TY - JOUR
AU - Veliyev,, Bezirgen
AB - Summary The main contribution of this paper is to establish the formal validity of Edgeworth expansions for realized volatility estimators. First, in the context of no microstructure effects, our results rigorously justify the Edgeworth expansions for realized volatility derived in Gonçalves and Meddahi (2009, Econometrica 77, 283–306). Second, we show that the validity of the Edgeworth expansions for realized volatility might not cover the optimal two‐point distribution wild bootstrap proposed by Gonçalves and Meddahi. Then, we propose a new optimal nonlattice distribution, which ensures the second‐order correctness of the bootstrap. Third, in the presence of microstructure noise, based on our Edgeworth expansions, we show that the new optimal choice proposed in the absence of noise is still valid in noisy data for the pre‐averaged realized volatility estimator proposed by Podolskij and Vetter (2009, Bernoulli 15, 634–658). Finally, we show how confidence intervals for integrated volatility can be constructed using these Edgeworth expansions for noisy data. Our Monte Carlo simulations show that the intervals based on the Edgeworth corrections have improved the finite sample properties relatively to the conventional intervals based on the normal approximation. 1. Introduction The increasing availability of complete transaction and quote records for financial assets has spurred a body of literature that seeks to exploit this information when estimating the current level of return volatility. An early popular estimator of integrated volatility is to compute the sum of squared increments of the log price process (i.e., the realized volatility).1 An important characteristic of high‐frequency financial data is the presence of market microstructure effects: prices are observed with contamination errors (the so‐called noise) due to the presence of bid‐ask bounce effects, rounding errors, etc., which contribute to a discrepancy between the latent efficient price process and the price observed by the econometrician. This issue has received a fair amount of attention in the recent literature. Indeed, realized volatility is not consistent for integrated volatility under the presence of market microstructure noise. This has motivated the development of alternative estimators. Currently, there are four main approaches to quadratic variation estimation: the linear combination of realized volatilities obtained by subsampling (Zhang et al., 2005, and Zhang, 2006), kernel‐based autocovariance adjustments (Barndorff‐Nielsen et al., 2008), the pre‐averaging method (Podolskij and Vetter, 2009, and Jacod et al., 2009), and the maximum likelihood‐based approach (Xiu, 2010). Recently, Gonçalves and Meddahi (2009) – henceforth GM2009 – have shown that under general conditions on the price and volatility processes (but excluding microstructure noise), the use of the bootstrap for inference on volatility could help towards better performance than standard asymptotic inference. In particular, GM2009 have proposed a theoretical justification for using bootstrap for realized volatility. Their simulations confirm the better behaviour of the bootstrap method than the asymptotic‐based approach. Based on Edgeworth expansions, they also provide higher‐order refinements of the bootstrap that explain these findings under a stricter set of assumptions, which rule out drift, leverage effects and market microstructure effects. However, they do not prove the theoretical validity of their Edgeworth expansions (see GM2009, footnote 3 on p. 289). In this paper, we establish the theoretical validity of their Edgeworth expansions. In addition, we show that the validity of the Edgeworth expansions for realized volatility might not cover the optimal two‐point distribution wild bootstrap proposed by GM2009. Then, we suggest a new optimal external random variable with a density that yields the second‐order accuracy of the bootstrap. Gonçalves et al. (2014) have shown that the wild bootstrap procedure applied on the non‐overlapping pre‐averaged returns (as originally proposed by Podolskij and Vetter, 2009) estimates the asymptotic variance as well as the asymptotic mixed normal distribution of the pre‐averaged realized volatility estimator. However, for this relatively simple statistic, we can simply use, for instance, the consistent variance estimator proposed by Podolskij and Vetter (2009). Hence, the additional effort required for the bootstrap is justified if the resulting approximation to the distribution of the statistic is better than the one relying on the asymptotic normality. With no noisy data, the wild bootstrap studied by GM2009 does indeed have this property. In this paper, we show that this is also true for the wild bootstrap method applied to the non‐overlapping pre‐averaged returns. Specifically, in the presence of microstructure noise, based on our Edgeworth expansions, we show that the new optimal external random variable with a nonlattice distribution proposed in the absence of noise is still valid in noisy data for the pre‐averaging estimator of Podolskij and Vetter (2009). The main reason for the second‐order correctness of the proposed bootstrap procedure in Gonçalves et al. (2014) is the asymptotically correct skewness of the bootstrap distribution. Indeed, an important characteristic of the pre‐averaged realized volatility estimator of Podolskij and Vetter (2009) – see also Jacod et al. (2009) – is that it entails an analytical bias correction term. Jacod et al. (2009) have shown that this bias correction is only important for the proper centring of the confidence intervals and does not affect the variance of the estimator. This has motivated Gonçalves et al. (2014) to resample the pre‐averaged returns and then to construct the bootstrap t‐statistic without any bias correction term; see also Hounyo et al. (2013) and Hounyo (2013) for closely related proposals for bootstrapping high‐frequency financial data. In this paper, we formally show that up to o(n−1/4) (where n is the sample size), the bias correction term does not affect the first three cumulants of the studentized statistic, in particular the skewness of the estimator. As a consequence, the bootstrap method in Gonçalves et al. (2014) does not suffer from the absence of a bias correction term in the bootstrap t‐statistic at least to consistently estimate the skewness, and more generally in its ability to match the first and third cumulants of pre‐averaged realized volatility up to o(n−1/4) (small enough error to yield a second‐order refinement). Building on Edgeworth expansions for studentized statistics based on the pre‐averaged realized volatility estimator, we also propose confidence intervals for integrated volatility that incorporate an analytical correction for skewness as an alternative method of inference. Our approach extends the results in Gonçalves and Meddahi (2008) – henceforth GM2008 – by allowing for microstructure noise. As in GM2008, we also find that in a framework where there exist market microstructure effects, and the computational burden imposed by the bootstrap is high, the use of Edgeworth expansions is superior to using the normal approximation derived by Podolskij and Vetter (2009). Our Monte Carlo simulations show that the bootstrap outperforms the Edgeworth corrected intervals. Recently, Zhang et al. (2011) have also allowed microstructure effects and have provided Edgeworth corrections of the normalized statistic (where the denominator equals variance of the estimator in population) rather than studentized statistics (where the denominator is a consistent estimator of the estimator's variance) for several realized measures, including the realized volatility and the noise robust two time‐scale realized volatility estimator as a mean to improve upon the first‐order asymptotics. The main reason why we only focus on studentized statistics is because, in practice, the variance of realized volatility estimators is usually unknown, and thus studentized statistics are more used. In addition, in the simple framework without market microstructure noise, GM2008 proved that Edgeworth corrections based on normalized statistics are worse than the asymptotic theory. Edgeworth expansions for realized volatility are also developed by Lieberman and Phillips (2008) for inference on long memory parameters. A nice side result, which might be useful in other contexts, is that we derive the second‐order Edgeworth expansion of a certain form of studentized statistic, where observations are independent but not identically distributed. In particular, observations have specific heterogeneity properties, which to the best of our knowledge are not covered by other works in the literature. This can be found in Proposition A.1 in the Appendix. The remainder of the paper unfolds as follows. In Section 2, we briefly introduce the theoretical framework and the main assumptions. We also review the existing asymptotic theory of realized volatility – in particular, the pre‐averaged realized volatility estimator of Podolskij and Vetter (2009). In Section 3, we establish the formal validity of Edgeworth expansions for realized volatility estimators. Section 4 contains Monte Carlo results, while Section 5 concludes. All proofs are relegated to the Appendix. 2. Framework and Review of the Literature We focus on a single asset traded in a liquid financial market. Let X denote the latent efficient log‐price process defined on a probability space (Ω,F,P) equipped with a filtration (Ft)t≥0 ⁠. We assume that the sample‐path of X is continuous and determined by the stochastic differential equation dXt=btdt+σtdWt,t≥0,(2.1) where σ=(σt)t≥0 is an adapted càdlàg volatility process, b=(bt)t≥0 is an adapted càdlàg drift process and W=(Wt)t≥0 is a standard Brownian motion. By assumption Wt and (σt,bt) are independent, which in particular excludes the leverage effect. The object of interest is the integrated volatility of X, i.e., the process Γt=∫0tσs2ds. Without loss of generality, we let t=1 and define Γ=Γ1=∫01σs2ds as the integrated volatility of X over the period [0, 1], which is thought of as a given day. The availability of market frictions, such as bid‐ask spreads, price discreteness, rounding errors, etc., hamper us from observing the efficient price process X. Instead, we observe a noisy price process Y, observed at time points t=i/n for i=0,...,n, via Yt=Xt+εt,(2.2) from which we compute n intra‐day returns given by ΔinY≡Yi/n−Y(i−1)/n,i=1,...,n.(2.3) where εt represents the noise term that collects all the market microstructure effects. We impose the following assumption. Assumption 2.1. (a) εt is independent and identically distributed (i.i.d.) with mean 0 and variance ω2, and also E[|εt|2(6+δ)]<∞ for some δ>0 ⁠; (b) εt is independent of the latent log‐price Xt ⁠. This assumption is standard in the literature related to the noise robust estimators of integrated volatility; see, among others, Zhang et al. (2005) and Barndorff‐Nielsen et al. (2008). However, empirically, a decomposition into independent components as in 2.2 and i.i.d. assumption on noise do not always describe the dynamics of the observed price processes. These assumptions might be too strong, especially at the highest frequencies; see, e.g., Hansen and Lunde (2006) and Aït‐Sahalia et al. (2011) for more on this issue. Most of what we do here could be extended to allow for dependent noise following the details discussed in Gonçalves et al. (2014). However, an exploration of this extended setting is left for future research. Next, we introduce an additional regularity condition on the volatility and drift processes. Assumption 2.2. (a) The volatility σ is a càdlàg process, bounded away from zero, and satisfies (pathwise) limn→∞n−1/2∑i=1n|σηir−σξir|=0 ⁠, for some r>0 and for any ηi ⁠, ξi with 0≤ξ1≤η1≤n−1≤ξ2≤η2≤2n−1≤...≤ξn≤ηn≤1 ⁠; (b) the drift b is a non‐negative càdlàg process and satisfies (pathwise) limn→∞n−1/2∑i=1n|bηir−bξir|=0 ⁠, for some r>0 and for any ηi ⁠, ξi with 0≤ξ1≤η1≤n−1≤ξ2≤η2≤2n−1≤...≤ξn≤ηn≤1. Assumption 2.2 is stronger than required to prove the central limit theorem for the integrated volatility estimator, but it is a convenient assumption to derive Edgeworth expansions. Relaxing this assumption is beyond the scope of this paper. We note that Assumption 2.2(a) was already used in Barndorff‐Nielsen and Shephard (2003) and GM2009, while we impose Assumption 2.2(b) to deal with the drift term. In our proof, we have to assume that the drift has the same sign, which explains the non‐negativity restriction. The non‐negativity restriction of b in Assumption 2.2(b) is necessary to be able to treat the integrals of type ∫(i−1)/ni/nbsds similar to the integrals of type ∫(i−1)/ni/nσs2ds ⁠. In view of footnote 2 in Barndorff‐Nielsen and Shephard (2003), we note that if Assumption 2.2 holds for some r>0 ⁠, then it holds for any r>0. In the following, we denote by Γ̂n a consistent estimator of the integrated volatility Γ such that a central limit theorem holds with the convergence rate of τn ⁠. In particular, we have as n→∞ Tn≡τn(Γ̂n−Γ)V̂n→dN(0,1),(2.4) where V̂n is a consistent estimator of the asymptotic variance V of τnΓ̂n ⁠. As statistics of interest in this paper, we focus on the realized volatility and the pre‐averaging estimator of Podolskij and Vetter (2009). First, we review the existing results. While in the paper we have also analysed finite sample behaviour of the pre‐averaging estimator based on data at the highest frequency, the setting of moderate frequencies serves as an important benchmark. We start with this benchmark case because of its relative simplicity. 2.1. Realized volatility estimator In this subsection, we consider the simple case where no market microstructure noise exists (⁠ ε≡0 ⁠). It follows that Y=X, where X follows 2.1. In applied work, this refers to a situation where the sampling frequencies are low enough for the effects of market microstructure to be negligible (e.g., 5, 10 or 30 minutes). In this relatively simple scenario, a popular consistent estimator of integrated volatility is the realized volatility; see e.g., Barndorff‐Nielsen and Shephard (2002). Barndorff‐Nielsen et al. (2006) derived a feasible central limit theorem for realized volatility defined by Γ̂n=∑i=1n(ΔinY)2.(2.5) They showed that, as n→∞ ⁠, 2.4 holds, under very general conditions that allow the presence of the leverage effect, for the statistic Tn defined as Tn=n(∑i=1n(ΔinY)2−Γ)(2/3)n∑i=1n(ΔinY)4.(2.6) We can use this feasible asymptotic distribution result to build confidence intervals for integrated volatility. In particular, the conventional 100(1−α)% level one‐sided confidence interval for Γ is given by IC feas ,1−αAT−1=(−∞,Γ̂n−τn−1V̂nzα),(2.7) whereas a two‐sided symmetric feasible 100(1−α)% level interval for Γ is given by IC feas ,1−αAT−2=(Γ̂n−z1−α/2τn−1V̂n,Γ̂n+z1−α/2τn−1V̂n),(2.8) 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, z0.05=−1.645 ⁠, z0.975=1.96 when α=0.05 ⁠. As GM2009 have shown, in a finite sample, this approach can lead to important coverage distortions. As a remedy, GM2008 suggested the use of Edgeworth corrected confidence intervals for realized volatility. We study these intervals in detail in Section 3.3. However, GM2009 proposed confidence intervals based on bootstrap methods for Γ̂n ⁠. Their wild bootstrap method for realized volatility resamples as follows: ΔinY*=ΔinY·vi,i=1,...,n.(2.9) Here, the external random variable vi is an i.i.d. random variable independent of the data, whose moments are given by aq*≡E*[|vi|q] ⁠. As usual in the bootstrap literature, P* ⁠, E* and Var* respectively denote 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 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* converges weakly to Z under P* ⁠, for all samples contained in a set with probability P converging to one. Then, based on bootstrap returns ΔinY* ⁠, GM2009 defined the bootstrap realized volatility analogue of Γ̂n as Γ̂n*=∑i=1n(ΔinY*)2 ⁠. They showed that, as n→∞ Tn*≡n(Γ̂n*−a2*Γ̂n)V̂n*→d*N(0,1),(2.10) where V̂n*=((a4*−a2*2)/a4*)n∑i=1n(ΔinY*)4 ⁠. Indeed, this result justifies constructing bootstrap percentile‐t (bootstrap studentized statistic) intervals. In particular, a 100(1−α)% one‐sided bootstrap percentile‐t interval for integrated volatility is given by ICperc‐t,1−α*B−1=(−∞,Γ̂n−τn−1V̂nzα*B−1),(2.11) whereas a 100(1−α)% symmetric bootstrap percentile‐t interval for integrated volatility is given by ICperc‐t,1−α*B−2=(Γ̂n−z1−α*B−2τn−1V̂n,Γ̂n+z1−α*B−2τn−1V̂n).(2.12) Here, zα*B−1 is the α‐quantile of the bootstrap distribution of Tn* whereas z1−α*B−2 is the (1−α)‐quantile of the bootstrap distribution of |Tn*| ⁠. Next, we review the existing results of the pre‐averaged realized volatility estimator of Podolskij and Vetter (2009) . 2.2. Pre‐averaged estimator and its asymptotic theory We now turn to the case where market microstructure effects are not negligible (⁠ ε≠0 ⁠). Given that Y=X+ε ⁠, we can write ΔinY=(Xi/n−X(i−1)/n)+(εi/n−ε(i−1)/n)≡ΔinX+Δinε, where ΔinX denotes the 1/n‐frequency return on the efficient price process. Under Assumption 2.1, the order of magnitude of Δinε is Op(1) ⁠. In contrast, ΔinX is asymptotically uncorrelated and heteroscedastic with (conditional) variance given by ∫(i−1)/ni/nσs2ds ⁠. Thus, its order of magnitude is 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, e.g., Zhang et al. (2005) and Bandi and Russell (2008). As mentioned in Section 1, there are several estimators of realized volatility that explicitly take microstructure noise effects into account. We consider the non‐overlapping pre‐averaging estimator of Podolskij and Vetter (2009). To describe this approach, let kn be a sequence of integers, which will denote the window length over which the pre‐averaging of returns is done. Similarly, let g be a weighting function on [0, 1] such that g(0)=g(1)=0 and ∫01g(s)2ds>0 ⁠, and assume g is continuous and piecewise continuously differentiable with a piecewise Lipschitz derivative g′ ⁠. An example of a function that satisfies these restrictions is g(x)=min(x,1−x) ⁠. We also introduce ψ1kn=kn∑i=1kngikn−gi−1kn2andψ2kn=1kn∑i=1kng2ikn.(2.13) These quantities have the following limits ψ1kn=ψ1+o(n−1/4)andψ2kn=ψ2+o(n−1/4),(2.14) where ψ1=∫01(g′(s))2dsandψ2=∫01(g(s))2ds. 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 1/n‐horizon returns over each block of size kn ⁠, i.e. Y¯i=∑j=1kngjknΔi+jnY. The aim of pre‐averaging is to control the stochastic orders of the pre‐averaged terms via kn ⁠. In particular, we obtain X¯i=∑j=1kngjkn(X(i+j)/n−X(i+j−1)/n)=Opknn, and ε¯i=∑j=1kngjkn(ε(i+j)/n−ε(i+j−1)/n)=Op1kn. Thus, the impact of the noise is reduced the larger kn is. We put the following condition on kn ⁠. Assumption 2.3. (a) There exists θ∈(0,∞) such that knn=θ+o(n−1/4);(2.15) (b) for any n≥1 ⁠, kn divides n. Assumption 2.3(a) is standard in the literature (see Jacod et al., 2009). This choice implies that the orders of the terms X¯i and ε¯i are balanced and equal to Op(n−1/4) ⁠. An example that satisfies 2.15 is kn=[θn] ⁠. Assumption 2.3(b) is imposed in this work to deal with the Edgeworth expansion. Podolskij and Vetter (2009) propose the following estimator of integrated volatility Γ̂n=1ψ2kn∑m=0dn−1Y¯mkn2︸RV‐likeestimator−ψ1kn2kn2ψ2kn∑i=1n(ΔinY)2︸biascorrectionterm,(2.16) where dn≡n/kn and ψ1kn,ψ2kn are as in 2.13. The pre‐averaging estimator is then simply the analogue of the realized volatility but based on pre‐averaged returns and an additional term to remove the bias due to noise. As discussed in Jacod et al. (2009) and Gonçalves et al. (2014), this bias term does not contribute to the asymptotic variance of Γ̂n ⁠. One of our contributions is to show that, at second order, this bias term does not affect the asymptotic distribution of Γ̂n but possibly at third order its impact might be important. Under Assumptions 2.1 and 2.3, Podolskij and Vetter (2009) show that Γ̂n given by 2.16 satisfies a central limit theorem as in 2.4 with τn=n1/4 ⁠. In particular, the t‐statistic is Tn=n1/4(Γ̂n−Γ)V̂n,(2.17) where the asymptotic conditional variance V and V̂n (an estimator of V) are respectively given by V=2θψ22∫01(θψ2σs2+ψ1θω2)2dsandV̂n=2n3(ψ2kn)2∑m=0dn−1Y¯mkn4.(2.18) Recently, Gonçalves et al. (2014) have shown that a wild bootstrap procedure applied to the non‐overlapping pre‐averaged returns Y¯mkn estimates the asymptotic variance V as well as the asymptotic mixed normal distribution of the pre‐averaged realized volatility estimator Γ̂n ⁠. More specifically, Gonçalves et al. (2014) suggested resampling as follows: Y¯mkn*=Y¯mkn·vm,m=0,...,dn−1. Here, the external random variable vm is an i.i.d. random variable independent of the data and whose moments are given by aq*=E*[|vm|q] ⁠. Then, based on bootstrap pre‐averaged returns Y¯mkn* ⁠, Gonçalves et al. (2014) defined the bootstrap pre‐averaged realized volatility estimator as Γ̂n*=1ψ2kn∑m=0dn−1Y¯mkn*2. They show that, as n→∞ Tn*≡n1/4(Γ̂n*−E*(Γ̂n*))V̂n*→d*N(0,1),(2.19) where E*(Γ̂n*)=a2*ψ2kn∑m=0dn−1Y¯mkn2 and V̂n*=(a4*−a2*2)a4*(ψ2kn)2n∑m=0dn−1Y¯mkn*4. This justifies constructing bootstrap percentile‐t intervals for integrated volatility in the presence of noise. Note that although in 2.16 Γ̂n contains a bias correction term, it is not the case for Γ̂n* ⁠. As they argue, this is because the bias correction term by definition does not affect at first order the asymptotic variance of Γ̂n ⁠. In the next section, we investigate the impact of the bias correction term on the first three cumulants of studentized statistic up to o(n−1/4) ⁠. We note that the work of Jacod et al. (2009) considers an estimator of integrated volatility using all pre‐averaged returns (i.e., overlapping blocks), while we study only the estimator using non‐overlapping pre‐averaged returns. The main reason is that the wild bootstrap method suggested above is not appropriate for the overlapping case due to the strong dependence of pre‐averaged returns. For further details on the failure of the wild bootstrap method in this context, see, e.g., Hounyo et al. (2013). In addition, the Edgeworth expansion for the overlapping estimator is more complicated as it falls into the framework of strongly dependent and heterogeneous data. 3. Edgeworth Expansion for Realized Volatility Here, we establish the validity of formal Edgeworth expansions for Γ̂n ⁠, where Γ̂n is given either by 2.5 or 2.16. Our results apply to the t‐statistic Tn and the bootstrap t‐statistic Tn* defined above. We start by studying the no‐noise case. 3.1. Edgeworth expansion without noise To describe the Edgeworth expansion, we need to introduce additional notation. To facilitate comparison, we keep the notation of GM2009 whenever possible. For any r,s>0 ⁠, we let Rr,s=Rr/Rsr/s and σr,s=σ¯r/(σ¯s)r/s where Rr=n(r/2)−1∑i=1n|ΔinY|randσ¯r=∫01σtrdt.(3.1) Similarly, we let ar,s=ar/asr/s where as=E[|U|s] such that U∼N(0,1) ⁠. Theorem 3.1. (a) Suppose 2.1 holds with b=0 ⁠. Under Assumption 2.2(a), conditionally on σ, the second‐order Edgeworth expansions of the studentized statistics Tn and Tn* defined in 2.6 and 2.10, respectively, are given by P[Tn≤x]=Φ(x)+n−1/2q1(x)φ(x)+o(n−1/2),(3.2) where q1(x)=A12−16(B1−3A1)(x2−1)σ6,4,(3.3) with A1=a6−a2a4a4(a4−a22)1/2=42andB1=a6−3a2a4+2a23(a4−a22)3/2=42. (b) In addition, suppose that {ΔjnY*=ΔjnY·vj,j=1,...,n} ⁠, where vj∼i.i.d. whose moments are given by as*=E*|vj|s with a2(6+δ)*<∞ for some δ>0 and vj satisfy the Cramer condition. That is, for all r>0, there exists Mr∈(0,1) such that |φn,j(t)|≤Mrforall∥t∥≥randn≥1,1≤j≤n,(3.4) where φn,j is the characteristic function of (n|ΔjnY|2vj2,n2|ΔjnY|4vj4)′ under P* ⁠. Then P*[Tn*≤x]=Φ(x)+n−1/2q1*(x)φ(x)+op(n−1/2),(3.5) where q1*(x)=A1*2−16(B1*−3A1*)(x2−1)R6,4, with A1*=a6*−a2*a4*a4*(a4*−a2*2)1/2andB1*=a6*−3a2*a4*+2a2*3(a4*−a2*2)3/2. We note that these results are derived under the same assumptions as in Proposition 4.1 of GM2009. Because we have shown the validity of our Edgeworth expansions in this paper, our results justify Proposition 4.1 of GM2009. In contrast to GM2009 (see footnote 3 on p. 289), we do not assume the existence of Edgeworth expansions derived in 3.2 and 3.5, rather we formally verify conditions under which these Edgeworth expansions exist (as some cumulants might be infinite). Unfortunately, the best existing choice of vj (i.e., the optimal two‐point distribution) suggested in Proposition 4.5 of GM2009 does not satisfy condition (3.4) in part (b) of Theorem 3.1 and hence it is unlikely that the second‐order Edgeworth expansions of the bootstrap studentized statistic Tn* exist for this choice. Thus, we suggest a distribution that has a density. Proposition 3.1. Let Tn and Tn* be defined as in (2.6) and (2.10), respectively. Moreover, v1,...,vn as defined in 2.9 are i.i.d. with vi=ηi ⁠, where ηi has the gamma density f(x)=βαΓ(α)xα−1exp(−βx)I(x>0) with α=β=25/6 ⁠. Suppose (2.1) holds with b=0 ⁠. Under Assumption 2.2(a), conditionally on σ, as n→∞ supx∈R|P*[Tn*≤x]−P[Tn≤x]|=op(n−1/2). The square‐root term in the optimal choice of the external random variable in Proposition 3.1 suggests the following modification of the wild bootstrap procedure proposed by GM2009. We propose to resample directly the square returns (ΔinY)2 instead of the raw returns ΔinY ⁠: (ΔinY*)2=(ΔinY)2·|ηi|,i=1,...,n.(3.6) Here, as before, the external random variable ηi is an i.i.d. random variable independent of the data, whose moments are given by aq*=E*[|ηi|q] ⁠. For the second‐order accuracy of the bootstrap, GM2009 imposed conditions on the first even moments (⁠ a2* ⁠, a4* and a6* ⁠) of the external random variable v, whereas with the new wild bootstrap we require conditions on the first three moments (⁠ a1* ⁠, a2* and a3* ⁠) of ηi ⁠. Then, the gamma distribution choice of ηi defined in Proposition 3.1 provides a second‐order asymptotic refinement. So far, we have focused on the case b=0 ⁠. In the following remark, we allow a non‐zero drift term. Remark 3.1. Under Assumption 2.2, conditionally on σ and b, the second‐order formal Edgeworth expansion of the studentized statistic Tn defined in 2.6 (assuming the corresponding Edgeworth expansion exists) is given by P[Tn≤x]=Φ(x)+n−1/2q1(x)φ(x)+o(n−1/2),(3.7) where q1(x)=A12−16(B1−3A1)(x2−1)σ6,4−b¯22σ¯4,(3.8) with b¯2=∫01bt2dt and A1, B1 defined as in Theorem 3.1, in particular A1=B1=4/2 ⁠. Assuming that the corresponding Edgeworth expansions exist, Remark 3.1 emphasized that the effect of the drift on Tn is not negligible at second order. In particular, a comparison of equations 3.3 and 3.8 shows that an additional term −(b¯2/2σ¯4) shows up in 3.8 when b≠0 ⁠. At first order, one can show that the effect of the drift on Tn is Op(n−1/2) ⁠, that is negligible. As highlighted in GM2009, the results in Theorem 3.1 are not special cases of Liu (1988). She derived the second‐order Edgeworth expansions of the studentized statistic defined by Tn=n(n−1∑i=1nZi−n−1∑i=1nE[Zi])V̂n, where Z1,...,Zn are a set of independent but not identical random observations with the sample variance V̂n=n−1∑i=1nZi2−(n−1∑i=1nZi)2 ⁠. She also showed the second‐order properties of the weighted bootstrap of Wu (1986), the so‐called wild bootstrap procedure. The differences between the work of Liu (1988) and the results in Theorem 3.1 are at least twofold. First, her results apply to t and bootstrap t‐statistics that are both studentized by the sample variance. In particular, in part (a) of Theorem 3.1, we would be able to use the results of Liu (1988) in the context of realized volatility (with no noise), if instead of using the studentized statistics t defined in 2.6 we have considered the following t‐statistic: Tn=n(R2−Γ)R4−R22.(3.9) Here, Rr (with r=2,4 ⁠) is given by 3.1. By letting Zi≡n|ΔinY|2 ⁠, we easily obtain that R2=n−1∑i=1nZi and V̂n=R4−R22=n−1∑i=1nZi2−(n−1∑i=1nZi)2 equals the sample variance estimator of nR2 ⁠. Unfortunately, we cannot use R4−R22 to studentize realized volatility when volatility is time‐varying. Second, the wild bootstrap of Liu (1988) is applied on centred observations. In particular, in order to use the second‐order Edgeworth expansions of Liu (1988) for the bootstrap t‐statistic, the wild bootstrap observations should be resampled as follows: Zi*=n−1∑i=1nZi−(Zi−n−1∑i=1nZi)vi,i=1,...,n. Here, Zi=n|ΔinY|2 and vi∼ i.i.d. with mean 0 and variance 1. We observe that this is different from the wild bootstrap method of GM2009 suggested for realized volatility. The t‐statistics defined in 2.6 and 2.10 are our statistics of interest here and these are not covered by results in Liu (1988). 3.2. Edgeworth expansion for the pre‐averaging estimator Let us introduce notations. For r,s>0 ⁠, we let R̃r,s=R̃r/R̃sr/s, σ̃r,s=σ̃r/(σ̃s)r/s where R̃r=1(ψ2kn)r/2n(r/4)−(1/2)∑m=0dn−1|Y¯mkn|randσ̃r=∫01σt2+ω2ψ1θ2ψ2r/2dt.(3.10) Furthermore, we denote si2≡∑j=1kn−1g2jkn∫((i−1)kn+j−1)/n((i−1)kn+j)/nσt2dt.(3.11) Note that, conditionally on σ, si2 is the expectation of (X¯(i−1)kn)2 ⁠. We also let Zdn,i=dnψ2kn(Y¯(i−1)kn)2,μdn,i=dnψ2knsi2+ψ1knω2kn,Bdn,i=ψ1kndn2kn2ψ2kn∑j=(i−1)kn+1ikn−1(|Δjnε|2−2ω2). To state our Edgeworth expansion results for pre‐averaged realized volatility, we require a slightly stronger condition on the volatility σ than Assumption 2.2(a) and a variant of the Cramer condition. Assumption 3.1. The volatility σ is a càdlàg process, bounded away from zero, and satisfies the following regularity condition. For some δ>0, we have 1ψ2kn∑i=1dnsi2−∫01σt2=O(n−1/2−δ). For g(x)=min(x,1−x) ⁠, examples of processes that satisfy Assumption 3.1 are those such that σt2=Ct+Jt ⁠, where Ct (the continuous part of σt2 ⁠) is twice continuously differentiable and Jt (the jump part of σt2 ⁠) allows jumps that can occur at points ikn/n ⁠. Assumption 3.2. For all r>0 ⁠, there exists Mr∈(0,1) such that |φdn,i(t)|≤Mr for all ∥t∥≥r and dn≥1,1≤i≤dn ⁠, where φdn,i is the characteristic function of (Zdn,i−μdn,i−Bdn,i,Zdn,i2−E[Zdn,i2])′ ⁠. Assumption 3.2 is a version of the Cramer condition for a triangular array of row‐wise independent R2‐valued random vectors; see equation (6.28) in Lahiri (2003) for a similar condition. In the framework of no noise as in Section 3.1 with no drift and constant volatility, Assumption 3.2 can be replaced by the classical Cramer condition for i.i.d. data; see, e.g., equation (6.31) in Lahiri (2003). Under above conditions, the following theorem holds true. Theorem 3.2. (a) Suppose 2.1 holds with b=0 ⁠. Under Assumptions 2.1, 2.3, 3.1 and 3.2 and conditionally on σ, the formal second‐order Edgeworth expansions of the studentized statistics Tn and Tn* defined in 2.17 and (2.19), respectively, are given by P[Tn≤x]=Φ(x)+n−1/4q1(x)φ(x)+o(n−1/4),(3.12) where q1(x)=A12−16(B1−3A1)(x2−1)σ̃6,4 with A1 and B1 defined as in Theorem 3.1, in particular A1=B1=4/2 ⁠. (b) In addition, suppose that {Y¯mkn*=Y¯mkn·vm,m=0,...,dn−1} ⁠, where vm∼i.i.d. whose moments are given by as*=E*[|vm|s] with a2(6+δ)*<∞ for some δ>0 and vm satisfy the Cramer condition. Namely, for all r>0, there exists Mr∈(0,1) such that |φdn,m(t)|≤Mr for all ∥t∥≥r and dn≥1,0≤m≤dn−1 ⁠, where φdn,m is the characteristic function of (dn|Y¯mkn|2vm2,dn2|Y¯mkn|4vm4)′ under P* ⁠. Then P*[Tn*≤x]=Φ(x)+n−1/4q1*(x)φ(x)+op(n−1/4),(3.13) where q1*(x)=A1*2−16(B1*−3A1*)(x2−1)R̃6,4, with A1* and B1* defined as in Theorem 3.1. Theorem 3.2 extends Proposition 4.1 of GM2009 to the noisy setting by utilizing the pre‐averaged realized volatility estimator of Podolskij and Vetter (2009). In contrast to the no‐noise case, we require the Cramer condition for the validity of Theorem 3.2 in addition to the regularity conditions on σ. The verification of the Cramer condition under even the i.i.d. noise assumption as in Assumption 2.1 may involve nontrivial technical work. The added challenge is readily illustrated by computing the distribution of the random variable Zm,i−Bm,i in a toy model where εt is i.i.d. N(0,ω2) ⁠. It is easy to see that in this case Zm,i=dμm,i·χ2(1) ⁠, where χ2(1) denotes the standard chi‐squared distribution with one degree of freedom, whereas Bm,i=dψ1kndn2kn2ψ2knω2·∑i=1kn−1Ũi2, where (Ũi)i=1kn−1 are one‐dependent standard normal random variables with Cov (Ũi,Ũi−1)=−1 ⁠. In addition, Zm,i and Bm,i are dependent. Thus, in this relatively simple context, one could ensure the validity of the Cramer condition by showing that Zm,i−Bm,i have a nonlattice distribution, something we have not attempted to prove in this paper. In the presence of noise, it would clearly be desirable to have a formal proof of the verification of the Cramer condition, but this is beyond the scope of this paper. Our approach in this section is similar to the approaches used, e.g., by Mammen (1993), Davidson and Flachaire (2008) and GM2009. Our main focus is on using formal Edgeworth expansions to explain the superior finite sample properties of the wild bootstrap procedure applied on the non‐overlapping pre‐averaged returns as recently studied by Gonçalves et al. (2014). Note, however, that in contrast to GM2009 (under no noise), we explicitly provide (high level) sufficient conditions that ensure the validity of our Edgeworth expansions in the noisy setting. Corollary 3.1. Let Tn and T̃n be defined as Tn=n1/4(Γ̂n−Γ)V̂nandT̃n=n1/4(Γ̂n+b̃n−(Γ+b̃))V̂n, where Γ̂n and V̂n are given by 2.16 and 2.18, respectively and b̃n=ψ1kn2kn2ψ2kn∑i=1n(ΔinY)2 and b̃=ψ1θ2ψ2ω2. Suppose (2.1) holds with b=0 ⁠. Under Assumptions 2.1, 2.3, 3.1 and 3.2, and conditionally on σ, the formal second‐order Edgeworth expansions of the studentized statistics Tn and T̃n are exactly the same. Remark 3.2. The bias term in the pre‐averaging estimator does not affect the second‐order Edgeworth expansion. Here, we provide the main idea behind this via a toy example involving normalized i.i.d. statistics. Let (Mn,i)i=1n and (Nn,i)i=1n be two triangular arrays of row‐wise i.i.d. random variables with mean zero and order O(1). Let σn2=E[Mn,12] and μn,3=E[Mn,13] ⁠. Define Sn=1σnn∑i=1nMn,iandUn=1σnn∑i=1nMn,i+1nNn,i. It is well known that, under the existence of third moments and the Cramer condition, the second‐order Edgeworth expansion of Sn is Φ(x)+1nμn,36σn3φ(x), where Φ(x) and φ(x) are the distribution and the density functions of the standard normal. It turns out that the term Un also has the same Edgeworth expansion if Mn,i and Nn,i are weakly correlated. Let us assume that E[Mn,1Nn,1]=O(n−1/2) ⁠. Then sn2≡ Var (Mn,1+n−1/2Nn,1)=σn2+2n−1/2E[Mn,1Nn,1]+n−1E[Nn,12]=σn2+O(n−1).(3.14) Now, we decompose Un as Un=1snn∑i=1nMn,i+1nNn,i+1σnn−1snn∑i=1nMn,i+1nNn,i≡Ûn+Rn. We note that the term Rn does not contribute to the second‐order Edgeworth expansion of Un due to 3.14. Also, Ûn has the same form as Sn (i.e., a normalized statistic) and hence possesses the same second‐order Edgeworth expansion in view of E[(Mn,1+n−1/2Nn,1)3]=μn,3+O(n−1/2)andsn3=σn3+O(n−1/2). Proposition 3.2. Let Tn and Tn* be defined as in 2.17 and (2.19), respectively. Suppose that vi has the same distribution as in Proposition 3.1 and (2.1) holds with b=0 ⁠. Under Assumptions 2.1, 2.3, 3.1 and 3.2, conditionally on σ, as n→∞ ⁠, we obtain supx∈R|P*[Tn*≤x]−P[Tn≤x]|=op(n−1/4). Proposition 3.2 shows the second‐order validity of the wild bootstrap method in the noisy setting and hence extends the result obtained in Proposition 3.1. 3.3. Edgeworth corrected interval for realized volatility estimators Our aim in this section is to explain how one can use the Edgeworth expansions derived in Sections 3.1 and 3.2 to construct valid confidence intervals for integrated volatility with improved coverage probabilities. Our approach follows Hall (1992); see also GM2008. In particular, based on Edgeworth expansions of Γ̂n ⁠, we define confidence intervals for Γ corrected by these Edgeworth expansions. Here, we consider one‐sided Edgeworth expansion corrected intervals for Γ. One can show that (see, e.g., Podolskij and Vetter, 2009), as n→∞ R̃r,s→par,sσ̃r,s. Thus, when the log price process follows (2.1) with b=0, we propose the following feasible (empirical) version of q1(x) ⁠: q̂1(x)=4(2x2+1)a6,4−1R̃6,462.(3.15) A one‐sided feasible Edgeworth expansion corrected 100(1−α)% level interval for Γ is given by IC feas ,1−αEE−1=−∞,Γ̂n−τn−1V̂nzα+τn−2V̂nq̂1(zα).(3.16) In contrast to the conventional intervals based on the normal approximation, this interval contains a skewness correction term equal to τn−2V̂nq̂1(zα) ⁠. Here, we do not pursue the derivation of a two‐sided symmetric feasible Edgeworth expansion corrected 100(1−α)% level interval for Γ. The main reason is because this interval, in contrast to IC feas ,1−αEE−1 ⁠, would involve in addition to a skewness term a kurtosis correction term, which is not available under results derived in Theorems 3.1 and 3.2.2 Remark 3.3. Our setting rules out leverage effects, which is the case when σ and W are correlated. Indeed, under no leverage assumptions, it is possible for us to condition on the path of σ and then use the independence of increments. However, if σ and W are correlated, we only have a martingale difference sequence instead of the independence property, and hence this approach breaks down. Recent work by Yoshida (2013) develops a general theory to deal with Edgeworth expansions involving mixed normal limits. We note that this work relies on very technical tools from Malliavin calculus which are beyond the scope of this paper. Podolskij and Yoshida (2016) apply this theory within the framework of power variations of diffusion processes. Although the last work allows leverage effects, it is assumed that σ is driven (only) by the original Brownian motion W, thereby excluding stochastic volatility models. While these works are limited to the setting of continuous volatility, our setting allows (in particular, in the no‐noise case) discontinuous volatility paths. 4. Monte Carlo Simulations Our aim here is to compare the finite sample performance of the Edgeworth expansion corrected intervals in comparison to the feasible asymptotic theory‐based intervals and the bootstrap method of Gonçalves et al. (2014) using the noisy diffusion model. The design of our Monte Carlo study is roughly identical to that used by Gonçalves et al. (2014) with some minor differences. In particular, we only consider the two‐factor stochastic volatility (SV2F) model analysed by Gonçalves et al. (2014) because it is more empirically relevant and exhibits overall larger coverage distortions than the one‐factor stochastic volatility model. Here we briefly describe the Monte Carlo design we use. To simulate log‐prices, we consider the following SV2F model, where3 dXt=bdt+σtdWt,σt=s‐exp(β0+β1τ1t+β2τ2t),dτ1t=α̃1τ1tdt+dB1t,dτ2t=α̃2τ2tdt+(1+φτ2t)dB2t, corr (dWt,dB1t)=ϕ1, corr (dWt,dB2t)=ϕ2. Our baseline model sets b=0 and ϕ1=ϕ2=0 ⁠, which is compatible with the assumption of no leverage and no drift. While the theory of the Edgeworth expansion developed in this paper does not allow the leverage effect, we have also studied this set‐up, which is nevertheless an obvious interest in practice and set b=0.03 and ϕ1=ϕ2=−0.3 ⁠. In both cases, we follow Huang and Tauchen (2005) and set β0=−1.2 ⁠, β1=0.04 ⁠, β2=1.5 ⁠, α̃1=−0.00137 ⁠, α̃2=−1.386 ⁠, and φ=0.25 ⁠. We initialize the two factors at the start of each interval by drawing the persistent factor from its unconditional distribution, τ10∼N(0,(−1/2α̃1)) ⁠, and by starting the strongly mean‐reverting factor at zero. For i=1,...,n ⁠, we let the market microstructure noise be defined as εi/n∼ i.i.d. N(0,ω2) ⁠. The size of the noise is an important parameter. We follow Barndorff‐Nielsen et al. (2008) and model the noise magnitude as ξ2=ω2/∫01σs4ds ⁠. We fix ξ2 equal to 0.0001, 0.001 and 0.01, and let ω2=ξ2∫01σs4ds ⁠. These values are motivated by the empirical study of Hansen and Lunde (2006), who investigate 30 stocks of the Dow Jones Industrial Average. 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 1/n‐horizon returns ΔinY≡Yi/n−Y(i−1)/n based on samples of size n. The pre‐averaging approach requires the choice of the window length kn=θn over which the pre‐averaging of returns is done. In our simulations, we follow Christensen et al. (2010) and use a conservative choice of kn (this corresponds to θ=1 ⁠). 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, Γ̂n=1−ψ1kn2kn2ψ2kn−11ψ2kn∑m=0dn−1Y¯mkn2−ψ1kn2kn2ψ2kn∑i=1n(ΔinY)2, where ψ1kn=kn∑i=1kngikn−gi−1kn2 and ψ2kn=(1/kn)∑i=1kng2(i/kn) ⁠. Table 1 gives the actual rates of 95% one‐sided confidence intervals of integrated volatility for the SV2F model, computed over 10,000 replications. Results are presented for eight different samples sizes: n= 23,400, 11,700, 7,800, 4,680, 1,560, 780, 390 and 195, corresponding to 1‐second, 2‐second, 3‐second, 5‐second, 15‐second, 30‐second, 1‐minute and 2‐minute frequencies, respectively. In our simulations, bootstrap intervals use 999 bootstrap replications for each of the 10,000 Monte Carlo replications. We consider one‐sided bootstrap percentile‐t interval computed at the 95% level given by 2.11. To generate the bootstrap data, we use three different external random variables, as follows. WB1:The two‐point distribution initially proposed by GM2009, where vj∼ i.i.d. such that vj=1531+186,withprobabilityp=12−3186−1531−186,withprobability1−p, for which we have μ2*=1 and μ4*=31/25 ⁠. WB2:A two‐point distribution vj∼ i.i.d. such that vj=231/4−1+52,withprobabilityp=5−125231/4−1−52,withprobability1−p=5+125, for which μ2*=22/3 and μ4*=10/3 ⁠. WB3:The new optimal nonlattice distribution vj∼ i.i.d. with the same distribution as in Proposition 3.1. Table 1. Coverage rate of nominal 95% n . No leverage and no drift . With leverage and drift . . CLT . EE‐est . WB1 . WB2 . WB3 . CLT . EE‐est . WB1 . WB2 . WB3 . ξ2=0.0001 195 67.98 76.32 77.56 86.87 80.06 68.77 76.23 77.98 86.84 80.45 390 76.07 82.83 85.71 90.01 85.17 76.01 83.05 86.16 90.24 85.50 780 78.44 85.11 88.65 90.64 86.13 77.98 84.51 88.22 90.33 87.10 1560 83.21 88.72 92.32 92.42 89.56 83.63 88.78 92.54 92.52 89.57 4680 84.73 89.37 93.37 92.19 90.20 84.65 89.47 93.50 92.46 90.41 7800 86.31 90.60 94.42 93.05 91.29 86.32 90.63 94.47 93.06 91.57 11700 87.07 91.44 95.17 93.47 91.87 87.70 91.69 95.10 93.79 92.05 23400 88.26 91.86 95.90 93.59 92.41 88.43 92.04 95.85 93.80 92.16 ξ2=0.001 195 68.20 76.71 78.13 86.98 80.42 68.76 76.60 78.67 87.08 80.42 390 76.21 83.20 85.86 89.88 85.36 76.06 83.40 86.19 90.11 85.58 780 78.71 85.19 88.85 90.69 86.16 77.86 84.59 88.30 90.28 86.97 1560 83.39 88.67 92.24 92.34 89.65 83.55 88.82 92.53 92.75 89.85 4680 84.82 89.51 93.59 92.27 90.09 84.69 89.77 93.58 92.65 90.36 7800 86.38 90.93 94.37 93.16 91.25 86.31 90.66 94.46 93.10 91.76 11700 87.16 91.39 95.10 93.32 91.94 87.66 91.65 95.10 93.79 92.15 23400 88.40 91.89 95.70 93.58 92.36 88.40 91.89 95.80 93.82 92.10 ξ2=0.01 195 70.55 78.70 81.07 87.61 81.11 70.21 78.66 80.65 87.09 81.25 390 77.63 84.36 87.64 90.46 86.24 77.35 84.14 87.88 90.42 86.65 780 79.84 86.12 89.98 91.16 86.56 79.21 85.22 89.29 90.13 87.13 1560 84.09 89.24 92.98 92.49 90.48 84.00 89.33 93.15 92.54 90.45 4680 85.31 90.18 94.18 92.87 90.54 85.43 90.20 94.35 93.02 90.88 7800 86.82 91.02 94.88 93.23 91.53 86.81 90.60 94.56 92.94 91.59 11700 87.59 91.35 95.23 93.35 92.20 88.05 91.72 95.05 93.50 92.38 23400 88.76 92.05 95.87 93.53 92.86 89.01 92.47 95.95 94.02 92.43 n . No leverage and no drift . With leverage and drift . . CLT . EE‐est . WB1 . WB2 . WB3 . CLT . EE‐est . WB1 . WB2 . WB3 . ξ2=0.0001 195 67.98 76.32 77.56 86.87 80.06 68.77 76.23 77.98 86.84 80.45 390 76.07 82.83 85.71 90.01 85.17 76.01 83.05 86.16 90.24 85.50 780 78.44 85.11 88.65 90.64 86.13 77.98 84.51 88.22 90.33 87.10 1560 83.21 88.72 92.32 92.42 89.56 83.63 88.78 92.54 92.52 89.57 4680 84.73 89.37 93.37 92.19 90.20 84.65 89.47 93.50 92.46 90.41 7800 86.31 90.60 94.42 93.05 91.29 86.32 90.63 94.47 93.06 91.57 11700 87.07 91.44 95.17 93.47 91.87 87.70 91.69 95.10 93.79 92.05 23400 88.26 91.86 95.90 93.59 92.41 88.43 92.04 95.85 93.80 92.16 ξ2=0.001 195 68.20 76.71 78.13 86.98 80.42 68.76 76.60 78.67 87.08 80.42 390 76.21 83.20 85.86 89.88 85.36 76.06 83.40 86.19 90.11 85.58 780 78.71 85.19 88.85 90.69 86.16 77.86 84.59 88.30 90.28 86.97 1560 83.39 88.67 92.24 92.34 89.65 83.55 88.82 92.53 92.75 89.85 4680 84.82 89.51 93.59 92.27 90.09 84.69 89.77 93.58 92.65 90.36 7800 86.38 90.93 94.37 93.16 91.25 86.31 90.66 94.46 93.10 91.76 11700 87.16 91.39 95.10 93.32 91.94 87.66 91.65 95.10 93.79 92.15 23400 88.40 91.89 95.70 93.58 92.36 88.40 91.89 95.80 93.82 92.10 ξ2=0.01 195 70.55 78.70 81.07 87.61 81.11 70.21 78.66 80.65 87.09 81.25 390 77.63 84.36 87.64 90.46 86.24 77.35 84.14 87.88 90.42 86.65 780 79.84 86.12 89.98 91.16 86.56 79.21 85.22 89.29 90.13 87.13 1560 84.09 89.24 92.98 92.49 90.48 84.00 89.33 93.15 92.54 90.45 4680 85.31 90.18 94.18 92.87 90.54 85.43 90.20 94.35 93.02 90.88 7800 86.82 91.02 94.88 93.23 91.53 86.81 90.60 94.56 92.94 91.59 11700 87.59 91.35 95.23 93.35 92.20 88.05 91.72 95.05 93.50 92.38 23400 88.76 92.05 95.87 93.53 92.86 89.01 92.47 95.95 94.02 92.43 Notes CLT refers to intervals based on the Normal. EE‐est refers to the value based on Edgeworth expansion corrected intervals. WB1 refers to wild bootstrap intervals based on the external random variable WB1. WB2 refers to wild bootstrap intervals based on the external random variable WB2. WB3 refers to wild bootstrap intervals based on the external random variable WB3. There were 10,000 Monte Carlo trials with 999 bootstrap replications each. Open in new tab Table 1. Coverage rate of nominal 95% n . No leverage and no drift . With leverage and drift . . CLT . EE‐est . WB1 . WB2 . WB3 . CLT . EE‐est . WB1 . WB2 . WB3 . ξ2=0.0001 195 67.98 76.32 77.56 86.87 80.06 68.77 76.23 77.98 86.84 80.45 390 76.07 82.83 85.71 90.01 85.17 76.01 83.05 86.16 90.24 85.50 780 78.44 85.11 88.65 90.64 86.13 77.98 84.51 88.22 90.33 87.10 1560 83.21 88.72 92.32 92.42 89.56 83.63 88.78 92.54 92.52 89.57 4680 84.73 89.37 93.37 92.19 90.20 84.65 89.47 93.50 92.46 90.41 7800 86.31 90.60 94.42 93.05 91.29 86.32 90.63 94.47 93.06 91.57 11700 87.07 91.44 95.17 93.47 91.87 87.70 91.69 95.10 93.79 92.05 23400 88.26 91.86 95.90 93.59 92.41 88.43 92.04 95.85 93.80 92.16 ξ2=0.001 195 68.20 76.71 78.13 86.98 80.42 68.76 76.60 78.67 87.08 80.42 390 76.21 83.20 85.86 89.88 85.36 76.06 83.40 86.19 90.11 85.58 780 78.71 85.19 88.85 90.69 86.16 77.86 84.59 88.30 90.28 86.97 1560 83.39 88.67 92.24 92.34 89.65 83.55 88.82 92.53 92.75 89.85 4680 84.82 89.51 93.59 92.27 90.09 84.69 89.77 93.58 92.65 90.36 7800 86.38 90.93 94.37 93.16 91.25 86.31 90.66 94.46 93.10 91.76 11700 87.16 91.39 95.10 93.32 91.94 87.66 91.65 95.10 93.79 92.15 23400 88.40 91.89 95.70 93.58 92.36 88.40 91.89 95.80 93.82 92.10 ξ2=0.01 195 70.55 78.70 81.07 87.61 81.11 70.21 78.66 80.65 87.09 81.25 390 77.63 84.36 87.64 90.46 86.24 77.35 84.14 87.88 90.42 86.65 780 79.84 86.12 89.98 91.16 86.56 79.21 85.22 89.29 90.13 87.13 1560 84.09 89.24 92.98 92.49 90.48 84.00 89.33 93.15 92.54 90.45 4680 85.31 90.18 94.18 92.87 90.54 85.43 90.20 94.35 93.02 90.88 7800 86.82 91.02 94.88 93.23 91.53 86.81 90.60 94.56 92.94 91.59 11700 87.59 91.35 95.23 93.35 92.20 88.05 91.72 95.05 93.50 92.38 23400 88.76 92.05 95.87 93.53 92.86 89.01 92.47 95.95 94.02 92.43 n . No leverage and no drift . With leverage and drift . . CLT . EE‐est . WB1 . WB2 . WB3 . CLT . EE‐est . WB1 . WB2 . WB3 . ξ2=0.0001 195 67.98 76.32 77.56 86.87 80.06 68.77 76.23 77.98 86.84 80.45 390 76.07 82.83 85.71 90.01 85.17 76.01 83.05 86.16 90.24 85.50 780 78.44 85.11 88.65 90.64 86.13 77.98 84.51 88.22 90.33 87.10 1560 83.21 88.72 92.32 92.42 89.56 83.63 88.78 92.54 92.52 89.57 4680 84.73 89.37 93.37 92.19 90.20 84.65 89.47 93.50 92.46 90.41 7800 86.31 90.60 94.42 93.05 91.29 86.32 90.63 94.47 93.06 91.57 11700 87.07 91.44 95.17 93.47 91.87 87.70 91.69 95.10 93.79 92.05 23400 88.26 91.86 95.90 93.59 92.41 88.43 92.04 95.85 93.80 92.16 ξ2=0.001 195 68.20 76.71 78.13 86.98 80.42 68.76 76.60 78.67 87.08 80.42 390 76.21 83.20 85.86 89.88 85.36 76.06 83.40 86.19 90.11 85.58 780 78.71 85.19 88.85 90.69 86.16 77.86 84.59 88.30 90.28 86.97 1560 83.39 88.67 92.24 92.34 89.65 83.55 88.82 92.53 92.75 89.85 4680 84.82 89.51 93.59 92.27 90.09 84.69 89.77 93.58 92.65 90.36 7800 86.38 90.93 94.37 93.16 91.25 86.31 90.66 94.46 93.10 91.76 11700 87.16 91.39 95.10 93.32 91.94 87.66 91.65 95.10 93.79 92.15 23400 88.40 91.89 95.70 93.58 92.36 88.40 91.89 95.80 93.82 92.10 ξ2=0.01 195 70.55 78.70 81.07 87.61 81.11 70.21 78.66 80.65 87.09 81.25 390 77.63 84.36 87.64 90.46 86.24 77.35 84.14 87.88 90.42 86.65 780 79.84 86.12 89.98 91.16 86.56 79.21 85.22 89.29 90.13 87.13 1560 84.09 89.24 92.98 92.49 90.48 84.00 89.33 93.15 92.54 90.45 4680 85.31 90.18 94.18 92.87 90.54 85.43 90.20 94.35 93.02 90.88 7800 86.82 91.02 94.88 93.23 91.53 86.81 90.60 94.56 92.94 91.59 11700 87.59 91.35 95.23 93.35 92.20 88.05 91.72 95.05 93.50 92.38 23400 88.76 92.05 95.87 93.53 92.86 89.01 92.47 95.95 94.02 92.43 Notes CLT refers to intervals based on the Normal. EE‐est refers to the value based on Edgeworth expansion corrected intervals. WB1 refers to wild bootstrap intervals based on the external random variable WB1. WB2 refers to wild bootstrap intervals based on the external random variable WB2. WB3 refers to wild bootstrap intervals based on the external random variable WB3. There were 10,000 Monte Carlo trials with 999 bootstrap replications each. Open in new tab Note that all of these choices of vj are asymptotically valid when used to construct bootstrap percentile‐t intervals. As we formally show in this paper, the choice of WB3 is still optimal to provide a second‐order asymptotic refinement for the wild bootstrap method applied on the non‐overlapping pre‐averaged returns. The wild bootstrap based on WB1 is able to match the first and third cumulants of pre‐averaged realized volatility, but as a lattice distribution might not satisfy the Cramer condition. Based on simulation results, Gonçalves et al. (2014) advocated the use of WB2. In Table 1, ‘CLT’ refers to the value predicted by the normal asymptotic, ‘EE‐est’ refers to the value based on Edgeworth expansion corrected intervals, whereas ‘WB1’, ‘WB2’ and ‘WB3’ refer to the values predicted by the bootstrap method based on external random variables WB1, WB2 and WB3, respectively. Starting with the baseline model no leverage and no drift, an inspection of Table 1 suggests that all intervals tend to under‐cover. The degree of under‐coverage is especially large for smaller values of n, when sampling is not too frequent. The results seem to be not very sensitive to the noise magnitude. One‐sided confidence intervals based on the asymptotic normal theory (without higher‐order correction) is not adequate to capture the skewness in the t‐statistics (as confirmed by simulations not reported here). Gonçalves et al. (2014) (see their Section 3) also found a similar pattern for symmetric two‐sided confidence intervals. See Gonçalves et al. (2014) for more results on the comparison between this model from the point of view of skewness and kurtosis. Overall, WB2 does very well for small samples (⁠ n=195 ⁠, 390 and 780) whereas WB1‐ and WB3‐based intervals do very well for large samples (⁠ n= 11,700 and 23,400). For instance, when ξ2=0.0001 ⁠, WB2 has a coverage probability equal to 86.87% when n=195 ⁠, whereas WB1 and WB3 cover integrated volatility only 77.56% and 80.06% of the time, respectively. These rates increase to 93.59%, 95.90% and 92.41%, respectively, for n= 23,400. The results also confirm that our expansion theory provides a good approximation of the small sample distribution of the pre‐averaged realized volatility estimator of Podolskij and Vetter (2009). In particular, for all sample sizes considered here, the intervals based on the Edgeworth corrections (EE‐est) have improved properties relative to the conventional intervals based on the normal approximation. Contrary to the bootstrap, the Edgeworth approach is an analytical approach that is easily implemented, without requiring any resampling of one's data. A comparison between the bootstrap (WB1, WB2 and WB3) and the Edgeworth expansion shows that the bootstrap outperforms the Edgeworth corrected intervals. For instance, when ξ2=0.001 ⁠, and we resample every five seconds (⁠ n= 4,680), the CLT‐based interval has a coverage probability equal to 84.82%, whereas the EE‐est‐based interval covers integrated volatility 89.51% of the time. For the bootstrap, these rates increase to 93.59%, 92.27% and 90.09% for WB1, WB2 and WB3, respectively. Notice, however, that results based on the WB1 and WB3 intervals are close, but slightly different especially for small samples (⁠ n=195 ⁠, 390 and 780). This observation suggests that the dominance of WB1 by WB2 for n small is not due to the possible non‐validity of the Edgeworth expansions for realized volatility based on WB1 (i.e., the optimal two‐point distribution wild bootstrap). The good performance of WB2 over WB1 and WB3 for smaller sample size is similar to the superior performance of the i.i.d. bootstrap over the optimal two‐point distribution WB1 in GM2009. Indeed, the Monte Carlo simulations in GM2009 show that despite the fact that the i.i.d. bootstrap does not theoretically provide an asymptotic refinement for one‐sided confidence intervals when the volatility is stochastic, this latter outperforms WB1. Accordingly, it would be useful to develop a new theory that provides a more reliable guide to gauge the finite sample performance of the bootstrap for financial high‐frequency data. Some initial simulation results (not reported here) confirmed that the same pattern is also observed in the absence of market microstructure noise effect in the toy model with constant volatility and no drift (⁠ σ=1 and b=0 ⁠; i.e., dYt=dXt=dWt ⁠). In particular, for very small sample sizes, the ad hoc choice used in Gonçalves et al. (2014) – that is, WB2 (lattice) – which cannot be explained by our theory, seems to dominate WB3 (non‐lattice), which rigorously verifies all our conditions. This suggests that a formal treatment of the good behaviour of WB2 (lattice) requires a different approach (e.g., the development of Edgeworth expansion for lattice distributions, where observations are heterogeneously distributed). This is beyond the scope of this paper. A similar pattern is observed for all intervals in the presence of drift and leverage effects. For all methods, results are robust to drift and leverage effects. In particular, despite the fact that our Edgeworth expansion corrected intervals do not theoretically take into account these effects, EE‐est‐based intervals outperform the CLT‐based intervals in the presence of drift and leverage effects. 5. Conclusion The main contribution of this paper has been to establish the theoretical validity of the Edgeworth expansions for realized volatility estimators. Furthermore, we propose a new optimal nonlattice distribution for the wild bootstrap suggested by GM2009, which is able to provide a second‐order asymptotic refinement. In the presence of microstructure noise, based on our Edgeworth expansions, we show that the new optimal choice proposed in the absence of noise is still valid in noisy data for the pre‐averaged realized volatility estimator proposed by Podolskij and Vetter (2009). Finally, we also propose confidence intervals for integrated volatility, which incorporate an analytical correction for skewness as alternative method of inference. Thus, we extend existing results in GM2008 by allowing for microstructure noise. The results of our Monte Carlo study show that the Edgeworth‐based coverage probabilities provide very accurate approximations to the sample ones compared to the normal‐based coverage probabilities. A comparison between the bootstrap and the Edgeworth expansion shows that the bootstrap‐based intervals outperform the Edgeworth corrected intervals. In the process of developing the expansions for realized volatility estimators, we also show how to derive the second‐order Edgeworth expansions of a certain form of studentized statistic, where observations are independent but not identically distributed with specific heterogeneity properties (Proposition A.1 in the Appendix). This result should have applications to other situations. Establishing the validity of the Edgeworth expansions for realized volatility estimators under general conditions that allow drift and leverage effects, as for instance in Barndorff‐Nielsen et al. (2006), is a promising extension of this work. Another important extension is to prove similar results for other existing noise and/or jump robust realized volatility measures. These extensions are left for future research. Acknowledgements We would like to thank Anders Bredahl Kock, Sílvia Gonçalves and Mark Podolskij for many useful comments and discussions on the first version of the paper. We acknowledge support from the Center for Research in Econometric Analysis of Time Series (CREATES), funded by the Danish National Research Foundation (DNRF78). References Aït‐Sahalia , Y. , P. A. Mykland and L. Zhang ( 2011 ). Ultra high frequency volatility estimation with dependent microstructure noise . Journal of Econometrics 160 , 160 – 75 . Google Scholar Crossref Search ADS WorldCat Andersen , T. G. , T. Bollerslev and F. X. Diebold ( 2010 ). Parametric and nonparametric volatility measurement . In L. P. Hansen and Y. Ait‐Sahalia (Eds.), Handbook of Financial Econometrics , 67 – 138 . Amsterdam : North‐Holland . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Andersen , T. G. , T. Bollerslev, F. X. Diebold and P. Labys ( 2001 ). The distribution of realized exchange rate volatility . Journal of the American Statistical Association 96 , 42 – 55 . Google Scholar Crossref Search ADS WorldCat Babu , G. J. and K. Singh ( 1983 ). Inference on means using the bootstrap . Annals of Statistics 11 , 999 – 1003 . Google Scholar Crossref Search ADS WorldCat Babu , G. J. and K. Singh ( 1984 ). On one term Edgeworth correction by Efron's bootstrap . Sankhyā , Series A 46 , 219 – 32 . OpenURL Placeholder Text WorldCat Bandi , F. M. and J. R. Russell ( 2008 ). Microstructure noise, realized variance, and optimal sampling . Review of Economic Studies 75 , 339 – 69 . Google Scholar Crossref Search ADS WorldCat Barndorff‐Nielsen , O. E. , S. E. Graversen, J. Jacod and N. Shephard ( 2006 ). Limit theorems for bipower variation in financial econometrics . Econometric Theory 22 , 677 – 719 . Google Scholar Crossref Search ADS WorldCat Barndorff‐Nielsen , O. E. , P. R. Hansen, A. Lunde and N. Shephard ( 2008 ). Designing realised kernels to measure the ex‐post variation of equity prices in the presence of noise . Econometrica 76 , 1481 – 536 . Google Scholar Crossref Search ADS WorldCat Barndorff‐Nielsen , O. E. and N. Shephard ( 2002 ). Econometric analysis of realized volatility and its use in estimating stochastic volatility models . Journal of the Royal Statistical Society , Series B 64 , 253 – 80 . Google Scholar Crossref Search ADS WorldCat Barndorff‐Nielsen , O. E. and N. Shephard ( 2003 ). Realized power variation and stochastic volatility models . Bernoulli 9 , 243 – 65 . Google Scholar Crossref Search ADS WorldCat Barndorff‐Nielsen , O. E. and N. Shephard ( 2007 ). Variation, jumps, market frictions and high frequency data in financial econometrics . In R. Blundell, P. Torsten and W. K. Newey (Eds.), Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress , Volume III, 328 – 372 . Cambridge : Cambridge University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Bhattacharya , R. N. and R. R. Rao ( 1986 ). Normal approximation and asymptotic expansions . Melbourne, FL : Robert E. Krieger Publishing . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Christensen , K. , S. Kinnebrock and M. Podolskij ( 2010 ). Pre‐averaging estimators of the ex‐post covariance matrix in noisy diffusion models with non‐synchronous data . Journal of Econometrics 159 , 116 – 33 . Google Scholar Crossref Search ADS WorldCat Comte , F. and E. Renault ( 1998 ). Long memory in continuous‐time stochastic volatility models . Mathematical Finance 8 , 291 – 323 . Google Scholar Crossref Search ADS WorldCat Davidson , R. and E. Flachaire ( 2008 ). The wild bootstrap, tamed at last . Journal of Econometrics 146 , 162 – 69 . Google Scholar Crossref Search ADS WorldCat Gonçalves , S. , U. Hounyo and N. Meddahi ( 2014 ). Bootstrap inference for pre‐averaged realized volatility based on non‐overlapping returns . Journal of Financial Econometrics 12 , 679 – 707 . Google Scholar Crossref Search ADS WorldCat Gonçalves , S. and N. Meddahi ( 2008 ). Edgeworth corrections for realized volatility . Econometric Reviews 27 , 139 – 62 . Google Scholar Crossref Search ADS WorldCat Gonçalves , S. and N. Meddahi ( 2009 ). Bootstrapping realized volatility . Econometrica 77 , 283 – 306 . Google Scholar Crossref Search ADS WorldCat Hall , P. ( 1992 ). The Bootstrap and Edgeworth Expansion , Springer Series in Statistics. New York, NY : Springer . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Hansen , P. R. and A. Lunde ( 2006 ). Realized variance and market microstructure noise . Journal of Business and Economic Statistics 24 , 127 – 61 . Google Scholar Crossref Search ADS WorldCat Hautsch , N. and M. Podolskij ( 2013 ). Pre‐averaging based estimation of quadratic variation in the presence of noise and jumps: theory, implementation, and empirical evidence . Journal of Business and Economic Statistics 31 , 165 – 83 . Google Scholar Crossref Search ADS WorldCat Hounyo , U. ( 2013 ). Bootstrapping realized volatility and realized beta under a local Gaussianity assumption . Working paper 2013‐30, CREATES, Aarhus University. Hounyo , U. , S. Gonçalves and N. Meddahi ( 2013 ). Bootstrapping pre‐averaged realized volatility under market microstructure noise . Working paper 2013‐28, CREATES, Aarhus University. Huang , X. and G. Tauchen ( 2005 ). The relative contribution of jumps to total price variance . Journal of Financial Econometrics 3 , 456 – 99 . Google Scholar Crossref Search ADS WorldCat Jacod , J. , Y. Li, P. A. Mykland, M. Podolskij and M. Vetter ( 2009 ). Microstructure noise in the continuous case: the pre‐averaging approach . Stochastic Processes and their Applications 119 , 2249 – 76 . Google Scholar Crossref Search ADS WorldCat Jacod , J. and P. E. Protter ( 1998 ). Asymptotic error distributions for the Euler method for stochastic differential equations . Annals of Probability 26 , 267 – 307 . Google Scholar Crossref Search ADS WorldCat Lahiri , S. N. ( 2003 ). Resampling Methods for Dependent Data , Springer Series in Statistics. New York, NY : Springer . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Lieberman , O. and P. C. B. Phillips ( 2008 ). Refined inference on long memory in realized volatility . Econometric Reviews 27 , 254 – 67 . Google Scholar Crossref Search ADS WorldCat Liu , R. Y. ( 1988 ). Bootstrap procedures under some non‐i.i.d. models . Annals of Statistics 16 , 1696 – 708 . Google Scholar Crossref Search ADS WorldCat Mammen , E. ( 1993 ). Bootstrap and wild bootstrap for high‐dimensional linear models . Annals of Statistics 21 , 255 – 85 . Google Scholar Crossref Search ADS WorldCat Meddahi , N. ( 2002 ). A theoretical comparison between integrated and realized volatility . Journal of Applied Econometrics 17 , 479 – 508 . Google Scholar Crossref Search ADS WorldCat Podolskij , M. and M. Vetter ( 2009 ). Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps . Bernoulli 15 , 634 – 58 . Google Scholar Crossref Search ADS WorldCat Podolskij , M. and N. Yoshida ( 2016 ). Edgeworth expansion for functionals of continuous diffusion processes . Forthcoming in Annals of Appled Probability . OpenURL Placeholder Text WorldCat Xiu , D. ( 2010 ). Quasi‐maximum likelihood estimation of volatility with high frequency data . Journal of Econometrics 159 , 235 – 50 . Google Scholar Crossref Search ADS WorldCat Yoshida , N. ( 2013 ). Martingale expansion in mixed normal limit . Stochastic Processes and their Applications 123 , 887 – 933 . Google Scholar Crossref Search ADS WorldCat Zhang , L. ( 2006 ). Efficient estimation of stochastic volatility using noisy observations: a multi‐scale approach . Bernoulli 12 , 1019 – 43 . Google Scholar Crossref Search ADS WorldCat Zhang , L. , P. A. Mykland and Y. Aït‐Sahalia ( 2005 ). A tale of two time scales: determining integrated volatility with noisy high‐frequency data . Journal of the American Statistical Association 100 , 1394 – 411 . Google Scholar Crossref Search ADS WorldCat Zhang , L. , P. A. Mykland and Y. Aït‐Sahalia ( 2011 ). Edgeworth expansions for realized volatility and related estimators . Journal of Econometrics 160 , 190 – 203 . Google Scholar Crossref Search ADS WorldCat Appendix: Proofs of the Validity of Edgeworth Expansion A.1 Auxiliary results The main goal of this section is to prove Proposition A.1. In Section A.2, we show that the main results of this paper belong to the framework of Proposition A.1. In this section, we deal with two‐dimensional random vectors. Transposes are denoted by ′ and for x=(x1,x2)′∈R2 and ν=(ν1,ν2)′∈N2 we use the notations ∥x∥=x12+x22andxν=(x1)ν1(x2)ν2. For the mean zero triangular array (Am,i) ⁠, m≥1 ⁠, 1≤i≤m and p≥2 ⁠, we denote ρm,p=m−1∑i=1mE[∥Am,i∥p]. Below, we provide sufficient conditions for the validity of the Edgeworth expansion. Assumption A.1. Let (Am,i)i=1m ⁠, m≥1 ⁠, be a row‐wise independent triangular array of two‐dimensional random vectors with mean zero such that: (a) for all m≥1 ⁠, we have (1/m)∑i=1mE[Am,iAm,i′]=I2 ⁠; (b) there exists δ>0 and C>0 such that E[∥Am,i∥3+δ]≤C for all i, m; (c) there exists M∈(0,1) such that |φm,i(t)|≤M for ∥t∥≥(16ρm,3)−1 and m≥1,1≤i≤m ⁠, where φm,i is the characteristic function of Am,i ⁠. We note that some initial Hermite polynomials are given by H0(x)=1,H1(x)=x,H2(x)=x2−1,H3(x)=x3−3x. For 1≤i≤m and ν∈N2 ⁠, we define χν,i=E[(Am,i)ν]andχ¯νm=m−1∑i=1mχν,i. We recall that the two‐dimensional polynomial appearing in the second‐order Edgeworth expansion is given by (see Section 7 in Bhattacharya and Rao, 1986) p1(t,s)=∑j=03χ¯(3−j,j)mH3−j(t)Hj(s)(3−j)!j!.(A.1) Now, we are ready to state the following classical result on Edgeworth expansions. Lemma A.1. Suppose that the two‐dimensional random vectors (Am,i)i=1m satisfy Assumption A.1. Let Sm=1m∑i=1mAm,i. Then, the second‐order Edgeworth expansion of Sm is given by P[Sm∈(−∞,y]×(−∞,z]]=∫−∞y∫−∞z(1+m−1/2p1(t,s))φ(s,t)dsdt+o(m−1/2) uniformly in x. Proof: The result follows from Theorem 6.2 in Lahiri (2003) as it is easy to show that Assumption A.1 satisfies the conditions of Theorem 6.2 in Lahiri (2003). □ Now, we are ready to describe the form of the t‐statistic. We define tm=(1/m)∑i=1m(Zm,i−μm,i−Bm,i)+(1/m)bmVm,(A.2) with Vm=a4−a22a41m∑i=1mZm,i2, where the structures of Zm,i ⁠, μm,i ⁠, Bm,i ⁠, bm and as are provided below. To prove the Edgeworth expansion for tm ⁠, we need the following conditions. Assumption A.2. We suppose that tm in A.2 satisfies the following. (a) For each m≥1 ⁠, the random vectors (Zm,i,Bm,i)i=1m are independent and Zm,i has the representation Zm,i=(αm,ium,i+βm,i)2 ⁠, where αm,i and βm,i are real triangular arrays and um,i∼U for all i, m. We denote as=E[|U|s] and impose a2>0 ⁠. In addition, we have E[Bm,i]=0 for all i, m. (b) Let us denote μm,i≡E[Zm,i] ⁠. There exists C>0 such that for all i and m, we obtain |E[Zm,i2]−a4a22μm,i2|+|E[Zm,i3]−a6a23μm,i3|+|E[Zm,iBm,i]|+|E[Zm,i2Bm,i]|≤Cm and E[|Zm,i|2(3+δ)]+E[|mBm,i|3+δ]≤C. (c) For all r>0 ⁠, there exists Mr∈(0,1) such that |φm,i(t)|≤Mr for all ∥t∥≥r and m≥1,1≤i≤m ⁠, where φm,i is the characteristic function of (Zm,i−μm,i−Bm,i,Zm,i2−E[Zm,i2])′ ⁠. (d) There exists ν>0 such that for all m≥1 we have ν≤1m∑i=1mE[Zm,i2]andν≤min(vm2,wm2−um2), where vm2=1m∑i=1m Var (Zm,i−μm,i−Bm,i),Em=1mvm∑i=1m(Zm,i−μm,i−Bm,i),Fm=a4−a22a41m∑i=1m(Zm,i2−E[Zm,i2]),um= Cov (Em,Fm),wm2= Var (Fm). (e) There exists b̂∈R and C>0 such that the real sequence (bm)m≥1 satisfies |bmvm−b̂|≤Cm. Note that when um,i∼N(0,1) ⁠, the first few even moments of um,i are given by a2=1 ⁠, a4=3 and a6=15 ⁠. We also use analogous results with the proposition below for the bootstrap, which can have different moments as ⁠. Before we state the main result, we need a final notation. For each p≥1 ⁠, we denote κm,p=1m∑i=1m(μm,i)p. Proposition A.1. Under Assumption A.2, we obtain P[tm≤x]=Φ(x)+m−1/2A12−16(B1−3A1)(x2−1)κm,3(κm,2)3/2−b̂φ(x)+o(m−1/2) uniformly in x, where A1=a6−a2a4a4(a4−a22)1/2andB1=a6−3a2a4+2a23(a4−a22)3/2. Proof: It will be convenient to write the studentized statistic in the following way: tm=Em+m−1/2bm/vmVm/vm2≡ẼmVm/vm2. Using the Taylor series for f(x)=x−1/2 of Vm/vm2 around 1, we obtain 1Vm/vm2=1−Vm−vm22vm2+38(ξm)5/2(Vm−vm2)2vm4, where ξm is between 1 and Vm/vm2 ⁠. Next, we observe that Assumption A.2(b) implies |(Vm−vm2)−1mFm|≤Cm,(A.3) for some C>0 ⁠. Using the above identities and Assumption A.2(e), we can decompose tm=Um+m−1/2b̂+Rm,(A.4) where the leading term is Um=Em−12mEmFmvm2, whereas the remainder term is given by Rm=−Ẽm2vm2Vm−vm2−1mFm+3Ẽm8(ξm)5/2(Vm−vm2)2vm4+1mbmvm−b̂−bmFm2mvm3≡Rm(1)+Rm(2)+Rm(3).(A.5) It suffices to show P[Um≤x]=Φ(x)+m−1/2A12−16(B1−3A1)(x2−1)κm,3(κm,2)3/2φ(x)+o(m−1/2),(A.6) P[tm≤x]=P[Um+m−1/2b̂≤x]+o(m−1/2),(A.7) uniformly in x, as the expansion in A.6 easily implies that the expansion of Um+m−1/2b̂ is the one stated in Proposition A.1. First, we prove A.6. Note that we cannot apply Lemma A.1 directly to (Em,Fm) ⁠, because it may not possess I2 covariance. For this purpose, we apply a certain transformation and denote Gm=−umEm+Fmwm2−um2. We want to use Lemma A.1 with (Em,Gm) and thus need to show that Assumption A.1 is satisfied. We easily observe that parts (a) and (b) are satisfied. For part (c), we note that the components of (Em,Gm) have the form C̃m(Zm,i−μm,i−Bm,i,Zm,i2−E[Zm,i2])′ ⁠, where C̃m=1vm0−umvmwm2−um2a4−a22a4wm2−um2. Also, given (3+δ) moments of Zmi2 and Bm,i in Assumption A.2(b), we obtain the result (16ρm,3)−1≥r1>0 ⁠, where ρm,3 belongs to (Em,Gm) ⁠. Now, let t12+t22≥r12 ⁠. The structure of the above matrix and Assumptions A.2(c) and (d) imply that it suffices to find some r2>0 such that (t1+t2ηm)2+γm2t22≥r22, where |ηm|≤η uniformly and |γm|≥γ>0 ⁠. We choose Δ such that 0<Δ<1 and Δη<1 ⁠. Then, r2=r1min(1−Δ2η2,Δγ) ⁠. This is easily seen by conditioning on |t2|≥Δr1 and |t2|<Δr1 ⁠. Thus, we obtain P[Em≤y,Gm≤z]=∫−∞y∫−∞z(1+m−1/2p1(t,s))φ(s,t)dsdt+o(m−1/2). Note that Um=Em+m−1/2[EmGm]Lm[EmGm]′, where Lm=−12vm2um(1/2)wm2−um2(1/2)wm2−um20≡cmbm/2bm/20. We obtain P[Um≤x]=∫∫{t+m−1/2(cmt2+bmts)≤x}1+p1(t,s)mφ(t,s)dsdt+o1m. To compute the above integral, we rely on Lemma 5 in Babu and Singh (1983); the proof of this result is provided on pp. 228–229 of Babu and Singh (1984). Although these results mention only the existence of a certain polynomial, a careful inspection of the proof yields an explicit polynomial in our setting. That is P[Um≤x]=∫−∞x∫1+p1(v,s)m1−2vcm+bmsm1+v(cmv2+bmvs)mφ(v,s)dsdv+o1m. Recalling A.1, we observe that several terms cancel in the above expression, which leads to P[Um≤x]=Φ(x)+φ(x)m2cm−cm(2+x2)+χ¯(3,0)m(1−x2)6+o1m=Φ(x)+φ(x)m−cmx2+χ¯(3,0)m(1−x2)6+o1m. Note that Assumption A.2 implies E[(Zm,i−μm,i−Bm,i)3]=(a6−3a2a4+2a23)μm,i3+O(m−1), Var (Zm,i−μm,i−Bm,i)=(a4−a22)μm,i2+O(m−1) uniformly in i. Hence, we obtain χ¯(3,0)m=1mvm3∑i=1mE[(Zm,i−μm,i−Bm,i)3]=B1κm,3(κm,2)3/2+O(m−1),cm= Cov (Em,Fm)−2vm2=−A12κm,3(κm,2)3/2+O(m−1). By inserting these values, we finish the proof of A.6. To prove A.7, we note that it suffices to show P[|Rm|≥m−a]=o(m−1/2)(A.8) for some a>1/2 (which will be chosen later). Indeed, using A.8 and the fact that the Edgeworth expansion of Um+m−1/2b̂ holds uniformly in x, we obtain P[tm≤x]≤P[Um+m−1/2b̂≤x+m−a]+P[|Rm|≥m−a]=P[Um+m−1/2b̂≤x]+o(m−1/2). Similarly, we show P[tm≤x]≥P[Um+m−1/2b̂≤x]+o(m−1/2), and thus obtain A.7. To prove A.8, we recall the decomposition in A.5: Rm=Rm(1)+Rm(2)+Rm(3). We observe that Rm(2) (and thus Rm ⁠) might not have moments. So, we cannot show A.8 with a plain application of the Markov inequality. Due to A.3 and Assumptions A.2(d) and (e), we obtain E[(Rm(1)+Rm(3))2]≤Cm2(A.9) for some C>0 ⁠. Recalling the constant ν in Assumption A.2(d), we define ν̃=((a4−a22)/a4)(ν/2) ⁠. Note that P[Vm<ν̃]=P1m∑i=1mZm,i2<ν2≤P1m∑i=1m(Zm,i2−E[Zm,i2])<−ν2≤P|1m∑i=1m(Zm,i2−E[Zm,i2])|>ν2≤Cm for some C>0 using the Markov inequality. With rm≡(ν̃/vm2)5/2<1 and κ>1 ⁠, this last result, together with the Hölder inequality and A.3, implies PRm(2)>12m−a≤P3Em4(Vm−vm2)2vm4>rmm−a+P(ξm)5/2<rm≤CmκaE|Em|κ|Vm−vm2|2κ+PVm<ν̃≤CmκaE|Em|κp1/pE|Vm−vm2|2κq1/q+o(m−1/2)≤O(m−κ(1−a))+o(m−1/2)=o(m−1/2) by choosing p=4 ⁠, q=4/3 ⁠, κ=9/8 and a∈(1/2,10/18) ⁠. The last result combined with A.9 implies A.8, which finishes the proof. □ A.2 Proofs of the main results Recalling 3.1 and 3.10, for r>0 ⁠, we denote σ¯nr=1n∑i=1n(n∫(i−1)/ni/nσt2dt)r/2,σ̃nr=1n/kn∑i=1n/knsi2nknψ2kn+ψ1knω2nψ2knkn2r/2, where si2 was defined in 3.11. Having defined the necessary notations, we state the following preliminary result. Lemma A.2. For r=2 ⁠, 4 and 6, we have σ¯nr−σ¯r=o(n−1/2),σ̃nr−σ̃r=o(n−1/4). Proof: The first result follows from Lemma 2 in Barndorff‐Nielsen and Shephard (2003) and is omitted. □ We apply similar arguments and prove the second result. In view of the binomial theorem and 2.14, it suffices to show 1n/kn∑i=1n/knsi2nknψ2knp−∑i=1n/kn∫((i−1)kn)/n(ikn)/nσt2p=o(n−1/4).(A.10) We define min=inf((i−1)kn)/n≤t≤(ikn)/nσtandMin=sup((i−1)kn)/n≤t≤(ikn)/nσt. By definitions of the related terms above, we easily observe that (min)2≤si2nknψ2kn≤(Min)2andknn(min)2p≤∫((i−1)kn)/n(ikn)/nσt2p≤knn(Min)2p. Then, we observe that the absolute value of the expression in A.10 can be bounded by Bn≡1n/kn∑i=1n/kn(Min)2p−(min)2p. Because σ is pathwise bounded, we easily obtain Bn≤Cn/kn∑i=1n/kn(Min−min). By definition of the supremum/infimum, there exists tin and sin in [(i−1)kn/n,ikn/n] such that (Min−min)≤|σtin−σsin|+2/n. Now, because kn divides n, Assumption 2.2(a) implies that Bn=o(kn/n) ⁠, which finishes the proof. □ We are now ready to prove the main theorems in this paper. Proof of Theorem 3.1: We start with the proof of part (a) of Theorem 3.1. We recall Y=X with the drift b=0 for this theorem and note that we can write the t‐statistic in 2.6 for the realized volatility as in A.2 by choosing m=n, Bn,i=0 and Zn,i=|nΔinX|2=(αm,ium,i+βm,i)2, where un,i∼N(0,1) ⁠, βm,i=0 and μn,i=E[Zn,i]=αn,i2=n∫(i−1)/ni/nσt2dt. We intend to utilize Proposition A.1 and observe that Assumptions A.2(a), (b), (d) and (e) are obviously satisfied under Assumption 2.2(a). Concerning A.2(c), we have (Zm,i−μm,i−Bm,i,Zm,i2−E[Zm,i2])=(αn,i2(un,i2−1),αn,i4(un,i4−3)). The result follows as αn,i2≥α>0 for all n and i under Assumption 2.2(a). Having verified the conditions, we look at the expansion in Proposition A.1 and observe that κn,3(κn,2)3/2=σ¯n6(σ¯n4)3/2. Then, we apply Lemma A.2 and finish the proof with σ¯n6(σ¯n4)3/2=σ6,4+o(n−1/2). Similar arguments apply to results in part (b) of Theorem 3.1 (i.e., the bootstrap part). Given that in the statement of part (b) of Theorem 3.1 we supposed that the Cramer condition is satisfied, we only need to verify Assumptions A.2(a), (b), (d) and (e). It is easy to see that it is the case. In particular, note that we can write the bootstrap t‐statistic in 2.10 as in A.2 form by choosing m=n, Bm,i=0 and Zm,i=(αm,ium,i+βm,i)2, where αm,i=nΔinY ⁠, um,i=vi and βm,i=0 such that the bootstrap external random variable vi∼ i.i.d. with moments given by as*=E*|vi|s ⁠. □ Proof of Proposition 3.1: Let vi=ηi∼ i.i.d. with the same distribution as in Proposition 3.1. Because ηi (and thus vi ⁠) has a density, the Cramer condition for v is satisfied. The discussion before Proposition 4.5 in GM2009 means that the following moment conditions are sufficient for a second‐order asymptotic refinement: E[v2]=1,E[v4]=3125andE[v6]=31253725. We note that E[v2r]=E[ηr] for r=1 ⁠, 2 and 3. For the gamma distribution with parameters α>0 and β>0 ⁠, it is well known that E[η]=αβ,E[η2]=α(α+1)β2andE[η3]=α(α+1)(α+2)β3. Solving these equations in α and β leads to α=β=25/6 ⁠. □ Proof of Remark 3.1: We proceed as in the proof of part (a) of Theorem 3.1. In particular, we recall Y=X for this case and note that we can write the t‐statistic in 2.6 for the realized volatility as in A.2 by choosing m=n ⁠, Bn,i=0 and Zn,i=|nΔinX|2. Because nΔinX is normally distributed, we obtain μn,i≡E[Zn,i]=n∫(i−1)/ni/nσt2dt+n∫(i−1)/ni/nbtdt2. We want to apply Proposition A.1 and observe that Assumptions A.2(a),(b) and (d) are obviously satisfied. Regarding Assumption A.2(e), note that bn=n∑i=1n(∫(i−1)/ni/nbtdt)2 and vn2=2σ¯n4+O(n−1/2) ⁠. Hence, Lemma A.2 for σ¯n4 and a variant of this lemma for bn under Assumption 2.2(b) yield b̂=b¯2/(2σ¯4) ⁠. Because we assume the existence of the expansions, we do not verify the Cramer condition stated in Assumption A.2(c). □ Next, we move to the proof of the result for the pre‐averaging estimator. Proof of Theorem 3.2: For the proof of part (a) of this theorem, our first aim is to write the main part of the pre‐averaging estimator in the form given by A.2. For this purpose, we denote m=n/kn and, recalling 3.11, we write Zm,i=nknψ2kn(Y¯(i−1)kn)2,μm,i=nknψ2knsi2+ψ1knω2kn,Bm,i=nknψ1kn2kn2ψ2kn∑j=(i−1)kn+1ikn−1(|Δjnε|2−2ω2). Then, Tn=tm+Rn where Tn and tm were defined in 2.17 and A.2, respectively, and the remainder term Rn is given by Rn=R̃n/V̂n,(A.11) with R̃n=n1/41ψ2kn∑i=1dnsi2−∫01σt2dt−n1/4ψ1kn2kn2ψ2kn∑i=1n(|ΔinY|2−|Δinε|2)+∑j=1dn(|Δjknnε|2−2ω2)≡R̃n(1)+R̃n(2). First, assume that we can apply Proposition A.1 for tm (we later show that Rn is negligible). Proceeding as in the proof of Theorem 3.1, we obtain κn,3(κn,2)3/2=σ̃n6(σ̃n4)3/2. Then, we will be finished due to Lemma A.2. Now, we verify that the assumptions of Proposition A.1 are satisfied. Clearly, Assumptions A.2(a), (c), (d) and (e) hold true. Next, we check Assumption A.2(b). Because X¯(i−1)kn and ε¯(i−1)kn are independent and have 0 means, we obtain E[(Y¯(i−1)kn)4]=E[(X¯(i−1)kn)4]+6E[(X¯(i−1)kn)2]E[(ε¯(i−1)kn)2]+E[(ε¯(i−1)kn)4]. Because the term X¯(i−1)kn is normally distributed with mean 0 and variance si2 ⁠, its moments are well known. However, the term ε¯(i−1)kn might not be normally distributed and needs careful treatment. To deal with the moments of the noise term, we define the variables h(j/kn)=g((j+1)/kn)−g(j/kn) for 1≤j≤kn−1 ⁠. It is easy to see that ε¯i=∑j=0kn−1−h(j/kn)ε(i+j)/n. For p=4 and 6, let us denote ψpkn=knp−1∑j=0kn−1h(j/kn)p. We note that as g is Lipschitz continuous, we have ψpn=O(1) for p=4 and 6. We also denote the pth absolute moment of εi/n with mp ⁠. A simple calculation shows that E[(ε¯i)4]=3(ψ1kn)2ω4kn2+(m4−3ω4)ψ4knkn3.(A.12) At this stage, we easily obtain E[Zm,i2]=n2(kn)2(ψ2kn)2E[(Y¯(i−1)knn)4]=3n2(kn)2(ψ2kn)2si2+ψ1knω2kn2+3n2(m4−3)ψ4knkn5(ψ2kn)2 and, due to m=n/kn ⁠, this leads to |E[Zm,i2]−a4a22μm,i2|≤Cm. Similarly, it is possible to show the assumption related to E[Zm,i3]=n3(kn)3(ψ2kn)3E[(Y¯(i−1)kn)6]. In this case, crucial steps are to use A.12 and E[(ε¯i)6]=15(ψ1kn)3ω4kn3+(m6−15ω4)ψ6knkn5+O(1)kn5. Concerning the condition for E[Zm,iBm,i] ⁠, we observe that E[Zm,iBm,i]=n2ψ1kn2kn4(ψ2kn)2∑j=(i−1)kn+1ikn−1E[(ε¯(i−1)kn)2((Δjnε)2−2ω2)]. To compute this expression, for each j in above range, we find that E[(ε¯(i−1)kn)2((Δjnε)2−2ω2)]=(h(j/kn)2+h((j−1)/kn)2)(m4−ω4)+4h(j/kn)h((j−1)/kn)ω4=O(1/kn2), where we exploited the Lipschitz continuity of g. This easily leads to the identity E[Zm,iBm,i]=O(1/kn)=O(1/m) ⁠. Lastly, we obtain E[Zm,i2Bm,i]=n3ψ1kn2kn5(ψ2kn)3∑j=(i−1)kn+1ikn−1×E[(6(X¯(i−1)kn)2(ε¯(i−1)kn)2+(ε¯(i−1)kn)4)((Δjnε)2−2ω2)]. In this case, the condition is verified by noting that the additional term satisfies E[(ε¯(i−1)kn)4((Δjnε)2−2ω2)]=O(1/kn3). Next, we show that the remainder term defined in A.11 does not influence the Edgeworth expansion of Tn ⁠. We deal with this term similarly to A.5. Assumption 3.1 implies R̃n(1)=O(n−1/4−δ) for some δ>0 ⁠. Moreover, we have E[(R̃n(2))2]≤C/n for some C>0 ⁠. Combining these results leads to P[|Rn|>n−1/4−δ/2]=o(n−1/4). This implies that Rn has no effect on the Edgeworth expansion of Tn ⁠. For the proof of results in part (b) of Theorem 3.2, similar arguments as in the bootstrap part of Theorem 3.1 (no‐noise case) apply. In particular, we can write the bootstrap t‐statistic in 2.19 as in A.2 form by choosing m=n/kn ⁠, Bm,i=0 and Zm,i=(αm,ium,i+βm,i)2, where αm,i=(n/(knψ2kn))1/2Y¯ikn ⁠, um,i=vi and βm,i=0 such that the bootstrap external random variable vi∼ i.i.d. with moments given by as*=E*|vi|s ⁠. In particular, we have μm,i=a2(n/(knψ2kn))Y¯ikn2 ⁠. This completes the proof. □ Proof of Corollary 3.1: Immediate given Proposition A.1 and the proof of part (a) of Theorem 3.2.□ Proof of Proposition 3.2: The result follows given part (a) of Theorem 3.2 in conjunction with Corollary 3.1 by applying Proposition 3.1. □ Footnotes " See, e.g., the early work by Andersen et al. (2001), Barndorff‐Nielsen and Shephard (2002), Comte and Renault (1998), Jacod and Protter (1998), Meddahi (2002) and Barndorff‐Nielsen et al. (2006). See also Andersen et al. (2010) and Barndorff‐Nielsen and Shephard (2007) for reviews. " We refer to Hall (1992), GM2008 and Zhang et al. (2011) for further details that explain why these intervals are expected to outperform the conventional intervals based on the normal approximation. In the context of no noise, GM2008 also derived a two‐sided symmetric feasible Edgeworth expansion corrected interval for Γ. " 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(x0)/x0)x0−x02+x2 if x>xo, with x0=log(1.5) ⁠. © 2016 Royal Economic Society.
TI - Validity of Edgeworth expansions for realized volatility estimators
JF - The Econometrics Journal
DO - 10.1111/ectj.12058
DA - 2016-02-01
UR - https://www.deepdyve.com/lp/oxford-university-press/validity-of-edgeworth-expansions-for-realized-volatility-estimators-p6QWR6Q5S3
SP - 1
VL - 19
IS - 1
DP - DeepDyve
ER -