are given by MSE(RV () ) = 2σ 4 [2λ λ 2 + (λ 2λ 2 ) + 1 ], 4

5 MSE[RV () ] = 2σ 4 [ 4λ 2 + 3(λ λ 2 ) λ ]. Let 0 and 1 be the optial sapling frequencies for RV() and RV (), respectively. It holds that 0 is given iplicitly as the real (positive) solution to = 1/(2λ 2 ) whereas 1 = 3 + λ/(2λ). It can be verified that 1 is several ties larger than 0, thus the optial RV() requires ore frequent sapling that the optial RV. This is quite intuitive, because RV () inforation in the data without being affected by a severe bias. can utilized ore 3. Epirical Analysis We analyze the Alcoa Inc. (AA) stock over a saple period that spans the five year fro January 2, 1998 to Deceber 31, The data are transaction prices fro the NYSE extracted fro the Trade and Quote (TAQ) database. The raw data were filtered for outliers and we discarded transactions outside period fro 9:30a to 4:00p, and days with less than five hours of trading were reoved fro the saple, which reduced the saple by 13 days. Thus we used the previoustick ethod to construct the RVs for a total of n = 1, 242 days and denoted these by RV () t RV (),t, t = 1,...,n. The RVs are calculated for the hours that the arket is open, approxiately 390 inutes per day (6.5 hours) for ost days. Fro Leas 2 and 3 it follows that 2ω 2 = E[RV () RV () ] such that ˆω 2 = 1 2 (RV() RV () ) is a natural estiator of ω 2 (under the assuptions of Corollary 4), where we define the saple averages, RV () n 1 n t=1 RV() t and RV () and n 1 n t=1 RV(),t. With = 390 (1- inute intraday returns) we find that RV () RV () = which leads to ˆω 2 = 0.657/(2 390) = , and since RV () = we obtain ˆλ = /4.762 = This leads to and 1 4, 890, which corresponds to intraday returns that are sapled approxiately every 2 inutes and every 5 seconds, respectively. 4 By plugging these nubers into the forulae of Corollary 4 we find the relative ean squared error to be MSE(RV ( 0 ) )/MSE(RV ( 1 ) ) 4.88, which (in theory) iplies that RV ( 1 ) is alost five tie ore efficient than RV ( 0 ) in ters of the ean squared error criterion. The ost coonly used sapling frequency is 5-inute sapling, which corresponds to = 78 in our application. As noted by Bandi & Russell (2003) this results in an additional loss of efficiency and theoretically we have that MSE(RV (78) ) is about 10 ties larger than MSE(RV ( 1 ) ). of 0. 4 Bandi & Russell (2003) reported optial saple frequencies for RV () (for several assets) that are quite siilar to our estiate 5

6 Fro Corollary 4 we observe that the root ean squared errors are proportional to σ 2, such that RMSE(RV () ) = σ 2 c RV () and RMSE(RV () ) = σ 2 c AC () where c 2 RV () 2[2λ λ 2 + (λ 2λ 2 ) + 1 ] and c2 AC () 2[4λ2 + 3(λ λ 2 ) + 3+λ ]. In the left panel of Figure 1 we have plotted c RV () and c AC () using our epirical estiate of λ. This reveals that the RV () doinate the RV () except at the lowest frequencies. The left panel also shows that the RV () is less sensitive to the choice of. This is also clear fro the right panel of Figure 1, where we have displayed the relative MSE of RV () to that of (the optial) RV ( 0 ) and the relative MSE of RV () to that of (the optial) RV ( 1 ). One aspect that can be read of Figure 1 is that the RV () continue to doinate the optial RV ( 0 ) for a wide ranges of frequencies, and not just in a sall neighborhood of the optial value, 1. [Figure 1 about here] The optial saple frequencies of Corollary 4 depend on paraeters that are likely to differ across days. So our estiates above should be viewed as approxiations for daily average values, in the sense that 0 = 200 is a sensible sapling frequency to use (on average), although different values are likely to be better on soe days. While 1 indicate that we should saple intraday returns every 5 seconds, we shall see that the iplications of the iid noise assuption do not hold in practice if intraday returns are sapled at high frequencies. In our application the iplications see to fail once intraday returns are sapled ore frequently than every 30 seconds Epirical Evidence against the IID Noise Assuption Under the iid noise assuption the RV () should be unbiased at any frequency. This can be understood fro the fact that the iid noise assuption causes the first-order autocorrelation of e i, (and hence y i, ) to be non-zero, whereas higher-order covariances are all zero. The RV () properly corrects for the first-order autocorrelation in y i,, which is the reason that the RV () is unbiased under the iid assuption. If higher-order autocorrelations of y i, are non-zero, which could be the case if the noise coponent, u(t), was dependent across tie (different fro iid noise), then the RV () would be biased (for large s). This proble is evident fro the signature plots in Figure 2 that show that the RV () is biased for sapling frequency above 30 seconds. For exaple, with 1-second sapling the bias is quite severe and close to that of the standard RV, however the RV () generally has a saller bias. [Figure 2 about here] 6

7 In spite of this shortcoing, we will still argue that the RV () is preferred to the standard RV. The volatility signature plot of RV AC1 indicate that the tie-dependence in u persists for less than 30 seconds, because the signature plot is quite constant for the frequencies that are below a 30-second sapling. So our estiate of λ (that is based on 1-inute returns) should not be affected by the tie dependence, and this value of λ suggests that the MSE of the RV (780) (30-seconds returns) is 58% saller than that of the optial RV ( 0 ), see Figure 1. Nevertheless, Figure 2 shows that there is a need to study the properties of the RV under a ore general specification for the noise process, such as the Ornstein Uhlenbeck specification that was analyzed in a related setting by Aït-Sahalia, Mykland & Zhang (2003). 4. Concluding Rearks We have derived the bias and variance properties of RV (), which equals the standard realized variance plus a bias correction that is given fro the first-order autocorrelation of intraday returns. The RV () copares favorable to the standard easure of RV in ters of the ean squared error criterion. Our epirical analysis showed that the MSE of RV () ay be 90% saller than the MSE of the ost coon easure of RV, provided that the arket icrostructure noise satisfies the iid assuption. Most of the existing theoretical studies of the RV in the presence of arket icrostructure effects are based on this assuption, however our epirical analysis revealed that this assuption does not hold in practice. While it ay be true (or approxiately true) for sapling at low frequencies, it does not hold when returns are sapled ore frequently than every 30 seconds in our epirical analysis. This followed directly fro the volatility signature plot of RV () in Figure 2. While the RV () is biased when sapling at high frequencies, its bias was less severe than that of the standard RV, and RV () was found to doinate the standard RV () when the forer is based on a less aggressive sapling, such as 30-second sapling. However, our analysis has revealed a need to study the properties of RV-easures under a ore general specification for the noise process. Soe preliinary results can be found in Hansen & Lunde (2003) who use a odel-free noise structure, and in Ooen (2002b) who use a odel-based approach. Acknowledgeents We thank Neil Shephard for valuable coents. Financial support fro the Danish Research Agency, grant no is gratefully acknowledged. All errors reain our responsibility. 7

8 Appendix of Proofs Proof of Lea 1. That (u(t 1 ),...,u(t k )) N(µ,ω 2 I k ) follows fro the definition of u, and since this is a well-defined (ultivariate) Gaussian distribution, the existence of u follows directly fro Kologorov s Existence Theore, see Billingsley (1995, chapter 7). As stated earlier, we condition on σ(t) in our analysis, thus without loss of generality we treat σ(t) as a deterinistic function in our derivations. Proof of Lea 2. The bias follows directly fro the decoposition y 2 i, = y 2 i, + e2 i, + 2y i, e i,, since E(e 2 i, ) = 2ω2. Siilarly, we see that var(rv () ) = var( y 2 i, ) + var( ei, 2 ) + 4 var( yi, e i,) because the three sus are uncorrelated. The first su involves uncorrelated ters such that var( yi, 2 ) = var(yi, 2 ) = 2 σ i, 4, where the last equality follows fro the Gaussian assuption. For the second su we find E(e 4 i, ) = E(u i, u i 1, ) 4 = E(u 2 i, + u2 i 1, 2u i,u i 1, ) 2 = E(u 4 i, + u4 i 1, + 4u2 i, u2 i 1, + 2u2 i, u2 i 1, ) + 0 = 6ω 4 + 6ω 4 = 12ω 4, E(e 2 i, e2 i+1, ) = E(u i, u i 1, ) 2 (u i+1, u i, ) 2 = E(u 2 i, + u2 i 1, 2u i,u i 1, )(u 2 i+1, + u2 i, 2u i+1,u i, ) = E(u 2 i, + u2 i 1, )(u2 i+1, + u2 i, ) + 0 = 6ω4. such that var(e 2 i, ) = 12ω4 [E(e 2 i, )]2 = 8ω 4 and cov(e 2 i,, e2 i+1, ) = 2ω4. Since cov(e 2 i,, e2 i+h, ) = 0 for h 2 it follows that var( ei, 2 ) = var(ei, 2 ) + cov(ei, 2, e2 i+h, ) = 8ω4 + 2( 1)2ω 4 = 12 ω 4 4ω 4. i, j=1 i = j The last su involves uncorrelated ters such that var( e i, yi, ) = var(e i, yi, ) = 2ω2 Finally, the asyptotic norality follows by the central liit theore for heterogeneous arrays with finite σ 2 i,. dependence, and the fact that 2 σ 4 i, + 2ω2 σ 2 i, 4ω4 = O(1). Proof of Lea 3. First we note that RV () = Y i, + U i, + V i, + W i,, where Y i, y i, (y i 1, + y i, + y i+1, ) U i, (u i, u i 1, )(u i+1, u i 2, ) V i, y i, (u i+1, u i 2, ) W i, (u i, u i 1, )(y i 1, + y i, + y i+1, ), 8

9 since y i, (y i 1, + y i, + y i+1, ) = (yi, + u i, u i 1, )(yi 1, + y i, + y i+1, + u i+1, u i 2, ) =Y i, + U i, + V i, + W i,. Thus the properties of RV () are given fro those of Y i,, U i,, V i,, and W i,. It follows directly that E(Y i, ) = σ i, 2, and E(U i,) = E(V i, ) = E(W i, ) = 0, which shows that E[RV () ] = σ i, 2, and the variance of RV() is given by var[rv () ] = var[ Y i, + U i, + V i, + W i, ] = (1) + (2) + (3) + (4) + (5), where (1) = var( Y i, ), (2) = var( U i, ), (3) = var( V i, ), (4) = var( W i, ), (5) = cov( V i,, W i, ), since all other sus are uncorrelated. Next, we derive derive the expressions of each of these five ters. 1. Y i, = yi, (y i 1, + y i, + y i+1, ) and given our assuptions it follows that E[y 2 i, y 2 j, ] = σ i, 2 σ 2 j, for i = j, and E[yi, 2 y 2 j, ] = E[y 4 i, ] = 3σ i, 4 for i = j, such that var(y i, ) = 3σ 4 i, + σ 2 i, σ 2 i 1, + σ 2 i, σ 2 i+1, [σ 2 i, ]2 = 2σ 4 i, + σ 2 i, σ 2 i 1, + σ 2 i, σ 2 i+1,. The first-order autocorrelation of Y i, is E[Y i, Y i+1, ] = E[y i, (y i 1, + y i, + y i+1, )y i+1, (y i, + y i+1, + y i+2, )] = E[y i, (y i, + y i+1, )y i+1, (y i, + y i+1, )] + 0 = 2E[y 2 i, y 2 i+1, ] = 2σ 2 i, σ 2 i+1,, such that cov(y i,, Y i+1, ) = σ 2 i, σ 2 i+1,, whereas cov(y i,, Y i+h, ) = 0 for h 2. Thus (1) = = 2 = 6 (2σ i, 4 + σ i, 2 σ i 1, 2 + σ i, 2 σ i+1, 2 ) + 1 σ i, 2 σ i 1, 2 + σ i, 2 σ i+1, 2 σ i, σ i, 2 σ i 1, σ i, 2 σ i+1, 2 σ 2 1, σ 2 0, σ 2, σ 2 +1, σ i, 4 2 σ i, 2 (σ i, 2 σ i 1, 2 ) + 2 σ i, 2 (σ i+1, 2 σ i, 2 ) σ 2 1, σ 2 0, σ 2, σ 2 +1, = 6 σ i, σ i, 2 (σ i, 2 σ i 1, 2 ) + 2 σ i, 2 (σ i+1, 2 σ i, 2 ) i=2 σ 2 1, σ 2 0, σ 2, σ 2 +1, 2σ 2 1, (σ 2 1, σ 2 0, ) + 2σ 2, (σ 2 +1, σ 2, ) 1 = 6 σ i, 4 2 (σ i+1, 2 σ i, 2 )2 2(σ 4 1, + σ 4, ) + σ 2 1, σ 2 0, + σ 2, σ 2 +1, i=2 2. U i, = (u i, u i 1, )(u i+1, u i 2, ) and fro E(U 2 i, ) = E(u i, u i 1, ) 2 E(u i+1, u i 2, ) 2 it follows that var(u 2 i, ) = 4ω4. The first and second order autocovariance are given by E(U i, U i+1, ) = E[(u i, u i 1, )(u i+1, u i 2, )(u i+1, u i, )(u i+2, u i 1, )] = E[u i 1, u i+1, u i+1, u i 1, ] + 0 = ω 4, and E(U i, U i+2, ) = E[(u i, u i 1, )(u i+1, u i 2, )(u i+2, u i+1, )(u i+3, u i, )] 9

10 = E[u i, u i+1, u i+1, u i, ] + 0 = ω 4, whereas E(U i, U i+h, ) = 0 for h 3. Thus, (2) = 4ω 4 + 2( 1)ω 4 + 2( 2)ω 4 = 8ω 4 6ω V i, = y i, (u i+1, u i 2, ) such that var(v 2 i, ) = σ 2 i, 2ω2 and E[V i, V i+h, ] = 0 for all h = 0. Thus (3) = var( V i, ) = 2ω 2 σ 2 i,. 4. W i, = (u i, u i 1, )(yi 1, + y i, + y i+1, ) such that var(w 2 i, ) = 2ω2 (σ i 1, 2 + σ i, 2 + σ i+1, 2 ). The first order autocovariance equals cov(w i,, W i+1, ) = E[ u 2 i, (y 2 i, + y 2 i+1, )] = ω2 (σ 2 i, + σ 2 i+1, ), while cov(w i,, W i+h, ) = 0 for h 2. Thus (4) = [2ω 2 (σ i 1, 2 + σ i, 2 + σ i+1, 2 ) 1 ω 2 (σ i, 2 + σ i 1, 2 ) ω 2 (σ i, 2 + σ i+1, 2 )] = ω 2 (σ i 1, 2 + σ i+1, 2 ) + ω2 [σ 2 1, + σ 2 0, + σ 2, + σ 2 +1, ] i=2 = 2ω 2 σ i, 2 + ω2 [σ 2 0, σ 2, + σ 2 +1, σ 2 1, ] + ω2 [σ 2 1, + σ 2 0, + σ 2, + σ 2 +1, ] = 2ω 2 σ i, 2 + 2ω2 [σ 2 0, + σ 2 +1, ]. 5. The autocovariances between the last two ters are given by E[V i, W i+h, ] = E[y i, (u i+1, u i 2, )(u i+h, u i 1+h, )(y i 1+h, + y i+h, + y i+1+h, )], showing that cov(v i,, W i±1, ) = ω 2 σ i, 2, while all other covariances are zero. Fro this we conclude that (5) = 2 ω 2 σ 2 i, ω2 [σ 2 1, + σ 2, ]. where By adding up the five ters we find 6 1 σ i, 4 2 (σ i+1, 2 σ i, 2 )2 2(σ 4 1, + σ 4, ) + σ 2 1, σ 2 0, + σ 2, σ 2 +1, + 8ω4 6ω 4 + 2ω 2 σ i, 2 + 2ω2 σ i, 2 + 2ω2 [σ 2 0, + σ 2 +1, ] + 2 σ i, 2 ω2 ω 2 [σ 2 1, + σ 2, ] = 8ω 4 + 6ω 2 ( σ i, 2 ω2 ) + 6 σ i, 4 + ω2 (σ 2 0, + σ 2 +1, ) + κ, κ = 2 (σ i+1, 2 σ i, 2 )2 2(σ 4 1, + σ 4, ) + σ 2 1, σ 2 0, + σ 2, σ 2 +1, +ω 2 (σ 2 0, σ 2 1, + σ 2 +1, σ 2, ). Since σ 2 (t) is Lipschitz, there exists an ɛ > 0 such that σ 2 (t) σ 2 (t + δ) ɛδ for all t and all δ. Thus if we define the interval, J i, [a + (i 1), a + i ], we have that σ i, 2 = J i, σ 2 (s)ds sup s Ji, σ 2 (s) = O( 1 ), since = (b a)/ = O( 1 ), and σ i, 2 σ i 1, J 2 = σ 2 (s) σ 2 (s )ds σ 2 (s) σ 2 (s ) ds i, J i, 10

11 sup s J i, σ 2 (s) σ 2 (s ) 2 ɛ = O( 2 ). Finally, (σ 2 i+1, σ 2 i, )2 ( ɛ ) 2 = O( 3 ), which proves that κ = O( 2 ). The asyptotic norality follows fro the CLT that applies to heterogeneous arrays with finite dependence, since y i, (y i 1, + y i, + y i+1, ) is a finite dependent (3-dependent) process for any. Proof of Corollary 4. The MSE s are given fro Leas 2 and 3. Setting the MSE(RV () )/ 4λ 2 + 6λ 2 2 equal to zero yields the first order condition of the corollary. Siilarly we find MSE(RV () )/ 4λ 2 (3 + λ) 2, which proves that 1 = 3 + λ/(2λ). References Aït-Sahalia, Y., Mykland, P. A. & Zhang, L. (2003), How often to saple a continuous-tie process in the presence of arket icrostructure noise, Working Paper w9611, NBER. Andersen, T. G. & Bollerslev, T. (1997), Intraday periodicity and volatility persistence in financial arkets, Journal of Epirical Finance 4, Andersen, T. G., Bollerslev, T. & Diebold, F. X. (2003), Paraetric and nonparaetric volatility easureent, in Y. Aït-Sahalia & L. P. Hansen, eds, forthcoing in Handbook of Financial Econoetrics, Vol. I, Elsevier-North Holland, Asterda. Andersen, T. G., Bollerslev, T., Diebold, F. X. & Ebens, H. (2001), The distribution of realized stock return volatility, Journal of Financial Econoics 61(1), Andersen, T. G., Bollerslev, T., Diebold, F. X. & Labys, P. (2000), Great realizations, Risk 13(3), Andreou, E. & Ghysels, E. (2002), Rolling-saple volatility estiators: Soe new theoretical, siulation, and epirical results, Journal of Business & Econoic Statistics 20(3), Bandi, F. M. & Russell, J. R. (2003), Microstructure noise, realized volatility, and optial sapling, Working paper, Graduate School of Business, The University of Chicago. Barndorff-Nielsen, O. E. & Shephard, N. (2002), Econoetric analysis of realised volatility and its use in estiating stochastic volatility odels, Journal of the Royal Statistical Society B 64, Billingsley, P. (1995), Probability and Measure, 3nd edn, John Wiley and Sons, New York. Bollen, B. & Inder, B. (2002), Estiating daily volatility in financial arkets utilizing intraday data, Journal of Epirical Finance 9, Corsi, F., Zubach, G., Müller, U. & Dacorogna, M. (2001), Consistent high-precision volatility fro high-frequency data, Econoic Notes 30(2), Dacorogna, M. M., Gencay, R., Müller, U., Olsen, R. B. & Pictet, O. V. (2001), An Introduction to High-Frequency Finance, Acadeic Press, London. French, K. R., Schwert, G. W. & Stabaugh, R. F. (1987), Expected stock returns and volatility, Journal of Financial Econoics 19(1),

12 Hansen, P. R. & Lunde, A. (2003), An optial and unbiased easure of realized variance based on interittent high-frequency data. Mieo prepared for the CIREQ-CIRANO Conference: Realized Volatility. Montreal, Noveber Meddahi, N. (2002), A theoretical coparison between integrated and realized volatility, Journal of Applied Econoetrics 17, Newey, W. & West, K. (1987), A siple positive sei-definite, heteroskedasticity and autocorrelation consistent covariance atrix, Econoetrica 55, Ooen, R. A. C. (2002a), Modelling realized variance when returns are serially correlated. anuscript, Warwick Business School, The University of Warwick. Ooen, R. A. C. (2002b), Statistical odels for high frequency security prices. anuscript, Warwick Business School, The University of Warwick. Wasserfallen, W. & Zierann, H. (1985), The behavior of intraday exchange rates, Journal of Banking and Finance 9, Zhang, L., Mykland, P. A. & Aït-Sahalia, Y. (2003), A tale of two tie scales: Deterining integrated volatility with noisy high frequency data, Working Paper w10111, NBER. Zhou, B. (1996), High-frequency data and volatility in foreign-exchange rates, Journal of Business & Econoic Statistics 14(1),

13 c RV () c AC () c 2 RV ()/c2 AC ( 1 ) c2 AC ()/c2 RV ( 0 ) RMSE Sapling Frequency in Seconds MSE ratio Sapling Frequency in Seconds Figure 1: Left: The MSEs of RV () and RV () as a function of the sapling frequency,. Right: Relative MSE of RV () to RV ( 1 ) where 1 is the optial sapling frequency for RV, and relative MSE of RV () to RV ( 0 ) where 0 is the optial sapling frequency for the standard RV. RV () RV () Average RV (AA) Sapling Frequency in Seconds Figure 2: Signature plots of the standard RV () and the bias corrected RV (). 13