Inference about Realized Volatility using In ll Subsampling

Transcription

1 Inference about Realized Volatility using In ll Subsapling Ilze Kalnina y and Oliver Linton z The London School of Econoics (PRELIMINARY AND INCOMPLETE) April 16, 7 Abstract We investigate the use of subsapling, Politis and Roano (1994), for conducting inference about quadratic variation of a discretely observed di usion process under an in ll asyptotic schee. Recently, subsapling has been used to do an additive bias correction to obtain a consistent estiator of the quadratic variation of a di usion process subject to easureent error. We show that the usual subsapling ethod is inconsistent when applied to the inference question we ention above. This is due to a high correlation between estiators on di erent subsaples. We discuss an alternative approach that does not have this correlation proble; however, it has a vanishing bias only under soothness assuptions on the volatility path. Finally, we propose a subsapling schee that delivers consistent inference without any soothness assuptions on the volatility path. This is a general ethod and can be potentially applied to conduct inference for quadratic variation in the presence of jups and/or icrostructure noise by subsapling appropriate consistent estiators. Keywords: Realised Volatility, Seiartingale, Subsapling JEL Classification: C1 We would like to thank Peter Bühlann and Nour Meddahi for helpful coents, as well as seinar participants at Oberwolfach (March 19 3, 7). This research was supported by the ESRC. y Departent of Econoics, London School of Econoics, Houghton Street, London WCA AE, United Kingdo. E-ail address: i.kalnina@lse.ac.uk z Departent of Econoics, London School of Econoics, Houghton Street, London WCA AE, United Kingdo. E-ail address: o.linton@lse.ac.uk. 1

2 1 Introduction Recently, estiation of integrated volatility of the price process has been a fruitful area of research. Applications include evaluation of variance forecasting odels, portfolio anageent, and asset pricing. Di erent estiators of integrated volatility have been proposed depending on the statistical odel used. The richer is the odel for the price process (e.g., with jups and/or arket icrostructure noise added to the di usion process), the ore coplicated estiators have been used. Also, if not the estiator itself, its asyptotic variance and hence inference changes depending on the assuptions on the price process. As an exaple, Aït-Sahalia, Mykland, and Zhang (6) and Kalnina and Linton (6) consider price being a di usion process containated by autocorrelated (forer) or endogenous (latter) easureent error. The result is ore coplicated expressions for the variance of the estiator. These variances cannot be estiated using the ethods that work with i.i.d., independent fro the latent price easureent error (or any other ethod known to us). The ai of this paper is to investigate the potential of the subsapling as a robust ethod for inference in the pure in ll asyptotic fraework, which could work in the presence of di erent assuptions on the observed price process. We choose the relatively siple setup where the underlying price process follows a di usion, as there are no subsapling results for inference in this fraework. Without jups or easureent error, a consistent estiator for quadratic variation of the price process is the widely used realised volatility. The question we want to answer is, can subsapling be used to conduct inference on realised volatility? The subsapling ethod of Politis and Roano (1994) has been shown to be useful in any situations as a way of conducting inference under weak assuptions and without utilizing knowledge of liiting distributions. The basic intuition it builds on goes as follows. Iagine the standard setting of discrete tie with long-span (also called increasing doain) asyptotics. Take soe general estiator b n (think of i.i.d. is, E(); b P n 1 i ;) with T bn d n! N (; V ) : In this paper we will be interested only in the standard case when n p n. Now, construct subsaples of consecutive observations ( << n), starting at di erent values (whether they are overlapping or not is irrelevant here). Then, the asyptotic distribution of ( b n;;j ) is the sae, bn;;j d! N (; V ) ; j 1; : : : ; K (1)

3 for each subsaple. Hence, we can estiate V as the saple variance of b n;;j (with centering around b n, the proxy for the true value ). This yields bv 1 K bn;;j b n ; () and we have bv p! V: This is our starting point. We show that in an in ll sapling schee a direct application of this ethod does not achieve the required consistency. The intuition behind this failure is straightforward. We do not have a stationary underlying process (in particular, we have a heteroscedasticity that does not vanish in the liit). Therefore, realised volatility on these subsaples ( b n;;j ) does not converge to quadratic variation (). That is, (1) does not hold. Recently, the word subsapling has been used in connection with the estiation of quadratic variation of a latent price process subject to arket icrostructure noise, see Zhang, Mykland, and Aït-Sahalia (5) and Barndor -Nielsen and Shephard (7). The subsapling schee in this setting is slightly di erent fro the usual one and is perhaps better called In ll Price subsapling. Zhang, Mykland, and Aït-Sahalia (5) use this In ll Price subsapling to de ne a bias correction ethod that achieves consistent estiation. For these type of subsaples, equation (1) holds, so we explore the possibility of their use for inference. We show that In ll Price subsapling does not deliver consistent inference for the realised volatility, due to high correlation between subsaples. We recover consistency by odifying the In ll Price subsapling in In ll Returns subsapling. However, this ethod requires restrictive assuptions on the volatility path to achieve consistency. Finally, we propose Subset Centered In ll subsapling, which does achieve the required consistency without any soothness assuptions on the volatility path. It builds on the basic principles of Politis and Roano (1994), but uses a di erent centering for each subsaple. The reainder of the paper is organized as follows. Section outlines the odel and soe basic facts about realized volatility. We describe the usual subsapling ethod and show its inconsistency in section 3. Section 4 introduces In ll Price subsapling and shows it is also inconsistent, albeit asyptotically unbiased. Section 5 introduces In ll Returns subsapling and shows its consistency, under soe soothness assuption on the volatility path. Section 6 introduces the Subset Centered In ll Subsapling and shows its consistency. Section 7 copares di erent ethods for inference in 3

4 a Monte Carlo siulation study. Section 8 concludes. We follow the notation of Politis, Roano, and Wolf (1999) for easier coparison with the subsapling literature. Model Suppose that we have a scalar Brownian di usion process d t t dt + t dw t (3) that is observed at ties t ; : : : ; t n on the interval [; T ]; where T is xed, so that our asyptotics are in ll (as n! 1). For the current version of the paper, we will ipose the assuption of no leverage, i.e., we assue independence of f t g and fw t g. We further assue that t is locally bounded and t is càdlàg. We are interested in the quadratic variation of the process over the observation interval QV Z T sds: Without loss of generality we can take the observations equally spaced and T 1. estiator of QV is the realized volatility The usual RV n n ti ti 1 ; which satis es p n (RVn QV ) ) MN(; V ) Z T V IQ 4 sds: The convergence follows fro Barndor -Nielsen and Shephard (). Note that we do not need to approxiate the whole distribution of RV n as it is known to be ixed noral, as well as the convergence rate is known to be p n. Therefore, we will only be concerned with the estiation of V. Barndor -Nielsen and Shephard () also show that p n (RVn QV ) p ev ) N(; 1); 4

5 where e V IQ n is a consistent estiator of V IQ: Here, IQ n is the realized quarticity, IQ n n 3 n ti ti 1 4 ; which successfully iics the structure of R T 4 sds. Note, however, that this estiator would not be consistent for the variance of realised volatility if we allowed for jups in (3). The purpose of this paper is to explore the use of subsapling as a eans of conducting inference without utilising the knowledge of the asyptotic variance of the estiator. Therefore, it can have the potential to be applied with di erent underlying assuptions, which lead to di erent, possibly very coplicated, asyptotic variances. Recently, Goncalvez and Meddahi (5) have proposed a bootstrap algorith. They use the i.i.d and the wild bootstrap. They showed that the i.i.d bootstrap is consistent and that the wild bootstrap achieves soe higher order re neents in a special case. Both bootstrap ethods exploit the independence of stock returns that holds conditional on the volatility path. In the leverage case where the processes f t g and fw t g are no longer independent, stock returns are dependent over tie and one cannot condition on the volatility path. be consistent. 1 In this case, the above ethods ay not Unlike the bootstrap ethod of Goncalvez and Meddahi (5), the subsapling ethod itself does not ipose the no-leverage structure and requires no rando nuber generator. Our ethod should also be robust to non-equally spaced data and to serial dependence since the data is used in blocks unlike the bootstrap ethod. Although we currently use no leverage assuption for our proofs, our Monte Carlo siulations indicate that our proposed subsapling ethod should also work in the absence of this restriction. 3 Regular subsapling We start our investigation of subsapling for the in ll asyptotic fraework with the well established ethod of Politis and Roano (1994), which is developed for the long span asyptotic fraework. In Politis, Roano, and Wolf (1999) there is soe discussion about subsapling for continuous paraeter rando elds, which includes continuous tie processes as a special case. Unfortunately, the discussion is suited to the continuous observation and increasing doain case where T! 1: 1 Although since the distribution theory for integrated volatility is the sae under leverage as under no leverage it ay be that the no leverage bootstrap is consistent. 5

6 Also, they assue strong stationarity, which we do not have. They use the following notation, which we have specialized for the scalar paraeter case. 3 Let E j;;h ft : (j 1)h < t (j 1)h + hg Y j f(t) : t E j;;h g b n;;j b (Y j ); where b (Y j ) is the estiator coputed with the data Y j ; in particular b (Yj ) te j;;h ( t t 1 ) : We have < j K; where K is the nuber of subsaples, K n + 1: See Figure 1 in the appendix for a graphical illustration of the subsaples. In our case where is only observed at a discrete set of points the data Y j consists of a nite set of points of lower cardinality than n: The estiator b n b (Y ) RV n ; when Y consists of all data. Assuption of Politis, Roano and Wolf (1999) is satis ed, i.e., the sapling distribution of n ( b n ) converges weakly. Therefore, in the setting of continuous observations and long-span asyptotics, the intuition laid out in the introduction applies and V should be approxiated by (), i.e., bv regular 1 K bn;k;j b n : However, in our setting, it is easy to see that b V is not a consistent estiator for V. This is due to b n;;j only estiating a part of b n and the asyptotic distribution of ( b n;;j in nity instead of being the sae as that of n ( b n Proposition 1. We have E bvregular + o () : ). As a result, we have ) diverging to Subsapling results have also been developed for heteroscedasticity that vanishes in the liit, i.e., is close to hooscedasticity. This is an unrealistic setting in in ll asyptotics. 3 We will present the version of regular subsapling that uses overlapping subsaples. All the results hold also with non-overlapping subsaples. However, for the cases discussed in Politis, Roano, and Wolf (1999), it achieves higher e ciency. 6

7 4 In ll price subsapling In this and the next section we will consider two subsapling schees that are otivated by the literature on estiating QV in the presence of easureent error. Both recover the principle that ( b n;;j ) has the sae asyptotic distribution as n ( b n ). First, we explore a subsapling schee that exactly irrors the construction of subsaples in the estiator of QV of Zhang, Mykland, and Aït-Sahalia (5). Let the subsaples consist of data that are K observations apart. In particular, write K ( + 1) T and construct the j th subsaple Y j and the estiator of on this subsaple as follows (see Figure in the appendix for a graphical illustration), E j; ft : t j + i(k + 1); i 1; : : : ; g ; Y j f(t) : t E j; g ; b n;;j b (Y j ): In each subsaple we have + 1 (log-)prices and hence returns. For j 1; : : : ; K b n;;j te j; ( t t 1 ) j+ik j+(i 1)K : We can easily see that ( b n;;j ) has the sae asyptotic distribution as n ( b n ). Therefore, we consider the estiator of V IQ as in (), bv In ll price 1 K bn;;j b n : Proposition. We have E bvin ll price! EV; Var bvin ll price O (1) : (4) This estiator is asyptotically unbiased, but it never converges in probability to the true V. Why do we have this proble? Note that any two subsaples fully overlap (in ters of tie covered by variances involved; see Figure ), apart fro end e ects. As a result, the asyptotic correlation between estiators on any two subsaples is one. We have Cov b n;;i; b n;;j Var b n;;i O ; 7

8 and so the failure of consistency follows. We next consider a odi cation that is otivated by this proble. In the following section we construct subsaples that do not overlap at all while preserving the property that ( b n;;j ) has the sae asyptotic variance as n ( b n ): 5 In ll Returns Subsapling To avoid the proble of in ll price subsapling, consider subsapling one-period returns r ti ti ti 1 instead of log-prices t. See Figure 3 in the appendix for a graphical illustration. Let the subsaples consist of returns that are K observations apart. In particular, write K ( + 1) n and construct the j th subsaple Y j and the estiator of on this subsaple as follows, E j; ft : t j + i(k + 1); i 1; : : : ; g Y j fr(t) : t E j; g In each subsaple we have returns. For j 1; : : : ; K b n;;j K te j; r t K r j+(i 1)K K j+(i 1)K j+(i 1)K 1 : De ne b n as before and bv In ll returns 1 K bn;;j b n : (5) As opposed to previous forulation, we now have cov( b n;;i ; b n;;j ) for i 6 j; 4 which leads us to Proposition 3. Assue that volatility paths are a.s. Hölder continuous of order larger than 1/. We have p bv In ll returns! V: (6) Without iposing soe soothness on the volatility path b V In ll returns is biased due to the large gaps we have in the subsaples (see Figure 3). For asyptotic unbiasedness of b V In ll returns, we need 4 This covariance is exactly zero if the drift is zero. It is negligible is the drift is non-zero. 8

9 the expected value of the estiator on each subsaple to converge to su ciently fast. If volatility path is càdlàg, we have, conditional on f t g, E b n;;j K Z [j+(i 1)K]n (u) du! [j+(i 1)K 1]n (u) du due to Rieann integrability of saple paths of f t g. However, what we need for asyptotic unbiasedness of V b In ll returns, is E( b n;;i ) o( 1 ). This is because of the scaling factor in (5), which appears because we are estiating the asyptotic variance. On the one hand, we have an in ll price subsapling that has a variance proble, and on the other hand, we have in ll returns subsapling that has a bias proble. The natural question is, can we nd an estiator that is consistent without soothness assuptions by just varying the size of the gaps? The answer sees to be no. One can however, achieve consistency without soothness assuptions by odifying the principle that ( b n;;j ) has the sae asyptotic variance as n ( b n ), for each j. We show this in the next section. 6 Subset Centered In ll Subsapling In regular subsapling, the proble was that we were centering our saple variance at "the wrong quantity". In the forula for V b regular ; bv regular 1 bn;k;j b n ; K b n is plays the role of, but the proble is that b n;k;j 9 so this estiator explodes. In the two previous sections we rede ned b n;k;j so as to recover the principle b n;k;j! and saw this does not work very well. Instead, consider centering estiators at j such that b n;k;j! j and then use the property of integrated variance that it can be added up over subsaples to recover the integrated variance over the full saple, P j j. Now, j is not observable and the best proxy for j we have is realised volatility over the subsaple, b n;k;j. Therefore, we have to use soething else to play the role of the estiator. So de ne b n;k;j;j to be realised volatility calculated using every J th observation of the j th subsaple of length (J << ). Since it has a slower rate of convergence than b n;k;j, the error of using b n;k;j instead of is negligible. Our estiator of V becoes 9

10 bv SC n KJ bn;j;k;k b n;k;k : k1 Here, the nuber of subsaples K depends on how uch di erent subsaples overlap. Increasing the aount of overlap does not change the rate of convergence of b V, but it decreases the asyptotic variance, so axiu overlap estiator is preferred. See the Monte Carlo siulations (Figure 11) for an illustration. Proposition 4. We have bv SC p! V: (7) The estiator is siilar in structure to Lahiri, Kaiser, Cressie, and Hsu (1999). They siilarly use two grids for subsapling to predict stochastic cuulative distribution function. However, they assue that the underlying process is stationary and their asyptotic fraework is ixed in ll and increasing doain. 7 Nuerical Work We do a siulation coparison of the above subsapling ethods, plus realized quarticity. siulate the Heston (1993) odel: We d t ( t v t ) dt + t db t dv t ( v t ) dt + v 1 t dw t ; where v t t, and B t ; W t are independent standard Brownian otions. We take paraeters fro Zhang, Mykland, and Aït-Sahalia (5): :5; 5; :4; :5: We set the length of the saple path to 5, which is a proxy for 34 that is the nuber of seconds in a business day. We set the tie between observations corresponding to one second when a year is one unit, and the nuber of replications to be 1,. For b V regular ; b V In ll price ; bv In ll returns ; and b V SC ; we use K: For b V SC we use J 15: We hold the volatility saple path constant across siulations for easier coparison. Appendix contains the following gures showing the results. 1

11 Table 1. Finite saple distributions of di erent ethods Figure 4 The volatility saple path Figure 5 Kernel density over siulations of V b 5 regular Figure 6 Kernel density over siulations of V b In ll price Figure 7 Kernel density over siulations of V b In ll returns Figure 8 Kernel density over siulations of V b SC Figure 9 Kernel density over siulations of IQ n Figure 1 Kernel densities over siulations of V b In ll price ; V b In ll returns ; V b SC ; and IQ n Figure 11 Boxplots of V b SC for di erent overlaps (and IQ n ) Fro the siulated volatility saple path (Figure 3) we can approxiate the true V ar(rv ) V IQ and we get 4: For ease of coparison, this nuber is reported in the titles of all gures in appendix. Also, this can be approxiated by the (scaled) variance of RV over siulations. Here is a brief suary coparing the eans of feasible ethods with infeasible bencharks: Table. Finite saple eans of di erent ethods Theoretical p IQ.1 infeasible Finite saple p Var(RV ). infeasible Square root of ean over siulations of V b regular.17 feasible Square root of ean over siulations of V b In ll price. feasible Square root of ean over siulations of V b In ll returns.1 feasible Square root of ean over siulations of V b SC (axiu overlap).194 feasible Square root of ean over siulations of IQ n.1 feasible We see that b V regular overestiates the true V by a large factor, thus supporting the asyptotic result. The other feasible ethods work well. We can also see that b V In ll price is asyptotically unbiased, whereas fro Figure 6 in the appendix we see that its distribution over siulations is strongly right-skewed. Fro Figure 1 we can see that, IQ n has the sallest variance, re ecting the fact that is has uch faster rate of convergence. Our proposed estiator b V SC has soe nite saple negative bias. One could do a nite saple correction and use J V b J 1 SC (recall J 15, so this 5 Overlapping and non-overlapping usual subsapling give close to identical estiates, so we only plot the overlapping version. 11

12 factor is non-negligible), see section A.4.. for theoretical justi cation. Adjusted estiator has a saller negative bias in siulations. J V b J 1 SC One of the paraeters to choose in our proposed estiator b V SC is the aount of overlap between di erent subsaples, which a ects the total nuber of subsaples K: We do a siulation exercise to see how exactly it a ects the nite saple properties. First, Figure 8 shows the nite saple distribution for three di erent scenarios. Also, Figure 1 has eleven di erent scenarios suarised in boxplots, ranging fro non-overlapping subsaples to axiu overlap. We know fro theory that the aount of overlap does not a ect the convergence rate. The siulations, however, indicate that it decreases the asyptotic variance, i.e., b VSC with the axiu aount of overlap (with K n + 1) gives the lowest variance. This phenoenon is also observed in the long span asyptotic fraework. 8 Conclusions and Extensions In this note we investigate the use of subsapling for conducting inference about quadratic variation. We show that the usual subsapling ethod as in Politis, Roano, and Wolf (1994) is inconsistent. We also show that subsapling that works for estiation of QV in the presence of noise, or odi cations thereof, do not work to estiate the asyptotic variance of QV. There is uch further work that can be done. Iportantly, one can generalize the process that can follow. For exaple, one could allow for leverage and jups, in which case this estiator would becoe copleentary to the estiators of Aït-Sahalia and Jacod (6) and Veraart (7), while the estiator IQ n would becoe inconsistent. Also, one could allow for easureent error that containates, in which case one has to consider subsapling T SRV or realized kernels instead of RV as estiators for QV. Exaples of cases where only the asyptotic theory has been developed, but not the tools for inference, include estiation of quadratic variation with autocorrelated easureent error (Aït-Sahalia, Mykland, and Zhang 6a) or with endogenous easureent error (Kalnina and Linton 6). 1

13 A Appendix Recall that for the current version of the paper we are assuing no leverage, so that all arguents are ade conditional on the volatility path. Also, all arguents are done for a zero drift. Extension to nonzero drift is straightforward (given that we assue a locally bounded drift, and hence bounded drift w.l.o.g. for the purposes of consistency). A.1 Proof of Proposition 1 The j th subsaple contains observations f j 1 ; j ; :::; +j 1 g. Then b (Yj ) ( j+i 1 j+i ) : In this case, the nuber of subsaples is K n + 1: Let z j;i R ((j+i 1)n (j+i )n udw u and y j;i ( R ((j+i 1)n (j+i )n udw u ) j;i; which are independent (across i for given j) and ean zero rando variables conditional on the process u; where j;i E[( Z ((j+i (j+i )n 1)n u dw u ) ] Z ((j+i (j+i )n 1)n udu O(1n): In above we use use local boundedness of u (which follows fro the assuption that paths of are càdlàg) to conclude orders of agnitude, i.e., Z ((j+i (j+i )n 1)n udu 1 n sup u O(1n): u We use this arguent of boundedness to conclude stochastic orders of agnitude throughout the appendix. We have var (y j;i ) E 4 3 var (z j;i ) E 4 Z ((j+i (j+i Z ((j+i (j+i O(1n ): )n )n 1)n 1)n Z ((j+i (j+i )n 1)n! 3 4 u dw u 5 E 4! 3 u dw u 5 j;i Z ((j+i (j+i! Z ((j+i 1)n udu udu 13 (j+i )n 1)n )n!! 3 u dw u 5

14 We show the following leas. Lea 1.1. We have " K # E b n;k;j udu + O : n Lea 1.. We have Then, E b n;k;j O n : bv 1 bn;;j b n K bn;;j + b n + bn;;j b n K K K bn;;j + Op 1 n 1 b n;k;j + 1 K b n;k;j + o p (1) and E( V b ) E( K b n;k;j) + 1 E( K b n;k;j ) + o(1) K O + 1 n K udu + O + o(1) n + o()! 1; using K n + 1 and n o (1) : 14

15 Proof of Lea 1.1. " K # E b n;k;j E zj;i Z ((j+i 1)n (j+i )n Z (+j 1)n (j 1)n udu udu + O n udu : Proof of Lea 1.. E b n;k;j E 4 E 4 z j;i! 3 5! 3 ( j;i + y j;i ) 5 Eyj;i + O : n Z ((j+i (j+i )n i 1 1)n j;i j;i udu! + i 1 j;i j;i A. Proof of Proposition A..1 Let Notation and preliinary calculations z j;i Z (j+ik)n (j+(i 1)K)n u dw u ; 15

16 and so b n;k;j where: zj;i j;i + y j;i ; j;i E[( R (j+ik)n (j+(i 1)K)n udw u ) ] R (j+ik)n (j+(i 1)K)n udu; and y j;i ( R (j+ik)n (j+(i 1)K)n udw u ) j;i; which are independent (across i for given j) and ean zero rando variables conditional on the process u: That is, E[z j;i z j;i ] E[y j;i y j;i ] whenever i 6 i : Furtherore, z j;i is conditionally noral and so satis es E[zj;i] 4 3E [zj;i]: Note that, for j < j, 8 R (infj;j >< g+ik)n (axfj;j g+(i 1)K)n udu 6 i i E [z j;i z j ;i ] R (j +i K)n (j+i >: K)n udu 6 i i + 1 ji i j > 1; i i 1: Furtherore, E[y j;i z j;i ] ; and var (z j;i ) E 4 Z (j+ik)n (j+(i 1)K)n z j;i;! 3 Z (j+ik)n u dw u 5 j;i udu O p (Kn) (j+(i 1)K)n var (y j;i ) E 4 Z (j+ik)n (j+(i 1)K)n! 3 4 u dw u 5 E 4 Z (j+ik)n (j+(i 1)K)n! 3 u dw u 5 K n Z (j+ik)n (j+(i 1)K)n Z (j+ik)n (j+(i 1)K)n udu! 4 udu + o p K n O p (K n ); since Z! (j+ik)n udu K (j+(i 1)K)n n We prove this result below. Z (j+ik)n (j+(i 1)K)n 4 udu + o(k n ): (8) By adding these results over all subsaples, we get that for each j and each j, cov( b n;k;j ; b n;k;j ) O 1 : (9) Proof of (8).Suppose that f is a bounded positive continuous function on [; 1]; say. Then for > ; g() R f(x)dx is of order as! ; because Z f(x)dx sup f(x): x[;] 16

17 Furtherore, g is di erentiable in with g () f(): Therefore, by the ean value theore g() g() + g () f() for soe : Therefore, Z f(x)dx f (): Furtherore, f is also a bounded continuous function on [; 1] and satis es Z for soe : By continuity of f at ; f (x)dx sup f (x) x[;] Z f (x)dx f ( ) li # f () f ( ) 1; and it follows that Z f(x)dx Z f (x)dx + o( ): A.. Main proof Write bv K K b n;k;j + b n b n 1 K b n;k;j + 1 K b n;k;j b n;k;j + o p (1); where the approxiation is valid by Barndor -Nielsen and Shephard (). We ake use of the following leas. Lea.1. " 1 E K # b n;k;j 1 K jn Kn+jn udu: 17

18 Lea.. " # 1 E b n;k;j K n jn jn 4 udu + 1 K jn Kn+jn 1 udu! + o : Lea.3. Var K b n;k;j 1 K! b n;k;j O(1): These leas iply that E b V K n E[ b n;k;j] + 1 K jn Kn+jn 1 udu K 4 udu + K jn E[ b n;k;j ] + o(1) Kn+jn jn Kn+jn udu + o(1): udu! + udu We have to show that R udu + 1 K 8 < : jn Kn+jn udu! udu jn Kn+jn 9 udu ; o(kn): (1) This is true because, write I R 1 udu; I jn R jn udu; and I 1 Kn+jn R 1 1 Kn+jn udu: Then R I + 1 K 1 K I I(I jn + I 1 Kn+jn ) + (I jn + I 1 Kn+jn ) I(I (I jn + I 1 Kn+jn )) (I jn + I 1 Kn+jn ) O(K n ): 18

19 Therefore, EV b n n jn Kn+jn 4 udu + o(1) 4 udu + o(1); 4 udu + o(1) and the rst part of (4) follows. The second part follows fro Lea 3. Proof of Lea.1. We have " # 1 E b n;k;j K 1 K 1 K 1 K E zj;i Z (j+ik)n jn (j+(i Kn+jn 1)K)n udu: udu Proof of Lea.. We have 1 E b n;k;j 1 E 4 K K 1 E 4 K! 3 zj;i 5! 3 ( j;i + y j;i ) 5 1 K n n jn Ey j;i + 1 K Z (j+ik)n (j+(i Kn+jn 1)K)n 4 udu + 1 K 19 i 1 j;i j;i! udu + 1 K jn jn Kn+jn jn udu! udu! + o 1 :

20 by (8). Proof of Lea.3. This can be seen fro the covariance between b n;k;j and b n;k;i : A.3 Proof of Proposition 3 We rst derive the ean and variance of b n;;j. We have E b n;;j KE j+(i 1)K j+(i 1)K 1 K Z (j+(i by Rieann integrability of u: Furtherore, 1)K)n (j+(i 1)K 1)n udu + o (1) udu Var b n;;j K Var j+(i 1)K j+(i 1)K 1 K Var j+(i 1)K j+(i 1)K 1 K Z (j+(i 4 udu + o 1)K)n (j+(i 1)K 1)n 1 : Now we calculate the expected value of the estiator. udu! E b V In ll returns E 1 K bn;;j b n E bn;;j E K b n;;j + R 4 udu + o (1) + R;

21 where R K K n E bn;;j E b n;;j E b o n;;j b n + K E no p 1 E b o n;;j b n + K E E b n;;j E E b n;;j b n : b n We know that b n O n 1 : Therefore, have R o (1) if E b n;;j o p 1 : For this, Rieann integrability is not enough. Therefore, assue Hölder continuity of order larger than 1. Now we calculate the variance of V b In ll returns. To facilitate calculations, introduce the usual notation: z j;i Z (j+(i 1)K)n (j+(i 1)K 1)n b n;;j K u dw u z j;i z j;i j;i + y j;i Z (j+(i j;i E[( y j;i ( 1)K)n (j+(i 1)K 1)n Z (j+(i 1)K)n (j+(i 1)K 1)n We rst do soe preliinary calculations. u dw u ) ] u dw u ) j;i: Z (j+(i 1)K)n (j+(i 1)K 1)n udu Ey j;i ; so that Var (y j;i ) Eyj;i 8 < Z! 9 (j+(i 1)K)n Eyj;i E : u dw u j;i (j+(i 1)K 1)n ; 8 < Z! 4 (j+(i 1)K)n E : u dw u + j;i (j+(i 1)K 1)n Z (j+(i 1)K)n (j+(i 1)K 1)n 9! u dw u j;i ; 3 j;i + j;i j;i j;i 1

22 Eyj;i 4 8 < E : 8 < E : Z (j+(i 1)K)n (j+(i 1)K 1)n Z (j+(i 1)K)n (j+(i 1)K 1)n u dw u! j;i 9 ; u dw u! 8 + j;i Z (j+(i 1)K)n (j+(i 1)K 1)n u dw u! 3 j;i 4 Z (j+(i 1)K)n (j+(i 1)K 1)n u dw u! 6 15 j;i 4 + j;i 4 4 j;i j;i j;i j;i 4 f g 6 j;i 4 : Z! 4 (j+(i 1)K)n j;i + 6 u dw u (j+(i 1)K 1)n 9 j;i ; 3 j;i j;i j;i Using the sae approxiation of b V In ll returns becoes Var b V In ll returns Var K " K as in Proposition 1, the variance of b V In ll returns bn;;j + op (1) Var bn;;j + o (1) : To show VarV b In ll returns o(1); we show that Var bn;;j Var b n;;j b n;;j o (K ). In order to do that, we proceed in three steps. That is to say, to calculate Var(x), we rst rst calculate E(x), then y x E(x), and nally E(y ): Step 1. We have E b ( E 4 8 < E n;;j K : K b n;;j j;i + y j;i ) j;i K!! + K y j;i + K 3 j;i + y j;i 5 j;i y j;i # 9 ; K j;i

23 K K j;i! + K! + K j;i Ey j;i K j;i K j;i j;i: Step. We have b n;;j b n;;j E b n;;j b n;;j ( ) K j;i + y j;i K j;i + y j;i E b n;;j b n;;j K K K j;i!! + K y j;i + K j;i K y j;i! K j;i j;i + K j;i y j;i j;i! K y j;i + K! K y j;i K j;i y j;i! K y j;i K j;i K + K y j;i j;i + op n 1 ; y j;i K K j;i! j;i because even without any soothness assuptions on we have K P j;i o (1) and K P y j;i O (Kn ) O (n 1 ) : Step 3. The only ter in the variance, whose negligibility needs to be shown (because p K > 3

24 n 1 ), is E 4K K 4 4E K 4 4E y j;i! K j;i 3 5! 4! y j;i + 4E j;i 4! 4! y j;i + 4 j;i 8 j;i E! 3 5 j;i! 3 y j;i 5 K 4 4E! 4 y j;i 4 K 4 4 Eyj;i K 4 46 K 4 "56 i 1;i 6i 4 j;i + 1! 3 5 j;i Eyj;iEy j;i 4 i 1;i 6i 4 j;i + 8 j;i j;i 4 i 1;i 6i! 3 5 j;i # j;i j;i O K 4 n 4 + O K 4 n 4 O o(k ):! 3 5 j;i This iplies Var b V! and so we have ean square convergence and so (6) follows by Chebyshev s inequality. A.4 Proof of Proposition 4 In the rst subsection of this proof we explain the notation, in the second we show that the bias of bv SC is negligible, i.e., E( V b SC ) V o(1), and in the third subsection we show that Var( V b SC ) o(1), so that Proposition 4 follows by Chebyshev s inequality. 4

25 A.4.1 Notation for b V SC Much of the notation is the sae as for the other estiators. K is the nuber of subsaples. is the nuber of high frequency returns in of each subsaple. J is the nuber of high frequency returns that t into one low frequency return (Figure in appendix coing soon!), 1 < J <. There are J nuber of low frequency in each subsaple. Take. J (i.e., divisible by J). The proof is for a general aount of overlap between subsaples, so introduce a variable s, for shift. If subsaples are constructed as observations in a window, we ove the window by sn to get every next subsaple. For exaple, s corresponds to no overlap between subsaples and s 1 corresponds to axiu possible overlap. Assue. s. As we show below, we have K ns s + 1. The subsaple copies of the RV estiator ( b n;;j in ()) will be b n;j;k;1 J P Ji J(i 1) ; b n;j;k; so that the copy corresponding to k th subsaple is J P Ji+s J(i 1)+s ; etc:; b n;j;k;k J P The nuber of subsaples is the k such that We have Ji+s(k 1) J(i 1)+s(k 1) : J J + s (k ax 1) n ) k ax K ns s + 1: E b n;j;k;k J P Z [Ji+s(k 1)]n [J(i 1)+s(k 1)]n udu Z [+s(k s(k 1)n 1)]n udu: The subsaple copies of the true paraeter QV ( in (), except that we have a di erent copy for each subsaple for our centering) will be We have b P n;k;1 ( i i 1 ) ; b P n;k; ( i+s i 1+s ) ; etc: b n;k;k P E b n;k;k P i+s(k 1) i 1+s(k 1) : Z [i+s(k 1)]n udu Z [+s(k [i 1+s(k 1)]n s(k 1)n 1)]n udu: 5

26 Recall that the estiator of V is A.4. Derivation of E( b V SC ) Introduce the following notation, bv SC n KJ bn;j;k;k b n;k;k : k1 E bn;j;k;k b n;k;k E b n;j;k;k + E b n;k;k A k + B k C k ; E b n;j;k;k b n;k;k so that we have E b V SC A k E b n;j;k;k E J P + P i >i J P P n K KJ k1 (A k + B k C k ) : Then, " J P Ji+s(k 1) J(i 1)+s(k 1) # E Ji+s(k 1) J(i 1)+s(k 1) 4 J P E Ji+s(k 1) J(i 1)+s(k 1) E Ji +s(k 1) J(i 1)+s(k 1) Z [Ji+s(k 1)]n [J(i 1)+s(k 1)]n u! + Z [+s(k s(k 1)n 1)]n udu! : Siilarly, B k E b P n;k;k E P Z [i+s(k 1)]n [i 1+s(k 1)]n i+s(k 1) i 1+s(k 1) u! + Z [+s(k s(k 1)n 1)]n udu! : In the third ter, we have covariances between realised volatilities on the two grids, which we 6

27 J P denote as corresponding suation over low frequency returns C k;i as follows, C k E b n;j;k;k b n;k;k Cov bn;j;k;k ; b n;k;k + E b n;j;k;k E b n;k;k Cov J P J P Ji+s(k 1) J(i 1)+s(k 1) ; P C k;i + E b n;j;k;k E b n;k;k : i+s(k 1) i 1+s(k 1)! + E b n;j;k;k E b n;k;k and We then notice that C k;1 is the variance of the realised volatility over the 1 st low frequency return, C k;1 Cov Cov " Cov + P JP J+s(k 1) s(k 1) ; P J+s(k 1) s(k 1) ; JP JP JP j >j P J Var J P j+s(k 1) j 1+s(k 1) i+s(k 1) i 1+s(k 1) i+s(k 1) i 1+s(k 1) j+s(k 1) j 1+s(k 1) j +s(k 1) j 1+s(k 1) ; i+s(k 1) i 1+s(k 1) Z [i+s(k i+s(k 1) i 1+s(k 1) 1)]n [i 1+s(k 1)]n Siilarly with other Ck;i s and so we have u! : C k J P J P B k : Z [i+s(k 1)]n [i 1+s(k 1)]n Z [i+s(k 1)]n u! + E b n;j;k;k E b n;k;k u! + Z [+s(k [i 1+s(k 1)]n s(k 1)n 1)]n udu! 7

28 Therefore, EV b SC n KP E bn;j;k;k b n n;k;k KJ k1 KJ 8 n KP < Z J KJ : P [Ji+s(k 1)]n u k1 [J(i 1)+s(k 1)]n KP (A k B k ) k1! J P Z [i+s(k 1)]n u [i 1+s(k 1)]n! 9 ; n KJ k1 J P Z [Ji+s(k 1)]n u [J(i 1)+s(k 1)]n! n KJ P k1 Z [i+s(k 1)]n [i 1+s(k 1)]n u! : We show just below that the rst ter converges to V: Notice that second ter is like J 1 V, which eans that one could do an easy nite saple bias correction by estiating V by J V b J 1 SC instead of bv SC : Note that in our siulation setup we have J 15, so the adjustent factor is non-negligible. We deal with suations in the rst ter by re-grouping the ters so that each group covers the interval [; 1] apart fro end-e ects, J Z! P [Ji+s(k 1)]n u k1 [J(i 1)+s(k 1)]n s (n )J Z! P Jin+s(p 1)n u p1 J(i 1)n+s(p 1)n s J n p1 J sn J udu + o n J sn udu + o Now we can conclude asyptotic unbiasedness, EV b SC Z n J 1 J KJ sn udu + o + O J 1 sn V + o(1) : by using the fact that K ns s + 1 ns + o(ns): 8

29 A.4.3 Derivation of Var( b V SC ) Var b V SC Var " n KJ n KJ n KJ n KJ # bn;j;k;k b n;k;k k1 K k 1 k1 bn;j;k;k Cov in(k;k+s) k ax(1;k s s) k1 bn;j;k;k Var k1 bn;j;k;k Cov b n;k;k ; bn;j;k;k b n;k;k b n;k;k ; b n;k;k ; bn;j;k;k b n;k;k where we use the fact that: 1) all these covariances ust be nonnegative, ) for a xed ter k in the rst suation, only the ters fro k to nonzero covariances. Now we calculate the agnitude of b n;j;k;k b n;k;k, h i Var bn;j;k;k b n;k;k " J P P Var Ji+s(k 1) J(i 1)+s(k 1) Var " J P s to k + s ters in the suation over k give rise J P Ji+s(k 1) J(i 1)+s(k 1) i+s(k 1) i 1+s(k 1) # JP j+j(i 1)+s(k 1) j 1+J(i 1)+s(k 1) # " J P Var " J P J P Var Ji+s(k 1) J(i 1)+s(k 1) JP Ji+s(k 1) J(i 1)+s(k 1) JP n Var Ji+s(k 1) J(i 1)+s(k 1) ) JP Var j+j(i 1)+s(k 1) j 1+J(i 1)+s(k 1) j+j(i 1)+s(k 1) j 1+J(i 1)+s(k 1) # j+j(i 1)+s(k 1) j 1+J(i 1)+s(k 1) # (11) 9

30 J P 8 < : O J Z [Ji+s(k 1)]n [J(i 1)+s(k 1)]n! J n J O n udu ;! J P Z [j+j(i 1)+s(k 1)]n [j 1+J(i 1)+s(k 1)]n udu! 9 ; where (11) follows by noticing that, in the expression of variance, the covariance between the rst and second ter equals the variance of the second ter. Using this, we can show that Var bvsc is negligible. We have and recall we have n o(1): Var bvsc n KJ s K J n4 K J n K J sn 4 Ks n ; Notice how the agnitude of Var( b V SC ) does not depend on the aount of overlap (i.e., it does not depend on the paraeter s). 3

31 References [1] Aït-Sahalia, Y., and J. Jacod (6). Testing for jups in a discretely observed process. Unpublished paper: Departent of Econoics, Princeton University. [] Aït-Sahalia, Y., P. Mykland, and L. Zhang (5). How Often to Saple a Continuous- Tie Process in the Presence of Market Microstructure Noise. Review of Financial Studies, 18, [3] Aït-Sahalia, Y., P. Mykland, and L. Zhang (6a). Ultra high frequency volatility estiation with dependent icrostructure noise. Unpublished paper: Departent of econoics, Princeton University [4] Aït-Sahalia, Y., P. Mykland, and L. Zhang (6b). Coents on Realized Variance and Market Microstructure Noise, by P. Hansen and A. Lunde, Journal of Business and Econoic Statistics. [5] Aït-Sahalia, Y., Zhang, L., and P. Mykland (5). Edgeworth Expansions for Realized Volatility and Related Estiators. Working paper, Princeton University. [6] Awartani, B., Corradi, V., and W. Distaso (4). Testing and Modelling Market Microstructure E ects with and Application to the Dow Jones Industrial Average. Working paper. [7] Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., and N. Shephard (6). Designing Realised Kernels to Measure the Ex-post Variation of Equity Prices in the Presence of Noise. Working Paper. [8] Barndorff-Nielsen, O. E. and Shephard, N. (). Econoetric analysis of realised volatility and its use in estiating stochastic volatility odels. Journal of the Royal Statistical Society B 64, [9] Barndorff-Nielsen, O. E. and Shephard, N. (7). Variation, jups, arket frictions and high frequency data in nancial econoetrics. Advances in Econoics and Econoetrics. Theory and Applications, Ninth World Congress, (edited by Richard Blundell, Persson Torsten and Whitney K Newey), Econoetric Society Monographs, Cabridge University Press. 31

32 [1] Gonçalves, S. and N. Meddahi (5). Bootstrapping Realized Volatility. Unpublished paper. [11] Kalnina, I. and O. B. Linton (6). Estiating Quadratic Variation Consistently in the Presence of Correlated Measureent Error. STICERD Working Paper, London School of Econoics [1] Karatzas, I. and S. E. Shreve (5). Brownian Motion and Stochastic Calculus, New York: Springer-Verlag. [13] Lahiri, S. N., M. S. Kaiser, N. Cressie, and N. Hsu (1999). Prediction of Spatial Cuulative Distribution Functions Using Subsapling. Journal of the Aerican Statistical Association, 94, [14] Podolskij, M. (6). New Theory on Estiation of Integrated Volatility with Applications. PhD Thesis, Bochu University [15] Politis, D. N. and J. P. Roano (1994), Large saple con dence regions based on subsaples under inial assuptions. Annals of Statistics, [16] Politis, D. N., J. P. Roano and M. Wolf (1999). Subsapling, Springer-Verlag, New York. [17] Veraart, A. (7). Feasible inference for realised variance in the presence of jups. Unpublished paper, Oxford university. [18] Zhang, L. (4). E cient estiation of stochastic volatility using noisy observations: a ultiscale approach. Bernoulli. Forthcoing. [19] Zhang, L., P. Mykland, and Y. Aït-Sahalia (5). A tale of two tie scales: deterining integrated volatility with noisy high-frequency data. Journal of the Aerican Statistical Association, 1, [] Zhou, B. (1996). High-frequency data and volatility in foreign-exchange rates. Journal of Business and Econoic Statistics 14,

33 All n observations 1/n /n First subsaple 1/n /n /n Second subsaple 1/n /n 3/n (+1)/n Third subsaple /n 3/n 4/n (+)/n Figure 1: Regular Subsapling All n observations 1/n /n First subsaple K/n K/n Second subsaple 1/n (1+K)/n (1+K)/n Third subsaple /n (+K)/n (+K)/n Figure : In ll Price Subsapling 33

34 All n observations 1/n /n First subsaple 1/n K/n (K+1)/n K/n (K+1)/n Second subsaple 1/n /n (K+1)/n (K+)/n (K+1)/n (K+)/n Third subsaple /n 3/n (K+)/n (K+3)/n (K+)/n (K+3)/n Figure 3: In ll Returns Subsapling 34

35 Figure 4: Volatility saple path 35

36 8 x x 1 4 Figure 5: Kernel density over siulations of b V regular. Note that V :41 4, so that this estiator very uch overestiates the true quantity. 36

37 .5 x x 1 5 Figure 6: Kernel density over siulations of b V in ll price. Note that V :4 1 5 : 37

38 9 x x 1 6 Figure 7: Kernel density over siulations of b V in ll returns. Note that V 4:5 1 6 : 38

39 1 x x 1 6 Figure 8: Kernel densities over siulations of V b SC estiator, for three di erent aounts of overlap between subsaples. Aount of overlap does not see to a ect the expected value of V b SC, but it decreases the variance. Maxiu overlap (solid line) has the sallest variance. V b SC with no overlap between subsaples (dashed line) has the largest variance. The iddle line (dots) belongs to V b SC in an interediate case (ore precisely, we have the shift paraeter s J, see section A.4.1 Notation for V b SC for explanation of s). True value is V 4:5 1 6 : 39

40 4.5 x x 1 6 Figure 9: Kernel density over siulations of IQ n. Note that V 4:5 1 6 : 4

41 4.5 x x 1 6 Figure 1: Kernel densities over siulations of b V in ll price (dotted); b V in ll returns (crosses), b V SC (solid) and IQ n (dashed). Note that V 4:5 1 6 : 41

42 Figure 11: Boxplot of V b SC, Subset Centered In ll Subsapling, over siulations for di erent overlaps (last boxplot is IQ n ). axis indicates by how any high frequency observations we "shift" the i th subsaple to obtain the (i + 1)st subsaple. Denote this nuber by s. See also Section A.4.1. Notation for V b SC : For exaple, s 1 denotes axiu overlap, and s ( 15 here) denotes no overlap. Nuber of subsaples K increases as we increase the aount of overlap according to K ns + s + 1 (provided this gives an integer). 4