Bootstrapping pre-averaged realized volatility under market microstructure noise∗ Ulrich Hounyo†, S´ılvia Gon¸calves‡, and Nour Meddahi§ Aarhus University, University of Western Ontario and Toulouse School of Economics July 4, 2016
Abstract The main contribution of this paper is to propose a bootstrap method for inference on integrated volatility based on the pre-averaging approach, where the pre-averaging is done over all possible overlapping blocks of consecutive observations. The overlapping nature of the pre-averaged returns implies that the leading martingale part in the pre-averaged returns are kn -dependent with kn growing slowly with the sample size n. This motivates the application of a blockwise bootstrap method. We show that the “blocks of blocks” bootstrap method is not valid when volatility is time-varying. The failure of the blocks of blocks bootstrap is due to the heterogeneity of the squared pre-averaged returns when volatility is stochastic. To preserve both the dependence and the heterogeneity of squared pre-averaged returns, we propose a novel procedure that combines the wild bootstrap with the blocks of blocks bootstrap. We provide a proof of the first order asymptotic validity of this method for percentile and percentile-t intervals. Our Monte Carlo simulations show that the wild blocks of blocks bootstrap improves the finite sample properties of the existing first order asymptotic theory. We use empirical work to illustrate its use in practice. Keywords: Block bootstrap, high frequency data, market microstructure noise, preaveraging, realized volatility, wild bootstrap. ∗
We would like to thank Ilze Kalnina, Kevin Sheppard and Neil Shephard for many useful comments and discussions. This work was supported by grants FQRSC-ANR and SSHRC. In addition, Ulrich Hounyo acknowledges support from CREATES - Center for Research in Econometric Analysis of Time Series (DNRF78), funded by the Danish National Research Foundation, as well as support from the Oxford-Man Institute of Quantitative Finance. Finally, Nour Meddahi benefited from the financial support of the chair “March´e des risques et cr´eation de valeur” Fondation du risque/SCOR. † Department of Economics and Business Economics, Aarhus University, 8210 Aarhus V., Denmark. Email:
[email protected]. ‡ Department of Economics, Faculty of Social Science, University of Western Ontario, 1151 Richmond Street N., London, Ontario, Canada, N6A 5C2. Tel: (519) 661-2111. Ext: 85232. Email:
[email protected]. § Toulouse School of Economics, 21 all´ee de Brienne -Manufacture des Tabacs-31000, Toulouse, France. Email:
[email protected].
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1
Introduction
Estimation of integrated volatility is complicated by the existence of market microstructure noise. This noise represents the discrepancy between the true efficient price of an asset and its observed counterpart and is caused by a multitude of market microstructure effects (such as bid-ask bounds, the discreteness of price changes and the existence of rounding errors, the gradual response of prices to a block trade, the existence of data recording errors such as prices entered as zero, misplaced decimal points, etc). In frictionless markets, and when the log-price process follows a continuous semimartingale, realized volatility computed as the sum of squared intraday returns converges to the integrated volatility as the sampling frequency goes to infinity (see e.g. Andersen, Bollerslev, Diebold, and Labys (2001), Barndorff-Nielsen and Shephard (2002)). See also related work discussed in Jacod and Protter (1998) and Barndorff-Nielsen and Shephard (2001). However, realized volatility is no longer consistent for integrated volatility under the presence of market microstructure noise. This has motivated the development of alternative estimators. One popular method is the pre-averaging approach first introduced by Podolskij and Vetter (2009) and further studied by Jacod et al. (2009). The basic underlying idea consists of first averaging out the noise by computing pre-averaged returns and then computing a realized volatility-like estimator using the pre-averaged returns. Although the pre-averaged realized volatility estimator is consistent for integrated volatility, its convergence rate is much slower than that of realized volatility (when there is no noise) and this can result in finite sample distortions that persist even at very large sample sizes. For this reason, the bootstrap is a useful alternative method of inference in this context. In this paper, we propose a bootstrap method that can be used to estimate the distribution and the variance of the pre-averaged realized volatility estimator of Jacod et al. (2009). Our proposal is to resample the pre-averaged returns instead of resampling the original noisy returns. To be valid, the bootstrap needs to mimic the dependence and heterogeneity properties of the (squared) pre-averaged returns. When pre-averaging occurs over overlapping blocks of returns, as in Jacod et al. (2009), the leading martingale part in the squared pre-averaged returns are kn -dependent, where kn denotes the block length of the interval over which the pre-averaging √ is done and n denotes the sample size. Since kn is proportional to n, kn → ∞ as n → ∞, which implies that the pre-averaged returns are strongly dependent. This suggests that a block bootstrap applied to the pre-averaged returns is appropriate and its application amounts to a “blocks of blocks” bootstrap, as proposed by Politis and Romano (1992) and further studied by B¨ uhlmann and K¨ unsch (1995) (see also K¨ unsch (1989)). Nevertheless, as we show here, such a bootstrap scheme is not valid when volatility is time-varying. The reason is that squared pre-averaged returns are heterogenously distributed (in particular, their mean and variance are
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time-varying) and this creates a bias term in the blocks of blocks bootstrap variance estimator when volatility is stochastic. Thus, to handle both the dependence and heterogeneity of the squared pre-averaged returns, we propose a novel bootstrap approach that combines the wild bootstrap with the blocks of blocks bootstrap. We name this novel approach the wild blocks of blocks bootstrap. One of our main contributions is to show that this method consistently estimates the variance and the entire distribution of the pre-averaged estimator of Jacod et al. (2009). We provide a proof of the first order asymptotic validity of this method for constructing bootstrap unstudentized (percentile) as well as bootstrap studentized (percentile-t) intervals. The pre-averaging approach can also be implemented with non-overlapping intervals, as in Podolskij and Vetter (2009). However, the overlapping method is expected to provide more precise estimates of the integrated variance. We provide intuition of this in Section 2.2. Gon¸calves, Hounyo and Meddahi (2014) study the consistency of the wild bootstrap for the non-overlapping estimator of Podolskij and Vetter (2009). The wild bootstrap exploits the asymptotic independence of the pre-averaged returns when these are computed over nonoverlapping intervals. This method is no longer valid when overlapping intervals are used to compute pre-averaged returns since these are strongly dependent. For this reason, a new bootstrap method is needed for the Jacod et al.’s (2009) approach. Although the wild blocks of blocks bootstrap that we propose here requires the choice of an additional tuning parameter (the block size), we suggest an empirical procedure to select the block size that performs well in our simulations. Other estimators of integrated volatility that are consistent under market microstructure noise include the subsampling approach of Zhang et al. (2005) (see also the multiscale realized volatility estimator of Zhang (2006)) and the realized kernel estimator of Barndorff-Nielsen et al. (2008) (the maximum likelihood-based estimator of Xiu (2010) is also a recent addition to this literature). The bootstrap could also be useful for inference in the context of these estimators. Indeed, Zhang et al. (2011) showed that the asymptotic normal approximation is often inaccurate for the subsampling realized volatility estimator,1 whose finite sample distribution is skewed and heavy tailed. They proposed Edgeworth corrections for this estimator as a way to improve upon the standard normal approximation. Unfortunately, Zhang et al. (2011) provided the Edgeworth corrections of the normalized statistic (where the denominator equals the variance of the estimator in population) rather than studentized statistic (where the denominator is a consistent estimator of the estimator’s variance), while Gon¸calves and Meddahi (2008) proved that Edgeworth corrections based on normalized statistic is worse than the asymptotic theory when there is no noise. The main reason why we focus on the pre-averaging approach here is that it naturally lends 1 Similarly, Bandi and Russell (2011) discussed the limitations of asymptotic approximations in the context of realized kernels and proposed an alternative solution.
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itself to the bootstrap. In particular, we resample the pre-averaged returns instead of the individual returns and exploit the dependence and heterogeneity properties of the pre-averaged returns to prove the consistency of the bootstrap. The rest of this paper is organized as follows. In the next section, we first introduce the setup, our assumptions and review the existing asymptotic theory of Jacod et al. (2009). Section 3 contains the bootstrap results. In Section 3.1 we show that the blocks of blocks bootstrap is consistent only when volatility is constant whereas Section 3.2 describes the wild blocks of blocks bootstrap and shows its consistency under stochastic volatility and i.i.d. noise. Section 4 discusses how to apply our bootstrap methods to the case of autocorrelated market microstructure noise (Section 4.1) and jumps (Section 4.2). Section 5 presents the simulation results whereas Section 6 contains an empirical application. Section 7 concludes. Two appendices are provided. Appendix A contains the tables with simulation results whereas Appendix B is a mathematical appendix with the proofs. A word on notation. In this paper, and as usual in the bootstrap literature, P ∗ (E ∗ and V ar∗ ) denotes the probability measure (expected value and variance) induced by the bootstrap resampling, conditional on a realization of the original time series. In addition, for a sequence ∗
of bootstrap statistics Zn∗ , we write Zn∗ = oP ∗ (1) in probability, or Zn∗ →P 0, as n → ∞, in probability, if for any ε > 0, δ > 0, limn→∞ P [P ∗ (|Zn∗ | > δ) > ε] = 0. Similarly, we write Zn∗ = OP ∗ (1) as n → ∞, in probability if for all ε > 0 there exists a Mε < ∞ such that ∗
limn→∞ P [P ∗ (|Zn∗ | > Mε ) > ε] = 0. Finally, we write Zn∗ →d Z as n → ∞, in probability, if conditional on the sample, Zn∗ weakly converges to Z under P ∗ , for all samples contained in a set with probability P converging to one.
2
Setup, assumptions and review of existing results
2.1
Setup and assumptions
Let X denote the latent efficient log-price process defined on a probability space (Ω0 , F 0 , P 0 ) equipped with a filtration (Ft0 )t≥0 . We model X as a Brownian semimartingale process defined by the equation Z Xt = X0 +
t
Z
0
t
σs dWs , t ≥ 0,
as ds +
(1)
0
where a = (at )t≥0 is an adapted c`adl`ag drift process, σ = (σt )t≥0 is an adapted c`adl`ag volatility process and W = (Wt )t≥0 a standard Brownian motion. The object of interest is the quadratic variation of X, i.e. the process Z t Ct = σs2 ds, 0
4
also known as the integrated volatility. Without loss of generality, we consider s ∈ [0, t] with R1 t = 1 and define C1 = 0 σs2 ds as the integrated volatility of X over a given time interval [0, 1], which we think of as a given day. The presence of market frictions such as price discreteness, rounding errors, bid-ask spreads, gradual response of prices to block trades, etc, prevent us from observing the true efficient price process X. Instead, we observe a noisy price process Y , observed at time points t = ni for i = 0, . . . , n, given by Yt = Xt + t ,
(2)
where t represents the noise term that collects all the market microstructure effects. In order to make both X and Y measurable with respect to the filtration, we define a new probability space Ω, (Ft )t≥0 , P , which accommodates both processes. To this end, we follow Jacod et al. (2009) and assume one has a second space Ω1 , (Ft1 )t≥0 , P 1 , where Ω1 denotes R[0,1] and F 1 the product Borel-σ-field on Ω1 . Next, for any t ∈ [0, 1], we define Qt ω (0) , dy to be the probability measure on R, which corresponds to the transition from Xt ω (0) to the observed process Yt . In the case of i.i.d. noise, this transition kernel is rather simple (see e.g. equation 1 (0) (1) (2.7) of Vetter (2008)), but it becomes more pronounced in a general framework. P ω , dω denotes the product measure ⊗t∈[0,1] Qt ω (0) , · . The filtered probability space Ω, (Ft )t∈[0,1] , P T on which the process Y lives is then defined with Ω = Ω0 ×Ω1 , F = F 0 ×F 1 , Ft = s>t Fs0 ×Fs1 , and P dω (0) , dω (1) = P 0 ω (0) P 1 ω (0) , dω (1) . We assume that t is centered and independent, conditionally on the efficient price process X. In addition, we assume that the conditional variance of t is c`adl`ag. Assumption 1 below collects these assumptions. Assumption 1. (i) E (t |X) = 0 and t and s are independent for all t 6= s, conditionally on X. (ii) αt = E (2t |X) is c`adl`ag and E (8t ) < ∞. Assumption 1 amounts to Assumption (K) in Jacod et al. (2009). As they explain, this assumption is rather general, allowing for time varying variances of the noise and dependence between X and . See Jacod et al. (2009) for particular examples of market microstructure noise that satisfy Assumption 1. However, empirically the conditional independence assumption on may be unrealistic especially at the highest frequencies (see e.g. Hansen and Lunde (2006)). We will investigate the impact of autocorrelated noise on the bootstrap performance in Sections 4 and 5.
5
2.2
The pre-averaged estimator and its asymptotic theory
We observe Y at regular time points returns at frequency n1 ,
i , n
for i = 0, . . . , n, from which we compute n intraday
ri ≡ Y i − Y i−1 , i = 1, . . . , n. n
n
Given that Y = X + , we can write ri = X i − X i−1 + i − i−1 ≡rie + ∆i , n
where rie = X i − X i−1 denotes the n
n
n
n
1 -frequency n
n
return on the efficient price process. Under
Assumption 1, the order of magnitude of ∆i ≡ i − i−1 is OP (1) . In contrast, the ex-post n n R i/n 2 e variation of ri is given by (i−1)/n σs ds. The order of magnitude of rie is then OP n−1/2 . This decomposition shows that the noise completely dominates the observed return process as n → ∞, implying that the usual realized volatility estimator is biased and inconsistent. See Zhang et al. (2005) and Bandi and Russell (2008). To describe the Jacod et al. (2009) pre-averaging approach, let kn be a sequence of integers which will denote the window length over which the pre-averaging of returns is done. Similarly, R1 let g be a weighting function on [0, 1] such that g (0) = g (1) = 0 and g (s)2 ds > 0, and assume 0
g is continuous and piecewise continuously differentiable with a piecewise Lipschitz derivative g 0 . An example of a function that satisfies these restrictions is g (x) = min (x, 1 − x) . We introduce the following additional notation. Let Z1 φ1 (s) =
g 0 (u) g 0 (u − s) du and φ2 (s) =
s
Z1 g (u) g (u − s) du, s
and for i = 1, 2, let ψi = φi (0) . For instance, for g (x) = min (x, 1 − x), we have that ψ1 = 1 and ψ2 = 1/12. For i = 0, . . . , n−kn +1, the pre-averaged returns Y¯i are obtained by computing the weighted sum of all consecutive n1 -horizon returns over each block of size kn , kn X j ¯ Yi = g ri+j . kn j=1 The effect of pre-averaging is to reduce the impact of the noise in the pre-averaged return. Specifically, as shown by Vetter (2008), kn X j ¯i = X i+j − X i+j−1 = OP X g n n kn j=1
6
r
kn n
! ,
and kn X j 1 ¯i = g i+j − i+j−1 = OP √ . n n k k n n j=1 Thus, the impact of the noise is reduced the larger kn is. To get the efficient n−1/4 rate of convergence, Jacod et al. (2009) propose to choose a sequence of integers kn such that the following assumption holds. Assumption 2. For θ ∈ (0, ∞), we have that k √n = θ + o n−1/4 . n
(3)
¯ i and ¯i ) are balanced and equal to This choice implies that the orders of the two terms (X √ OP n−1/4 . An example that satisfies (3) is kn = [θ n]. Based on the pre-averaged returns Y¯i , Jacod et al. (2009) propose the following estimator of integrated volatility, n−k n n +1 X 1 ψ1 X 2 2 ¯ P RVn = Yi − r , ψ2 kn i=0 2nθ2 ψ2 i=1 i
(4)
where ψ1 and ψ2 are as defined above. The first term of the right-hand side of (4) is an average of realized volatility-like estimators based on pre-averaged returns of length kn whereas the second term is a bias correction term. As discussed in Jacod et al. (2009), this bias term does not contribute to the asymptotic variance of P RVn . In order to give the central limit theorem for P RVn , we introduce the following numbers that are associated with g, Z1
Z1 φi (s) φj (s) ds, and Ψij = −
Φij = 0
sφi (s) φj (s) ds. 0
For the simple function g (x) = min (x, 1 − x), Φ11 = 1/6, Φ12 = 1/96 and Φ22 = 151/80640. Under Assumption 1 and (kn , θ) satisfying (3), Jacod et al. (2009) show that as n → ∞, R1 n1/4 P RVn − 0 σs2 ds √ →st N (0, 1), V where →st denotes stable convergence, and Z 1 4 σs2 αs αs2 4 V = 2 Φ22 θσs + 2Φ12 + Φ11 3 ds ψ2 0 θ θ
7
(5)
(6)
is the asymptotic variance of n1/4 P RVn . To estimate V consistently, Jacod et al. (2009) propose n−2k n−kn +1 i+2kn −1 n +1 X 4Φ22 X 4 Φ12 Φ22 ψ1 4 2 X ˆ Vn = Yi + 3 − Yi rj2 3θψ24 i=0 nθ ψ23 ψ24 i=0 j=i+kn n−2k +1 n X Φ11 Φ12 ψ1 Φ22 ψ12 1 2 − 2 + + ri2 ri+2 . (7) 2 3 4 3 nθ ψ2 ψ2 ψ2 i=0 Together with the CLT result (5), we have that R1 n1/4 P RVn − 0 σs2 ds p →st N (0, 1). Tn ≡ Vˆn We can use this feasible asymptotic distribution result to build confidence intervals for inteR1 grated volatility. In particular, a two-sided feasible 100(1 − α)% level interval for 0 σs2 ds is given by: q q a −1/4 −1/4 Vˆn , P RVn + z1−α/2 n Vˆn , (8) ICF eas,1−α = P RVn − z1−α/2 n where z1−α/2 is such that Φ z1−α/2 = 1 − α/2, and Φ (·) is the cumulative distribution function of the standard normal distribution. For instance, z0.975 = 1.96 when α = 0.05. Note that the pre-averaging approach can also be implemented with non-overlapping intervals, as in Vetter (2008) and Podolskij and Vetter (2009). By Theorem 3.7 of Vetter (2008), we can also build asymptotic confidence intervals for integrated volatility based on the non-overlapping pre-averaged realized volatility estimator. In particular, a two-sided feasible R1 100(1 − α)% level interval for 0 σs2 ds based on non-overlapping intervals, is given by: q q PV PV −1/4 P V −1/4 ICF eas,1−α = P RVn − z1−α/2 n VˆnP V , P RVn + z1−α/2 n VˆnP V , (9) where P RVnP V and VˆnP V are the non-overlapping pre-averaged realized volatility estimator and a consistent estimator of the asymptotic variance, respectively. Following Corollary 3.3 and 3.6 of Vetter (2008), we have that P RVnP V
n −1c b knn −1c √ b kX n n X ψ n 1 X 2 1 2 2 PV ¯ ˆ = Yikn − r , and Vn = Y¯ik4 n . ψ2 i=0 2nθ2 ψ2 i=1 i 3ψ22 i=0
It is well known that the pre-averaged estimator based on overlapping blocks is more efficient than the pre-averaged estimator based on non-overlapping returns. In particular, for the parametric model of zero drift, constant volatility σ > 0 and constant conditional variance of t , αt = α > 0, if we choose for example the weight function g (x) = min (x, 1 − x), some simple derivations show that the ”optimal” choice of θ which minimizes the asymptotic vari√ ance V given in (6) is approximately equal to 4.777 α/σ. Under this special setting, using
8
√ θ = 4.777 α/σ, the non-overlapping pre-averaged estimator of Podolskij and Vetter (2009) has √ an asymptotic variance that is approximately equal to 20.11σ 3 α, whereas the overlapping es√ timator of Jacod et al. (2009) has a variance equal to 8.545σ 3 α (this is very close to the lower √ bound 8σ 3 α derived by Gloter and Jacod (2001)). Therefore, based on efficiency, we should choose the overlapping approach of Jacod et al. (2009) rather than the non-overlapping one. However, as shown by our simulation results (see Appendix A), we find that the pre-averaging approach leads to important coverage probability distortions when returns are not sampled too frequently. This motivates Gon¸calves, Hounyo and Meddahi (2014) to propose the wild bootstrap as alternative method of inference for the Podolskij and Vetter’s (2009) estimator. Here, we focus on bootstrapping the more efficient pre-averaged realized volatility estimator of Jacod et al. (2009).
3
The bootstrap
The goal of this section is to propose a bootstrap method that can be used to consistently R1 1/4 P RVn − 0 σs2 ds as well as of the studentized statistic estimate the distribution of n p R1 n1/4 P RVn − 0 σs2 ds / Vˆn . This justifies the construction of bootstrap percentile and percentilet confidence intervals for integrated volatility, respectively. Gon¸calves and Meddahi (2009) proposed bootstrap methods for realized volatility in the absence of market microstructure noise. In their ideal setting, intraday returns ri (conditionally on the path of the volatility σ and the drift a) are uncorrelated, but possibly heteroskedastic due to stochastic volatility, thus motivating the use of a wild bootstrap method. When intraday returns are contaminated by market microstructure noise, they are no longer conditionally uncorrelated, as in Gon¸calves and Meddahi (2009). This implies that the wild bootstrap is no longer valid when applied to ri . Instead, a block bootstrap method applied to the intraday returns would seem appropriate. One complication arises in this context: the statistic of interest is not symmetric in the observations and the block bootstrap generates blocks of observations that are conditionally independent. In particular, since the first term in P RVn is an average of the squared preaveraged returns Y¯i2 , it depends on all the products of intraday returns inside blocks of size kn . If we generate block bootstrap intraday returns, these will be independent between blocks, implying that the bootstrap statistic may look at many pairs of intraday returns that are independent in the bootstrap world. This not only renders the analysis very complicated but can induce biases in the bootstrap estimator. To avoid this problem when dealing with statistics that are not symmetric in the underlying observations, K¨ unsch (1989), Politis and Romano (1992) and B¨ uhlmann and K¨ unsch (1995) studied the “blocks of blocks” bootstrap, where one applies the block bootstrap to appropriately pre-specified blocks of observations. In 9
our context, the blocks of blocks bootstrap consists of applying a traditional block bootstrap to the squared pre-averaged returns Y¯i2 . As we will see next, this approach is not valid when volatility is time-varying. The reason is that when volatility is stochastic, squared pre-averaged returns are not only dependent but also heterogeneous. The block bootstrap does not capture this heterogeneity unless volatility is constant2 . In order to capture both the time dependence and the heterogeneity in Y¯i2 , we propose a novel bootstrap procedure that combines the wild bootstrap with the block bootstrap. Although the consistent estimator of integrated volatility is P RVn , only the first term in P RVn drives the variance of the limiting distribution of P RVn . In particular, as Jacod et al. (2009) have shown, the second term is a bias correction term which does not contribute to the asymptotic variance (it only ensures that the estimator is well centered at the integrated volatility). For this reason, our proposal is to bootstrap only the first contribution to P RVn , P] RV n =
n−k n +1 X 1 Y¯i2 . ψ2 kn i=0
This statistic depends only on the pre-averaged returns, to which we apply a particular boot strap scheme. More specifically, let Y¯i∗ : i = 0, 1, . . . , n − kn + 1 denote a bootstrap sample from Y¯i : i = 0, 1, . . . , n − kn + 1 . The bootstrap analogue of P RVn is P RVn∗
∗
= P] RV n −
n ψ1 X 2 r , 2nθ2 ψ2 i=1 i
where3 ∗
P] RV n =
n−k n +1 X 1 Y¯i∗2 . ψ2 kn i=0
Since the (conditional) expected value of n1/4 (P RVn∗ − P RVn ) induced by the bootstrap resampling methods considered in this paper is not always zero, we center P RVn∗ around E ∗ (P RVn∗ ) . Thus, we use the bootstrap distribution of ∗ ∗ RV n − E ∗ P] RV n n1/4 (P RVn∗ − E ∗ (P RVn∗ )) = n1/4 P] R1 as an estimator of the distribution4 of n1/4 P RVn − 0 σs2 ds . 2
See Gon¸calves and White (2002) for a discussion of the impact of mean heterogeneity on the validity of the block bootstrap for the sample mean. 3 This implies that our bootstrap statistic actually contains the bias term. Nevertheless, since this term is evaluated on the original sample rather than on the bootstrap data, our bootstrap method does not capture the added uncertainty caused by estimation of this term. Our simulations show that despite this, the bootstrap is very accurate, outperforming the asymptotic normal approximation. ∗ 4 ^ In particular, we can explicitly compute the bootstrap expectation of P RV n (and we do so in (10) and (17)), for the blocks of blocks bootstrap and the wild blocks of blocks bootstrap, respectively. For instance, under the wild blocks of blocks bootstrap scheme, using an external random variable η with mean 1, it follows
10
Next, we consider the blocks of blocks bootstrap approach applied to P] RV n and show that it is asymptotically invalid when volatility is time-varying. This motivates a new bootstrap method that combines the wild bootstrap with the block bootstrap, which we study in the last subsection.
3.1
The blocks of blocks bootstrap
To describe this approach, let Nn = n − kn + 2 denote the total number of pre-averaged returns and let bn denote the block size. We suppose that Nn = Jn ·bn , so that Jn denotes the number of blocks of size bn one needs to draw to get Nn = n − kn + 2 bootstrap observations. The blocks of ∗ blocks bootstrap generates a bootstrap resample Y¯i−1 : i = 1, . . . , Nn by applying the moving blocks bootstrap of K¨ unsch (1989) to the scaled pre-averaged returns Y¯i−1 : i = 1, . . . , Nn . Letting I1 , . . . , IJn be i.i.d. random variables distributed uniformly on {0, 1, . . . , Nn − bn }, we set ∗ Y¯i−1+(j−1)b = Y¯i−1+Ij for 1 ≤ j ≤ Jn and 1 ≤ i ≤ bn . n
The bootstrap analogue of P] RV n is ∗
P] RV n =
Nn Jn bn 1 X 1 X 1 X Nn 1 ¯ 2 ∗2 Y¯i−1 Y = Ij +i−1 , ψ2 kn i=1 Jn j=1 bn i=1 kn ψ2 {z } | ≡ZIj +i
¯ i + ¯i = OP n−1/4 given that Note that in our setup, Y¯i = X √ 2 kn is such that kn / n = θ + o n−1/4 . This implies that Y¯i−1 = OP n−1/2 and therefore 2 is OP (1). Zi = n−kknn +2 ψ12 Y¯i−1 We can easily show that ! ! Jn bn NX bn n −bn X X X ∗ 1 1 1 1 ∗ ∗ E P] RV n = E ZI +i = Zj+i . (10) Jn j=1 bn i=1 j Nn − bn + 1 j=0 bn i=1 where we let Zi ≡
Nn 1 ¯ 2 Y . kn ψ2 i−1
Similarly,
Vn∗
Jn bn X X ∗ √ ∗ 1 1 ∗ 1/4 ] ≡ V ar n P RV n = nE RV n ZIj +i − E ∗ P] Jn j=1 bn i=1 !2 bn ∗ √ 1 ∗ 1 X ∗ = n E RV n ZI1 +i − E P] Jn bn i=1 !2 NX bn n −bn ∗ √ bn 1 1 X n Zj+i − E ∗ P] RV n . = Nn Nn − bn + 1 j=0 bn i=1
∗
that E ∗ (P RVn∗ ) = P RVn .
11
!2
(11)
Our next result studies the convergence of Vn∗ when bn = (p + 1) kn , and p ≥ 1 is either fixed as n → ∞ or p → ∞ after n → ∞ (which we denote by writing (n, p)seq → ∞). To emphasize ∗ the dependence of Vn∗ on p we write Vn,p .
Lemma 3.1 Suppose Assumption 1 holds and kn → ∞ as n → ∞ such that Assumption 2 ∗ ∗ ∗ holds. Let Vn,p ≡ V ar∗ n1/4 P] RV n denote the moving blocks bootstrap variance of n1/4 P] RV n based on a block length equal to bn = (p + 1) kn , where p ≥ 1. Then, a) For any fixed p ≥ 1, as n → ∞, P
∗ Vn,p −→ Vp + Bp ,
where 1
Z
γ 2 (p)t dt
Vp = 0
with 4 γ (p)t = 2 ψ2 2
Φ22 +
2 2 1 1 1 σ t αt αt 4 Ψ22 θσt + 2 Φ12 + Ψ12 + Φ11 + Ψ11 , p+1 p+1 θ p+1 θ3
and "Z Bp = θ (p + 1) 0
1
ψ1 σt2 + 2 αt θ ψ2
2
Z
1
dt − 0
2 # ψ 1 . σt2 + 2 αt dt θ ψ2 P
b) When σt = σ and αt = α are constants, Bp = 0 for any p ≥ 1 and Vp −→ V ≡ P ∗ limn→∞ V ar n1/4 P RVn as p → ∞. In this case, Vn,p −→ V as (n, p)seq → ∞. P
∗ c) More generally, when σt and/or αt are stochastic, Vn,p −→ ∞ as (n, p)seq → ∞.
Part a) of Lemma 3.1 shows that when the bootstrap block size bn is a fixed proportion of the pre-averaging block size kn , the blocks of blocks bootstrap variance converges in probability to Vp + Bp , where Bp is a bias term due to the fact that volatility is time-varying. When both the volatility σt and αt , the conditional variance of t , are constants, Bp is equal to zero for any P
value of p. If p → ∞ (i.e. if bn /kn → ∞ as n → ∞), then Vp −→ V , the asymptotic variance of P
∗ n1/4 P RVn . Therefore, under these conditions, Vn,p −→ V as (n, p) → ∞ sequentially. Although ∗ this result does not necessarily imply the consistency of Vn,p towards V as (n, p) → ∞ jointly
(because sequential convergence does not by itself imply joint convergence), it is a first step in that direction (see in particular Lemma 6 of Phillips and Moon, 1999). We do not pursue the ∗ derivation of the joint limit of Vn,p here because that would distract us from the main message of Lemma 3.1, which is the invalidity of the blocks of blocks bootstrap variance estimator when
σt and/or αt are time varying. In this more general and practically relevant case, part c) of ∗ Lemma 3.1 shows that Vn,p diverges to ∞ in probability as (n, p)seq → ∞. The main reason for
12
P
this inconsistency result is that Bp −→ ∞ as p → ∞. Notice that even though the limit derived in part c) is sequential, we can conclude that the same result holds as (n, p) → ∞ jointly. The P ∗ argument is as follows. Suppose it was the case that Vn,p −→ V ≡ lim V ar n1/4 P RVn , as (n, p) → ∞ jointly. Then by Lemma 5 of Phillips and Moon (1999), we should have that P ∗ Vn,p −→ V sequentially as (n, p)seq → ∞, which is in contradiction with the result of part c). ∗ Hence, Vn,p cannot converge in probability to V , as (n, p) → ∞ jointly. More generally, we can ∗ exists, then by the same argument, it must coincide with the show that if the joint limit of Vn,p P
∗ sequential limit. Since we actually proved that Vn,p −→ ∞ sequentially as (n, p)seq → ∞, this ∗ implies Vn,p must diverge as (n, p) → ∞ jointly.
Lemma 3.1 suggests that the blocks of blocks bootstrap is consistent for the variance of P RVn only under constant volatility, constant conditional variance of noise and if we let the bootstrap block size bn grow at a faster rate than the pre-averaging block size kn . This result is related to a consistency result of the blocks of blocks bootstrap established in B¨ uhlmann and K¨ unsch (1995). As they showed, when the statistic of interest is an average of smooth functions of blocks of consecutive stationary strong mixing observations of size kn , where kn tends to infinity, the crucial condition for the block bootstrap to be valid is that the block size bn grows at a faster rate than kn . This is because the blocks over kn observations (which in our case correspond to the pre-averaged returns) are strongly dependent for |i − j| ≤ kn , where kn → ∞, and bn must be large enough to capture this dependence. B¨ uhlmann and K¨ unsch (1995) consider observations generated from a stationary strong mixing process and therefore they do not find any bias problem related to heterogeneity. Nevertheless, this becomes a problem in our context when volatility is stochastic. Therefore, a different bootstrap method is required to handle both the time dependence and the heterogeneity of pre-averaged returns. Note that the inconsistency of the blocks of blocks bootstrap variance estimator for the asymptotic variance of P RVn when the volatility is time-varying is not in contrast to the i.i.d. bootstrap results in Gon¸calves and Meddahi (2009) for realized volatility (in the absence of noise). In particular, the i.i.d. bootstrap variance estimator of Gon¸calves and Meddahi (2009) (cf. page 287) for the asymptotic variance of the realized volatility is given by !2 Z 1 2 Z 1 n n X X 4 2 P 4 2 e e → 3 σt dt − σt dt , n (ri ) − (ri ) 0 0 i=1 i=1 | {z } ≡VGM <∞
which is equal to 2
R1 0
σt4 dt (i.e. the asymptotic conditional variance of the realized volatility)
only when the volatility is constant. This means that even in the absence of noise, when the volatility is time-varying we would not use the i.i.d. bootstrap method of Gon¸calves and Meddahi (2009) to compute standard errors of statistics based on functionals of realized volatility. However, note that although the 13
i.i.d. bootstrap method in Gon¸calves and Meddahi (2009) does not consistently estimate the asymptotic variance of realized volatility, their bootstrap method is still asymptotically valid for studentized (percentile-t) bootstrap intervals. This is not necessarily the case for the blocks of blocks bootstrap method applied to P RVn . The main reason is that when the volatility is time-varying, and the bootstrap block size bn grows faster than kn (i.e., the more realistic case P ∗ of choice of bn ), Vn,p −→ ∞ as (n, p) → ∞ jointly.
3.2
The wild blocks of blocks bootstrap
In this section, we propose and study the consistency of a novel bootstrap method for preaveraged returns based on overlapping blocks of kn intraday returns. It combines the blocks of blocks bootstrap with the wild bootstrap5 and in this manner gets rid of the bias term Bp associated with the blocks of blocks bootstrap variance Vn∗ in (11). Here, let bn be a sequence of integers such that bn ∝ n δ ,
(12)
where δ ∈ (0, 1), and assume that Jn is such that Jn · bn = Nn . Let υ1 , . . . , υJn be i.i.d. random variables whose distribution is independent of the original sample. Denote by µ∗q = E ∗ υjq its q-th order moments. For j = 1, . . . , Jn , let bn X ¯j = 1 Y¯ 2 B bn i=1 i−1+(j−1)bn 2 denote the block average of the squared pre-averaged returns Y¯i−1+(j−1)b for block j, we also n
let ηj = vj2 . We then generate the bootstrap pre-averaged squared returns as follows, ∗2 ¯j+1 + Y¯ 2 ¯j+1 ηj , for 1 ≤ j ≤ Jn − 1 and for 1 ≤ i ≤ bn . Y¯i−1+(j−1)b = B − B i−1+(j−1)b n n
(13)
¯j+1 is not available and therefore we let For the last block j = Jn , B ∗2 ¯j + Y¯ 2 ¯j ηj , for 1 ≤ i ≤ bn . Y¯i−1+(j−1)b = B − B i−1+(j−1)b n n
(14)
Our method is related to the wild bootstrap approach of Wu (1986) and Liu (1988). More ¯ n , where Xi is indepenspecifically, in Wu (1986) and Liu (1988), the statistic of interest is X dently but heterogeneously distributed with mean µi and variance σi2 . Their wild bootstrap generates Xi∗ as ¯ n + Xi − X ¯ n ηi , for 1 ≤ i ≤ n, Xi∗ = X 5
A possible alternative bootstrap method is the local block bootstrap introduced by Paparoditis and Politis (2002) and Dowla et al. (2003) (see also the review paper by Kreiss and Paparoditis (2011)), where one resamples only blocks of observations close to the given data point. Proving the asymptotic validity of this method in our context is an interesting question which we leave for future research.
14
√ ¯∗ ¯ where ηi is i.i.d. (0, 1). Liu (1988) shows that the bootstrap distribution of n X n − Xn is P √ ¯ n Xn − µ ¯n , where µ ¯n = n−1 ni=1 µi , provided consistent for the distribution of Pn 1 ¯n )2 → 0 (and some other regularity conditions). i=1 (µi − µ n Our bootstrap method can be seen as a generalization of the wild bootstrap of Wu (1986) and Liu (1988) to the kn -dependent case. In particular, here the statistic of interest is an average of blocks of observations of size kn , Nn 1 X P] RV n = Zi , Nn i=1
where Zi ≡
Nn 1 ¯ 2 Y kn ψ2 i−1
has time-varying moments and is kn -dependent (conditionally on X), i.e.
Zi is independent of Zj for all |i − j| > kn . To preserve the serial dependence, we divide the data into Jn non-overlapping blocks of size bn and generate the bootstrap observations within a given block j using the same external random variable ηj . This preserves the dependence within each block. When there is no dependence, we can take bn = 1, in which case our bootstrap method amounts to Liu’s wild bootstrap with one difference: instead of centering each bootstrap observation Zi∗ around the overall mean P] RV n , we center Zi∗ around Zi+1 . The reason for the new centering is that P ¯n )2 → 0 (unless volatility is µi in our context does not satisfy Liu’s condition n1 ni=1 (µi − µ constant). Hence centering around P] RV n does not work here. Instead, we show that centering around Zi+1 yields an asymptotically valid bootstrap method for P] RV n even when volatility is stochastic. The bootstrap data generating process (13) and (14) yields a bootstrap sample Y¯0∗2 , . . . , Y¯N∗2n −1 which we use to compute ∗
P] RV n =
Nn 1 X Y¯ ∗2 , ψ2 kn i=1 i−1
(15)
the wild blocks of blocks bootstrap analogue of P] RV n . Let bn X ¯∗ = 1 B Y¯ ∗2 j bn i=1 i−1+(j−1)bn
¯j . Given (13), we have that for j = 1, . . . , Jn − 1, be the bootstrap analogue of B ¯∗ = B ¯j+1 + B ¯j − B ¯j+1 ηj , B j
15
(16)
¯j for j = Jn . This implies that we can write ¯j∗ = B whereas from (14), B ∗
P] RV n =
Jn bn Jn −1 bn X bn X 1 X ∗2 ¯j∗ + bn B ¯J∗ = Y¯i−1+(j−1)b B n ψ2 kn j=1 bn i=1 ψ2 kn j=1 ψ2 kn n
Jn −1 bn X ¯J . ¯j+1 + B ¯j − B ¯j+1 ηj + bn B = B ψ2 kn j=1 ψ2 kn n
We can now easily obtain the bootstrap mean and variance of P RVn∗ . In particular, ! JX Jn −1 n −1 ∗ bn X b n ∗ ¯ ¯ ¯j − B ¯j+1 E ∗ (ηj ) , ] Bj+1 + BJn + E P RV n = B ψ2 kn j=1 ψ2 kn j=1
(17)
and Vn∗
≡ V ar
∗
n
1/4
Jn −1 ∗ n1/2 b2n X ¯j − B ¯j+1 2 V ar∗ (ηj ) . ] B P RV n = 2 2 ψ2 kn j=1
Our next result studies the convergence of Vn∗ when bn satisfies (12) such that 1/2 < δ < 2/3. To prove the consistency of Vn∗ for V we impose the following additional condition. Assumption 3. σt is locally bounded away from zero and is a continuous semimartingale. This assumption rules out jumps in σt and is common in the realized volatility literature (e.g., equation (3) of Barndorff-Nielsen et al. (2008) or equation (3) of Gon¸calves and Meddahi (2009)). We can prove the following results. Lemma 3.2 Suppose Assumptions 3 hold and the block size bn satisfies (12) such that 1, 2 and ∗ 1/4 ] ∗ ∗ 1/2 < δ < 2/3. Let Vn ≡ V ar n P RV n denote the wild blocks of blocks bootstrap variance ∗
of n1/4 P] RV n based on a block length equal to bn and external random variables ηj ∼ i.i.d. with mean E ∗ (ηj ) and variance V ar∗ (ηj ) = 1/2. Then, p lim Vn∗ = V ≡ lim V ar n1/4 P RVn , n→∞
n→∞
This result shows that if we let δ > 1/2, i.e., bn grow faster than kn (i.e., bn /kn → ∞) but such that bn /n → 0 and V ar∗ (ηj ) = 1/2, the wild blocks bootstrap variance estimator is consistent for the asymptotic variance of P RVn under Assumptions 1, 2 and 3. Given the consistency of the bootstrap variance estimator, and the fact that it is possible to obtain an exact and explicit formula of V ∗ , one may simply use V ∗ in place of Vˆn given by (7) as n
n
alternative consistent estimator of V . Together with the CLT result (5), we have that R1 n1/4 P RVn − 0 σs2 ds p →st N (0, 1). ∗ Vn
16
As alternative method of inference (which does not require any resampling of one’s data), we can use this feasible asymptotic distribution result to build confidence intervals for integrated R1 volatility. In particular, a two-sided feasible 100(1 − α)% level interval for 0 σs2 ds is given by: p p ICFb eas,1−α = P RVn − z1−α/2 n−1/4 Vn∗ , P RVn + z1−α/2 n−1/4 Vn∗ , (18) where Vn∗ =
Jn −1 n1/2 b2n X ¯j − B ¯j+1 2 , B 2 2 2ψ2 kn j=1
(19)
z1−α/2 is such that Φ z1−α/2 = 1 − α/2, and Φ (·) is the cumulative distribution function of the standard normal distribution. The structure of the wild blocks of blocks bootstrap method is related to the ideas in the recent paper of Mykland and Zhang (2014). To see this, it may be helpful to rewrite Vn∗ given by (19) as follows # JX n −1 2 1 P] RV j+1 − P] RV j , Vn∗ = n1/2 · 2 j=1 "
(20)
where P] RV j =
bn 1 X Y¯ 2 ψ2 kn i=1 i−1+(j−1)bn
denotes the analogue of P] RV n computed for the block j. Given results in equations (7) and (11) in Mykland and Zhang (2014), it is easy to see that under some regularity conditions R1 the asymptotic variance (AVAR) of many estimators, say Θ = 0 θ˜t dt, in the high-frequency literature can be estimated based on n −1 2 1 JX ˆ ˆ ˆ \ Θj+1 − Θj , (21) AVAR Θ − Θ = 2 j=1 ˆ j is the estimator Θ calculated on the j-th block and such that Θj = where Θ
R jbn /n
˜
θ dt. (j−1)bn /n t
6
More precisely, under some regularity conditions (including negligible edge effect and continuous spot process θet equations (7) and (11) in Mykland and Zhang (2014) amount to, ! JX JX n −1 n −1 2 2 b 2 h i n e θe ˆ j+1 − Θ ˆj = 2 ˆ j − Θj + Θ AVAR Θ θ, (1 + op (1)) , (22) 3 n 1 j=1 j=1 In our context we assume that the spot parameter process θ˜t is a continuous semi1 martingale. Without loss of generality one may let θ˜t ≡ σt2 + θψ Note that 2 ψ αt . 2 R1 2 R 1 2 ψ1 ^ AVAR P RV n − 0 σt + θ2 ψ2 αt dt =AVAR P RVn − 0 σt dt , thus in order to estimate the asymp R1 1 ˆ −Θ=P ^ dt. totic variance of n1/4 P RVn , one can simply focus on Θ RV n − 0 σt2 + θψ 2 ψ αt 2 6
17
and
ˆ −Θ = AVAR Θ
JX n −1
! ˆ j − Θj AVAR Θ (1 + op (1)) ,
(23)
j=1
h
i e e respectively, where θ, θ is the total quadratic variation of the spot process θet over the whole 1 interval from 0 to 1. Given (22) and (23), it follows that n −1 2 4 b 2 h i 1 JX n e θe (1 + op (1)) . ˆ j+1 − Θ ˆj − ˆ −Θ = θ, (24) Θ AVAR Θ 2 j=1 3 n 1 Thus, if bnn can be taken hto be i small enough, then one can simply use (21), i.e., a one scale e e estimator by ignoring the θ, θ term. Note that given the normalization of AVAR in Mykland 1 ˆ is such that and Zhang(2014) (cf. footnote 1), we have AVAR=AVARn = n−2α V , where Θ ˆ − Θ →st N (0, V ), for some α > 0. Thus, in our context, the one scale estimator formula nα Θ ˆ = P] applied to Θ RV n with α = 1/4, gives JX n −1 2 ˆ −Θ = 1 \ Θ P] RV j+1 − P] RV j = n−1/2 Vn∗ . AVAR 2 j=1
We emphasize that the paper by Mykland and Zhang (2014) goes much further in developing the asymptotic variance estimator, including estimators with hard edge effect and allowing non continuous spot process. In particular, Mykland and Zhang (2014) show that by subsampling and averaging one can still use a result akin to (22) when θet is a general semimartingale. In addition, they argue that subsampling and averaging can at the same time help to deal with hard edge effect which can lead the additivity in (23) to fail. Thus, the final estimator is more complicated and is based on two- or multi-scale construction. Their approach aims to avoid using ˆ (for instance, no need to know the closed form the information on the asymptotic variance of Θ of the AVAR). However, this is not without consequences for their method. For example they have to introduce an additional layer of blocks to implement the bias-correction term. The ˆ can also go negative in finite samples, which is not the asymptotic variance estimator of Θ case of the bootstrap. This relationship with Mykland and Zhang (2014), in particular the way both method managed blocks of adjacent summands suggests that our wild blocks of blocks bootstrap approach may be applied very generally in the field of nonparametric estimation with infill asymptotics. The exploration of this is beyond the scope of this paper. ∗ ∗ Our next result proves the consistency of the bootstrap distribution of n1/4 P] RV n − E ∗ P] RV n . 4(2+ε) Theorem 3.1 Suppose Assumptions 1, 2 and 3 hold such that for any ε ≥ 2, E t < ∞, ∗
and the block size bn satisfies (12) such that 1/2 < δ < 2/3. Let P] RV n be the pre-averaged realized volatility estimator based on a block length equal to bn and an external random variable 18
ηj ∼ i.i.d. (E ∗ (ηj ) , V ar∗ (ηj )) such that V ar∗ (ηj ) = 12 , and for some ε ≥ 2 E ∗ |ηj |2+ε ≤ ∆ < ∞. Then Z 1 ∗ 1/4 ∗ ∗ 2 1/4 ∗ ] ] P RVn − σs ds ≤ x →P 0 as n → ∞. ≤x −P n P RV n − E P RV n sup P n x∈R 0
Theorem 3.1 justifies using the wild blocks of blocks bootstrap to construct bootstrap percentile intervals for integrated volatility. Specifically, a 100 (1 − α) % symmetric bootstrap percentile interval for integrated volatility based on the bootstrap is given by ∗ ICperc,1−α = P RVn − n−1/4 p∗1−α , P RVn + n−1/4 p∗1−α , (25) ∗ ∗ where p∗1−α is the 1−α quantile of the bootstrap distribution of n1/4 P] RV n − E ∗ P] RV n . Next, we propose a consistent bootstrap variance estimator that allows us to form bootstrap percentile-t intervals. More specifically, we can show that the following bootstrap variance estimator consistently estimates Vn∗ for any choice of the external random variable ηj : Jn −1 n1/2 b2n V ar∗ (η) X ∗ ˆ ¯j∗ − B ¯j+1 2 . Vn = 2 2 B ψ2 kn E ∗ (η 2 ) j=1
Our proposal is to use this estimator to construct a bootstrap studentized statistic, ∗ ∗ RV n n1/4 P] RV n − E ∗ P] q , Tn∗ ≡ ∗ ˆ Vn the bootstrap analogue of Tn . Theorem 3.2 Suppose Assumptions 1, 2 and 3 hold such that for any ε ≥ 2, E
4(2+ε) t
< ∞,
∗
and the block size bn satisfies (12) such that 1/2 < δ < 2/3. Let P] RV n be the pre-averaged realized volatility estimator based on a block length equal to bn and an external random variable ηj ∼ i.i.d. (E ∗ (ηj ) , V ar∗ (ηj )) such that for some ε ≥ 2 E ∗ |ηj |2+ε ≤ ∆ < ∞. Then sup |P ∗ (Tn∗ ≤ x) − P (Tn ≤ x)| →P 0 as n → ∞. x∈R
Theorem 3.2 justifies constructing bootstrap percentile-t intervals. In particular, a 100 (1 − α) % symmetric bootstrap percentile-t interval for integrated volatility is given by q q −1/4 ∗a ∗ ∗ −1/4 ˆ ICperc−t,1−α = P RVn − q1−α n Vn , P RVn + q1−α n Vˆn ,
(26)
or alternatively we can use p p ∗b ∗ ∗ ICperc−t,1−α = P RVn − q1−α n−1/4 Vn∗ , P RVn + q1−α n−1/4 Vn∗ , ∗ where q1−α is the (1 − α)-quantile of the bootstrap distribution of |Tn∗ |.
19
(27)
4
Possible extensions
Two important assumptions underlying our results so far are that the market microstructure noise is conditionally independent and that asset prices evolve continuously over the time interval [0, 1]. Our goal in this section is to discuss the applicability of the bootstrap when we relax these assumptions. Specifically, we discuss two possible extensions of our results. The first extension is to autocorrelated market microstructure noise whereas the second is to the presence of jumps in the log-price process.
4.1
Autocorrelated noise
Empirically, the conditional independence noise assumption is somewhat unrealistic for ultrahigh frequency data (see, among others, Hansen and Lunde (2006)). Hautsch and Podolskij (2013) relax this assumption to allow for q-dependent noise, at the cost of not allowing for time varying variances of the noise process and dependence between X and . As it turns out, the main consistency result for Jacod et al. (2009) pre-averaged estimators (cf. their Theorem 3.1) still holds under dependent noise. The key difference is that the limit of the required bias-correction term now depends on the higher order autocorrelations of the noise process instead of depending on αt = E (2t |X) (in particular, αt is replaced by the long run variance P ρ2 = ρ (0) + 2 qk=1 ρ (k), where ρ (k) = Cov (1 , 1+k ) , and q is the order of dependence of the noise process (i )i≥0 ). The main implication is that the bias correction for pre-averaged realized volatility must depend on an estimator of ρ2 . Hautsch and Podolskij (2013) discuss an estimator of ρ2 given by ρ2n
= ρn (0) + 2
q X
ρn (k) ,
k=1
where ρn (0) , . . . , ρn (q) are obtained by a simple recursion, ρn (q) = −γn (q + 1) , ρn (q − 1) = −γn (q) + 2ρn (q) , ρn (q − 2) = −γn (q − 1) + 2ρn (q − 1) − ρn (q) , n 1X where γn (k) = ri ri+k , k = 0, . . . , q + 1. n i=1 This implies the following consistent estimator of integrated volatility under a q-dependent autocorrelated noise process: P RVnq
1 = ψ2 kn
n−k n +1 X
Y¯i2
i=1
|
{z
}
RV -like estimator
20
−
ψ1 2 ρ θ2 ψ2 n | {z }
new bias correction term
.
(28)
To obtain a feasible asymptotic procedure, Hautsch and Podolskij (2013) also propose the following consistent estimator of V q ≡ limn→∞ V ar n1/4 P RVnq : n−2k n−kn +1 n +1 X Φ12 Φ22 ψ1 4Φ22 X 8ρ2n 4 2 q ˆ − Vn = Y i + 2√ Yi 4 3 4 3θψ2 i=0 ψ2 θ n ψ2 i=0 2 4 Φ12 ψ1 Φ22 ψ1 4ρn Φ11 −2 3 + . + 2 3 θ ψ2 ψ2 ψ24
(29)
We conjecture that the wild blocks of blocks bootstrap remains valid when we relax the conditional independence assumption on i provided we use it to approximate the distribution of P RVnq . Indeed, the conditional independence noise assumption used in our proof in Appendix B is not essential to guarantee the consistency of the wild blocks of blocks bootstrap variance since we do not use any prior knowledge on i apart from the kn -dependence of ¯i . If i is a q-dependent sequence, then ¯i becomes (kn + q)-dependent, and the result of Lemma 3.2 still holds, although higher order autocorrelations of appear in the limit. So long as E (t |X) = 0, ¯i admits −1/2 asymptotic normality at the usual rate kn , (see e.g. the proof of Lemma 1 of Hautsch and Podolskij (2013)), and if we let the block size bn grow faster than kn +q and set V ar∗ (ηj ) = 1/2, then the wild blocks of blocks bootstrap variance estimator will remain consistent for V q . Moreover, by using the wild blocks of blocks bootstrap, a stationarity condition on is not required, since by construction it is robust to the heterogeneity of square pre-averaged returns. These facts lead us to conjecture that the wild blocks of blocks bootstrap is valid when applied to the new bias adjusted pre-averaged volatility estimator under autocorrelated noise. Although we do not provide a detailed proof of this result, we explore the finite sample properties of the wild bootstrap under autocorrelation in i in the Monte Carlo simulations section. Our results show that this extension of the boostrap is robust to autocorrelated market microstructure noise.
4.2
Jumps
It is well known that the presence of jumps in the log-price process invalidates the usual estimators of integrated volatility. Two popular approaches to handle jumps are the use of multipower variation measures and thresholding techniques, first proposed by Barndorff-Nielsen and Shephard (2004, 2006) and Mancini (2009), respectively, under the no-noise assumption. When we allow for jumps and market microstructure noise at the same time, estimation of integrated volatility becomes even harder and the pre-averaged realized volatility estimator of Jacod et al. (2009) is no longer a consistent estimator. A natural approach in this context is to apply a method that handles jumps (such as thresholding) to the pre-averaged returns. This idea was recently explored by Jing et al. (2014), who prove the consistency of a pre-averaging21
threshold (PT) estimator in the presence of jumps and market microstructure noise. The PT estimator is a thresholded version of Jacod et al. (2009)’s estimator where we first pre-average the intraday returns to reduce the impact of market microstructure noise and then truncate the pre-averaged returns in order to remove the contribution of jumps to the quadratic variation process. Formally, this estimator is given by P RVnJ
n−k n n +1 X 1 ψ1 X 2 2 ˇ = Yi − r , ψ2 kn i=0 2nθ2 ψ2 i=1 i
(30)
where Yˇi is the truncated pre-averaged return given by Yˇi = Y¯i 1{|Y¯i |
(30). Interestingly, Jing et al. (2014) show that the asymptotic variance of the PT estimator is identical to that derived by Jacod et al. (2009) in the continuous case. This suggests that applying the wild blocks of blocks bootstrap method to the truncated pre-averaged returns is a valid procedure when the price process has jumps and is contaminated by market microstructure noise. In a recent paper, Hounyo (2016) shows that this is indeed the case. In particular, Hounyo (2016) extends the wild blocks of blocks bootstrap proposed in the current paper to multivariate measures of volatility under market microstructure noise and the possible presence of jumps. Our goal in this section is to describe how to apply the wild blocks of blocks bootstrap when used to estimate the distribution of the PT estimator of Jing et al. (2014). For j = 1, . . . , Jn , let bn 1 X ˇ Bj = Yˇ 2 bn i=1 i−1+(j−1)bn 2 denote the block average of the truncated squared pre-averaged returns Yˇi−1+(j−1)b for block j. n
The wild blocks of blocks bootstrap pseudo-observations (analogues of (13) and (14)) become: ∗2 ˇj+1 + Yˇ 2 ˇ Yˇi−1+(j−1)b =B (32) i−1+(j−1)bn − Bj+1 ηj , for 1 ≤ j ≤ Jn − 1 and for 1 ≤ i ≤ bn , n and ∗2 ˇj + Yˇ 2 ˇ Yˇi−1+(J =B i−1+(Jn −1)bn − BJn ηJn , for 1 ≤ i ≤ bn , n −1)bn
(33)
respectively, where ηi are i.i.d. random variables whose distribution is independent of the original sample as in Section 4. Given the bootstrap sample Yˇ0∗2 , . . . , YˇN∗2n −1 , we can compute 22
∗
¯j∗ (see (15) and (16)) by replacing Y¯i∗2 by Yˇi∗2 . Similarly, we can the analogues of P] RV n and B compute the analogues of the bootstrap confidence intervals given by (25), (26) and (27) by replacing P RVn by P RV J and Vˆn by a jump-robust consistent estimator of V . For instance, n
following Jing et al. (2014), one may simply use the truncated version of (7) where we replace Y¯i2 and Y¯i4 by Yˇi2 and Yˇi4 , respectively. For the theoretical justification and finite sample properties of this method, see Hounyo (2016).
5
Monte Carlo results
In this section, we compare the finite sample performance of the bootstrap with the feasible asymptotic theory for confidence intervals of integrated volatility in the case of i.i.d. and autocorrelated market microstructure noise. We consider two data generating processes in our simulations. First, following Zhang et al. (2005), we use the one-factor stochastic volatility (SV1F) model of Heston (1993) as our data-generating process, i.e. dXt = (µ − νt /2) dt + σt dBt , and dνt = κ (e α − νt ) dt + γ (νt )1/2 dWt , where νt = σt2 , and we assume Corr(B, W ) = ρ. The parameter values are all annualized. In particular, we let µ = 0.05/252, κ = 5/252, α e = 0.04/252, γ = 0.05/252, ρ = −0.5. For i = 1, . . . , n, we let the market microstructure noise be defined as i ∼ i.i.d.N (0, α). The size n of the noise is an important parameter. We follow Barndorff-Nielsen et al. (2008) and model qR 1 the noise magnitude as ξ 2 = α/ 0 σs4 ds. We fix ξ 2 equal to 0.0001, 0.001 and 0.01 and let qR 1 4 α = ξ2 σ ds. These values are motivated by the empirical study of Hansen and Lunde 0 s (2006), who investigate 30 stocks of the Dow Jones Industrial Average. We also consider a more realistic two-factor stochastic volatility (SV2F) model analyzed by Barndorff-Nielsen et al. (2008), where7 dXt = adt + σt dWt , σt = s-exp (β0 + β1 τ1t + β2 τ2t ) , dτ1t = α e1 τ1t dt + dB1t , dτ2t = α e2 τ2t dt + (1 + φτ2t ) dB2t , corr (dWt , dB1t ) = ϕ1 , corr (dWt , dB2t ) = ϕ2 . 7
The function s-exp is the usual exponential function with a linear growth function splined in at high values of its argument: s-exp(x) = exp(x) if x ≤ x0 and s-exp(x) = √ exp(x20 ) 2 if x > xo , with x0 = log(1.5). x0 −x0 +x
23
We follow Huang and Tauchen (2005) and set a = 0.03, β0 = −1.2, β1 = 0.04, β2 = 1.5, α e1 = −0.00137, α e2 = −1.386, φ = 0.25, ϕ1 = ϕ2 = −0.3. We initialize the two factors at the start of each interval by drawing the persistent factor from its unconditional distribution, τ10 ∼ −1 N 0, 2eα1 , and by starting the strongly mean-reverting factor at zero. We simulate data for the unit interval [0, 1] and normalize one second to be 1/23400, so that [0, 1] is thought to span 6.5 hours. The observed Y process is generated using an Euler scheme. We then construct the n1 -horizon returns ri ≡ Yi/n − Y(i−1)/n based on samples of size n. We use two different values of θ: θ = 1/3, as in Jacod et al. (2009), and θ = 1, as in Christensen, Kinnebrock and Podolskij (2010). The latter value corresponds to a conservative choice of kn . We also follow the literature and use the weight function g (x) = min (x, 1 − x) to compute the pre-averaged returns. In order to reduce finite sample biases associated with Riemann integrals, we follow Jacod et al. (2009) and Hautsch and Podolskij (2013) and use the finite sample adjustments version of the pre-averaged realized volatility estimator, ! −1 n−k n n +1 kn kn X X n 1 ψ ψ 1 1 P RVna = 1 − Y¯i2 − ri2 , kn kn kn 2 2 n − k + 2 2nθ ψ2 ψ2 kn i=0 2nθ ψ2 i=1 n where
ψ1kn
2 kn P i g kn − g i−1 and ψ2kn = = kn kn i=1
1 kn
kn P i=1
g
2
i kn
. Similarly, Vˆn as defined in
(7) replaces Φ11 , Φ12 and Φ22 by their Riemann approximations, ! ! kn kn X X 1 1 1 2 2 Φk11n = kn φk1n (j) − φk1n (0) , Φk12n = φk1n (j) φk2n (j) − φk1n (0) φk2n (0) , and 2 kn i=1 2 i=1 ! kn 2 1 kn 2 1 X kn Φ22 = 3 , φk2n (j) − φ2 (0) kn i=1 2 where φk1n
kX n −1
i−1 i i−j−1 i−j (j) = kn g −g g −g , and k k k k n n n n i=j+1 kX n −1 i i−j kn φ2 (j) = g g . kn kn i=j+1
Tables 1 through 5 give the actual rates of various 95% confidence intervals of integrated volatility computed over 10,000 replications. Results are presented for eight different samples sizes: n = 23400, 11700, 7800, 4680, 1560, 780, 390 and 195, corresponding to “1-second”, “2-second”, “3-second”, “5-second”, “15-second”, “30-second”, “1-minute” and “2-minute” frequencies. In our simulations, bootstrap intervals use 999 bootstrap replications for each of the 10,000 Monte Carlo replications. To generate the bootstrap data using the wild blocks of blocks
24
bootstrap we use a two point distribution ηj = vj2 with vj ∼ i.i.d. such that: ( 1/4 √ √ 5−1 −1+ 5 1 √ , with prob p = 2 2√ 2 5√ vj = , 1/4 −1− 5 1 √ , with prob 1 − p = 25+1 2 2 5 √ for which µ∗2 = 2 and µ∗4 = 5/2, implying that V ar∗ (ηj ) = 1/2. This choice of ηj is asymptotically valid when used to construct bootstrap percentile as well as percentile-t intervals. The choice of the bootstrap block size is critical. We follow Politis, Romano and Wolf (1999) and use the minimum volatility method to choose the bootstrap block. Details of the algorithm are given in Appendix A.
5.1
i.i.d. noise
In this subsection, we simulate results for the case of i.i.d. market microstructure noise. For the CLT-based intervals and the wild blocks of blocks bootstrap-based intervals, Tables 1 and 2 show that for the two models, all intervals tend to undercover. The degree of undercoverage is especially large for smaller values of n, when sampling is not too frequent. The SV2F model exhibits overall larger coverage distortions than the SV1F model, for all sample sizes. Results are sensitive to the value of the tuning parameter θ. When θ = 1/3, larger market microstructure effects induce larger coverage distortions. In particular, the coverage distortions are very important when ξ 2 = 0.01 in comparison to the case where market microstructure effects are moderate or negligible (ξ 2 = 0.001 and ξ 2 = 0.0001). This reflects the fact that for this value of θ, kn is not sufficiently large to allow pre-averaging to remove the market microstructure bias. The pre-averaged estimator is biased in finite samples and this explains the finite sample distortions. In contrast, for the conservative choice of kn , results are not very sensitive to the noise magnitude. The reason is that the larger is the block size over which the pre-averaging is done, the smaller is the impact of the noise. In all cases, the wild blocks of blocks bootstrap outperforms the existing first order asymptotic theory. As expected, the average chosen block size is larger for larger sample sizes, but our results show that it is not sensitive to the noise magnitude. This is because the noise magnitude is almost irrelevant for the intensity of the autocorrelation of the square pre-averaged returns (as confirmed by simulations not reported here). In Table 3 we compare the overlapping pre-averaging approach of Jacod et al. (2009) with the non-overlapping approach of Podolskij and Vetter (2009). We focus on the SV model with 2 factors (SV2F), for which distortions are larger, and show coverage rates as well as average confidence interval length for intervals based on these two approaches when θ = 1. For the overlapping case, we include results for the usual approach based on the CLT of Jacod et al. (2009) (called CLT1 in Tables 1 and 2) and the best performing bootstrap percentile-t interval
25
from Table 2 (cf. Boot3); the coverage rate results are a subset of the results in Table 2 but the interval length results are new. We compare these two methods with two other methods that rely on the non-overlapping estimator of Podolskij and Vetter (2009). One is a CLTbased method that relies on equation (9) in the main text (CLTP V in Table 3). The other is a bootstrap percentile-t method based on the wild bootstrap approach of Gon¸calves, Hounyo, and Meddahi (2014) that uses the optimal nonlattice distribution of Hounyo and Veliyev (2016) for generating the external random variable (BootP V in the table); as it turns out, this choice dominates the best performing two-point distribution considered by Gon¸calves, Hounyo, and Meddahi (2014). As the comparison between CLT1 and CLTP V shows, intervals based on the efficient overlapping preaveraging approach of Jacod et al. (2009) have better coverage rates than intervals based on the non-overlapping CLT based approach of Podolskij and Vetter (2009). They are also shorter on average than the CLTP V invervals, showing the importance of using more efficient estimators of integrated volatility. In both cases, the bootstrap helps reduce the coverage distortions associated with the use of critical values obtained from a mixed Gaussian distribution. However, Boot3, the wild blocks of blocks bootstrap interval applied to the overlapping estimator of Jacod et al. (2009), has smaller coverage distortions than BootP V , the wild bootstrap interval based on the non-overlapping estimator of Podolskij and Vetter (2009). Moreover, the latter is longer on average than the first due to the use of a less efficient estimator of integrated volatility. Therefore, although the wild blocks of blocks bootstrap involves the choice of an extra tuning parameter (the block size) when compared to the wild bootstrap, its finite sample properties are superior to those of the wild bootstrap of Gon¸calves, Hounyo, and Meddahi (2014). Given this evidence, our recommendation is to use the wild blocks of blocks bootstrap method proposed in the current paper.
5.2
Autocorrelated noise
In a second set of experiments, we look at the case where the market microstructure noise is autocorrelated. In particular, we follow Kalnina (2011) and let the market microstructure noise be generated as an M A(1) process (for a given frequency of the observations): α i = u i + λu i−1 , u i ∼ i.i.d.N 0, , (34) n n n n 1 + λ2 so that V ar () = α. Three different values of λ are considered, λ = −0.3, qR −0.5, and λ = −0.9. 1 4 We chose α as in the i.i.d. case discussed above, i.e. we let α = ξ 2 σ ds. We let θ = 1 0 s (conservative choice of kn ). Our aim here is to evaluate by Monte Carlo simulation the performance of the wild blocks of blocks bootstrap when applied to the statistic that relies on the new bias correction of Hautsch and Podoskij (2013), which is robust to noise autocorrelation. We consider five types of intervals 26
(two types of intervals based on the asymptotic normal distribution under the label CLT1 and CLT2 and three types of intervals based on the wild blocks of blocks bootstrap under the label Boot1, Boot2 and Boot3), computed at the 95% level. More specifically, for the asymptotic theory-based approach we consider the following intervals, q q q q −1/4 q −1/4 P RVn − 1.96n Vˆn , P RVn + 1.96n Vˆnq , and p p P RVnq − 1.96n−1/4 Vn∗ , P RVnq + 1.96n−1/4 Vn∗ .
(35) (36)
For the bootstrap, we consider P RVnq − n−1/4 p∗0.95 , P RVnq + n−1/4 p∗0.95 ,
(37)
q q q q ∗ −1/4 q ∗ −1/4 ˆ P RVn − q0.95 n Vn , P RVn + q0.95 n Vˆnq , and
(38)
∗ P RVnq − q0.95 n−1/4
p p ∗ Vn∗ , P RVnq + q0.95 n−1/4 Vn∗ .
(39)
Whereas (37) corresponds to bootstrap percentile intervals, (38) and (39) correspond to bootstrap percentile-t intervals. Note that for the bootstrap based-intervals, the bootstrap quantile ∗ p∗0.95 and q0.95 are computed exactly as in the i.i.d. noise case (it is based q on the absolute value ∗ ∗ ∗ ∗ of n1/4 P] RV n − E ∗ P] RV n and n1/4 P] RV n − E ∗ P] RV n / Vˆn∗ , respectively, whose
form is unaffected by the new bias adjustment used in P RVnq ). Tables 4 and 5 contains the results. We only report results for the SV2F model, since it is more empirically relevant and it exhibits overall larger coverage distortions than the SV1F model. Two sets of results are presented. First, we present results for intervals based on P RVn , the non-robust pre-averaged estimator discussed for the uncorrelated noise case (Table 4). Then, we present results for intervals based on P RVnq , the robust estimator based on the new bias correction of Hautsch and Podolskij (2013) (Table 5). The results show that intervals based on P RVn are more distorted when market microstructure effects are moderate or high (ξ 2 = 0.001 and ξ 2 = 0.01) and there is autocorrelation in i than otherwise. The main reason for the distortions is the fact that P RVn is not correctly centered and standardized under autocorrelation. For instance, when λ = −0.3, n = 195, and ξ 2 = 0.01 the CLT1-based interval has a coverage probability (from Table 4) equal to 72.98% under autocorrelated noise whereas its coverage rate is equal to 83.32% under uncorrelated noise (from Table 2). Although the difference is not very large for the smaller |λ| (intensity of autocorrelation), it gets much bigger for larger values of |λ|. For λ = −0.5 and −0.9, and (n = 195, and ξ 2 = 0.01) these rates equal 67.75% and 63.04%, respectively. Thus, the distortions increase with |λ|. Also for high effects of noise (ξ 2 = 0.01), the degree of undercoverage becomes especially large for larger values of n, when sampling is frequent. For instance when λ = −0.5, and (n = 195 and n = 23400) they 27
are equal to 67.75% and 28.97%, respectively. This confirms the invalidity of intervals based on P RVn under correlated noise. A similar pattern is observed for the CLT2-based intervals. We also see that these are close to the (percentile) Boot1-based intervals. However, if we rely on P RVnq as a point estimator of integrated volatility, the corresponding intervals (both asymptotic and bootstrap) are better centered and standardized and the distortions are smaller and closer to their values under the uncorrelated noise case. For instance, for n = 195, and ξ 2 = 0.01 the CLT1 intervals now have coverage rates equal to 84.33% and 84.56% when λ = −0.3 and λ = −0.5, respectively. A similar pattern is observed for larger sample sizes, although the rates are overall larger. For instance, for n = 23400 they are equal to 93.59% and 93.77%, respectively. When the wild blocks of blocks bootstrap method is used to compute critical values for the t-test based on P RVn and the error is MA(1), for high effects of noise (ξ 2 = 0.01), coverage rates are usually smaller than those obtained when the noise is uncorrelated (and therefore distortions are larger). As for the CLT-based intervals, the larger differences occur for the larger values of |λ|. For the smaller values of |λ|, the difference in coverage probability between the two types of errors is almost negligible. As for the CLT-based intervals, using the wild blocks of blocks bootstrap to compute critical values for the t-statistic based on P RVnq essentially eliminates the difference in coverage probabilities observed between the uncorrelated and the MA(1) errors. In summary, the results in Tables 4 and 5 show that under autocorrelated noise the statistic based on the bias correction of Hautsch and Podolskij (2013) works well and that the coverage rates of 95% nominal level intervals based on either the asymptotic mixed Gaussian distribution or the wild blocks of blocks bootstrap proposed in this paper are similar to those obtained under uncorrelated noise. In particular, the bootstrap (percentile-t) outperforms the asymptotic theory. Whereas, the results based on CLT2 and the (percentile) Boot1 intervals are close, but slightly different.
6
Empirical results
In this section, we implement the wild blocks of blocks bootstrap on high frequency data and compare it to the existing feasible asymptotic procedure of Jacod et al. (2009). The data consists of transaction log prices of General Electric (GE) shares carried out on the New York Stock Exchange (NYSE) in October 2011. We also consider transaction log prices of Microsoft (MSFT) in December 2010, taken from Thomson Reuter’s Tick History. GE represents highly liquid stocks with approximately 27 trade arrivals per minute. Conversely, MSFT is significantly less liquid with approximately 6 trade arrivals per minute. Our procedure for cleaning the data is exactly identical to that used by Barndorff-Nielsen et al. (2008) (for further details see Barndorff-Nielsen et al. (2009)). For each day, we consider data from the regular exchange 28
opening hours from time stamped between 9:30 a.m. until 4 p.m. We implement the pre-averaged realized volatility estimator of Jacod et al. (2009) on returns recorded every S transactions, where S is selected each day so that for GE and MSFT there are approximately 1493 and 82 observations a day, respectively. This means that on average, for GE and MSFT, these returns are recorded roughly every 15 seconds and 5 minutes, respectively. Table 6 in the Appendix provides the number of transactions per day, the sample size for the preaveraged returns, and the dependent-noise robust version of the pre-averaged realized volatility estimator using (28) (for q = 0, 1 and 2). We also report the optimal value of q (the number of non-vanishing covariances) using the decision rule proposed by Hautsch and Podolskij (2013). To implement the pre-averaged realized volatility estimator, we select the tuning parameter θ √ by following the conservative rule (θ = 1, implying that kn = n). To choose the block size bn , we follow Politis, Romano and Wolf (1999) and use the minimum volatility method (see Appendix A for details). As illustrated below, these stocks represent different empirical features and thus allow to gain valuable insights into the empirical performance of the wild blocks of blocks bootstrap method. For GE, Figure 1 in Appendix A shows daily 95% confidence intervals (CIs) for integrated volatility using both methods, the wild blocks of blocks bootstrap and the existing feasible asymptotic procedure of Jacod et al. (2009). In the latter case CIs are computed using (35) whereas for the bootstrap we use (39). The confidence intervals based on the bootstrap method are usually wider than the confidence intervals using the feasible asymptotic theory.8 This is especially true in periods with large volatility. To gain further insight on the behavior of our intervals for these periods, we implemented the test for jumps of Barndorff-Nielsen and Shephard (2006) using a moderate sample size (2-minute sampling intervals). It turns out that these days often correspond to days on which there is evidence for jumps (in particular for the 13, 17, 20 and 26 of October 2011). Since neither of the two types of intervals are valid in the presence of jumps, further analysis should be pursued for these particular days. In particular, we should rely on estimation methods that are robust to jumps such as the pre-averaged multipower variation method proposed by Podolskij and Vetter (2009) or the quantile estimation method of Christensen, Oomen, and Podolskij (2010). Similarly for MSFT (the less liquid stock) Figure 2 in Appendix A shows daily 95% confidence intervals for integrated volatility. The same patterns also emerges as for GE. The confidence intervals based on the wild blocks of blocks bootstrap method are usually wider than the confidence intervals using the feasible asymptotic theory. In contrast to GE, for MSFT we have 8
Nevertheless, as our Monte Carlo simulations showed, the latter typically have undercoverage problems whereas the bootstrap intervals have coverage rates closer to the desired level. Therefore if the goal is to control the coverage probability, shorter intervals are not necessarily better. The figures also show a lot of variability in the daily estimate of integrated volatility.
29
found no evidence of jumps at 5% significance level for days with large volatility. Importantly, the bootstrap based confidence sets of these days are larger than those based on the asymptotic theory, as suggested by the simulation study, which highlights the importance of using the bootstrap in these volatile days.
7
Concluding remarks
In this paper, we propose the bootstrap as a method of inference for integrated volatility in the context of the pre-averaged realized volatility estimator proposed by Jacod et al. (2009). We show that the “blocks of blocks” bootstrap method suggested by Politis and Romano (1992) is not valid when volatility is time-varying. This is due to the heterogeneity of the squared pre-averaged returns when volatility is stochastic. To simultaneously handle the dependence and heterogeneity of the squared pre-averaged returns, we propose a novel bootstrap procedure that combines the wild and the blocks of blocks bootstrap. We provide a set of conditions under which this method is asymptotically valid to first-order. Both percentile and percentile-t bootstrap intervals are considered. Our Monte Carlo simulations show that the wild blocks of blocks bootstrap improves the finite sample properties of the existing first-order asymptotic theory.
Appendix A: Simulation and empirical results Here we describe the Minimum Volatility Method algorithm of Politis, Romano and Wolf (1999, Chapter 9) for choosing the block size bn for a two-sided confidence interval. Algorithm: Choice of the bootstrap block size by minimizing confidence interval volatility
(i) For b = bsmall to b = bbig compute a bootstrap interval for IV at the desired confidence level, this resulting in endpoints ICb,low and ICb,up . (ii) For each b compute the volatility index V Ib as the standard deviation of the interval endpoints in a neighborhood of b. More specifically, for a smaller integer d, let V Ib equal to the standard deviation of the endpoints {ICb−d,low , . . . , ICb+d,low } plus the standard deviation of the endpoints {ICb−d,up , . . . , ICb+d,up }, i.e. v v u u d d X X u 1 u 1 2 t t ¯ ¯ up 2 , V Ib ≡ ICb+i,low − IC low + ICb+i,up − IC 2d + 1 i=−d 2d + 1 i=−d ¯ low = where IC
1 2d+1
Pd
i=−d
¯ up = ICb+i,low and IC 30
1 2d+1
Pd
i=−d
ICb+i,up .
(iii) Pick the value b∗ corresponding to the smallest volatility index and report {ICb∗ ,low , ICb∗ ,up } as the final confidence interval. To make the algorithm more computationally efficient, we have skipped a number of b values in regular fashion between bsmall and bbig . We have considered only the values of b such that b = pkn where p is a fixed integer. We employ bsmall = 2kn , bbig = min(θ N4n , 12kn ) and d = 2. Tables 1, 2, 3, 4 and 5 report the actual coverage rates for the feasible asymptotic theory approach and for our bootstrap methods using the optimal block size by minimizing confidence interval volatility. In Table 6 we provide some statistics of GE and MSFT shares in October 2011 and December 2010, respectively.
31
32
95.00 94.08 93.35 93.58 94.04 93.91 93.85 94.29
90.91 91.79 93.17 93.45 94.32 94.38 94.09 94.45
84.80 82.88 86.94 87.72 90.61 91.69 92.22 92.97
ξ 2 = 0.001 195 390 780 1560 4680 7800 11700 23400
ξ 2 = 0.01 195 390 780 1560 4680 7800 11700 23400 93.99 91.97 91.56 91.23 92.27 92.52 92.79 93.38
95.04 94.04 93.39 93.35 94.10 94.08 93.71 94.16
94.62 93.88 93.02 93.34 93.85 93.71 93.78 94.31
88.19 86.25 89.24 89.86 91.95 92.59 92.93 93.59
93.77 94.11 94.78 94.71 95.24 95.06 94.76 94.85
94.05 94.68 95.31 95.25 95.34 95.37 95.27 95.04
Boot2
96.17 94.09 93.40 92.70 93.48 93.20 93.48 93.77
96.61 95.78 94.97 94.72 94.95 94.68 94.27 94.68
96.63 95.61 94.74 94.82 94.80 94.35 94.29 94.60
Boot3
11.5 19.8 30.2 56.7 128.8 151.9 210.2 303.6
11.5 20.1 30.3 59.9 135.1 161.0 213.9 318.2
11.5 20.1 30.7 60.8 137.9 165.5 218.2 323.4
11.5 20.0 30.4 57.4 129.7 156.5 212.6 307.1
11.5 20.2 30.9 60.5 134.6 162.8 214.5 321.5
11.5 20.1 31.1 61.6 138.1 166.3 217.1 325.2
Avg. block size CLT2 Boot1
11.4 19.9 31.9 55.0 122.4 164.3 216.6 318.4
11.5 19.8 32.2 57.3 129.2 170.2 217.4 328.5
11.5 20.0 32.1 59.3 132.3 172.9 219.4 330.5
Boot2
11.3 19.3 29.2 50.7 113.3 144.6 202.2 281.4
11.3 19.5 29.8 55.0 122.6 154.1 203.1 299.5
11.4 19.6 30.3 55.9 126.9 158.0 207.9 304.6
Boot3
84.13 81.78 85.52 87.72 90.94 91.77 92.03 92.92
88.27 89.66 91.25 92.31 93.39 93.96 93.81 93.98
88.58 89.90 91.75 92.78 93.87 94.24 94.23 94.30
95.12 92.72 91.36 92.10 92.99 92.86 92.85 93.71
93.90 93.12 92.98 94.49 95.10 94.94 94.58 94.93
93.47 92.90 92.78 94.59 95.21 95.18 95.25 95.08
94.96 92.55 91.28 92.06 92.86 92.87 92.73 93.69
93.77 93.02 92.88 94.23 94.93 94.81 94.37 94.55
93.42 92.90 92.58 94.38 94.99 94.86 94.65 94.74
SV2F Coverage rate 95% CLT1 CLT2 Boot1
88.30 85.81 88.57 90.17 92.40 92.79 92.88 93.62
91.93 92.85 94.04 94.77 95.73 95.64 95.44 95.28
92.39 93.49 94.41 95.52 95.99 96.04 95.89 95.82
Boot2
96.90 94.85 93.38 93.59 93.88 93.63 93.63 94.06
96.44 95.54 95.18 95.87 95.90 95.84 95.27 95.28
96.16 95.26 94.83 96.03 96.16 96.09 95.69 95.72
Boot3
11.5 20.2 31.4 60.8 135.8 165.2 219.6 319.8
11.7 21.7 34.6 73.1 167.3 197.9 251.7 372.9
11.8 21.8 34.8 74.5 171.7 201.1 257.3 382.0
11.6 20.4 31.5 59.9 133.6 164.6 217.5 315.7
11.7 21.7 34.3 70.6 156.9 187.2 236.2 343.2
11.8 21.9 34.7 71.4 159.3 189.1 240.7 346.6
Avg. block size CLT2 Boot1
11.4 19.8 31.8 57.1 126.5 165.2 213.1 315.0
11.5 20.3 32.3 62.9 139.5 167.7 210.7 316.6
11.6 20.5 32.5 64.0 142.8 171.3 213.8 319.5
Boot2
11.3 19.6 30.3 54.4 121.2 152.0 203.3 288.0
11.5 20.4 32.1 62.9 141.0 169.1 214.0 316.6
11.6 20.5 32.2 64.3 143.6 172.6 217.3 318.8
Boot3
Notes: CLT1 and CLT2-intervals based on the Normal using equations (8) and (18), respectively; Boot1, Boot2 and Boot3-intervals based on the wild blocks of blocks bootstrap. Boot1 is for bootstrap percentile interval using equation (25), whereas Boot2 and Boot3 are for percentile-t intervals given in equations (26) and (27), respectively. 10,000 Monte Carlo trials with 999 bootstrap replications each.
93.94 92.13 91.69 91.28 92.28 92.59 92.65 93.61
94.80 93.98 93.04 93.20 93.91 93.66 93.97 94.10
ξ 2 = 0.0001 195 91.54 390 92.44 780 93.38 1560 93.83 4680 94.39 7800 94.60 11700 94.62 23400 94.67
n
SV1F Coverage rate 95% CLT1 CLT2 Boot1
Table 1. Coverage rates of nominal 95% intervals under i.i.d. noise with θ = 1/3
33
87.26 90.55 88.58 91.12 92.98 93.17 93.23 93.87
89.35 90.96 91.92 93.07 93.97 94.20 94.35 94.74
88.92 90.23 92.11 92.96 93.62 93.90 94.26 94.63
ξ 2 = 0.001 195 390 780 1560 4680 7800 11700 23400
ξ 2 = 0.01 195 390 780 1560 4680 7800 11700 23400 86.79 89.98 88.76 91.30 92.86 93.29 93.45 93.60
86.45 90.02 88.27 90.62 92.58 92.97 93.46 93.68
86.37 89.87 88.16 90.62 92.42 92.96 93.22 93.68
94.29 94.60 95.65 96.45 96.30 95.97 95.92 96.10
94.92 95.04 95.61 96.65 96.29 96.28 96.05 96.12
94.87 94.96 95.76 96.52 96.46 96.35 96.15 96.10
Boot2
93.83 94.97 94.20 95.63 95.22 95.19 95.22 95.32
93.40 94.52 93.87 95.05 95.35 95.13 95.30 95.07
93.42 94.50 94.08 95.04 95.44 95.19 95.10 95.21
Boot3
35.9 66.8 138.1 291.6 615.6 805.1 980.2 1523.4
36.0 67.0 138.3 292.8 615.8 807.0 979.3 1519.1
36.0 66.9 138.5 293.1 617.4 806.4 976.5 1518.9
35.6 65.1 130.5 274.3 565.7 747.7 891.6 1343.9
35.6 65.4 131.1 274.4 565.1 753.5 894.2 1335.5
35.6 65.4 131.4 275.1 567.7 751.9 897.4 1336.4
Avg. block size CLT2 Boot1
33.2 55.2 101.1 226.9 433.7 570.5 669.4 987.9
33.3 55.2 102.6 231.7 443.7 575.0 672.5 980.4
33.3 55.4 102.8 233.5 444.8 577.5 677.1 987.1
Boot2
33.2 60.4 117.7 251.4 491.4 686.4 816.2 1220.5
33.4 60.8 118.7 254.2 500.5 687.6 821.0 1220.1
33.4 60.7 119.0 254.2 501.4 689.7 823.1 1220.7
Boot3
83.32 85.96 88.06 89.69 91.62 92.40 93.01 93.75
83.32 86.18 88.03 89.53 91.67 92.46 92.63 93.55
83.45 86.07 87.99 89.57 91.68 92.42 92.93 93.62
85.17 88.66 89.01 92.68 94.72 94.93 94.94 95.77
84.73 88.42 89.26 92.55 94.70 94.93 95.08 95.56
84.74 88.30 89.27 92.48 94.88 95.15 95.18 95.70
84.28 87.97 88.73 92.13 94.07 94.23 94.48 94.91
83.66 87.90 88.61 92.07 93.96 94.40 94.16 94.62
83.80 87.81 88.67 91.91 94.08 94.21 94.32 94.89
SV2F Coverage rate 95% CLT1 CLT2 Boot1
90.51 91.50 93.28 94.62 95.38 95.69 95.85 95.97
91.02 91.72 93.69 95.00 95.64 95.80 95.87 96.16
91.05 91.89 93.75 95.09 95.55 95.76 95.94 96.24
Boot2
91.97 93.19 94.03 95.99 96.12 96.30 96.11 96.16
92.10 93.18 94.47 96.22 96.53 96.40 96.37 96.31
91.87 93.37 94.53 96.32 96.57 96.43 96.26 96.30
Boot3
36.3 68.4 140.3 295.8 622.6 818.6 995.7 1533.6
36.5 68.6 140.9 297.3 625.9 820.3 992.5 1533.8
36.4 68.7 141.1 296.8 625.1 820.0 993.4 1530.0
35.9 66.5 133.9 283.9 580.7 745.0 877.7 1286.4
36.1 67.0 134.8 286.4 586.4 747.2 873.4 1279.0
36.1 66.9 135.1 286.8 585.5 745.0 870.7 1277.3
Avg. block size CLT2 Boot1
34.5 60.1 117.8 268.2 534.5 650.0 770.5 1124.1
34.7 61.3 121.4 276.5 545.6 666.7 784.0 1143.2
34.8 61.3 121.6 277.6 549.1 665.5 782.4 1137.8
Boot2
Notes: CLT1 and CLT2-intervals based on the Normal using equations (8) and (18), respectively; Boot1, Boot2 and Boot3-intervals based on the wild blocks of blocks bootstrap. Boot1 is for bootstrap percentile interval using equation (25), whereas Boot2 and Boot3 are for percentile-t intervals given in equations (26) and (27), respectively. 10,000 Monte Carlo trials with 999 bootstrap replications each.
87.45 90.67 89.04 91.66 93.13 93.44 93.93 93.83
87.17 90.39 88.66 91.07 92.94 93.10 93.46 93.86
ξ 2 = 0.0001 195 89.41 390 91.01 780 91.96 1560 92.97 4680 93.95 7800 94.38 11700 94.52 23400 94.64
n
SV1F Coverage rate 95% CLT1 CLT2 Boot1
Table 2. Coverage rates of nominal 95% intervals under i.i.d. noise with θ = 1
34.4 62.1 125.3 267.7 522.0 666.3 794.7 1158.1
34.8 63.0 127.2 275.4 535.5 674.8 795.7 1161.2
34.8 63.1 127.7 276.6 537.4 673.4 799.2 1161.4
Boot3
34
92.10 93.18 94.47 96.22 96.53 96.40 96.37 96.31
83.32 86.18 88.03 89.53 91.67 92.46 92.63 93.55
83.32 85.96 88.06 89.69 91.62 92.40 93.01 93.75
ξ 2 = 0.001 195 390 780 1560 4680 7800 11700 23400
ξ 2 = 0.01 195 390 780 1560 4680 7800 11700 23400 75.09 81.81 83.60 88.17 88.81 90.21 91.34 91.67
73.67 80.53 83.08 86.94 88.63 90.20 90.95 91.25
73.64 80.37 82.82 86.67 88.38 90.03 90.86 91.26
87.08 90.51 90.54 92.78 92.62 93.29 93.74 93.74
86.65 90.12 90.66 92.22 92.50 93.40 93.52 93.42
86.82 89.92 90.68 92.18 92.48 93.29 93.55 93.25
BootP V
1.396 1.188 1.019 0.868 0.669 0.587 0.533 0.449
1.333 1.133 0.969 0.826 0.635 0.558 0.505 0.425
1.327 1.128 0.964 0.822 0.631 0.555 0.503 0.423
3.163 2.310 2.037 1.811 1.188 0.935 0.790 0.628
3.178 2.279 2.091 1.832 1.172 0.922 0.770 0.607
3.161 2.298 2.077 1.831 1.178 0.921 0.771 0.604
Avg. CI length CLT1 Boot3
1.723 1.564 1.337 1.188 0.907 0.825 0.752 0.633
1.572 1.432 1.227 1.086 0.834 0.762 0.695 0.584
1.559 1.420 1.218 1.075 0.827 0.755 0.690 0.580
CLTP V
4.623 3.551 2.601 2.049 1.303 1.136 0.993 0.760
4.562 3.468 2.551 1.951 1.247 1.079 0.947 0.719
4.577 3.459 2.549 1.933 1.241 1.074 0.941 0.716
BootP V
Notes: CLT1-intervals based on the Normal using equations (8); Boot3-intervals based on the wild blocks of blocks bootstrap using percentilet method given in equations (27). CLTP V -intervals based on the Normal using equations (9), whereas BootP V is for percentile-t intervals based on Gon¸calves et al. (2014) wild bootstrap approach using non-overlapping pre-averaged returns with external random variable given by the optimal nonlattice distribution of Hounyo and Veliyev (2016) (cf. Proposition 3.1), respectively. 10,000 Monte Carlo trials with 999 bootstrap replications each.
91.97 93.19 94.03 95.99 96.12 96.30 96.11 96.16
91.87 93.37 94.53 96.32 96.57 96.43 96.26 96.30
Coverage rate 95% CLT1 Boot3 CLTP V
ξ 2 = 0.0001 195 83.45 390 86.07 780 87.99 1560 89.57 4680 91.68 7800 92.42 11700 92.93 23400 93.62
n
Table 3. Summary results for Gon¸calves et al. (2014) wild bootstrap approach and the wild block of blocks bootstrap method under i.i.d. noise with θ = 1
35
72.98 73.11 72.86 72.18 67.48 63.85 60.28 52.45
ξ 2 = 0.01 195 390 780 1560 4680 7800 11700 23400
i=1
n P
76.32 78.90 77.64 80.03 78.15 74.71 71.11 63.98
83.10 87.03 87.87 90.99 92.94 93.01 93.23 93.27
83.33 82.29 83.02 82.87 77.94 73.86 70.08 61.37
90.36 91.09 92.81 94.22 94.42 94.70 94.44 94.73
90.96 91.79 93.66 95.00 95.47 95.67 95.78 96.09
Boot2
87.04 87.39 87.29 89.46 85.28 81.50 77.79 71.08
91.59 92.81 93.78 95.85 95.70 95.66 95.09 95.14
91.91 93.34 94.46 96.30 96.49 96.33 96.18 96.27
Boot3
67.75 65.89 64.07 61.43 51.82 45.94 40.16 28.97
81.89 84.34 86.13 87.45 89.30 89.93 90.13 90.45
83.29 85.88 87.83 89.46 91.44 92.18 92.69 93.40
73.53 74.97 72.33 74.85 69.41 63.92 58.24 48.05
83.67 87.17 87.97 91.27 93.31 93.30 93.46 93.77
84.64 88.13 89.08 92.45 94.77 94.94 95.07 95.49
λ = −0.5 CLT1 CLT2
72.52 73.71 71.13 73.28 68.09 61.25 54.76 43.60
82.91 86.50 87.39 90.63 92.42 92.40 92.72 92.62
83.75 87.66 88.48 91.51 94.11 94.08 94.24 94.70
Boot1
79.85 76.94 76.49 74.85 64.85 57.49 50.89 38.58
89.93 90.64 92.28 93.67 93.86 93.94 93.83 93.96
90.99 91.66 93.65 94.92 95.41 95.71 95.73 96.04
Boot2
i=1
n P
ri2
84.59 83.57 83.10 84.58 77.46 70.07 64.43 52.35
91.30 92.52 93.57 95.52 95.40 95.11 94.67 94.56
91.86 93.36 94.41 96.26 96.46 96.31 96.13 96.16
Boot3
63.04 60.59 57.45 52.59 40.04 32.84 26.34 15.55
81.71 83.92 85.64 86.89 88.59 89.35 89.44 89.42
83.34 85.79 87.83 89.43 91.39 92.14 92.62 93.35
70.37 71.29 67.25 68.41 60.58 52.84 45.75 32.86
83.55 86.86 87.87 90.78 93.02 92.63 93.09 93.10
84.64 88.07 89.10 92.42 94.77 94.80 95.03 95.45
λ = −0.9 CLT1 CLT2
69.19 69.73 65.98 66.44 58.69 50.79 42.75 29.88
82.74 86.12 87.17 90.12 92.16 91.96 92.29 91.97
83.79 87.50 88.42 91.53 94.06 94.10 94.39 94.57
Boot1
76.69 72.02 71.45 68.29 54.39 44.97 36.98 23.83
89.73 90.33 92.00 93.43 93.56 93.33 93.29 93.20
90.92 91.60 93.56 94.91 95.40 95.61 95.67 95.91
Boot2
82.82 80.75 79.97 80.55 70.15 60.77 52.86 38.94
91.41 92.24 93.49 95.30 95.36 94.87 94.15 93.94
91.98 93.39 94.36 96.25 96.42 96.31 96.12 96.19
Boot3
ri2 is a consistent estimator of the bias term in P RVn under uncorrelated noise. CLT1 and CLT2-intervals based
77.21 79.89 78.47 81.18 79.40 76.32 74.01 67.76
83.99 87.72 88.40 91.83 93.85 93.85 93.96 94.36
83.72 87.73 88.52 91.72 94.14 94.06 94.37 94.80
Boot1
ψ1 2nθ 2 ψ2
on the Normal using equations (8) and (18), respectively; Boot1, Boot2 and Boot3-intervals based on the wild blocks of blocks bootstrap. Boot1 is for bootstrap percentile interval using equation (25), whereas Boot2 and Boot3 are for percentile-t intervals given in equations (26) and (27), respectively. 10,000 Monte Carlo trials with 999 bootstrap replications each.
ψ1 2nθ2 ψ2
82.24 84.90 86.71 88.18 90.17 90.67 90.97 91.79
ξ 2 = 0.001 195 390 780 1560 4680 7800 11700 23400
ˆ = Notes: Bias
83.37 85.88 87.87 89.51 91.51 92.24 92.75 93.44
ξ 2 = 0.0001 195 390 780 1560 4680 7800 11700 23400
84.76 88.17 89.15 92.49 94.79 94.97 95.14 95.50
λ = −0.3 CLT1 CLT2
n
ˆ = Table 4. Coverage rates of nominal 95% intervals under correlated noise based on Bias
36
84.71 87.15 88.48 89.90 91.86 92.48 92.91 93.59
84.71 87.16 88.69 89.85 91.81 92.67 92.84 93.72
84.33 86.53 88.23 89.49 91.86 92.45 92.77 93.59
ξ 2 = 0.0001 195 390 780 1560 4680 7800 11700 23400
ξ 2 = 0.001 195 390 780 1560 4680 7800 11700 23400
ξ 2 = 0.01 195 390 780 1560 4680 7800 11700 23400 84.02 88.08 88.63 91.90 93.99 94.36 94.27 94.82
84.62 88.57 89.10 92.28 94.22 94.35 94.37 94.80
84.48 88.42 88.94 92.05 94.27 94.21 94.59 95.01
Boot1
90.90 91.83 93.71 94.64 95.74 95.65 95.94 96.08
91.78 92.51 94.14 95.23 95.67 95.98 95.91 96.33
91.77 92.54 94.07 95.31 95.74 95.91 95.89 96.28
Boot2
91.86 93.17 94.34 96.15 96.37 96.25 96.31 96.19
92.41 93.62 94.68 96.39 96.50 96.52 96.27 96.27
92.48 93.81 94.66 96.57 96.57 96.46 96.36 96.36
Boot3
84.56 86.75 88.28 89.51 91.94 92.58 93.00 93.77
85.00 87.22 88.63 89.88 91.95 92.64 93.00 93.73
84.70 87.12 88.40 89.95 91.90 92.52 92.93 93.62
84.85 88.86 89.37 92.52 94.97 95.06 95.23 95.63
85.59 89.16 89.75 92.84 95.03 95.12 95.28 95.72
85.39 88.99 89.53 92.74 94.91 95.07 95.29 95.67
λ = −0.5 CLT1 CLT2
83.88 88.15 88.77 91.82 94.19 94.26 94.36 94.70
84.63 88.58 89.16 92.26 94.21 94.26 94.38 94.90
84.41 88.46 88.95 91.92 94.32 94.26 94.55 94.99
Boot1
91.13 92.17 93.78 94.67 95.73 95.67 96.00 96.12
91.83 92.57 94.22 95.37 95.70 95.86 96.03 96.33
91.76 92.49 94.11 95.33 95.60 95.92 95.95 96.31
Boot2
ψ1 ρ2 θ 2 ψ2 n
91.89 93.09 94.55 95.97 96.65 96.42 96.57 96.42
92.46 93.68 94.63 96.45 96.53 96.52 96.31 96.48
92.48 93.79 94.61 96.55 96.59 96.47 96.32 96.33
Boot3
84.39 87.03 88.40 89.85 92.07 92.51 93.30 93.54
85.09 87.22 88.66 90.00 91.95 92.52 93.00 93.75
84.73 87.16 88.48 89.99 91.92 92.45 92.93 93.65
84.62 88.67 89.44 92.53 94.91 95.07 95.43 95.69
85.60 89.12 89.72 92.74 95.04 95.14 95.25 95.63
85.42 88.91 89.52 92.72 94.97 95.12 95.27 95.63
λ = −0.9 CLT1 CLT2
83.66 88.05 89.15 91.98 94.17 94.32 94.63 94.81
84.61 88.57 89.21 91.95 94.36 94.29 94.66 94.93
84.48 88.39 89.00 91.88 94.36 94.30 94.65 94.89
Boot1
91.33 92.35 93.69 94.82 95.77 95.96 96.28 96.32
91.83 92.36 94.28 95.43 95.72 95.98 96.05 96.23
91.80 92.34 94.12 95.38 95.69 95.87 96.00 96.20
Boot2
91.89 93.37 94.48 96.29 96.70 96.68 96.37 96.43
92.49 93.68 94.67 96.47 96.58 96.53 96.26 96.34
92.34 93.82 94.68 96.55 96.58 96.53 96.30 96.32
Boot3
1 ˆ = ψ Notes: Bias ρ2 is a consistent estimator of the bias term in P RVnq under autocorrelated noise. CLT1 and CLT2-intervals based on θ2 ψ2 n the Normal using equations (35) and (36), respectively; Boot1, Boot2 and Boot3-intervals based on the wild blocks of blocks bootstrap. Boot1 is for bootstrap percentile interval using equation (37), whereas Boot2 and Boot3 are for percentile-t intervals given in equations (38) and (39), respectively. 10,000 Monte Carlo trials with 999 bootstrap replications each.
84.85 88.79 89.15 92.41 94.66 94.94 95.05 95.68
85.64 89.13 89.77 92.79 95.00 94.98 95.15 95.68
85.36 89.03 89.55 92.66 94.92 95.07 95.26 95.66
λ = −0.3 CLT1 CLT2
n
ˆ = Table 5. Coverage rates of nominal 95% intervals under correlated noise noise based on Bias
Table 6. Summary statistics Days Trans n S
P RVnq · 103 q=0 q=1 q=2
q∗
P RVnq · 103 q = q∗
GE 3 Oct 4 Oct 5 Oct 6 Oct 7 Oct 10 Oct 11 Oct 12 Oct 13 Oct 14 Oct 17 Oct 18 Oct 19 Oct 20 Oct 21 Oct 24 Oct 25 Oct 26 Oct 27 Oct 28 Oct 31 Oct
12613 13782 10628 9991 9785 10660 8588 11160 8649 9261 8530 8751 9023 9251 12513 11642 10919 9249 14598 9405 8871
1402 1532 1519 1428 1398 1523 1432 1595 1442 1544 1422 1459 1504 1542 1565 1456 1365 1542 1622 1568 1500
9 9 7 7 7 7 6 7 6 6 6 6 6 6 8 8 8 6 9 6 6
0.903 1.705 0.721 0.688 0.686 0.720 1.498 0.727 1.499 1.556 1.498 1.507 1.545 1.556 0.833 0.791 0.775 1.556 1.776 1.557 1.559
1.113 1.734 0.722 0.742 0.687 0.830 1.499 0.727 1.499 1.556 1.499 1.582 1.644 1.557 0.941 0.839 0.776 1.557 1.778 1.633 1.667
1.121 1.735 0.723 0.858 0.688 0.951 1.499 0.729 1.499 1.556 1.499 1.584 1.645 1.557 0.942 0.840 0.776 1.557 1.779 1.699 1.669
1 1 0 2 0 2 0 0 0 0 0 1 1 0 1 1 0 0 0 4 1
1.113 1.734 0.721 0.858 0.686 0.951 1.498 0.727 1.499 1.556 1.498 1.582 1.644 1.556 0.941 0.839 0.775 1.556 1.776 1.746 1.667
MSFT 1 Dec 2 Dec 3 Dec 6 Dec 7 Dec 8 Dec 9 Dec 10 Dec 13 Dec 14 Dec 15 Dec 16 Dec 17 Dec 20 Dec 21 Dec 22 Dec 23 Dec 27 Dec 28 Dec 29 Dec 30 Dec 31 Dec
2177 1520 2530 1717 1847 1473 1851 1375 1469 2558 2304 1872 3385 3827 4105 3742 3716 2010 1676 1555 1572 1887
78 77 80 79 81 78 78 77 78 82 80 79 89 93 95 92 93 80 79 78 79 79
28 20 32 22 23 19 24 18 19 32 29 24 39 42 44 41 40 26 22 20 20 24
0.112 0.079 0.077 0.072 0.063 0.061 0.071 0.084 0.083 0.074 0.101 0.069 0.096 0.174 0.483 0.355 0.318 0.071 0.096 0.079 0.053 0.069
0.124 0.087 0.088 0.097 0.087 0.083 0.083 0.101 0.100 0.090 0.120 0.084 0.114 0.351 0.554 0.400 0.357 0.098 0.120 0.087 0.079 0.080
0.133 0.088 0.088 0.098 0.089 0.084 0.083 0.112 0.106 0.091 0.121 0.088 0.115 0.366 0.556 0.401 0.361 0.113 0.124 0.088 0.085 0.081
0 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0
0.112 0.079 0.077 0.097 0.087 0.083 0.071 0.101 0.083 0.074 0.101 0.069 0.096 0.351 0.483 0.355 0.318 0.098 0.096 0.079 0.079 0.069
“Trans” denotes the number of transactions, n is the sample size used to calculate the pre-averaged realized volatility, we have sampled every Sth transaction price, so the period over which returns are calculated for GE and MSFT are roughly 15 seconds and 5 minutes, respectively. P RVnq is the dependent-noise robust version of the pre-averaged realized volatility estimator, q is the order of autocorrelation, q ∗ is the optimal value of q selected using the decision rule proposed by Hautsch and Podolskij (2013).
37
Figure 1: 95% Confidence Intervals (CI’s) for the daily IV, for each regular exchange opening days for GE in October 2011, calculated using the asymptotic theory of Jacod et al. (2009) based on (35) (CI’s with bars), and the wild blocks of blocks bootstrap method based on (39) (CI’s with lines). The pre-averaging realized volatility estimator is the middle of all CI’s by construction. Days on the x-axis.
Figure 2: 95% Confidence Intervals (CI’s) for the daily IV, for each regular exchange opening days for MSFT in December 2010, calculated using the asymptotic theory of Jacod et al. (2009) based on (35) (CI’s with bars), and the wild blocks of blocks bootstrap method based on (39) (CI’s with lines). The pre-averaging realized volatility estimator is the middle of all CI’s by construction. Days on the x-axis.
Appendix B: Proofs As in Jacod et al. (2009), we assume throughout this Appendix that the processes a, σ and X are bounded processes satisfying (1) with a and σ adapted c`adl`ag processes. As Jacod et al. (2009) explain, this assumption simplifies the mathematical derivations without loss of generality (by a standard localization procedure detailed in Jacod (2008)). Formally, we derive our results under the following assumption. 38
Assumption 4. X satisfies equation (1) with a and σ adapted c`adl`ag processes such that a, σ, and X are bounded processes (implying that α is also bounded).
Notation In the following, K denotes a constant which changes from line to line. Moreover, we follow Jacod et al. (2009) and use the following additional notation. We let kn kn X X j j ¯ X i+j − X i+j−1 , ¯i = g i+j − i+j−1 , Xi = g n n n n kn kn j=1 j=1 ¯ i + ¯i . In addition, we let and note that Y¯i = X 2 Z i+j kn X n j ci = g σt2 dt; i+j−1 k n j=1 n 2 kX n −1 j+1 j 2 Ai = E ¯i |X = g −g α(i+j)/n ; and kn kn j=0 Yei = Y¯i2 − Ai − ci . Following Jacod et al. (2009), we also introduce the following random variables. For j = 1, . . . , Jn , we let 1 √ ζ(p)(j−1)(p+1)kn , with ζ (p)j = η (p)j = θψ2 n
j+(p+1)kn −1
X
Yei ,
i=j
where p ≥ 1 is a fixed integer; η (p)j is the normalized sum of squared pre-averaged returns Y˜i n , over a block of size bn = (p + 1) kn . Note that η (p)j is measurable with respect to Fj(p+1)k n 0 the sigma algebra generated by all Fj(p+1)kn /n -measurable random variables plus all variables Ys , with s < j (p + 1) kn . Finally, we let β(p)i = sups,t∈[ i , i+(p+1)kn ] (|as − at | + |σs − σt | + |αs − αt |) , n
(40)
n
and 2 2 1 1 σt αt 1 αt 4 Φ22 + Ψ22 θσt + 2 Φ12 + Ψ12 + Φ11 + Ψ11 . p+1 p+1 θ p+1 θ3 (41) Our bootstrap estimators depend crucially on 4 γ (p)t = 2 ψ2 2
bn 1 X 1 2 ¯ Bj ≡ Y¯i−1+(j−1)b = n bn i=1 bn
jbn −1
X
Y¯i2 , for j = 1, . . . , Jn ,
i=(j−1)bn
where Jn = Nn /bn is the number of non-overlapping blocks of size bn out of Nn = n − kn + 2 observations on pre-averaged returns. Our first result is instrumental in proving our bootstrap results.
39
Lemma B.1 Suppose Assumptions 2 and 4 hold. Then, for all integer p ≥ 1, and each q > 0, we have that P Jn q a1) √1n E β (p) j=1 (j−1)(p+1)kn → 0. a2)
√1 n
PJ n
β (p)q(j−1)(p+1)kn →P 0. P Jn q n a3) √1n E E β (p) |F → 0. j=1 (j−1)(p+1)kn (j−1)(p+1)kn
a4)
√1 n
PJ n
a5)
√1 n
PJ n
a6)
√1 n
PJ n
a7)
√1 n
PJ n
j=1
q n P E β (p) |F j=1 (j−1)(p+1)kn → 0. (j−1)(p+1)kn q P n E β (2p + 1) |F j=1 (j−1)(p+1)kn → 0. (j−1)(p+1)kn r 2 n E β (p)(j−1)(p+1)kn |F(j−1)(p+1)kn →P 0. j=1 r 2 n E β (2p + 1)(j−1)(p+1)kn |F(j−1)(p+1)kn →P 0. j=1
Proof of Lemma B.1. Part a1). Given the definition of β (p)(j−1)(p+1)kn we can write β (p)(j−1)(p+1)kn ≤ sups,t∈[ (j−1)(p+1)kn , (j−1)(p+1)kn +(p+1)kn ] (|as − at |) n
n
+ sups,t∈[ (j−1)(p+1)kn , (j−1)(p+1)kn +(p+1)kn ] (|σs − σt |) n
n
+ sups,t∈[ (j−1)(p+1)kn , (j−1)(p+1)kn +(p+1)kn ] (|αs − αt |) n
n
≡ Γ (a, p)(j−1)(p+1)kn + Γ (σ, p)(j−1)(p+1)kn + Γ (α, p)(j−1)(p+1)kn . Given that Γ (a, p)(j−1)(p+1)kn , Γ (σ, p)(j−1)(p+1)kn and Γ (α, p)(j−1)(p+1)kn are strictly positive, for any q > 0, using the c-r inequality, we can write q q q q β (p)(j−1)(p+1)kn ≤ K Γ (σ, p)(j−1)(p+1)kn + Γ (a, p)(j−1)(p+1)kn + Γ (α, p)(j−1)(p+1)kn . It follows that n−1/2 E
Jn X
! β (p)q(j−1)(p+1)kn
≤ Kn−1/2 E
j=1
Jn X
! Γ (σ, p)q(j−1)(p+1)kn
j=1
+Kn−1/2 E
Jn X
! Γ (a, p)q(j−1)(p+1)kn
j=1
+Kn−1/2 E
Jn X
! Γ (α, p)q(j−1)(p+1)kn
= o (1) ,
j=1
where we use Lemma 5.3 of Jacod, Podolskij and Vetter (2010) to show that each of the terms above are o (1) (given that a, σ and α are c`adl`ag bounded processes).
40
Proof of Lemma B.1. Part a2). Note that given the result of part a1) of Lemma B.1, P 2 Jn q 1 it is sufficient to show that n E → 0. By the c-r inequality, j=1 β (p)(j−1)(p+1)kn 1 E n
Jn X j=1
!2 β
(p)q(j−1)(p+1)kn
Jn ≤ E n
Jn X
! β
(p)2q (j−1)(p+1)kn
j=1
1 ≤ K√ E n
Jn X
! β
(p)2q (j−1)(p+1)kn
,
j=1
√ which is o (1) by part a1) of Lemma B.1 and given that Jn = O ( n) . Proof of Lemma B.1. Part a3). Given the law of iterated expectations, the result follows directly from part a1) of Lemma B.1. Proof of Lemma B.1. Part a4). The proof follows similarly as in part a2) of q n Lemma B.1, where we now consider the variable E β (p)(j−1)(p+1)kn |F(j−1)(p+1)kn in place of β (p)q(j−1)(p+1)kn . Proof of Lemma B.1. Part a5). Given the definition of β(p)i , for any p ≥ 1, such that bn = (p + 1) kn we can write Jn [X Jn 2 ] 1 X 1 q n n √ E β (2p + 1)(j−1)bn |F(j−1)bn = √ E β (2p + 1)q2(j−1)bn |F2(j−1)b n n j=1 n j=1
[ J2n ] 1 X n E β (2p + 1)q(2(j−1)+1)bn |F(2(j−1)+1)b , +√ n n j=1 which is oP (1) given part a4) of Lemma B.1. Proof of Lemma B.1. Part a6). Here, the proof contains two steps. Step 1. We show r P Jn n show that √1n E E β (p)2(j−1)(p+1)kn |F(j−1)(p+1)k → 0. Step 2. We show show that j=1 n r PJn 2 1 n V ar E β (p) |F → 0. Note that using the first expression in j=1 (j−1)(p+1)kn (j−1)(p+1)kn n equation (5.47) of Jacod et al. (2009), the result of step 1 follows directly. Given this result, to r 2 P J 2 n n show step 2, it is sufficient to show that n1 E E β (p)(j−1)(p+1)kn |F(j−1)(p+1)k → 0. j=1 n We have that ! Jn r Jn 2 J X 1 X n 2 2 n n E β (p)(j−1)(p+1)kn |F(j−1)(p+1)kn ≤ E E β (p)(j−1)(p+1)kn |F(j−1)(p+1)kn n j=1 n j=1 ! Jn Jn X Jn X 1 = E β (p)2(j−1)(p+1)kn ≤ K √ E β (p)2(j−1)(p+1)kn , n j=1 n j=1 √ which is o (1) given equation (5.47) of Jacod et al. (2009) and the fact that Jn = O ( n) under our assumptions. Proof of Lemma B.1. Part a7). The proof follows similarly as part a5) and therefore we omit the details. Our next result is crucial to the proofs of Lemmas 3.1 and 3.2.
41
Lemma B.2 Under Assumptions 1, 2, and 4, if bn = (p + 1) kn where p ≥ 1 is fixed, then 2 √ 2X Z 1 J ψ1 nbn n ¯ 2 P 2 σs + 2 αs ds. B → Vp + θ (p + 1) kn2 ψ22 j=1 j θ ψ2 0 ¯j , we have that Proof of Lemma B.2. Given the definition of B ¯j = 1 B bn
jbn −1
X i=(j−1)bn
1 Y¯i2 = bn
where Ai ≡ E (¯2i |X) and ci =
jbn −1
X i=(j−1)bn
1 Y¯i2 − Ai − ci + {z } bn | ≡Y˜i
jbn −1
X
(Ai + ci )
i=(j−1)bn
2 R i+j kn P 2 n g kjn i+j−1 σt dt. It follows that n
j=1
√
Jn nb2n X ¯ 2 = B1n + B2n + B3n , B kn2 ψ22 j=1 j
where B1n
Jn √ X n ≡ j=1
B2n
B3n
1 √
θψ2 n
jbn −1
X
2 Yei =
Jn √ X η (p)2j , n j=1
i=(j−1)bn
jbn −1 Jn X 2 X ≡ η (p)j (Ai + ci ) ; and θψ2 j=1 i=(j−1)bn 2 jbn −1 Jn X X 1 (Ai + ci ) . ≡ 2 2√ θ ψ2 n j=1 i=(j−1)bn
P
R1
γ2 0 t
P
P
We show that (1) B1n → (p) dt; (2) B2n → 0, and that (3) B3n → (p + 1) θ Starting with (1), write Z 1 Jn √ X 2 n η (p)j − γt2 (p) dt = B1.1n + B1.2n + B1.3n , with j=1
R1 0
σt2
+
ψ1 α θ2 ψ2 t
0
B1.1n
Jn √ X n = n η (p)2j − E η (p)2j |F(j−1)(p+1)k , n j=1
B1.2n B1.3n
Jn Jn N 1 X √ X n 2 n n = E η (p)j |F(j−1)(p+1)kn − γ(p)2j−1 , Jn n Jn j=1 j=1 Z 1 Jn Nn 1 X 2 γt2 (p) dt. = γ(p) j−1 − Jn n Jn j=1 0
We show that each of B1.`n →P 0 for ` = 1, 2, 3. For ` = 1, by Lenglart’s inequality (see e.g. Jn P n Lemma 4.4 of Vetter (2008)), it is sufficient to show that n E η (p)4j |F(j−1)(p+1)k →P 0, n j=1
42
2
dt.
which follows immediately by using equation (5.57) of Jacod et al. (2009). Next, to show that B1.2n →P 0, note that Jn N 1 X √ n 2 2 n nE η (p) |F(j−1)(p+1)k − B1.2n ≤ γ(p) j−1 j n Jn n Jn j=1 Jn X √ √ 1 1 2 n 2 = nE ζ (p) |F − nγ(p) (p + 1) θ j−1 (j−1)(p+1)kn (j−1)(p+1)kn Jn θ2 ψ22 n n j=1 √ X Jn n 2 n 3 2 2 = 2 2 E ζ (p) |F − θ ψ (p + 1) γ(p) j−1 (j−1)(p+1)kn 2 (j−1)(p+1)kn Jn θ ψ2 n j=1 J
n X K χ(p)(j−1)(p+1)kn ≤ 2 2√ θ ψ2 n j=1
√ where we use the fact that Nn /Jn = (p + 1) kn with kn = θ n and rely on equation (5.41) of Jacod et al. (2009) to bound the term in absolute value, where r −1/4 n χ(p)(j−1)(p+1)kn = n + E β (p)2(j−1)(p+1)kn |F(j−1)(p+1)k n and β (p)i is as defined in (40). It follows that Jn Jn Jn r 1 X 1 X 1 X 2 −1/4 n √ √ √ χ(p)(j−1)(p+1)kn ≤ n + E β (p)(j−1)(p+1)kn |F(j−1)(p+1)kn →P 0, n j=1 n j=1 n j=1 where the first term is of order O n−1/4 and the second term is oP (1) given part a6) of Lemma B.1. Finally, B1.3n →P 0 follows immediately by Riemann’s integrability of σ, the fact that Nn → 1 and Jn → ∞ as n → ∞. n jbP jbP n −1 n −1 ¯ i2 − ci . We can write X (Ai + ci ) and ζ (X, p)j = To show (2), let ϕj ≡ i=(j−1)bn
i=(j−1)bn
B2n
Jn 2 X = ϕj · η (p)j = B2.1n + B2.2n , with θψ2 j=1
B2.1n = B2.2n
Jn 2 X n ϕj η (p)j − E ϕj η (p)j |F(j−1)(p+1)k , and n θψ2 j=1
Jn 2 X n = E ϕj η (p)j |F(j−1)(p+1)k . n θψ2 j=1
We show that each of√B2.`n →P 0 for ` = 1, 2. Note that given the√definitions of Ai , ci , and the fact that kn = θ n, Assumption 4 implies that Ai + ci ≤ K/ n uniformly in i. Given that bn = (p + 1) kn , it follows that ϕj ≤ K uniformly in j. Starting with ` = 1, by Lenglart’s
43
inequality, it is sufficient to show that
Jn P
n →P 0. We can write E ϕ2j η (p)2j |F(j−1)(p+1)k n
j=1 Jn X
E
ϕ2j η
(p)2j
n |F(j−1)(p+1)k n
≤ K
j=1
Jn X
n E η (p)2j |F(j−1)(p+1)k n
j=1
!! Jn Jn N 1 X 1 √ X n 2 n 2 γ(p) j−1 = K √ n E η (p)j |F(j−1)(p+1)kn − Jn n Jn j=1 n j=1 ! ! Z 1 Z 1 Jn 1 Nn 1 X 1 γ(p)2j−1 − +K √ γt2 (p) dt + √ γt2 (p) dt Jn n n Jn j=1 n 0 0 Z 1 1 1 1 γ 2 (p) dt ≡ K √ B1.2n + √ B1.3n + √ n n n 0 t 1 1 1 = √ oP (1) + √ oP (1) +OP √ = oP (1) , n n n R1 where in particular we use the fact that B1.2n =oP (1) and B1.3n = oP (1) , and 0 γt2 (p) dt = OP (1) . It follows that B2.1n →P 0. Next, to show that B2.2n →P 0, note that we can write ! Jn X 2K 1 1/4 n −1/4 B2.2n ≤ n E η (p) |F = O n oP (1) = oP (1) , P (j−1)(p+1)kn j θψ2 n1/4 j=1 given that ϕj ≤ K, and given equation (5.49) of Jacod et al. (2009). Finally, to show (3), note that given the definitions of Ai and ci , and by using equations (5.23) and (5.36) of Jacod et al. (2009), we can write jbn −1 jbn −1 X X θψ2 2 p ψ1 √ α(j−1)bn /n + √ σ(j−1)bn /n + O √ + pβ(p)(j−1)bn . (Ai + ci ) = θ n n n i=(j−1)bn
i=(j−1)bn
(42) It follows that B3n ≡
1 √ θ2 ψ22 n
Jn X
X
j=1
2
jbn −1
(Ai + ci ) = Ln + Rn ,
i=(j−1)bn
where the leading term is 2 2 Z 1 Jn Nn 1 X ψ1 ψ1 2 P 2 Ln = (p + 1) θ α(j−1)bn /n + σ(j−1)bn /n → (p + 1)θ σt + 2 αt dt. n Jn j=1 θ2 ψ2 θ ψ2 0 (43) The remainder is such that J
Rn = K · OP
n 1 1 X √ +√ β(p)2(j−1)bn n n j=1
by using Lemma (5.4) of Jacod et al. (2009).
44
! →P 0
Proof of Lemma 3.1. Part a) Given the definition of Vn∗ , we can write √ nNn bn ∗ ∗ ∗ Vn = V1n − V2n , (Nn − bn + 1)2 where ∗ V1n =
∗ V2n =
1 bn 1 bn
bX n −1
√
∗ ∗ v1n,t , with v1n,t ≡
t=0 bX n −1
∗2 ∗ v2n,t , with v2n,t
t=0
n (Nn − bn + 1) Nn
n −t [ NX bn ]
bX n +t
j=1
i=t+1
!2 Zi+(j−1)bn
, and
n −t [ NX bn ] bn +t X 1 ≡ Zi+(j−1)bn . Nn j=1 i=t+1
2 R1 ∗ →P Vp +θ (p + 1) 0 σs2 + θψ2 ψ1 2 αs ds We now proceed in two steps. In Step 1, we show that v1n,t R 2 1 ψ1 2 ∗2 + →P σ α ds , also uniformly in uniformly in t. In Step 2, we show that v2n,t s 2 s θ ψ2 0 √
n bn t. This together with the fact that (N nN 2 → (p + 1) θ as n → ∞ when bn = (p + 1) kn n −bn +1) and kn satisfies iAssumption 2 imply the result. Proof of Step 1. For t = 0, . . . , bn − 1 and h Nn −t j = 1, . . . , bn , let
bn bn 1 X kn ψ2 1 X 2 ¯ ¯ Bj,t ≡ = Y Zi+t+(j−1)bn , bn i=1 i−1+t+(j−1)bn Nn bn i=1 2 ¯j,t are averages of non-overlapping blocks for given t. where Zi ≡ Nknn ψ12 Y¯i−1 and note that the B With this notation, we have that
∗ v1n,t
n −t ] √ 2 [ NX b Nn2 nbn n ¯ 2 = B , (Nn − bn + 1) Nn kn2 ψ22 j=1 j,t 2
where we can show that (Nn −bNnn+1)Nn → 1 under the condition that bn = (p + 1) kn . Using arguments similar to those used to prove Lemma B.2, we can show that n −t ] 2 √ 2 [ NX Z 1 b ψ1 nbn n ¯ 2 P 2 B → Vp + θ (p + 1) σs + 2 αs ds kn2 ψ22 j=1 j,t θ ψ2 0
uniformly in t. The proof of Step 2 relies on the consistency result in Theorem 1 of Christensen, ∗ Kinnebrock and Podolskij (2010). Indeed v2n,t is the main term in Jacod et al. (2009) preaveraged realized volatility estimator without the bias corrected term, with starting point t. Part b). Follows directly from part a) of Lemma 3.1 when replacing σt by a constant for all t. Part c). Follows directly from part a) of Lemma 3.1. Proof of Lemma 3.2. Given the definition of Vn∗ , we can write Jn −1 ∗ n1/2 b2 X ¯j − B ¯j+1 2 V ar∗ (ηj ) . Vn∗ = V ar∗ n1/4 P] RV n = 2 2n B ψ2 kn j=1
45
Let, Ξj =
bn ¯ Bj , ψ2 kn
by adding and substracting appropriately and given V ar∗ (η) = 1/2, it follows that ! JX JX n −1 n −1 n1/2 2 ΞJn − Ξ21 . Vn∗ = n1/2 Ξ2j − Ξj Ξj+1 + 2 j=1 j=1 {z } |
(44)
en ≡L
¯j and Ξj we can write Note that given the definition of B n1/2 2 n1/2 b2n ¯ 2 ¯ 2 ΞJn − Ξ21 = B1 + BJn 2 ψ22 kn2 2 ! bn = OP n3/4 = oP (1) , ¯j = OP (1/√n) uniformly in j, and the last equality where the second equality follows since B holds so long as δ < 3/4, which is verify under our assumptions. Thus, given (44) the rest of en →P V. The proof of this claim follows closey that for Theorem the proof can be reduced to L 4.1 of Christensen et al. (2013), however for completeness, we present here the relevant details. Following Christensen et al. (2013), we introduce two approximating version of Ξj first, namely bn 1 X 2 e Zj = Yei−1+(j−1)b , n ψ2 kn i=1 Zbj =
bn 1 X 2 Yei−1+jb , n ψ2 kn i=1
kn ¯ i , with W ¯i = P g t i+t − W i+t−1 where we have set Y˜ i =¯i + σ jbn W W , for jbn ≤ i ≤ kn Nn
n
t=1
n
(j + 1) bn − 1. Indeed we will show that the error due to replacing Y¯i by Y˜ i is small and will not affect our theoretical results, since σ is assumed to be an Ito semimartingale itself. We have that, for jbn ≤ i ≤ (j + 1) bn − 1 Z i+j Z i+j kn kn X X n n j j ¯ ˜ as ds+ g σs − σ jbn dWs E Yi −Y i = E g Nn i+j−1 i+j−1 kn kn j=1 j=1 n n 2 1/2 Z i+j kn X j kn n σs − σ jbn dWs ≤ K + g2 E Nn i+j−1 n kn j=1
≤ K
kn + n
kn bn n n
n
1/2
! ≤K
46
(kn bn )1/2 . n
Note also that E (|Zj |) ≤ K bnn , thus it follows that − 12 ! (kn bn )1/2 1 √ n kn 3/2 bn K , n 3/2 . So by using the fact that δ < 2/3 we ≤ K bnn
E Zj − Zej ≤ Kbn ≤ b b similarly for Zj , we have E Zj − Zj en − L bn = oP (1) , where obtain L bn = L
n −1 √ JX n Zbj2 − Zbj Zej+1 .
j=1
Then it is simple to deduce that 3/2 n −1 √ JX bn 2 2 n b b E Zj − E Zj |F (j−1)bn ≤ K n , n n α=1 3/2 n −1 √ JX bn n b e b e n , E Zj Zj+1 − E Zj Zj+1 |F (j−1)bn ≤ K n n j=1 by conditional independence, and now we are left with bn = L
n −1 √ JX 2 n b b e n E Zj − Zj Zj+1 |F (j−1)bn + oP (1) . n
j=1
From the same arguments as in Lemma 7.3 and Lemma 7.5 of Christensen et al. (2013) plus using δ > 1/2, we obtain √ nE Zbj2 − Zbj Zej+1 |F n(j−1)bn n Z αbn n bn = ς (s) ds + o , (α−1)bn n n uniformly in j, where we use Z 1 σs2 αs 4 αs2 4 V = ς (s) ds, with ς (s) = 2 Φ22 θσs + 2Φ12 + Φ11 3 , ψ2 θ θ 0 thus we have Z
1
en = L
ς (s) ds + oP (1) 0
and the proof is complete. Proof of Theorem 3.1 Let
Sn∗
= n
1/4
∗ ∗ ∗ ] ] P RV n − E P RV n =
47
bn ψ2 kn
Jn P j=1
zj∗ , where zj∗ =
¯j ∗ − E ∗ B ¯j n1/4 ψb2nkn B
∗
Jn P
. It follows that E ∗
! z ∗j
= 0, and
j=1 Jn X
Vn∗ ≡ V ar∗
! P
zj∗
→ V.
j=1
Since z1∗ , · · · , zJ∗n are conditionally independent, by the Berry-Esseen bound, for some small ε > 0 and for some constant C > 0 which changes from line to line, Jn √ X 2+ε sup P ∗ (Sn∗ ≤ x) −Φ x/ V ≤ C E ∗ zj∗ , x∈<
j=1
which converges to zero in probability as n → ∞. We have Jn Jn X X 2+ε ∗ ∗ ¯∗ ∗ ∗ 2+ε ∗ 1/4 bn ¯ Bj − E Bj E zj = E n ψ k 2 n j=1 j=1 2+ε X Jn ∗ 2+ε (2+ε) bn ¯ 4 ≤ 2n E ∗ B j ψ2 kn j=1 ∗
(2+ε) 4
2+ε
n
2+ε
≤ CE |η1 |
n
(2+ε) 4
kn−(2+ε) b(2+ε) n
Jn X 2+ε ¯j B j=1
≤ CE ∗ |η1 |
n
(2+ε) 4
kn−(2+ε) b(2+ε) n
1
Jn X
(2+ε)
bn
b(2+ε)−1 n
bn X
! 2(2+ε) Y¯i−1+(j−1)bn
i=1
j=1
!
Nn (2+ε) ε εX 2(2+ε) 2 ≤ CE ∗ |η1 |2+ε n 4 − 2 kn−(2+ε) b(1+ε) n Y¯i n | {z }| {z } i=1 =O(1) | {z } (2+ε) +(δ −1)(1+ε) ∝n
= Op n
1
4
(2+ε) +(δ1 −1)(1+ε) 4
=Op (1)
= op (1) ,
since for any ε ≥ 2, so long as δ < 2/3, we have 2+ε + (δ − 1) (1 + ε) < 0, and given that by 4 Theorem 3.3 of Jacod, Podolskij and Vetter (2010) 2+ε Z 1 Nn X ε 1 2(2+ε) P 2 ¯ 2 n Yi → µ2(2+ε) θψ2 σs + ψ1 αs ds, θ 0 i=1 ∗ ∗ which is bounded given Assumption 3, and E ∗ |ηj |2+ε ≤ ∆ < ∞. It follows that n1/4 P] RV n − E ∗ P] RV n N (0, V ) in probability. d d∗ Proof of Theorem 3.2 Given that Tn → N (0, 1), it suffices to show that Tn∗ → N (0, 1) in probability. Let ∗ ∗ 1/4 ∗ ] ] n P RV n − E P RV n p Hn∗ = , Vn∗
48
and note that s Tn∗ = Hn∗
Vn∗ , Vˆ ∗ n
∗
d where Vˆn∗ is defined in the main text. Theorem 3.1 proved that Hn∗ → N (0, 1) in probability. ∗ ∗ P ˆ∗ Thus, it suffices to show that V n − Vn → 0 in probability. In particular, we show that (1) P Bias∗ Vˆn∗ = 0, and (2) V ar∗ Vˆn∗ → 0. It is easy to verify that (1) holds by the definition of Vˆ ∗ and V ∗ . To prove (2), note that n
n
!2 JX n −1 2 n1/2 b2 V ar∗ (η) 2 2 n ¯j − B ¯j+1 E∗ B ηj2 −E ∗ η 2 V ar∗ Vˆn∗ = E ∗ Vˆn∗ − Vn∗ = ψ22 kn2 E ∗ (η 2 ) j=1 1/2 2 2 n −1 JX n bn V ar∗ (η) 2 2 2 ∗ ∗ ¯j − B ¯j+1 4 = η B η −E E ψ22 kn2 E ∗ (η 2 ) j=1 ! 1/2 2 2 JX JX n −1 n −1 ∗ n b V ar (η) 2 n 4 ¯j4 + ¯j+1 ≤ 23 E ∗ η 2 −E ∗ η 2 B B ψ22 kn2 E ∗ (η 2 ) j=1 j=1 ! 1/2 2 2 J n X n bn V ar∗ (η) ∗ 2 ∗ 2 2 4 4 4 3 ¯j − B ¯1 + B ¯J E η −E η B 2 = 2 n ψ22 kn2 E ∗ (η 2 ) j=1 ! !! 2 1/2 2 Jn bn ∗ X X V ar (η) 1 n b 2 n ¯4 + B ¯4 E ∗ η 2 −E ∗ η 2 Y¯ 8 − B ≤ 23 2 1 Jn ψ22 kn2 E ∗ (η 2 ) bn j=1 i=1 i−1+(j−1)bn
≤ 23
n1/2 b2n V ar∗ (η) ψ22 kn2 E ∗ (η 2 )
2 {z
|
=O
= OP
b3n n2
+ OP
b4n n3
E ∗ η 2 −E ∗ η
b4 n n
2 2
! ! Nn X 1 2 8 4 4 ¯ ¯ ¯ Yi − B1 + BJn bn n n | {z } i=1 } | {z } =OP ( 12 ) =OP (1)
n
= oP (1) ,
where we have used the fact that ηj is i.i.d. to justify the third equality and Theorem 3.3 of P n ¯8 Jacod, Podolskij and Vetter (2010) to justify the fact that n N i=1 Yi = OP (1).
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