Bootstrapping Realized Volatility

Sılvia GoncalvesDepartement de sciences economiques, CIREQ and CIRANO, Universite de Montreal∗

and

Nour MeddahiDepartement de sciences economiques, CIREQ and CIRANO, Universite de Montreal†

April 25, 2005

Preliminary and Incomplete - Comments WelcomePlease Do not Circulate Without the Authors’ Permission

Abstract

In this paper, we propose bootstrap methods for statistics evaluated on high frequency data suchas realized volatility. The bootstrap is as an alternative inference tool to the first-order asymptotictheory recently derived by Barndorff-Nielsen and Shephard (henceforth BN-S) in a series of papers.We consider the i.i.d. bootstrap and the wild bootstrap (WB) and prove their first-order asymptoticvalidity under conditions similar to those used by BN-S. We then use formal Edgeworth expansionsand Monte Carlo simulations to compare the accuracy of the bootstrap with the feasible asymptotictheory of BN-S. Our Edgeworth expansions show that the i.i.d. bootstrap provides an asymptoticrefinement when volatility is constant. Under stochastic volatility, the i.i.d. bootstrap is not ableto match the cumulants through third order and therefore the i.i.d. bootstrap error has the samerate of convergence as the error implied by the standard normal approximation. Nevertheless,we show through simulations and using Edgeworth expansions that the i.i.d. bootstrap is stillable to provide a smaller error than that of the standard normal approximation. For the possiblytime-varying volatility case, the WB provides an asymptotic refinement, provided we choose theexternal random variable used to construct the wild bootstrap pseudo data appropriately. MonteCarlo simulations suggest that both the i.i.d. bootstrap and the appropriately chosen wild bootstrapimprove upon the asymptotic theory of BN-S in finite samples.

Keywords: Realized volatility, i.i.d. bootstrap, wild bootstrap, Edgeworth expansions.

∗C.P.6128, succ. Centre-Ville, Montreal, QC, H3C 3J7, Canada. Tel: (514) 343 6556. Email: sil-via.goncalves@umontreal.ca.

†C.P.6128, succ. Centre-Ville, Montreal, QC, H3C 3J7, Canada. Tel: (514) 343 2399. Email:nour.meddahi@umontreal.ca.

1

1 Introduction

The increasing availability of high frequency financial data has contributed to the popularity of realized

volatility as a measure of volatility in empirical finance. Realized volatility is simple to compute (it

is equal to the sum of squared high frequency returns) and it is a consistent estimator of integrated

volatility under general nonparametric conditions (see e.g. Andersen, Bollerslev and Diebold (2002)

for a survey of the properties of realized volatility).

Recently, Barndorff-Nielsen and Shephard (henceforth BN-S) have developed an asymptotic theory

for realized volatility-like measures. In particular, for a rather general stochastic volatility model, BN-

S (2002, 2004c,d) establish a central limit theorem for realized volatility over a fixed interval of time,

for instance a day, as the number of intraday returns increases to infinity. Similarly, BN-S (2003,

2004b) show that a CLT applies to empirical measures based on powers of intraday returns (realized

power variation) and products of powers of absolute returns (e.g. bipower variation). More recently,

BN-S (2004e) provide a joint asymptotic distribution theory for the realized volatility and the realized

bipower variation, and show how to use this distribution to test for the presence of jumps in asset

prices.

In this paper, we propose bootstrap methods for statistics evaluated on high frequency data such

as realized volatility. Our main motivation for using the bootstrap is to improve upon the asymptotic

mixed normal approximations derived by BN-S. The bootstrap can be particularly valuable in the

context of high frequency data based measures. Current practice is to use a moderate number of

intraday returns, e.g. 30-minute returns, in computing realized volatility to avoid microstructure

biases. Sampling at long horizons may limit the value of the asymptotic approximations derived under

the assumption of an infinite number of intraday returns. For instance, the Monte Carlo simulations

in BN-S (2004a) show that the raw feasible asymptotic theory for realized volatility can be a poor

guide to the finite sample distribution of the standardized realized volatility. BN-S (2004a) propose a

log version of the raw statistic and show it has improved finite sample properties. Similarly, Huang

and Tauchen (2004) show that jump tests based on the (scaled) difference between realized volatility

and bipower variation can have potential size problems for certain data generating processes.

1

We focus on realized volatility and ask whether we can improve upon the existing first-order

asymptotic theory by relying on the bootstrap for inference on integrated volatility in the absence

of microstructure noise. Since the effects of microstructure noise are more pronounced at very high

frequencies, we expect the bootstrap to be a useful tool of inference based on realized volatility when

sampling at moderate frequencies such as 30 minutes horizon, as is often done in practice.1

Recently, a number of papers has studied the impact of microstructure noise on realized volatil-

ity, including Aıt-Sahalia, Mykland and Zhang (2004), Bandi and Russell (2004), Hansen and Lunde

(2004a,b), Zhang, Mykland and Aıt-Sahalia (2004), Barndorff-Nielsen, Hansen, Lunde and Shephard

(2004), and Zhang (2004). In particular, these papers propose alternative estimators of integrated

volatility that are robust to microstructure noise and that do not coincide with realized volatility.

Bootstrapping such measures is an interesting extension of our results, which we will consider else-

where.

We propose and analyze two bootstrap methods for realized volatility: an i.i.d. bootstrap and a

wild bootstrap. The i.i.d. bootstrap (cf. Efron, 1979) generates bootstrap pseudo intraday returns by

resampling with replacement the original set of intraday returns. The wild bootstrap observations are

generated by multiplying each original intraday return by an i.i.d. draw from a distribution that is

completely independent of the original data. The wild bootstrap was introduced by Wu (1986), and

further studied by Liu (1988) and Mammen (1993), in the context of cross-section linear regression

models subject to unconditional heteroskedasticity in the error term. Both methods are well known in

the bootstrap literature.2 We are the first to the best of our knowledge to propose their application to

realized volatility and to study their theoretical properties under a general stochastic volatility model.3

Our application of the i.i.d. bootstrap in this context is motivated by a benchmark model in

which the volatility is constant, implying that intraday returns are i.i.d. in this simple case. Although

this approach remains valid under time-varying volatility (as we will show here), the wild bootstrap1For instance, in their seminal paper, Andersen, Bollerslev, Diebold and Labys (2003) consider an empirical application

based on 30 minutes intraday returns for three major spot exchange rates.2Goncalves and Kilian (2004) apply both methods in the context of autoregressions subject to conditional het-

eroskedasticity of unknown form.3Zhang, Mykland and Aıt-Sahalia (2004) consider an application of the subsampling method to realized volatility

under stochastic volatility. In particular, they use subsampling plus averaging to bias correct the realized volatilitymeasure when microstructure noise is present. Our main goal here is to use the bootstrap to estimate the distribution(as opposed to the bias) of realized volatility.

2

(WB) is an alternative approach that explicitly takes into account the conditional heteroskedasticity

underlying stochastic volatility models. We show that both methods are first order asymptotically

valid under conditions similar to BN-S, when the bootstrap statistic is appropriately centered and

standardized. In particular, the i.i.d. bootstrap is asymptotically valid even when volatility is time-

varying and stochastic if we center and scale the realized volatility measure appropriately.4 In practice,

volatility is highly persistent, especially over a daily horizon, implying that it is at least locally nearly

constant. Therefore we expect the i.i.d. bootstrap to provide a good approximation even under

stochastic volatility.

A popular bootstrap for serially dependent data is the block bootstrap. In our context, intraday

returns are (conditionally on the volatility path) independent, and this implies that blocking is not

necessary for asymptotic refinements of the bootstrap. The issue here is heteroskedasticity and not

serial correlation.

We use Monte Carlo simulations and formal Edgeworth expansions to compare the accuracy of the

bootstrap and the normal approximations. Our Edgeworth expansions show that the i.i.d. bootstrap

provides an asymptotic refinement when volatility is constant, as expected given that returns are i.i.d.

under this simple model. Our simulations confirm this result. These simulations also suggest that

the i.i.d. bootstrap outperforms the asymptotic normal approximation under more general stochastic

volatility models. Based on our Edgeworth expansions, we argue that although the rate of convergence

of the i.i.d. bootstrap error is the same as that of the error of the normal approximation when

volatility is stochastic, the absolute magnitude of the coefficients describing the i.i.d. bootstrap error

is smaller than that of the coefficients entering the first term of the Edgeworth expansion for the

original statistic. This can explain the good finite sample behavior of the i.i.d. bootstrap in our

simulations. Davidson and Flachaire (2001) use a similar argument to explain the good finite sample

performance of a bootstrap method for cross-section regressions with unconditional heteroskedasticity.

Our formal Edgeworth expansions for the WB statistic show that it provides an asymptotic refinement

when volatility is heterogeneous if we choose the external random variable used to construct the wild4Recently, Goncalves and Vogelsang (2004) show the validity of the i.i.d. bootstrap for t-tests based on heteroskedastic

and autocorrelation consistent (HAC) variance estimators when data are serially dependent. There the i.i.d. bootstrapis applied in a naive fashion, without any centering or scaling correction.

3

bootstrap observations appropriately. We propose an appropriate choice for this external random

variable. Our Monte Carlo simulations show that the WB implemented with this choice outperforms

the first-order asymptotic normal approximation. The comparison between the this WB and the i.i.d.

bootstrap favors the i.i.d. bootstrap, which is the preferred method in the context of our study.

The remainder of this paper is organized as follows. In Section 2, we introduce the setup, review

the existing first order asymptotic theory of BN-S and their regularity conditions, and then present the

new Edgeworth expansions for the realized volatility statistic. In Section 3, we study the theoretical

properties of the i.i.d. bootstrap. In the first subsection, we establish the asymptotic validity of

the i.i.d. bootstrap under the general context of stochastic volatility models of BN-S. In the second

subsection, we give the formal Edgeworth expansions for the i.i.d. bootstrap statistic and discuss

its ability to provide asymptotic refinements. Section 4 deals with the wild bootstrap. Section 5

contains Monte Carlo simulation results and Section 6 concludes. In Appendix A we present the

tables containing the Monte Carlo results. The proofs of the results in Section 2 appear in Appendix

B, whereas Appendix C contains the proofs of the bootstrap results.

2 Asymptotic properties of realized volatility

2.1 Setup

We consider the following continuous-time model for the log price process {log St : t ≥ 0}:

d log St = µtdt + σtdWt, (1)

where Wt denotes a standard Brownian motion, µt denotes a drift term, and σt a volatility term. For

simplicity, we will assume that µt = 0 for all t. The drift term is of order dt, which is smaller than

the order (dt)1/2 of the volatility term in (1) (see e.g. Andersen, Bollerslev and Diebold (2002) for

a discussion of this result). Thus, the drift term is negligible at high frequencies. Our model is thus

given as

d log St = σtdWt, (2)

where σt > 0 is in general a time-varying stochastic process. We will assume throughout this paper

the independence between the stochastic volatility process σt and the Brownian motion Wt, i.e. we

4

assume no leverage effects. A benchmark model useful for comparisons is the time-invariant diffusion

model where σt = σ for all t > 0.

Given (2), the daily return for any day t is defined as

rt ≡ log St − log St−1 =∫ t

t−1σudWu, t = 1, 2, . . . .

Since t is fixed in our analysis, we let t = 1 throughout without loss of generality. We can define

intraday returns (for any given day) at horizon h as follows:

ri ≡ log Sih − log S(i−1)h =∫ ih

(i−1)hσudWu, for i = 1, . . . , 1/h,

with 1/h an integer. To simplify notation, we omit the dependence of intraday returns on the horizon

h. When σ is constant, intraday returns are i.i.d. N(0, σ2h

), i.e. we have that

ri =∫ ih

(i−1)hσudWu = σ

(Wih −W(i−1)h

) ≡ σui ∼ i.i.d. N(0, σ2h

),

where ui ≡ Wih − W(i−1)h ∼ i.i.d. N (0, h) for i = 1, . . . , 1/h. When volatility is time-varying and

stochastic, intraday returns are (conditionally on the path of the volatility process σ) independent but

heteroskedastic, i.e. we can write ri = σiui, where σ2i ≡

∫ ih(i−1)h σ2

udu, and ui ∼ i.i.d. N (0, 1) . In this

case, ri ∼ N(0, σ2

i

)for i = 1, . . . , 1/h.

The parameter of interest is the integrated volatility over a day,

IV =∫ 1

0σ2

udu,

which we assume to be finite. A simple estimator of the integrated volatility is the sum of squared

intraday returns, known as realized volatility:

RV =1/h∑

i=1

r2i .

This estimator is under certain assumptions (including absence of microstructure noise) a consistent

estimator of IV when the number of intraday observations increases to infinity (i.e. if h → 0). This

result is theoretically justified by the theory of quadratic variation.

5

2.2 Review of the first-order asymptotic theory of Barndorff-Nielsen and Shep-hard

Our goal is to perform inference on the integrated volatility, e.g., we would like to form a confidence

interval for the IV. One approach is to rely on asymptotic theory. This has been the standard approach

in the realized volatility literature, based on the asymptotic theory derived by Barndorff-Nielsen and

Shephard in a series of papers. We describe this approach here and state the regularity conditions

used by BN-S.

Following BN-S (2004b,e), we consider a stochastic volatility model in which the volatility process

σ satisfies the following assumption:

Assumption (V) The volatility process σ is (pathwise) cadlag, bounded away from zero, and

satisfies the following regularity condition:

limh→0

h1/2

1/h∑

i=1

∣∣∣σrηi− σr

ξi

∣∣∣ = 0,

for some r > 0 (equivalently for every r > 0) and for any ηi and ξi such that 0 ≤ ξ1 ≤ η1 ≤ h ≤ ξ2 ≤

η2 ≤ 2h ≤ · · · ≤ ξ1/h ≤ η1/h ≤ 1.

As Barndorff-Nielsen, Jacod and Shephard (2004) note in their Remark 1, the cadlag assumption

implies that all powers of σ are locally integrable with respect to Lebesgue measure, so that in partic-

ular∫ 10 σq

udu < ∞ for any q > 0. Under Assumption (V), σ can exhibit jumps, intra-day seasonality

and long-memory. Processes for {log St} satisfying (2) and Assumption (V) are a special case of the

continuous stochastic volatility semimartingales.

For any q > 0, define

Rq = h−q/2+1

1/h∑

i=1

|ri|q .

Note that for q = 2, R2 = RV. For general q > 0, Rq is known as the realized q-th order power

variation (cf. BN-S (2004b)). Similarly, for any q > 0, define the integrated power volatility

σq ≡∫ 1

0σq

udu.

Under Assumption (V), BN-S (2004b, Theorem 1) prove that as h → 0 realized power variation

converges in probability to a scaled version of integrated power volatility. BN-S (2002, 2003, 2004b)

6

also prove a CLT for realized volatility. We summarize their results in the following theorem.

Theorem 2.1 Assume (2), Assumption (V), and suppose σ is independent of W . Then, as h → 0,

RqP→ µqσ

q, (3)

where µq = E |Z|q, Z ∼ N (0, 1) , and

√h−1 (RV − IV )√

2∫ 10 σ4

udu→d N (0, 1) , (4)

and

Th ≡√

h−1 (RV − IV )√23h−1

∑1/hi=1 r4

i

→d N (0, 1) . (5)

The result given in (4) shows that the realized volatility centered at IV and appropriately stan-

dardized is asymptotically normal. This result is not immediately useful in practice because the

standardization factor depends on the unobserved quantity∫ 10 σ4

udu, known as the integrated quartic-

ity. Expression (5) is a feasible version of (4) that replaces∫ 10 σ4

udu with a consistent estimator given

by 13h−1

∑1/hi=1 r4

i . The finite sample performance of the feasible asymptotic theory given in (5) can

be poor, as simulations by BN-S (2002, 2004a) show. As a way of improving upon (5), BN-S (2002)

suggest to use a logarithmic version of this result (see Section 5 for more details on this approach).

Although the logarithmic transformation of realized volatility has improved finite sample properties

compared to the raw version, some coverage probability distortions remain for small sample sizes. Its

extension to the multivariate case is also unclear. Here, we suggest using a bootstrap approximation

instead.

2.3 Higher-order asymptotic properties of realized volatility

In order to prove that the bootstrap offers an asymptotic refinement for the realized volatility-based

statistic Th, we first develop a formal first-order Edgeworth expansion of the distribution of Th. In later

sections we will provide analogous expansions for the bootstrap distributions of our bootstrap statistics.

We will not provide a proof of the validity of the Edgeworth expansions we develop, which are in this

sense only formal expansions. Proving the validity of our Edgeworth expansions would be a valuable

7

contribution in itself, which we defer for future research. Here our focus is on using formal expansions

to theoretically explain the superior finite sample properties of the bootstrap. Our approach follows

Mammen (1993) and Davidson and Flachaire (2001), who also rely on formal Edgeworth expansions

for studying the accuracy of the bootstrap in the context of linear regression models. Finally, all of

our results are valid conditionally on the path of the stochastic process σ.

As is well known, the coefficients of the polynomials entering a first-order Edgeworth expansion are

a function of the first three cumulants of Th (cf. Hall, 1992). Next we provide asymptotic expansions

for these cumulants. Write

Th =

√h−1 (RV − IV )√

V,

where

V =23h−1

1/h∑

i=1

r4i ≡

23R4

is a consistent estimator of the asymptotic variance of RV given by 2∫ 10 σ4

udu. For any positive integer i,

let κi (Th) denote the ith cumulant of Th. In particular, recall that κ1 (Th) = E (Th), κ2 (Th) = V ar (Th)

and κ3 (Th) = E (Th −E (Th))3 .

Theorem 2.2 Consider DGP (2). Suppose Assumption (V) holds and σ is independent of W . Then,

conditionally on σ, as h → 0,

κ1 (Th) =√

h

−1

2c1

σ6

(σ4

)3/2

+ O (h) , (6)

κ2 (Th) = 1 + O (h) , (7)

κ3 (Th) =√

h

(c3,1 − 3

2c3,2 +

32c1

)σ6

(σ4

)3/2

+ O (h) , (8)

where, letting µq = E |Z|q, Z ∼ N (0, 1) ,

c1 =µ6 − µ2µ4

µ4

(µ4 − µ2

2

)1/2=

4√2

c3,1 =µ6 − 3µ2µ4 + 2µ3

2(µ4 − µ2

2

)3/2=

4√2

c3,2 = 3µ6 − µ2µ4

µ4

(µ4 − µ2

2

)1/2=

12√2.

8

The proof of Theorem 2.2 is reported in Appendix B. For the purpose of writing the Edgeworth

expansion, it is convenient to define the coefficients of the terms of order O(√

h)

associated with

κ1 (Th) and κ3 (Th) as follows:

κ1,2 ≡ −12c1

σ6

(σ4

)3/2, (9)

κ3,1 ≡(

c3,1 − 32c3,2 +

32c1

)σ6

(σ4

)3/2. (10)

Based on the cumulants approximations, we can define a formal first-order Edgeworth expansion

for the distribution of Th as follows.

Corollary 2.1 Under the assumptions of Theorem 2.2, and conditional on the path of the volatility

process, as h → 0

P (Th ≤ x) = Φ (x) +√

hq1 (x) φ (x) + o(√

h)

, (11)

uniformly over x ∈ R, where Φ(·) is the standard normal cdf and φ (x) is the standard normal pdf.

The function q1 (x) is defined as

q1 (x) = −(−c1

2+

16

(c3,1 − 3

2c3,2 +

32c1

) (x2 − 1

)) σ6

(σ4

)3/2

=16

(2x2 + 1

) 4√2

σ6

(σ4

)3/2≡ −1

3(2x2 + 1

)κ1,2.

When σ is constant, q1 (x) simplifies to q1 (x) = 16

4√2

(2x2 + 1

). When σ is stochastic, q1 (x) is a

function of the path of σ through the integrals σ6 and σ4. In this case, (11) describes an asymptotic ex-

pansion of the distribution of Th conditional on the volatility path. The leading term of the Edgeworth

expansion is the standard normal approximation Φ (x). This is as expected, given that BN-S (2002)

show that the first-order asymptotic distribution of Th is the standard normal distribution. Here we

provide a rate of convergence for the error committed by the first-order asymptotic approximation.

In particular, Corollary 2.1 implies that the error of the standard normal approximation is of order

O(√

h)

:

supx∈R

|P (Th ≤ x)− Φ(x)| = O(√

h)

.

9

If the bootstrap error if of a smaller order of magnitude (in h), the bootstrap is said to provide an

asymptotic refinement for Th. Our goal in the remainder of the paper is to investigate this possibility

for two bootstrap methods: the i.i.d. bootstrap and the wild bootstrap.

3 The i.i.d. bootstrap

3.1 First-order asymptotic validity of the i.i.d. bootstrap

To motivate the i.i.d. bootstrap, consider the benchmark model in which volatility is constant, i.e.

σt = σ > 0 for all t. As we discussed before, in this case intraday returns at horizon h are i.i.d.

N(0, σ2h

), which suggests the use of an i.i.d. bootstrap. Indeed, in this constant volatility case, the

i.i.d. bootstrap error is of order o(√

h) and it does offer an asymptotic refinement for Th, as we will

show in the next subsection. When volatility is not constant, the i.i.d. bootstrap error is of the same

order of magnitude in h as the standard normal error, and usual arguments suggest the i.i.d. bootstrap

does not provide an asymptotic refinement. Nevertheless, a less standard argument shows that the

i.i.d. bootstrap will still improve on the normal approximation when the volatility is heterogeneous,

consistent with our Monte Carlo simulation results in Section 5.

In this section, we show that the i.i.d. bootstrap is asymptotically valid for general stochastic

volatility models satisfying Assumption (V). This implies in particular that the i.i.d. bootstrap remains

first-order asymptotically valid even when the volatility is not constant. We denote the bootstrap

intraday h−period returns as r∗i . For the i.i.d. nonparametric bootstrap, we have that r∗i = rIi , where

Ii ∼ i.i.d. uniform on{1, . . . , 1

h

}. This amounts to resampling with replacement the sample of 1

h

intraday h−period returns. As usual in the bootstrap literature, we reserve the asterisk to denote

bootstrap quantities. We let P ∗ denote the probability measure induced by the bootstrap, conditional

on the original sample. Similarly, we let E∗ (and V ar∗) denote expectation (and variance) with respect

to the bootstrap data, conditional on the original sample.5

The bootstrap realized volatility is the usual realized volatility, but evaluated on the bootstrap5Note that once we condition on the original intraday returns, adding the volatility path to the information set does

not change the bootstrap probability measure. Thus, P ∗ can also be interpreted as being the probability measure inducedby the bootstrap, conditional on the original sample and the volatility path.

10

intraday returns:

RV ∗ =1/h∑

i=1

r∗2i .

Next, we characterize the first two bootstrap population moments of RV ∗.

Lemma 3.1 Consider DGP (2) and assume Assumption (V) holds. Let {r∗i : i = 1, . . . , 1/h} denote

an i.i.d. bootstrap sample of intraday returns. Then

a) E∗ (RV ∗) = RV for any h.

b) V ∗ ≡ V ar∗(√

h−1RV ∗)

= R4 − (RV )2 is such that V ∗ − V ∗0

P→ 0 as h → 0, where V ∗0 =

3σ4 −(σ2

)2> 0.

Appendix C contains the proof of this result and all other results in this and the next sections.

The first result shows that the i.i.d. bootstrap is an unbiased estimator of the realized volatility for

any sampling frequency h. The second result derives the limit (in probability) of the i.i.d. bootstrap

variance as the sampling frequency increases or h → 0. When the volatility is constant, σu = σ for

all u, and σq ≡ ∫ 10 σq

udu = σq for any q > 0, implying that(σ2

)2= σ4 = σ4. In this case, it follows

that V ∗0 = 2σ4 = 2

∫ 10 σ4du, i.e. the i.i.d. bootstrap variance V ∗ consistently estimates the asymptotic

variance of the realized volatility when volatility is constant. When volatility is heterogeneous, V ∗0 does

not coincide with 2∫ 10 σ4

udu, which shows that the i.i.d. bootstrap variance is not a consistent estimator

of 2∫ 10 σ4

udu in this more general case. However, this does not prevent the i.i.d. bootstrap from being

asymptotically valid when volatility is time-varying and stochastic. Indeed, we show next that if we

center and studentize the bootstrap realized volatility measure appropriately, the i.i.d. bootstrap is

asymptotically valid in the sense that the i.i.d. bootstrap realized volatility statistic converges to a

standard normal random variable as h → 0, in probability.

Let

V ∗ = h−1

1/h∑

i=1

r∗4i −

1/h∑

i=1

r∗2i

2

≡ R∗4 −RV ∗2,

where for any q > 0 we define R∗q as R∗

q = h−q/2+1∑1/h

i=1 |r∗i |q . In Appendix C we show that V ∗ is a

11

consistent estimator of the bootstrap variance V ∗. The i.i.d. bootstrap analogue of Th is given by

T ∗h ≡√

h−1 (RV ∗ −RV )√V ∗

.

Note that although we center the bootstrap realized volatility around the sample realized volatility,

the standard error that we propose to studentize the bootstrap statistic is not of the same form as that

used to studentize Th. In particular, it is not given by 23h−1

∑1/hi=1 r∗4i , which would be the bootstrap

analogue of V .

Theorem 3.1 Consider DGP (2) and assume Assumption (V) holds. Let {r∗i : i = 1, . . . , 1/h} denote

an i.i.d. bootstrap sample of intraday returns. Then, as h → 0,

supx∈R

|P ∗ (T ∗h ≤ x)− Φ(x)| P→ 0, (12)

where Φ(x) = P (Z ≤ x), with Z ∼ N (0, 1).

Theorem 3.1 establishes the first-order asymptotic validity of the i.i.d. bootstrap for general

stochastic volatility models satisfying Assumption (V). In particular, (11) and (12) imply that as

h → 0

P ∗ (T ∗h ≤ x)− P (Th ≤ x) = oP (1) ,

uniformly in x ∈ R. This result provides a theoretical justification for using the bootstrap distribution

of T ∗h to estimate the quantiles of the distribution of Th in the general context studied by BN-S.

3.2 Asymptotic refinements of the i.i.d. bootstrap

In this section we compare the accuracy of the i.i.d. bootstrap and the standard normal approximation.

The comparison is based on the formal Edgeworth expansions for the original and the bootstrap

statistic.

In order to obtain the bootstrap Edgeworth expansion, we first expand the first three cumulants

κ∗i (T ∗h ) of the bootstrap statistic T ∗h up to order OP (h) as follows.

Theorem 3.2 Consider DGP (2) and assume Assumption (V) holds. Let {r∗i : i = 1, . . . , 1/h} denote

12

an i.i.d. bootstrap sample of intraday returns. Then, as h → 0,

κ∗1 (T ∗h ) =√

h

(−1

2R6 − 3R4RV + 2RV 3

(R4 −RV 2)3/2

)+ OP (h)

κ∗2 (T ∗h ) = 1 + OP (h)

κ∗3 (T ∗h ) =√

h

(−2R6 + 6R4RV − 4RV 3

(R4 −RV 2)3/2

)+ OP (h) .

As expected, the coefficients entering the asymptotic expansions of the bootstrap cumulants are

a function of the intraday returns. The leading term of the variance of T ∗h is equal to 1, reflecting

the fact that T ∗h is a studentized statistic. The leading terms of the expansions for the first and third

cumulants are defined as

κ∗1,2,h ≡ −12

R6 − 3R4RV + 2RV 3

(R4 −RV 2)3/2

κ∗3,1,h ≡ −2R6 + 6R4RV − 4RV 3

(R4 −RV 2)3/2.

κ∗1,2,h and κ∗3,1,h are the bootstrap analogues of the coefficients κ1,2 and κ3,1 defined previously. As

we noted before, κ3,1 = 4κ1,2. Here it is also the case that κ∗3,1,h = 4κ∗1,2,h. Hence the i.i.d. bootstrap

mimics the relationship that exists between κ1,2 and κ3,1.

Given Theorem 3.2, we can write a formal Edgeworth expansion for the bootstrap statistic T ∗h to

order o(√

h)

. We state this result formally as follows.

Corollary 3.1 Under the assumptions of Theorem 3.2, as h → 0, we have that

P ∗ (T ∗h ≤ x) = Φ (x) +√

hq∗1 (x) φ (x) + oP

(√h)

,

where

q∗1 (x) =16

(2x2 + 1

) R6 − 3R4RV + 2RV 3

(R4 −RV 2)3/2≡ −1

3(2x2 + 1

)κ∗1,2,h.

The bootstrap error in estimating the distribution of Th (conditional on σ) is given by

P ∗ (T ∗h ≤ x)− P (Th ≤ x) =√

h

[plimh→0

q∗1 (x)− q1 (x)]

φ (x) + oP

(√h)

,

uniformly in x ∈ R. By Corollaries 2.1 and 3.1,

plimh→0

q∗1 (x)− q1 (x) = −13

(2x2 + 1

) (plimh→0

κ∗1,2,h − κ1,2

).

13

uniformly in x ∈ R. The following result characterizes formally the i.i.d. bootstrap error.

Theorem 3.3 Under the assumptions of Theorem 3.2, as h → 0,

plimh→0

q∗1 (x)− q1 (x) =16

(2x2 + 1

)

15σ6 − 9σ4 σ2 + 2(σ2

)3

(3σ4 −

(σ2

)2)3/2

− 4√2

σ6

(σ4

)3/2

, (13)

conditionally on σ. When σt = σ for all t, then

plimh→0

q∗1 (x)− q1 (x) = 0. (14)

In the general case, we have that uniformly in x ∈ R,

∣∣∣∣plimh→0

q∗1 (x)− q1 (x)∣∣∣∣ ≤ |q1 (x)| . (15)

An immediate consequence of (14) is that under constant volatility the error of the bootstrap

approximation is of order oP

(√h). This is of a smaller order of magnitude than the error of the

standard normal approximation which we showed before to be of order O(√

h). Thus, the i.i.d. boot-

strap provides an asymptotic refinement over the feasible asymptotic theory of BN-S under constant

volatility. In this case, it is even possible that the order of the asymptotic refinement of the bootstrap

is equal to OP (h). This would require showing that κ∗1,2,h − κ1,2 = OP

(√h)

. We do not pursue this

possibility any further here.

When volatility is heterogeneous, plimh→0 q∗1 (x) − q1 (x) 6= 0. Thus, the rate of convergence of

the bootstrap error is in this case of order OP

(√h), the same as that of the feasible asymptotic

theory of BN-S. The i.i.d. bootstrap is not able to match the cumulants of the original statistic when

volatility is time-varying and this explains why it does not provide an asymptotic refinement for the

distribution of Th. Nevertheless, our simulations show that even when volatility is stochastic the

i.i.d. bootstrap reduces the error in coverage probability by comparison with the asymptotic standard

normal approximation. We propose the following explanation. To order O(√

h), the bootstrap error

is determined by the difference√

h [plimh→0 q∗1 (x)− q1 (x)]φ (x) . Similarly, the error of the first-order

asymptotic normal approximation is determined by√

hq1 (x) φ (x) . (15) implies that the absolute

magnitude of the i.i.d. bootstrap contribution of order√

h to the error in approximating the true

14

sampling distribution of Th is smaller than that of the standard normal approximation. This suggests

a theoretical explanation for why the i.i.d. bootstrap outperforms the first-order asymptotic theory of

BN-S even in the presence of stochastic volatility. A similar argument has been proposed by Davidson

and Flachaire (2001) to explain the superior performance of a certain wild bootstrap in the context of

a cross-section linear regression model with unconditional heteroskedastic errors.

The ratio |(plimh→0 q∗1 (x)− q1 (x)) /q1 (x)| is a function of the volatility path and can be quantified

for a given stochastic model by simulation. Table 2 and Figure 1 contain results for the baseline models

considered in Section 5. The results suggest that this ratio is very small and close to zero for two

of the three models considered (namely for the log-normal and GARCH(1,1) diffusions), and slightly

higher for a two-factor diffusion model. This finding is consistent with the good performance of the

i.i.d. bootstrap for these models.

4 The wild bootstrap

4.1 First-order asymptotic validity of the wild bootstrap

As we argued previously, under stochastic volatility intraday returns are independent but heteroskedas-

tic, conditional on the volatility path. This motivates our application of the WB in this context.

Consider a sequence of i.i.d. external random variables ηi with moments given by µ∗q = E∗ |ηi|q, where

E∗ (·) denotes the expectation with respect to the distribution of ηi. The WB intraday returns are

generated as r∗i = riηi, i = 1, . . . , 1/h.

For applications we need to choose the distribution of ηi. As we will show here, this choice is

not important for the first order asymptotic validity of the WB as long as we carefully center and

studentize the bootstrap realized volatility statistic. The choice of ηi implies a specific centering and

studentization. Nevertheless, in order to prove an asymptotic refinement for the WB we need to choose

the distribution of ηi appropriately. In the next subsection we will provide an appropriate choice of

ηi and show that it delivers an asymptotic refinement for the distribution of Th.

The following result is the wild bootstrap analogue of Lemma 3.1.

Lemma 4.1 Consider DGP (2) and assume Assumption (V) holds. Let {r∗i : i = 1, . . . , 1/h} denote

a WB sample of intraday returns obtained with external random variables ηi ∼ i.i.d. such that µ∗q =

15

E∗ |ηi|q, q > 0. Then

a) E∗ (RV ∗) = µ∗2RV for any h.

b) V ∗ ≡ V ar∗(√

h−1RV ∗)

=(µ∗4 − µ∗22

)R4 is such that V ∗ − V ∗

0P→ 0 as h → 0, where V ∗

0 =

3(µ∗4 − µ∗22

)σ4 > 0.

Under stochastic volatility, the WB variance of the bootstrap realized volatility (scaled by√

h−1)

overestimates 2∫ 10 σ4

udu by a factor of 32

(µ∗4 − µ∗22

). To get a consistent bootstrap estimator of the

variance of√

h−1RV we need to choose ηi such that µ∗4 − µ∗22 = 23 . If we choose ηi such that µ∗2 = 1

(so that the bootstrap realized volatility is an unbiased estimator of RV ) then we should let µ∗4 = 53 .

Since the consistency of the bootstrap variance V ∗ for 2σ4 is not necessary for the first-order (or

higher-order) asymptotic validity of the wild bootstrap, we will not pursue the choice of ηi along this

dimension any further here.

We propose the following consistent estimator of V ∗ :

V ∗ =(

µ∗4 − µ∗22µ∗4

) h−1

1/h∑

i=1

r∗4i

≡

(µ∗4 − µ∗22

µ∗4

)R∗

4, (16)

and define the WB studentized statistic T ∗h as

T ∗h =

√h−1 (RV ∗ − µ∗2RV )√

V ∗. (17)

Suppose for instance that we choose ηi such that µ∗2 = 1 and µ∗4 = 3, e.g. we let ηi ∼ N (0, 1). Then

T ∗h =

√h−1 (RV ∗ −RV )√

V ∗,

V ∗ =23R∗

4,

so that for this choice of ηi, the statistic T ∗h is of the same exact form as the original statistic Th, with

the bootstrap data replacing the original data. However, for other choices of ηi this is not necessarily

the case. The bootstrap standard error and the centering of the bootstrap realized volatility depend

on the particular choice of distribution for ηi through the moments µ∗2 and µ∗4.

As long as we carefully center and studentize the wild bootstrap realized volatility statistic accord-

ing to (16) and (17), the choice of ηi is not important for the first-order asymptotic validity of the

16

wild bootstrap, as the following theorem shows.

Theorem 4.1 Consider DGP (2) and assume Assumption (V) holds. Let {r∗i : i = 1, . . . , 1/h} denote

a WB sample of intraday returns obtained with external random variables ηi ∼ i.i.d. such that µ∗q =

E∗ |ηi|q < ∞ for q = 2 and 2 (2 + ε) for some small ε > 0. Then, as h → 0,

supx∈R

|P ∗ (T ∗h ≤ x)− Φ(x)| P→ 0, (18)

where Φ (x) = P (Z ≤ x), with Z ∼ N (0, 1), and T ∗h is the statistic defined in equations (16) and (17).

4.2 Asymptotic refinements of the wild bootstrap

In this subsection we obtain a formal Edgeworth expansion for the WB statistic T ∗h . We use this

expansion to propose a choice for the external random variable ηi for which the WB provides an

asymptotic refinement over the standard normal approximation.

Theorem 4.2 Consider DGP (2) and assume Assumption (V) holds. Let {r∗i : i = 1, . . . , 1/h} denote

a WB sample of intraday returns obtained with external random variables ηi ∼ i.i.d. such that µ∗q =

E∗ |ηi|q < ∞ for q ≤ 10. Then, as h → 0,

κ∗1 (T ∗h ) =√

h

(−c∗1

2R6

R3/24

)+ OP (h)

κ∗2 (T ∗h ) = 1 + OP (h)

κ∗3 (T ∗h ) =√

h

((c∗3,1 −

32c∗3,2 +

32c∗1

)R6

R3/24

)+ OP (h) ,

where the constants c∗1, c∗3,1 and c∗3,2 are defined as

c∗1 =µ∗6 − µ∗2µ

∗4

µ∗4(µ∗4 − µ∗22

)1/2

c∗3,1 =µ∗6 − 3µ∗2µ

∗4 + 2µ∗32(

µ∗4 − µ∗22)3/2

c∗3,2 = 3(µ∗6 − µ∗2µ

∗4)

µ∗4(µ∗4 − µ∗22

)1/2.

17

As before, it is convenient to define the O(√

h)

order terms as follows.

κ∗1,2,h = −c∗12

R6

R3/24

(20)

κ∗3,1,h =(

c∗3,1 −32c∗3,2 +

32c∗1

)R6

R3/24

. (21)

The following result is a corollary of Theorem 4.2.

Corollary 4.1 Under the assumptions of Theorem 4.2, as h → 0, we have that

P ∗ (T ∗h ≤ x) = Φ (x) +√

hq∗1 (x) φ (x) + oP

(√h)

,

where

q∗1 (x) = −(

κ∗1,2,h +16κ∗3,1,h

(x2 − 1

))= −

(−c∗1

2+

16

(c∗3,1 −

32c∗3,2 +

32c∗1

) (x2 − 1

)) R6

R3/24

.

The contribution of order√

h to the formal first-order Edgeworth expansion for the distribution

of T ∗h depends on the polynomial q∗1 (x) which is now a function of κ∗1,2,h and κ∗3,1,h as given in (20)

and (21). Similarly to what we observed for the i.i.d. bootstrap, the wild bootstrap error (up to order

oP

(√h)) depends on the difference

√h [plimh→0 q∗1 (x)− q1 (x)]φ (x).

Lemma 4.2 Under the assumptions of Theorem 4.2, as h → 0, we have that

plimh→0

q∗1 (x)− q1 (x) = −[(

plimh→0

κ∗1,2,h − κ1,2

)+

16

(plimh→0

κ∗3,1,h − κ3,1

) (x2 − 1

)],

where

plimh→0

κ∗1,2,h − κ1,2 = −12

σ6

(σ4

)3/2

(5√3c∗1 − c1

)

plimh→0

κ∗3,1,h − κ3,1 =σ6

(σ4

)3/2

[(5√3c∗3,1 − c3,1

)− 3

2

(5√3c∗3,2 − c3,2

)+

32

(5√3c∗1 − c1

)]

with c1 = c3,1 = 4√2

and c3,2 = 12√2.

This result shows that the choice of ηi (which dictates the value of the constants c∗1, c∗3,1 and c∗3,2

through its moments) influences the magnitude of the wild bootstrap error. For instance, if we choose6

6Given that returns are (conditionally on σ) normally distributed, choosing ηi ∼ N (0, 1) could be a natural choice.Moreover, this is a first-order asymptotically valid choice that implies a WB statistic T ∗h whose form is exactly that ofTh but with the bootstrap data replacing the original data, as we argued above.

18

ηi ∼ N (0, 1), then c∗1 = c1, c∗3,1 = c3,1 and c∗3,2 = c3,2. This implies that

plimh→0

κ∗1,2,h − κ1,2 =(

5√3− 1

)κ1,2 6= 0

plimh→0

κ∗3,1,h − κ3,1 =(

5√3− 1

)κ3,1 6= 0.

Thus, if ηi ∼ N (0, 1), it follows that

plimh→0

q∗1 (x)− q1 (x) =(

5√3− 1

)q1 (x) ≈ 1.89q1 (x) ,

showing that this choice of ηi does not deliver an asymptotic refinement over the standard normal

approximation. It also shows that in absolute terms the contribution of the term O(√

h)

to the

bootstrap error is almost twice as large as the contribution of q1 (x) that is associated with the error

incurred by using the standard normal approximation. We conclude that ηi ∼ N (0, 1) is not a good

choice for the wild bootstrap. This is confirmed by our Monte Carlo simulations in the next section.

Our next result provides conditions on the external random variable ηi that ensure plimh→0 q∗1 (x)−

q1 (x) = 0, implying an asymptotic refinement of the WB over the standard normal approximation.

Theorem 4.3 Suppose ηi is i.i.d. with moments µ∗q = E∗ |ηi|q for q = 2, 4 and 6 such that

µ∗6 − µ∗2µ∗4

µ∗4(µ∗4 − µ∗22

)1/2=

√3

54√2

µ∗6 − 3µ∗2µ∗4 + 2µ∗32(

µ∗4 − µ∗22)3/2

=√

35

4√2.

Then under the assumptions of Theorem 4.2, as h → 0,

P ∗ (T ∗h ≤ x)− P (Th ≤ x) = oP

(√h)

,

uniformly in x ∈ R, where T ∗h is the statistic defined in equations (16) and (17).

The first equation in Theorem 4.3 is a rewriting of c∗1 =√

35 c1 as a function of µ∗2, µ∗4 and µ∗6,

whereas the second equation is equal to c∗3,1 =√

35 c3,1. According to this result, any choice of ηi with

moments µ∗2, µ∗4 and µ∗6 satisfying these two conditions delivers an asymptotic refinement of the WB.

Note that since c∗1 = 3c∗3,2 and c1 = 3c3,2, c∗3,2 =√

35 c3,2 is implied by c∗1 =

√3

5 c1. There is thus an

infinite number of possible solutions for µ∗2, µ∗4 and µ∗6 (and hence of choices of ηi) for which the WB

19

can deliver a refinement. In particular, we can show that the solution is of the form µ∗2 = γ2, µ∗4 = 3125γ4

and µ∗4 = 3125

3725γ6 for any γ 6= 0. Since the value of T ∗h is invariant to the choice of γ, we can choose

γ = 1 without loss of generality, implying µ∗2 = 1 (which ensures the WB realized volatility is an

unbiased estimator of realized volatility), µ∗4 = 3125 = 1.24, and µ∗6 = 31

253725 = 1.8352. Next, we propose

a two point distribution for ηi that matches these three moments and thus implies an asymptotic

refinement for the WB.

Corollary 4.2 Let ηi be i.i.d. such that

ηi =

{15

√31 +

√186 ≈ 1.33 with prob p = 1

2 − 3√186

≈ 0.28

−15

√31−√186 ≈ −0.83 with prob 1− p.

Define

T ∗h =

√h−1 (RV ∗ −RV )√

V ∗,

V ∗ =631

h−1

1/h∑

i=1

r∗4i

.

Under the assumptions of Theorem 4.2, as h → 0,

P ∗ (T ∗h ≤ x)− P (Th ≤ x) = oP

(√h)

,

uniformly in x ∈ R.

5 Monte Carlo results

In this section we assess the accuracy of the bootstrap in comparison with the feasible asymptotic

theory of BN-S when computing 95% symmetric confidence intervals for the integrated volatility. Our

Monte Carlo design is inspired by Andersen, Bollerslev and Meddahi (2004). In particular, we consider

the following stochastic volatility model

d log St = µdt + σt

[ρ1dW1t + ρ2dW2t +

√1− ρ2

1 − ρ22dW3t

],

where W1t, W2t and W3t are three independent standard Brownian motions. Our baseline models fix

µ = ρ1 = ρ2 = 0, implying that d log St = σtdW3t and no drift nor leverage effects exist.

We consider three different models for σt. The first model is the log-normal diffusion reported in

20

Andersen, Benzoni and Lund (2002) where σt is such that

d log σ2t = −0.0136

[0.8382 + log σ2

t

]dt + 0.1148dW1t.

Our second model is the GARCH(1,1) diffusion studied by Andersen and Bollerslev (1998):

dσ2t = 0.035

(0.636− σ2

t

)dt + 0.144σ2

t dW1t.

Finally, we consider the two-factor diffusion model analyzed by Chernov et al. (2003) (and recently

studied in the context of the nonparametric jump statistic test by Huang and Tauchen (2004)):

σt = s-exp(−1.2 + 0.04σ2

1t + 1.5σ22t

)

dσ21t = −0.00137σ2

1tdt + dW1t

dσ22t = −1.386σ2

2tdt +(1 + 0.25σ2

2t

)dW2t.

According to this model, the stochastic volatility factor σ22t has a feedback term in the diffusion effect.

The function s-exp is the usual exponential function with a polynomial function splined in at high

values of its argument. This diffusion model has continuous sample paths but can imply sample paths

for the price process that look like jumps.

Our baseline models assume no drift and no leverage effects and satisfy our regularity conditions.

Tables 1 through 3 (in Appendix A) report results for these models. Although our theory does not

apply to stochastic volatility models with drift and/or leverage effects, we include in the Monte Carlo

simulation three models for which µ 6= 0 and for which leverage effect exists. The results are reported

in Table 4 (Appendix A). Following Andersen, Bollerslev and Meddahi (2004), for the one-factor

log-normal and GARCH(1,1) diffusions we consider

d log St = 0.0314dt + σt

[−0.576dW1t +

√1− 0.5762dW3t

],

whereas for the two-factor diffusion model of Chernov et. al. (2003) we have that

d log St = 0.030dt + σt

[−0.30dW1t − 0.30dW2t +

√1− 0.302 − 0.302dW3t

].

The drift and leverage parameters are as in Huang and Tauchen (2004).

21

We study the finite sample performance of two-sided 95% level intervals. The 95% level confidence

interval for IV based on the feasible asymptotic theory of BN-S is given by:

RV ± 1.961√h−1

√V ,

where

V =23h−1

1/h∑

i=1

r4i .

This interval is symmetric about RV because the normal distribution is symmetric. For the bootstrap,

we consider both symmetric and equal-tailed intervals. The 95% level symmetric bootstrap confidence

intervals for IV are of the form,

RV ± q∗0.95

1√h−1

√V ,

where q∗0.95 is the 95% percentile of the bootstrap distribution of |T ∗h |, i.e. instead of using the standard

normal distribution to compute the critical value 1.96 we use the bootstrap. In the tables presented in

Appendix A we refer to the confidence intervals just described as “raw”. We consider three different

bootstrap methods for computing q∗0.95: the i.i.d. bootstrap and two wild bootstrap methods, one based

on ηi ∼ N (0, 1) and another based on the two-point distribution for which asymptotic refinements

are to be expected. Notice that the bootstrap statistics T ∗h on which q∗0.95 are based differ according

to the bootstrap method in question. In particular, except in the wild bootstrap based on the normal

distribution, T ∗h is not of the same form as Th.

We also report results for confidence intervals based on a logarithmic version of the statistic Th,

following BN-S. These are referred to as “log” and are of the following form:

log(RV )± q0.951√h−1

√V

(RV )2,

where q0.95 = 1.96 if the interval is based on the feasible asymptotic theory of BN-S. For the bootstrap

intervals, q0.95 = q∗0.95 where q∗0.95 denotes the 95% percentile of the bootstrap distribution of the

(absolute value of the) logarithmic version of each T ∗h . For the i.i.d. bootstrap, this is equal to

√h−1 (log RV ∗ − log RV )√

V ∗(RV ∗)2

,

22

with V ∗ = R∗4 −RV ∗2. For the wild bootstrap, it is equal to

√h−1 (log RV ∗ − log µ∗2RV )√

V ∗(RV ∗)2

,

where V ∗ =(

µ∗4−µ∗22µ∗4

)R∗

4. We note that the theory in this paper only provides the first-order asymp-

totic validity of the bootstrap “log” intervals (based on an application of the delta method, given

Theorems 3.1 and 4.1). Our Edgeworth expansions do not apply to the log versions of T ∗h . Thus, we

cannot use these expansions to make any predictions on the second-order correctness of the bootstrap

“log” intervals. We include these only for comparison purposes with the feasible asymptotic theory of

BN-S based on the logarithmic version of the statistic Th.

The 95% level equal-tailed bootstrap intervals are of the form

(RV −

√h√

V p∗0.975, RV −√

h√

V p∗0.025

),

where p∗α is the α-th percentile of the bootstrap distribution of T ∗h . We compute p∗α with the i.i.d.

bootstrap and the two WB methods as above. Log versions of these bootstrap intervals are also

considered.

We compute the actual coverage probabilities of the confidence intervals using each method for

each of the stochastic volatility models described above. We report the results for the models in

Tables 1 through 4 (presented in Appendix A) across 10,000 replications for six different sample sizes:

1/h = 1152, 576, 288, 96, 48 and 12, corresponding to “1.25-minute”, “2.5-minute”, “5-minute”, “15-

minute”, “half-hour”, “2-hour” returns. The bootstrap methods rely on 999 bootstrap replications for

each Monte Carlo replication. Tables 1 and 3 contain results for the baseline models, for symmetric

and equal-tailed intervals, respectively. Table 4 contains results for symmetric intervals for the models

with drift and leverage.

We summarize the results as follows: 1) All intervals tend to undercover, with the exception of

the wild bootstrap intervals based on ηi ∼ N (0, 1). 2) The degree of undercoverage is larger for the

feasible asymptotic-theory based intervals than for the bootstrap methods; it is larger the smaller the

sample size (i.e. the larger is h), and it is larger for the “raw” version of the intervals than for the “log”

23

version. 3) All methods do worst for the two-factor diffusion model of Chernov et. al. (2003). 4) The

i.i.d. bootstrap does remarkably well across all models, despite the fact that the volatility is stochastic

and hence time-varying. It essentially eliminates the coverage distortions associated with the BN-S

intervals for small values of 1/h for the log-normal and the GARCH(1,1) diffusions. The coverage

probability of the i.i.d. bootstrap intervals deteriorates for the two-factor model, but it remains very

competitive relatively to the other methods. 5) The WB intervals based on the normal distribution

tend to overcover across all models, with the degree of overcoverage being smaller for larger values of

the sample size. 6) The WB based on the two-point distribution tends to undercover, but less than the

feasible asymptotic theory-based intervals of BN-S. 7) The WB based on the two-point distribution is

generally worse than the i.i.d. bootstrap. Our Edgeworth expansions do not provide a justification for

why this is so. It is possible that the contribution of the h term is numerically more important than

the contribution of the√

h and that the i.i.d. bootstrap implies a smaller contribution for this term

as compared to the WB based on the two point distribution we proposed. 8) Equal-tailed intervals

tend to outperform symmetric intervals. This is true for all bootstrap methods, but especially so for

the WB based on the normal external random variable.

6 Conclusions

In this paper we propose two bootstrap methods for realized-volatility based statistics. One is the i.i.d.

bootstrap and the other is the wild bootstrap. We show that these methods are first-order asymptoti-

cally valid under quite general conditions, similar to those used recently by BN-S in a series of papers.

In particular, they are valid under stochastic volatility. Next, we study the accuracy of these bootstrap

methods in comparison to the standard normal approximation using formal Edgeworth expansions and

Monte Carlo simulations. The standard arguments based on Edgeworth expansions suggest that the

i.i.d. bootstrap offers an asymptotic refinement when volatility is constant but not otherwise. How-

ever, our simulations show that the i.i.d. bootstrap outperforms the standard normal approximation

even when volatility is not constant. We offer a possible explanation based on somewhat less standard

arguments. Davidson and Flachaire (2001) use similar arguments in a completely unrelated context.

We use formal Edgeworth expansions for the wild bootstrap to propose an appropriate choice of the

24

external random variable and show it outperforms the normal approximation in finite samples. Our

simulations show that the i.i.d. bootstrap is the preferred method of inference in this context.

Our focus here is on the bootstrap for realized volatility. Establishing the (first- and higher-order)

validity of the bootstrap for this simple statistic is an important step towards establishing its (first- and

higher-order) validity for more complicated statistics based on high-frequency data. For instance, an

interesting application of the bootstrap is to the nonparametric jump tests studied by BN-S (2003c),

Huang and Tauchen (2004) and Andersen, Bollerslev and Diebold (2004). Similarly, we can apply

the bootstrap for inference on integrated volatility in the presence of microstructure noise, relying on

more robust measures of volatility. These extensions are the subject of ongoing research.

25

Appendix A: Tables and Figures

Table 1. Coverage rates of nominal 95% symmetric percentile-t intervals for IV

Baseline volatility models: no leverage and no drift

Bootstrap Wild BootstrapBN-S i.i.d. ηi ∼ N (0, 1) ηi ∼ 2 point

h raw log raw log raw log raw logLog-normal diffusion

1/12 86.07 90.40 93.72 95.86 98.49 97.95 87.49 88.371/48 92.32 93.62 94.86 95.47 98.31 97.44 93.84 94.691/96 93.25 94.23 94.95 95.28 98.09 97.22 94.50 94.921/288 94.55 94.71 95.27 95.09 97.04 96.40 95.16 95.121/576 94.56 94.74 94.98 95.13 96.21 95.81 94.69 94.911/1152 94.83 94.88 94.98 95.06 95.63 95.41 94.85 94.84

GARCH(1,1) diffusion

1/12 86.08 90.40 93.75 95.86 98.51 97.96 87.49 88.301/48 92.32 93.64 94.87 95.46 98.32 97.42 93.83 94.661/96 93.21 94.22 95.00 95.27 98.12 97.22 94.44 94.931/288 94.57 94.70 95.18 95.11 97.05 96.38 95.17 95.131/576 94.52 94.79 94.99 95.15 96.24 95.81 94.72 94.871/1152 94.81 94.85 94.97 94.99 95.69 95.43 94.88 94.86

Two-factor diffusion

1/12 78.94 85.90 90.13 93.32 96.52 96.14 78.92 80.251/48 87.95 90.85 92.83 93.97 96.92 96.50 89.79 90.951/96 90.58 92.51 94.00 94.78 97.26 96.74 92.16 93.191/288 92.83 93.59 94.59 94.88 97.25 96.78 93.98 94.271/576 94.52 94.70 95.48 95.59 97.29 96.89 95.15 95.141/1152 94.64 94.77 95.20 95.11 96.52 96.08 94.89 94.92

Note: 10,000 replications, with 999 bootstrap replications each.

Table 2. Descriptive statistics for the ratio |(q∗1 (x)− q1 (x)) /q1 (x)| for the i.i.d. bootstrap

Baseline volatility models: no leverage and no drift

Log-normal GARCH(1,1) Two-factorMean 0.00160 0.00247 0.08885Minimum 1.4e− 004 1.8e− 004 0.0138925th percentile 0.00069 0.00109 0.07006Median 0.00116 0.00180 0.0888375th percentile 0.00204 0.00314 0.10680Maximum 0.01692 0.02436 0.21943

Note: 10,000 replications.

26

Table 3. Coverage rates of nominal 95% equal-tailed percentile-t intervals for IV

Baseline volatility models: no leverage and no drift

Bootstrap Wild BootstrapBN-S i.i.d. ηi ∼ N (0, 1) ηi ∼ 2 point

h raw log raw log raw log raw logLog-normal diffusion

1/12 86.07 90.40 95.94 95.89 94.33 96.34 86.65 87.921/48 92.32 93.62 95.57 95.37 94.17 95.78 94.08 94.231/96 93.25 94.23 95.36 95.33 94.48 95.59 94.64 94.771/288 94.55 94.71 95.13 95.07 94.72 95.25 94.86 94.831/576 94.56 94.74 95.09 95.08 94.86 95.18 94.77 94.861/1152 94.83 94.88 95.14 95.17 94.62 94.96 94.92 94.99

GARCH(1,1) diffusion

1/12 86.08 90.40 95.91 95.88 94.32 96.36 86.56 87.851/48 92.32 93.64 95.54 95.43 94.14 95.79 94.07 94.241/96 93.21 94.22 95.37 95.33 94.46 95.64 94.60 94.741/288 94.57 94.70 95.11 95.12 94.70 95.24 94.85 94.801/576 94.52 94.79 95.11 95.07 94.86 95.20 94.71 94.841/1152 94.81 94.85 95.13 95.13 94.63 94.98 94.96 94.96

Two-factor diffusion

1/12 78.94 85.90 93.79 93.89 94.31 95.86 78.69 80.321/48 87.95 90.85 94.38 94.32 93.51 95.64 90.57 91.201/96 90.58 92.51 94.73 94.75 93.77 95.43 92.67 92.941/288 92.83 93.59 94.77 94.85 93.97 95.25 93.96 94.141/576 94.52 94.70 95.22 95.30 94.49 95.37 94.72 94.711/1152 94.64 94.77 94.82 94.90 94.27 95.04 94.80 94.88Note: See Table 1.

27

Table 4. Coverage rates of nominal 95% symmetric percentile-t intervals for IV

Volatility models with leverage and constant Drift

Bootstrap Wild BootstrapCLT i.i.d. ηi ∼ N (0, 1) ηi ∼ 2 point

h raw log raw log raw log raw logLog-normal diffusion

1/12 85.86 90.80 93.92 95.90 98.38 97.92 87.60 88.901/48 92.60 93.76 95.24 95.68 98.70 97.86 94.20 94.741/96 93.66 94.34 95.10 95.44 98.22 97.42 94.56 95.221/288 94.58 94.68 95.34 95.18 96.90 96.32 95.18 94.981/576 94.40 94.62 94.68 94.68 96.06 95.58 94.64 94.841/1152 94.98 94.78 95.04 94.92 95.82 95.42 95.10 94.82

GARCH(1,1) diffusion

1/12 85.72 90.48 93.69 95.70 98.36 97.93 87.22 88.291/48 92.35 93.65 94.97 95.55 98.57 97.70 93.92 94.661/96 93.44 94.23 94.99 95.41 98.25 97.29 94.50 95.111/288 94.41 94.56 95.15 95.09 96.84 96.19 94.94 94.801/576 94.62 94.95 94.94 95.10 96.29 95.88 94.91 95.151/1152 95.04 95.10 95.13 95.16 96.05 95.59 95.13 95.15

Two-factor diffusion

1/12 79.52 86.09 90.87 93.50 96.75 96.34 79.55 80.401/48 87.81 90.76 92.89 94.08 97.05 96.57 89.69 90.821/96 90.31 92.04 93.57 94.43 97.04 96.51 91.99 92.791/288 93.14 93.76 94.81 94.99 97.30 96.68 94.08 94.361/576 94.25 94.46 95.15 95.24 97.20 96.67 94.86 95.001/1152 94.27 94.47 94.81 94.88 96.33 95.84 94.56 94.74

Note: See Table 1.

28

Figure 1: Kernel density estimate for the ratio |(q∗1 (x)− q1 (x)) /q1 (x)|for the i.i.d. bootstrap in the baseline models

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2−100

0

100

200

300

400

500

600Garch(1,1)log−normal2−factor

Appendix B: proof of results in Section 2.

Let σ2i ≡

∫ ih(i−1)h σ2

udu < ∞ and for any q > 0 define

σqh ≡ h1−q/2

1/h∑

i=1

(σ2

i

)q/2 ≡ h1−q/2

1/h∑

i=1

σqi ,

where σqi ≡

(σ2

i

)q. Note that for q = 2 IV = σ2h =

∫ 10 σ2

udu ≡ σ2, but in general σqh 6= σq ≡ ∫ 1

0 σqudu,

as defined in the main text. Let µq = E |Z|q, where Z ∼ N (0, 1) and q > 0. Since µ2 = 1, we can

write IV = µ2IV , which will be convenient later for proving the results for the wild bootstrap. The

following lemma will be useful for proving Theorem 2.2.

Lemma A.1 Consider DGP (2) and assume Assumption (V) holds, where σ is independent of W .

It follows that (conditionally on σ)

29

a) E (rqi ) = µqσ

qi .

b) V ≡ V ar(h−1/2RV

)=

(µ4 − µ2

2

)σ4

h.

c) E[(RV − µ2IV )

(V − V

)]= h

(µ4−µ22)(µ6−µ2µ4)

µ4σ6

h.

d) E[(RV − µ2IV )2

(V − V

)]= h2 µ4−µ2

2µ4

(µ8 − µ2

4 − 2µ2µ6 + 2µ22µ4

)σ8

h.

e) E[(RV − µ2IV )3

(V − V

)]= 3h2 (µ4−µ2

2)2(µ6−µ2µ4)

µ4σ4

hσ6h + O

(h3

).

f) E[(RV − µ2IV )3

]= h2

(µ6 − 3µ2µ4 + 2µ3

2

)σ6

h.

Proof of Theorem 2.2. Define

Sh ≡√

h−1 (RV − µ2IV )√V

and Uh ≡√

h−1(V − V

)

V.

We can write

Th = Sh

(V

V

)−1/2

= Sh

(1 +

√hUh

)−1/2.

For a given fixed value of k, a first-order Taylor expansion of f (x) = (1 + x)−k/2 around 0 yields

f (x) = 1− k2x + O

(x2

). Thus, provided Uh = OP (1), we have that

Th = Sh

(1− 1

2

√hUh

)+ O (h) .

The first three cumulants of Th are defined as

κ1 (Th) = E (Th) , κ2 (Th) = E(T 2

h

)− [E (Th)]2

κ3 (Th) = E(T 3

h

)− 3E(T 2

h

)E (Th) + 2 [E (Th)]3 .

Up to order O (h), we have that

E (Th) = 0−√

h12E (ShUh) + O (h)

E(T 2

h

)= 1−

√hE

(S2

hUh

)+ O (h)

E(T 3

h

)= E

(S3

h

)−√

h32E

(S3

hUh

)+ O (h) ,

where we have used the fact that E (Sh) = 0 and E(S2

h

)= 1 by construction. Next we compute

E (ShUh) , E(S2

hUh

), E

(S3

h

)and E

(S3

hUh

)(up to order O (h)). Given Lemma A.1, we have that

E (ShUh) =h−1

(V )3/2E

[(RV − µ2IV )

(V − V

)]=

(µ6 − µ2µ4)

µ4

(µ4 − µ2

2

)1/2

σ6h(

σ4h

)3/2≡ c1

σ6h(

σ4h

)3/2,

30

where c1 = 4√2

since µ2 = 1, µ4 = 3, and µ6 = 15. This implies that

κ1 (Th) =√

h

−c1

2σ6

h(σ4

h

)3/2

+ O (h) .

Under Assumption (V), BN-S (2004b) prove that σqh − σq = o

(h1/2

), which allows us to write

κ1 (Th) =√

h

−1

2c1

σ6

(σ4

)3/2

+ O (h) ,

proving (6). Next,

E(S2

hUh

)=

h−3/2

V 2E

[(RV − µ2IV )2

(V − V

)]=√

hµ4 − µ2

2

µ4V2

(µ8 − µ2

4 − 2µ2µ6 + 2µ22µ4

)σ8

h = O(√

h)

,

since V is bounded away from zero, and σ8h = O (1) under our assumptions. This implies that

E(T 2

h

)= 1 + O (h), and therefore κ2 (Th) = 1 + O (h) , proving (7). To show (8), note that

E(S3

h

)=

h−3/2

V 3/2E

((RV − µ2IV )3

)=√

hµ6 − 3µ2µ4 + 2µ3

2(µ4 − µ2

2

)3/2

σ6h(

σ4h

)3/2≡√

hc3,1σ6

h(σ4

h

)3/2,

where

c3,1 ≡ µ6 − 3µ2µ4 + 2µ32(

µ4 − µ22

)3/2=

4√2

= c1.

Also

E(S3

hUh

)=

h−2

V 5/2E

[(RV − µ2IV )3

(V − V

)]= 3

µ6 − µ2µ4

µ4

(µ4 − µ2

2

)1/2

σ6h(

σ4h

)3/2+O (h) ≡ c3,2

σ6h(

σ4h

)3/2+O (h) ,

where

c3,2 ≡ 3µ6 − µ2µ4

µ4

(µ4 − µ2

2

)1/2=

12√2.

Thus, to order O (h), we have that

κ3 (Th) =√

h

(c3,1 − 3

2c3,2 +

32c1

)σ6

h(σ4

h

)3/2+ O (h) =

√h

(c3,1 − 3

2c3,2 +

32c1

)σ6

(σ4

)3/2+ O (h) ,

where the last equality holds because σqh − σq = o

(h1/2

)under our assumptions.

Proof of Corollary 2.1. The formal Edgeworth expansion follows from Hall (1992, p. 48), where

q1 (x) is given by

q1 (x) = −(

κ1,2 +16κ3,1

(x2 − 1

)).

31

Replacing κ1,2 and κ3,1 by the appropriate expressions given in (9) and (10) yields the result.

Proof of Lemma A.1. a) follows trivially from noting that ri = σiui, where ui ∼ i.i.d. N (0, 1).

For b), note that RV =∑1/h

i=1 r2i , where r2

i is (conditional on σ) independent with V ar(r2i

)= E

(r4i

)−(E

(r2i

))2 = µ4σ4i −

(µ2σ

2i

)2 =(µ4 − µ2

2

)σ4

i , with σ4i ≡

(σ2

i

)2. For the remainder of the proof, note

that we can write

RV − µ2IV =1/h∑

i=1

(r2i − µ2σ

2i

)and V − V =

µ4 − µ22

µ4

h−1

1/h∑

i=1

(r4i − µ4σ

4i

),

where for any q > 0, rqi − µqσ

qi are independent with zero mean. Thus,

E[(RV − µ2IV )

(V − V

)]=

µ4 − µ22

µ4

h−1E

1/h∑

i=1

(r2i − µ2σ

2i

) 1/h∑

j=1

(r4j − µ4σ

4j

) .

Given the independence between r2i − µ2σ

2i and r4

j − µ4σ4j when i 6= j, the only term contributing to

the expectation above is when i = j, in which case it is equal to

E

1/h∑

i=1

(r2i − µ2σ

2i

) (r4i − µ4σ

4i

) =

1/h∑

i=1

E(r6i − r2

i µ4σ4i − µ2σ

2i r

4i + µ2µ4σ

6i

)= h2 (µ6 − µ2µ4) σ6

h.

Thus, E[(RV − µ2IV )

(V − V

)]= h

µ4−µ22

µ4(µ6 − µ2µ4) σ6

h, proving c). For d), note that

I1 ≡ E

1/h∑

i=1

1/h∑

j=1

1/h∑

k=1

(r2i − µ2σ

2i

) (r2j − µ2σ

2j

) (r4k − µ4σ

4k

)

=1/h∑

i=1

E[(

r2i − µ2σ

2i

)2 (r4i − µ4σ

4i

)]= h3

(µ8 − µ2

4 − 2µ2µ6 + 2µ22µ4

)σ8

h.

Hence

E((RV − IV )2

(V − V

))=

µ4 − µ22

µ4

h−1I1 = h2 µ4 − µ22

µ4

(µ8 − µ2

4 − 2µ2µ6 + 2µ22µ4

)σ8

h,

proving d). Next, we prove e). Consider

I2 ≡ E

1/h∑

i=1

1/h∑

j=1

1/h∑

k=1

1/h∑

l=1

(r2i − µ2σ

2i

) (r2j − µ2σ

2j

) (r2k − µ2σ

2k

) (r4l − µ4σ

4l

) .

Now, when i = j = k = l, we get

E[(

r2i − µ2σ

2i

)3 (r4i − µ4σ

4i

)]=

(µ10 − µ4µ6 − 3µ2µ8 + 3µ2µ

24

)σ10

i .

32

When i = k 6= j = l, of which there are 3 different combinations, we get

E[(

r2i − µ2σ

2i

)2 (r2j − µ2σ

2j

) (r4j − µ4σ

4j

)]

= E(r4i − 2µ2r

2i σ

2i + µ2

2σ4i

)E

(r6j − r2

j σ4jµ4 − µ2σ

2jr

4j + σ6

jµ2µ4

)

= σ4i σ

6j

(µ4 − µ2

2

)(µ6 − µ2µ4) .

All other combinations of indices yield a zero contribution to the expectation in I2. Thus,

I2 =(µ10 − µ4µ6 − 3µ2µ8 + 3µ2µ

24

) 1/h∑

i=1

σ10i + 3

(µ4 − µ2

2

)(µ6 − µ2µ4)

1/h∑

i6=j

σ4i σ

6j .

We can write

h−3

1/h∑

i6=j

σ4i σ

6j =

h−1

1/h∑

i=1

σ4i

h−2

1/h∑

i=1

σ6i

− h

h−4

1/h∑

i=1

σ10i

= σ4

hσ6h − hσ10

h ,

and therefore

I2 = h4(µ10 − µ4µ6 − 3µ2µ8 + 3µ2µ

24

)σ10

h + 3h3(µ4 − µ2

2

)(µ6 − µ2µ4)

(σ4

hσ6h − hσ10

h

).

It follows that

E[(RV − µ2IV )3

(V − V

)]=

µ4 − µ22

µ4

h−1I2

= h3 µ4 − µ22

µ4

(µ10 − 4µ4µ6 − 3µ2µ8 + 6µ2µ

24 + 3µ2

2µ6 − 3µ32µ4

)σ10

h + 3h2 µ4 − µ22

µ4

(µ4 − µ2

2

)(µ6 − µ2µ4) σ4

hσ6h

= 3h2 µ4 − µ22

µ4

(µ4 − µ2

2

)(µ6 − µ2µ4) σ4

hσ6h + O

(h3

),

since σ10h = O (1) under our assumptions. Finally, for f), note that

I3 ≡ E[(RV − µ2IV )3

]= E

1/h∑

i=1

1/h∑

j=1

1/h∑

k=1

(r2i − µ2σ

2i

) (r2j − µ2σ

2j

) (r2k − µ2σ

2k

) .

The only non zero contribution to I3 is when i = j = k, in which case we get

E[(

r2i − µ2σ

2i

)3]

=(µ6 − 3µ2µ4 + 2µ3

2

)σ6

i

and

I3 = h2(µ6 − 3µ2µ4 + 2µ3

2

)σ6

h,

proving f).

33

Appendix C: proofs of bootstrap results in Sections 3 and 4.

Throughout this Appendix K denotes a generic constant that may change from one usage to the

next. Following the bootstrap literature, for any bootstrap statistic T ∗h , we will write T ∗h →d∗ T in

probability (or in prob-P ) when the conditional distribution of T ∗h (conditional on the original sample)

converges to the distribution of T for all samples except those in a set of probability P converging to

zero. We will write T ∗h →P ∗ T in prob-P when for any ε > 0 P ∗ (|T ∗h − T | > ε) = oP (1) . By Markov’s

inequality, if E∗ (|T ∗h − T |p) = oP (1), then T ∗h →P ∗ T in prob-P . When this is true we say that T ∗h is

a consistent estimator of T .

For any bootstrap scheme, we define the following bootstrap statistic

S∗h ≡√

h−1 (RV ∗ − E∗ (RV ∗))√V ∗ ,

where V ∗ = V ar∗(h−1/2RV ∗). By construction, E∗ (S∗h) = 0 and V ar∗ (S∗h) = 1. T ∗h (defined in the

main text for the i.i.d. bootstrap and the wild bootstrap) is the corresponding studentized statistic

which replaces V ∗ with a consistent bootstrap estimator V ∗. Recall the definition of the bootstrap

realized q-th order power variation

R∗q = h1−q/2

1/h∑

i=1

|r∗i |q .

Next we present two lemmas which will be useful to prove the results of Sections 3 and 4. Lemma A.2

is the i.i.d. bootstrap analogue of Lemma A.1, whereas Lemma A.3 is its WB version. The proof of

these results will be presented at the end of this Appendix.

Lemma A.2 Consider DGP (2) and assume Assumption (V) holds, where σ is independent of W .

For the i.i.d. bootstrap, where for any given i, P ∗ (r∗i = rj) = h for any j = 1, . . . , 1/h, we have that

for any q > 0,

a) E∗ (r∗qi

)= hq/2Rq and E∗ (R∗q) = Rq.

b) V ∗ ≡ V ar∗(h−1/2RV ∗) = R4 −RV 2.

c) E∗[(RV ∗ −RV )

(V ∗ − V ∗

)]= h

[R6 − 3R4RV + 2RV 3

]+ OP

(h2

).

d) E∗[(RV ∗ −RV )2

(V ∗ − V ∗

)]= OP

(h2

).

e) E∗[(RV ∗ −RV )3

(V ∗ − V ∗

)]= h2

[3

(R4 −RV 2

) (R6 − 3R4RV + 2RV 3

)]+ OP

(h3

).

f) E∗[(RV ∗ −RV )3

]= h2

(R6 − 3R4RV + 2RV 3

).

34

Lemma A.3 Consider DGP (2) and assume Assumption (V) holds, where σ is independent of W .

For the wild bootstrap where r∗i = riηi with ηi i.i.d. such that E∗ (|ηi|q) = µ∗q, we have that for any

q > 0,

a) E∗ (r∗qi

)= µ∗qr

qi and E∗ (R∗q) = µ∗qRq.

b) V ∗ ≡ V ar∗(h−1/2RV ∗) =

(µ∗4 − µ∗22

)R4.

c) E∗[(RV ∗ − µ∗2RV )

(V ∗ − V ∗

)]= h

(µ∗4−µ∗22 )(µ∗6−µ∗2µ∗4)µ∗4

R6.

d) E∗[(RV ∗ − µ∗2RV )2

(V ∗ − V ∗

)]= h2 µ∗4−µ∗22

µ∗4

(µ∗8 − µ∗24 − 2µ∗2µ

∗6 + 2µ∗22 µ∗4

)R8.

e) E∗[(RV ∗ − µ∗2RV )3

(V ∗ − V ∗

)]= 3h2 (µ∗4−µ∗22 )2(µ∗6−µ∗2µ∗4)

µ∗4R4R6 + OP

(h3

).

f) E∗[(RV ∗ − µ∗2RV )3

]= h2

(µ∗6 − 3µ∗2µ

∗4 + 2µ∗32

)R6.

g) E∗(V ∗ − V ∗

)2= h

(µ∗4−µ∗22

µ∗4

)2 (µ∗8 − µ∗24

)R8.

Proof of Lemma 3.1. This follows trivially from Lemma A.2 a). For the second result, note that

V ∗ = h−1V ar∗

1/h∑

i=1

r∗2i

= h−1

1/h∑

i=1

V ar∗(r∗2i

)= h−2V ar∗

(r∗21

).

But V ar∗(r∗21

)= E∗ (

r∗41

) − (E∗ (

r∗21

))2 = h2R4 − (hRV )2 . Thus, V ∗ = R4 − RV 2. (3) now implies

that V ∗ P→ µ4σ4 −

(σ2

)2= V ∗

0 , given that µ4 = 3. That V ∗0 > 0 follows by noticing that we can

write V ∗0 = 2σ4 +

(σ4 −

(σ2

)2)

, where σ4 −(σ2

)2≥ 0 (by Jensen’s inequality) and σ4 > 0 given

Assumption (V).

Proof of Theorem 3.1. The proof contains two steps. Step 1: We show that the desired result

is true for the normalized i.i.d. bootstrap statistic S∗h. Step 2: We show that V ∗ →P ∗ V ∗ in prob-P.

Proof of Step 1. Given Lemma A.2 3., we can write

S∗h =

√h−1 (RV ∗ −RV )√

V ∗ =1/h∑

i=1

r∗2i −E∗ (r∗2i

)√

hV ∗ ≡1/h∑

i=1

z∗i ,

where z∗i ≡ r∗2i −E∗(r∗2i )√hV ∗

are (conditionally on the original sample) i.i.d. with E∗ (z∗i ) = 0 and

V ar∗ (z∗i ) = h2V ∗hV ∗ = h such that V ar∗

(∑1/hi=1 z∗i

)= h−1h = 1. Thus, by Katz’s (1963) Berry-Esseen

bound, for some small ε > 0 and some constant K,

supx∈R

∣∣∣∣∣∣∣∣P ∗

∑1/hi=1 z∗i√

V ar∗(∑1/h

i=1 z∗i) ≤ x

− Φ(x)

∣∣∣∣∣∣∣∣≤ K

1/h∑

i=1

E∗ |z∗i |2+ε . (22)

35

We show that the RHS of (22) converges to zero in probability. Notice that this amounts to verifying

Liapunov’s condition in the CLT for independent arrays (see, e.g. Theorem 27.3 of Billingsley, 1995;

BN-S (2004d, p. 35) also use this CLT). We have that

1/h∑

i=1

E∗ |z∗i |2+ε = h−1E∗ |z∗1 |2+ε = h−1h−2+ε2 |V ∗|− 2+ε

2 E∗(∣∣∣r∗2i − E∗ |r∗i |2

∣∣∣2+ε

)

≤ 2 |V ∗|− 2+ε2 h−1h−

2+ε2 E∗ |r∗i |2(2+ε) = 2 |V ∗|− 2+ε

2 h−1h−2+ε2 h2+εR2(2+ε)

= 2 |V ∗|− 2+ε2 h

ε2 R2(2+ε) = OP

(h

ε2

),

given that V ∗ P→ V ∗0 > 0 and R2(2+ε)

P→ µ2+εσ2(2+ε) = O (1) (along any path of σ, given Assumption

(V)). As h → 0, OP

(h

ε2

)= oP (1). Proof of Step 2.

Proof of Theorem 3.2. We can write

T ∗h = S∗h

(V ∗

V ∗

)−1/2

= S∗h(1 +

√hU∗

h

)−1/2, where U∗

h ≡√

h−1(V ∗ − V ∗

)

V ∗ .

Following the proof of Theorem 2.2, we have that

E∗ (T ∗h ) = 0−√

h12E∗ (S∗hU∗

h) + OP (h)

E∗ (T ∗2h

)= 1−

√hE∗ (

S∗2h U∗h

)+ OP (h)

E∗ (T ∗3h

)= E∗ (

S∗3h

)−√

h32E

(S∗3h U∗

h

)+ OP (h) ,

where E∗ (S∗h) = 0 and E∗ (S∗2h

)= 1 by construction. We compute E∗ (S∗hU∗

h) , E∗ (S∗2h U∗

h

), E∗ (

S∗3h

)

and E∗ (S∗3h U∗

h

)(up to order OP (h)). Given Lemma A.2 c), we have that

E∗ (S∗hU∗h) =

h−1

(V ∗)3/2E∗

[(RV ∗ −RV )

(V ∗ − V ∗

)]=

R6 − 3R4RV + 2RV 3

(R4 −RV 2)3/2.

Since κ∗1 (T ∗h ) = E∗ (T ∗h ) , this implies that

κ∗1 (T ∗h ) =√

h

(−1

2R6 − 3R4RV + 2RV 3

(R4 −RV 2)3/2

)+ O (h) .

For κ∗2 (T ∗h ), note that κ∗2 (T ∗h ) = E∗ (T ∗2h

)− (E∗ (T ∗h ))2. Given Lemma A.2 d), we have that

E∗ (S∗2h U∗

h

)=

h−3/2

V ∗2 E∗[(RV ∗ −RV )2

(V ∗ − V ∗

)]= h−3/2OP

(h2

)= OP

(√h)

,

since V ∗ is bounded away from zero in probability. This implies that E∗ (T ∗2h

)= 1 + OP (h), and

therefore κ∗2 (T ∗h ) = 1 + OP (h). For κ∗3 (T ∗h ), recall that κ∗3 (T ∗h ) = E∗ (T ∗3h

) − 3E∗ (T ∗2h

)E∗ (T ∗h ) +

36

2 [E∗ (T ∗h )]3 , where E∗ (T ∗3h

)depends on E∗ (

S∗3h

)and E

(S∗3h U∗

h

)as defined above. We have that

E∗ (S∗3h

)=

h−3/2

V ∗3/2E∗

((RV ∗ −RV )3

)=√

hR6 − 3R4RV + 2RV 3

(R4 −RV 2)3/2,

given Lemma A.2 f). Also, from Lemma A.2 e),

E∗ (S∗3h U∗

h

)=

h−2

V ∗5/2E∗

[(RV ∗ −RV )3

(V ∗ − V ∗

)]= 3

(R4 −RV 2

) (R6 − 3R4RV + 2RV 3

)

(R4 −RV 2)5/2+ OP (h)

= 3R6 − 3R4RV + 2RV 3

(R4 −RV 2)3/2+ OP (h) .

Thus, to order OP (h), we have that

κ∗3 (T ∗h ) =√

hR6 − 3R4RV + 2RV 3

(R4 −RV 2)3/2− 3

2

√h3

(R6 − 3R4RV + 2RV 3

(R4 −RV 2)3/2

)

−3

(−1

2R6 − 3R4RV + 2RV 3

(R4 −RV 2)3/2

)+ OP (h)

=√

h−2R6 + 6R4RV − 4RV 3

(R4 −RV 2)3/2+ OP (h)

which concludes the proof of Theorem 3.2.

Proof of Corollary 3.1. The result follows by replacing κ∗1,2,h and κ∗3,1,h in the polynomial q∗1 (x) =

−(κ∗1,2,h + 1

6κ∗3,1,h

(x2 − 1

))given in Hall (1992, p. 48).

Proof of Theorem 3.3. (13) follows by an application of Lemma 2.1. (15) follows trivially when

σ is constant because (σq)p = σqp for any q, p > 0. We prove (14) next. Define C = 4σ6√2(σ4)3/2

and

C∗ = 15σ6−9σ4 σ2+2(σ2)3

(3σ4−(σ2)2)3/2, and note that C > 0. Proving (14) is equivalent to proving |C − C∗| ≤ |C|,

which in turn is equivalent to proving 0 ≤ C∗ ≤ 2C. Next we show that C∗ ≥ 0. The Jensen’s

inequality implies that σ4 ≥ (σ2)2, and since σ4 > 0, it follows that the denominator of C∗ is positive.

For the numerator of C∗, note we can write

15σ6 − 9σ4 σ2 + 2(σ2)3 ≥ 15σ6 − 9(σ4)3/2 + 2(σ2)3 ≥ 9((σ4)3/2 − (σ4)3/2) + 6σ6 + 2(σ2)3,

using −(σ2)2 ≥ −σ4. Since the function ψ(x) = x3/2 for x > 0 is convex, we have that (σ4)3/2 −(σ4)3/2 ≥ 0, which implies 15σ6 − 9σ4 σ2 + 2(σ2)3 ≥ 6σ6 + 2(σ2)3 > 0, proving that the numerator of

C∗ is also positive. Next we prove C∗C ≤ 2. We can write

C∗

C=

15σ6 − 9σ4 σ2 + 2(σ2)3

8σ6

2√

2(σ4)3/2

(3σ4 − (σ2)2)3/2≡ C1 × C2.

37

We show that C1 ≤ 2 and C2 ≤ 1. First, note that

15σ6 − 9σ4 σ2 + 2(σ2)3

8σ6≤ 2 ⇐⇒ 15σ6−9σ4 σ2+2(σ2)3 ≤ 16σ6 ⇐⇒ 0 ≤ σ6+7σ4 σ2+2σ2

(σ4 − (σ2)2

),

which proves the result since(σ4 − (σ2)2

)≥ 0 and 0 ≤ σ6 + 7σ4 σ2. Finally, we have that

2√

2(σ4)3/2

(3σ4 − (σ2)2)3/2≤ 1 ⇐⇒ 8(σ4)3 ≤

(3σ4 − (σ2)2

)3⇐⇒ 8(σ4)3 ≤

(2σ4 +

(σ4 − (σ2)2

))3,

which holds true since(σ4 − (σ2)2

)≥ 0.

Proof of Lemma 4.1. From Lemma A.3, E∗ (r∗2i

)= µ∗2r

2i , which implies (i). Similarly,

V ar∗

1/h∑

i=1

r∗2i

=

1/h∑

i=1

V ar∗(r2i η

2i

)=

1/h∑

i=1

r4i V ar∗

(η2

i

)=

1/h∑

i=1

r4i

(µ∗4 − µ∗22

)=

(µ∗4 − µ∗22

) 1/h∑

i=1

r4i .

Thus V ∗ =(µ∗4 − µ∗22

)R4

P→ 3(µ∗4 − µ∗22

)σ4 > 0 by Lemma 2.1.

Proof of Theorem 4.1. We proceed as in the proof of Theorem 3.1. Proof of Step 1. We can

write

S∗h =1/h∑

i=1

r2i

(η2

i − µ∗2)

√hV ∗ ≡

1/h∑

i=1

x∗i .

Notice that xi is an array of independent random variables with E∗ (x∗i ) = 0 and V ar∗ (x∗i ) =r2i

hV ∗V ar∗(η2

i

)= (µ∗4−µ∗22 )r2

i

hV ∗ (so x∗i is heteroskedastic). Thus, V ar∗(∑1/h

i=1 x∗i)

=∑1/h

i=1 V ar∗ (x∗i ) =(µ∗4 − µ∗22

) ∑1/hi=1 r4

ihV ∗ =

(µ∗4 − µ∗22

)R4V ∗ = 1, given the definition of V ∗. It suffices to verify Lyapunov’s

condition (22) using the properties of the wild bootstrap. In particular, we can show that

1/h∑

i=1

|x∗i |2+ε = |V ∗|− 2+ε2 h

ε2 R2(2+ε)E

∗ ∣∣η2i − µ∗2

∣∣2+ε ≤ K |V ∗|− 2+ε2 h

ε2 R2(2+ε)

(µ∗2(2+ε) + µ∗2+ε

2

)= OP

(h

ε2

),

arguing as in the proof of Theorem 3.1. Proof of Step 2. We have that E∗(V ∗

)= µ∗4−µ∗22

µ∗4E∗ (R∗

4) = V ∗,

given the definition of V ∗. Thus, it suffices to show that V ar∗(V ∗

)= oP (1). Using Lemma A.3.g),

we have that V ar∗(V ∗

)= E∗

(V ∗ − V ∗

)2= OP (h) , which proves the result since h → 0.

Proof of Theorem 4.2. Note that we can write

T ∗h = S∗h

(V ∗

V ∗

)−1/2

= S∗h(1 +

√hU∗

h

)−1/2, where U∗

h ≡√

h−1(V ∗ − V ∗

)

V ∗ .

We then follow the proof of Theorem 2.2 but use Lemma A.3 instead of Lemma A.1.

Proof of Corollary 4.1. Follows from Hall (1992, p.48).

Proof of Lemma 4.2. Apply Lemma 2.1.

38

Proof of Theorem 4.3. The conditions on the bootstrap moments µ∗2, µ∗4 and µ∗6 are a restatement

of the equations 5√3c∗1 = c1 and 5√

3c∗3,1 = c3,1 (the first equation implies that 5√

3c∗3,2 = c3,2 since

c3,2 = 3c1 and c∗3,2 = 3c∗1). Thus, plimh→0 q∗1 (x) = q1 (x), implying the result.

Proof of Theorem 4.2. We seek ηi such that its moments are equal to µ∗2 = 1, µ∗4 = 3125 , and

µ∗6 = 3125

3725 . Let

ηi ={

a1 with prob pa2 with prob 1− p

.

We determine a1, a2 and p such that Eη2i = 1, Eη4

i = 3125 and Eη6

i = 3125

3725 . In particular, we can

show that a1 = 15

√31 +

√186, a2 = −1

5

√31−√186, and p = 1

2 − 3√186

solve the following system of

equations

a21p + a2

2 (1− p) = 1

a41p + a4

2 (1− p) =3125

a61p + a6

2 (1− p) =3125

3725

.

Proof of Lemma A.2. a) follows trivially from the properties of the i.i.d. bootstrap. b) is part

of Lemma 3.1. For the remaining results, note that given b) and the definition of V ∗ for the i.i.d.

bootstrap, we can write

V ∗ − V ∗ = R∗4 −R4 −

[(RV ∗ −RV )2 + 2RV (RV ∗ −RV )

], (23)

where

RV ∗ −RV =1/h∑

i=1

(r∗2i − hRV

)and R∗

4 −R4 = h−1

1/h∑

i=1

(r∗4i − h2R4

),

where for any q > 0{r∗qi − hq/2Rq

}are (conditionally on the sample) i.i.d. with zero mean. We start

with c). We have that

E∗[(RV ∗ −RV )

(V ∗ − V ∗

)]= I1 − I2 − I3,

where

I1 ≡ E∗ [(RV ∗ −RV ) (R∗4 −R4)] , I2 ≡ E∗

[(RV ∗ −RV )3

]and I3 ≡ 2RV E∗

[(RV ∗ −RV )2

].

We analyse I1, I2 and I3 separately. First, using the independence between r∗2i −hRV and r∗4j −h2R4

39

when i 6= j, the fact they have mean zero, and part a) of this lemma,

I1 = h−1E∗

1/h∑

i=1

1/h∑

j=1

(r∗2i − hRV

) (r∗4j − h2R4

) = h−1

1/h∑

i=1

E∗ [(r∗2i − hRV

) (r∗4i − h2R4

)]

= h−2E∗ (r∗6i − h2R4r

∗2i − hRV r∗4i + h3R4RV

)

= h−2[h3R6 − h3R4RV − h3RV R4 + h3RV R4

]= h (R6 −R4RV ) .

Similarly,

I2 = E∗

1/h∑

i=1

(r∗2i − hRV

)

3

=1/h∑

i=1

E∗[(

r∗2i − hRV)3

]

= h−1E∗(r∗6i − 3r∗4i hRV + 3r∗2i (hRV )2 − h3RV 3

)

= h−1(h3R6 − 3h3R4RV + 3h3RV 3 − h3RV 3

)

= h2(R6 − 3R4RV + 2RV 3

),

and I3 = 2hRV(R4 −RV 2

), since E∗

[(RV ∗ −RV )2

]= V ar∗ (RV ∗) = h

(R4 −RV 2

)by part b).

Thus,

E∗[(RV ∗ −RV )

(V ∗ − V ∗

)]= h

[(R6 −R4RV )− 2RV

(R4 −RV 2

)]− h2(R6 − 3R4RV + 2RV 3

).

(24)

Under our assumptions, by Lemma 2.1,

R6 − 3R4RV + 2RV 3 P→ µ6σ6 − 3µ4µ2σ

4σ2 + 2µ32

(σ2

)3,

where the limit is bounded along any path of σ. Thus, the last term in (24) is OP

(h2

), proving c).

Next we prove d). Given 23, we have that

E∗[(RV ∗ −RV )2

(V ∗ − V ∗

)]= J1 − J2 − J3,

where

J1 ≡ E∗[(RV ∗ −RV )2 (R∗

4 −R4)], J2 ≡ E∗

[(RV ∗ −RV )4

]and J3 ≡ 2RV E∗

[(RV ∗ −RV )3

].

40

Starting with J1, we have that

J1 = h−1E∗

1/h∑

i=1

1/h∑

j=1

1/h∑

k=1

(r∗2i − hRV

) (r∗2j − hRV

) (r∗4k − h2R4

)

= h−1

1/h∑

i=1

E∗[(

r∗2i − hRV)2 (

r∗4i − h2R4

)]

= h−2E∗ [(r∗4i − 2r∗2i hRV + h2RV 2

) (r∗4i − h2R4

)]

= h−2E∗ (r∗8i − r∗4i h2R4 − 2r∗6i hRV + 2r∗2i h3RV R4 + h2RV 2r∗4i − h4RV 2R4

)

= h2(R8 −R2

4 − 2R6RV + 2R4RV 2).

Next, consider

J2 =1/h∑

i=1

1/h∑

j=1

1/h∑

k=1

1/h∑

l=1

E∗ [(r∗2i − hRV

) (r∗2j − hRV

) (r∗2k − hRV

) (r∗2l − hRV

)].

If i = j = k = l, we have that

1/h∑

i=1

E∗((

r∗2i − hRV)4

)=

1/h∑

i=1

E∗ (r∗8i − 4r∗6i hRV + 6r∗4i h2RV 2 − 4r∗2i h3RV 3 + h4RV 4

)

= h−1h4(R8 − 4R6RV + 6R4RV 2 − 4RV 4 + RV 4

)= h3OP (1) = OP

(h3

),

where we have used Lemma 2.1 to claim that the quantity in parenthesis is OP (1) . Similarly, if

i = j 6= k = l, of which there are 3 different combinations, we get

31/h∑

i6=k

E∗[(

r∗2i − hRV)2

]E∗

[(r∗2k − hRV

)2]

= 3(h−2 − h−1

) [h2

(R4 −RV 2

)]2 = 3h2(R4 −RV 2

)2+OP

(h3

),

given that E∗[(

r∗2i − hRV)2

]= h2

(R4 −RV 2

)and that

∑1/hi 6=k 1 = h−2 − h1. All other cases imply

zero expectation. Thus,

J2 ≡ E∗[(RV ∗ −RV )4

]= 3h2

(R4 −RV 2

)2 + OP

(h3

)= OP

(h2

).

Using our computations for I2, we have that J3 = OP

(h2

). This concludes the proof of d). Next, we

prove e). Given 23, we have that

E∗[(RV ∗ −RV )3

(V ∗ − V ∗

)]= M1 −M2 −M3,

where

M1 ≡ E∗[(RV ∗ −RV )3 (R∗

4 −R4)], M2 ≡ E∗

[(RV ∗ −RV )5

]and M3 ≡ 2RV E∗

[(RV ∗ −RV )4

].

41

Using arguments similar to those used above, we can show that M2 = OP

(h3

), which is smaller than

O(h2

)and therefore asymptotically negligible to this order. Given J2, we have that

M3 = 2h2RV[3

(R4 −RV 2

)2]

+ OP

(h3

).

Next we compute M1. We have that

M1 = h−1∑

i,j,k,l

E∗ [(r∗2i − hRV

) (r∗2j − hRV

) (r∗2k − hRV

) (r∗4l − h2R4

)]

= h−1

1/h∑

i=1

E∗[(

r∗2i − hRV)3 (

r∗4i − h2R4

)]+ 3h−1

1/h∑

i6=k

E∗((

r∗2i − hRV)2

)E∗ ((

r∗2k − hRV) (

r∗4k − h2R4

))

≡ M1,1 + M1,2.

With some algebra we can show that M1,1 = OP

(h3

), which is smaller than the order OP

(h2

)of

interest here. Since in particular E∗ [(r∗2i − hRV

) (r∗4i − h2R4

)]= h3 (R6 −R4RV ), we can show

that

M1,2 = 3h−1(h−2 − h−1

) (h2

(R4 −RV 2

)h3 (R6 −R4RV )

)= 3h2

(R4 −RV 2

)(R6 −R4RV )+OP

(h3

).

Collecting all terms of order OP

(h2

), we conclude that

E∗[(RV ∗ −RV )3

(V ∗ − V ∗

)]= h2

[3

(R4 −RV 2

) ((R6 −R4RV )− 2RV

(R4 −RV 2

))]+ OP

(h3

)

= h2[3

(R4 −RV 2

) (R6 − 3R4RV + 2RV 3

)]+ OP

(h3

),

proving e). Finally, f) follows from the derivation of I2.

Proof of Lemma A.3. Since r∗i = riηi, where ηi ∼ i.i.d. N (0, 1) , the proof follows very closely

that of Lemma A.1. Note that we can write

RV ∗ − µ∗2RV =1/h∑

i=1

(r∗2i − µ∗2r

2i

)and V ∗ − V ∗ =

µ∗4 − µ∗22µ∗4

h−1

1/h∑

i=1

(r∗4i − µ∗4r

4i

),

where for any q > 0, r∗qi − µ∗qrqi are (conditionally on the sample) independent with zero mean. Thus,

for instance,

E∗[(RV ∗ − µ∗2RV )

(V ∗ − V ∗

)]=

µ∗4 − µ∗22µ∗4

h−1E∗

1/h∑

i=1

(r∗2i − µ∗2r

2i

) 1/h∑

j=1

(r∗4j − µ∗4r

4j

) .

Given the independence between r∗2i − µ∗2r2i and r∗4j − µ∗4r

4j when i 6= j, the only term contributing to

42

the expectation above is when i = j, in which case it is equal to

E∗

1/h∑

i=1

(r∗2i − µ∗2r

2i

) (r∗4i − µ∗4r

4i

) =

1/h∑

i=1

E∗ (r∗6i − r∗2i µ∗4r

4i − µ∗2r

2i r∗4i + µ∗2µ

∗4r

6i

)= h2 (µ∗6 − µ∗2µ

∗4) R6,

which implies c). The other results follow similarly.

References

[1] Aıt-Sahalia, Y., Mykland, P.A. and L. Zhang, 2004. How often to sample a continuous-time

process in the presence of market microstructure nois, Review of Financial Studies, forthcoming.

[2] Andersen, T.G., L. Benzoni and J. Lund, 2002. An Empirical Investigation of Continuous-Time

Equity Return Models, Journal of Finance, 57, 1239-1284.

[3] Andersen, T.G. and T. Bollerslev, 1998. Answering the Skeptics: Yes, Standard Volatility Models

Do Provide Accurate Forecasts, International Economic Review, 39, 885-905.

[4] Andersen, T.G., T. Bollerslev, and F.X. Diebold, 2002. Parametric and Nonparametric Measure-

ments of Volatility, in Y. Aıt-Sahalia and L.P. Hansen (Eds.), Handbook of Financial Economet-

rics, forthcoming.

[5] Andersen, T.G., T. Bollerslev, and N. Meddahi, 2004. Correcting the errors: volatility forecast

evaluation using high-frequency data and realized volatilities. Econometrica, forthcoming.

[6] Bandi, F.M. and J.R. Russell, 2004. Microstructure noise, realized volatility, and optimal sam-

pling, manuscript, Graduate School of Business, University of Chicago.

[7] Barndorff-Nielsen, O. 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, 253-280.

[8] Barndorff-Nielsen, O. and N. Shephard, 2003. Realised power variation and stochastic volatility

models Bernoulli, volume 9, 2003, 243-265 and 1109-1111.

[9] Barndorff-Nielsen, O. and N. Shephard, 2004a. How accurate is the asymptotic approximation to

the distribution of realised variance?, in Identification and Inference for Econometric Models, ed.

by D.F. Andrews, J.L. Powel, and J.H. Stock. Cambridge University Press, forthcoming.

[10] Barndorff-Nielsen, O. and N. Shephard, 2004b. Power and bipower variation with stochastic

volatility and jumps, Journal of Financial Econometrics, 2, 1-48.

43

[11] Barndorff-Nielsen, O. and N. Shephard, 2004c. A feasible central limit theory for realised volatility

under leverage, manuscript, University of Oxford.

[12] Barndorff-Nielsen, O. and N. Shephard, 2004d. Econometric analysis of realised covariation: high

frequency based covariance, regression and correlation in financial economics, Econometrica, 72,

885-925.

[13] Barndorff-Nielsen, O. and N. Shephard, 2004e. Econometrics of testing for jumps in financial

economics using bipower variation, manuscript, University of Oxford.

[14] Barndorff-Nielsen, O., J. Jacod, and N. Shephard, 2004. Limit theorems for bipower variation in

financial econometrics, manuscript. University of Oxford.

[15] Barndorff-Nielsen, O. and N. Shephard, 2005. Power variation and time change, Theory of prob-

ability and its applications, forthcoming.

[16] Barndorff-Nielsen, O., P.R. Hansen, A. Lunde and N. Shephard, 2004. Regular and modified

kernel-based estimators of integrated variance: the case with independent noise, manuscript,

University of Oxford.

[17] Barndorff-Nielsen, O., S.E. Graversen, and N. Shephard, 2004. Power variation and stochastic

volatility: a review and some new results, Journal of Applied Probability, 41, 133-143.

[18] Billingsley, P., 1995, Probability and measure, Wiley, New York.

[19] Chernov, M., R. Gallant, E. Ghysels, and G. Tauchen, 2003. Alternative models for stock price

dynamics, Journal of Econometrics, 116, 225-257.

[20] Davidson, R. and Flachaire, E., 2001. The wild bootstrap, tamed at last. Working Paper, Darp58,

STICERD, London School of Economics.

[21] Efron,B., 1979. Bootstrap methods: another look at the jackknife. Annals of Statistics 7, 1-26.

[22] Goncalves, S. and L. Kilian, 2004. Bootstrapping Autoregressions with Conditional Heteroskedas-

ticity of Unknown Form. Journal of Econometrics, 2004, 123, 89-120.

[23] Goncalves, S. and T. Vogelsang, 2004. Block Bootstrap Puzzles in HAC Robust Testing: The

Sophistication of the Naive Bootstrap, manuscript, Universite de Montreal.

[24] Hall, P., 1992. The bootstrap and Edgeworth expansion. Springer-Verlag, New York.

44

[25] Hansen, P.R. and A. Lunde, 2004a. Realized variance and market microstructure, manuscript,

Stanford University.

[26] Hansen, P.R. and A. Lunde, 2004b. An unbiased measure of realized variance, manuscript, Stan-

ford University.

[27] Huang, X. and G. Tauchen, 2004. The relative contribution of jumps to total price variance,

manuscript, Duke University.

[28] Katz, M.L., 1963. Note on the Berry-Esseen theorem. Annals of Mathematical Statistics 34, 1107-

1108.

[29] Liu, R.Y., 1988. Bootstrap procedure under some non-i.i.d. models. Annals of Statistics 16, 1696-

1708.

[30] Mammen, E., 1993. Bootstrap and wild bootstrap for high dimensional linear models. Annals of

Statistics 21, 255-285.

[31] Wu, C.F.J., 1986. Jackknife, bootstrap and other resampling methods in regression analysis.

Annals of Statistics 14, 1261-1295.

[32] Zhang, L, P.A. Mykland, and Y. Ait-Sahalia, 2004. A tale of two time-scales: determining inte-

grated volatility with noisy high frequency data, manuscript, Princeton University.

[33] Zhang, L., 2004. Efficient estimation of stochastic volatility using noisy observations: a multi-scale

approach, manuscript, Carnegie Mellon University, Department of Statistics.

45