Bootstrapping realized volatility and realized beta under a local Gaussianity assumption∗ Ulrich Hounyo
†
Aarhus University and CREATES May 4, 2016
Abstract The main contribution of this paper is to propose a new bootstrap method for statistics based on high frequency returns. The new method exploits the local Gaussianity and the local constancy of volatility of high frequency returns, two assumptions that can simplify inference in the high frequency context, as recently explained by Mykland and Zhang (2009). Our main contributions are as follows.
First, we show that the local Gaussian bootstrap is
rst-order consistent when used to estimate the distributions of realized volatility and realized betas.
Second, we show that the local Gaussian bootstrap matches accurately the rst four cu-
mulants of realized volatility up to
o (h)
(where
h−1
denotes the sample size), implying that this
method provides third-order re nements. This is in contrast with the wild bootstrap of Gonçalves and Meddahi (2009), which is only second-order correct. Third, we show that the local Gaussian bootstrap is able to provide second-order re nements for the realized beta, which is also an improvement of the existing bootstrap results in Dovonon, Gonçalves and Meddahi (2013) (where the pairs bootstrap was shown not to be second-order correct under general stochastic volatility). Lastly, we provide Monte Carlo simulations and use empirical data to compare the nite sample accuracy of our new bootstrap con dence intervals for integrated volatility with the existing results.
JEL Classi cation: C15, C22, C58 Keywords: High frequency data, realized
volatility, realized beta, bootstrap, Edgeworth expan-
sions.
1
Introduction
Realized measures of volatility have become extremely popular in the last decade as higher and higher frequency returns are available. Despite the fact that these statistics are measured over large samples, their nite sample distributions are not necessarily well approximated by their asymptotic mixedGaussian distributions.
This is especially true for realized statistics that are not robust to market
microstructure noise since in this case researchers usually face a trade-o between using large sample sizes and incurring in market microstructure biases. This has spurred interest in developing alternative ∗
I would like to thank Nour Meddahi for very stimulating discussions, which led to the idea of this paper.
I am
especially indebted to Sílvia Gonçalves for her valuable comments. I am also grateful to Asger Lunde, Peter Exterkate and Matthew Webb, the co-editor Eric Renault and anonymous referees for helpful advice, comments and suggestions. I acknowledge support from CREATES - Center for Research in Econometric Analysis of Time Series (DNRF78), funded by the Danish National Research Foundation, as well as support from the Oxford-Man Institute of Quantitative Finance. † Department of Economics and Business, Aarhus University, 8210 Aarhus V., Denmark. Email:
[email protected].
1
approximations based on the bootstrap. In particular, Gonçalves and Meddahi (2009) have recently proposed bootstrap methods for realized volatility whereas Dovonon, Gonçalves and Meddahi (2013) have studied the application of the bootstrap in the context of realized regressions. The main contribution of this paper is to propose a new bootstrap method that exploits the local Gaussianity framework described in Mykland and Zhang (2009). As these authors explain, one useful way of thinking about inference in the context of realized measures is to assume that returns have constant variance and are conditionally Gaussian over blocks of consecutive
M
observations. Roughly
speaking, a high frequency return of a given asset is equal in law to the product of its volatility (the spot volatility) multiplied by a normal standard distribution. Mykland and Zhang (2009) show that this local Gaussianity assumption is useful in deriving the asymptotic theory for the estimators used in this literature by providing an analytic tool to nd the asymptotic behaviour without calculations being too cumbersome. This approach also has the advantage of yielding more e cient estimators by varying the size of the block (see Mykland and Zhang (2009) and Mykland, Shephard and Sheppard (2012). The main idea of this paper is to see how and to what extent this local Gaussianity assumption can be explored to generate a bootstrap approximation. In particular, we propose and analyze a new bootstrap method that relies on the conditional local Gaussianity of intraday returns. The new method (which we term the local Gaussian bootstrap) consists of dividing the original data into non-overlapping blocks of
M
observations and then generating the bootstrap observations at each frequency within a
block by drawing a random draw from a normal distribution with mean zero and variance given by the realized volatility over the corresponding block. Using Mykland and Zhang's (2009) blocking approach, one can act as if the instantaneous volatility is constant over a given block of consecutive observations. In practice, the volatility of asset returns is highly persistent, especially over a daily horizon, implying that it is at least locally nearly constant. We focus on two realized measures in this paper: realized volatility and realized regression coe cients. The latter can be viewed as a smooth function of the realized covariance matrix. Our proposal in this case is to generate bootstrap observations on the vector that collects the intraday returns that enter the regression model by applying the same idea as in the univariate case. Speci cally, we generate bootstrap observations on the vector of variables of interest by drawing a random vector from a multivariate normal distribution with mean zero and covariance matrix given by the realized covariance matrix computed over the corresponding block. Our ndings for realized volatility are as follows. When
M
is xed, the local Gaussian bootstrap
is asymptotically correct but it does not oer any asymptotic re nements. More speci cally, the rst four bootstrap cumulants of the
t-statistic
estimator that is based on a block size of
based on realized volatility and studentized with a variance
M
do not match the cumulants of the original
to higher order (although they are consistent). Note that when
M = 1,
the new bootstrap method
coincides with the wild bootstrap of Gonçalves and Meddahi (2009) based on a
2
t-statistic
N (0, 1) external random
variable. As Gonçalves and Meddahi (2009) show, this is not an optimal choice, which is in line with our results. Therefore, our result generalizes that of Gonçalves and Meddahi (2009) to the case of a xed
M > 1.
However, if the block length
M → ∞,
as
h→0
such that
M h → 0,
(where
h−1
denotes
the sample size), then the local Gaussian bootstrap is able to provide an asymptotic re nement. In particular, we show that the rst and third bootstrap cumulants of the corresponding cumulants at the rate
o h
1/2
t-statistic
converge to the
, which implies that the local Gaussian bootstrap oers
a second-order re nement. In this case, the local Gaussian bootstrap is an alternative to the optimal two-point distribution wild bootstrap proposed by Gonçalves and Meddahi (2009). More interestingly, we also show that the local Gaussian bootstrap is able to match the second and fourth order cumulants up to
o (h),
small enough error to yield a third-order asymptotic re nement.
This is contrast to
the optimal wild bootstrap methods of Gonçalves and Meddahi (2009), which can not deliver thirdorder asymptotic re nements. We also show that the local Gaussian bootstrap variance is a consistent estimator of the (conditional) asymptotic variance of realized volatility. Then, we provide a proof of the rst-order asymptotic validity of the local Gaussian bootstrap method for unstudentized (percentile) intervals. This is contrast to the best existing bootstrap methods of Gonçalves and Meddahi (2009), which can not consistently estimate the asymptotic covariance matrix of realized covolatility. For the realized regression estimator proposed by Mykland and Zhang (2009), the local Gaussian bootstrap matches the cumulants of the
t-statistics
through order
o h1/2
when
M → ∞.
Thus, this
method can promise second-order re nements. This is contrast with the pairs bootstrap studied by Dovonon, Gonçalves and Meddahi (2013), which is only rst-order correct. Our Monte Carlo simulations suggest that the new bootstrap method we propose improves upon the rst-order asymptotic theory in nite samples and outperforms the existing bootstrap methods. The rest of this paper is organized as follows. In the next section, we rst introduce the setup, our assumptions and describe the local Gaussian bootstrap. In Sections 3 and 4 we establish the consistency of this method for realized volatility and realized beta, respectively. Section 5 contains the higher-order asymptotic properties of the bootstrap cumulants. Section 6 contains simulations, while the results of an empirical application are presented in Section 7. We conclude in Section 8 by describing how the local Gaussian bootstrap can be extended to a setting which allow for irregularly spaced data, nonsynchronicity, market microstructure noise and/or jumps in the log-price process. Three appendices are provided. Appendix A contains the tables with simulation results whereas Appendix B and Appendix C contain the proofs.
2
Framework and the local Gaussian bootstrap
The statistics of interest in this paper can be written as smooth functions of the realized multivariate volatility matrix. Here we describe the theoretical framework for multivariate high frequency returns and introduce the new bootstrap method we propose.
3
Sections 3 and 4 will consider in detail the
theoretical properties of this method for the special cases of realized volatility and realized beta, respectively. We follow Mykland and Zhang (2009) and assume that the log-price process of a
d-dimensional vector of assets is de ned on a probability space (Ω, F, P ) equipped with a ltration
(Ft )t≥0 .
We model
X
as a Brownian semimartingale process that follows the equation,
t
Z Xt = X0 +
Z
where
µ = (µt )t≥0
is a
d-dimensional
dimensional Brownian motion and
Σt ≡ σt σt0
t
σs dWs , t ≥ 0,
µs ds + 0
that
(1) (d) 0 Xt = Xt , . . . , Xt
(1)
0
predictable locally bounded drift vector,
σ = (σt )t≥0
is an adapted càdlàg
is the spot covariance matrix of
X
W = (Wt )t≥0
d-
is
d × d locally bounded process such
t.
at time
We follow Barndor-Nielsen et al. (2006) and assume that the matrix process
σt
is invertible and
satis es the following assumption
t
Z σt = σ0 +
as ds +
and
Z
a, b,
and
v
Z
t
bs dWs +
0 where
t
Z
vs dZs ,
0
a
are all adapted càdlàg processes, with
is a vector Brownian motion independent of
(2)
0 also being predictable and locally bounded,
W.
The representation in (1) and (2) is rather general as it allows for leverage and drift eects. Assumption 2 of Mykland and Zhang (2009) or equation (1) of Mykland and Zhang (2011) also impose a Brownian semimartingale structure on the instantaneous covolatility matrix
σ.
Equation (2) rules out
jumps in volatility, but this can be relaxed (see Assumption H1 of Barndor-Nielsen et al. (2006) for a weaker assumption on Suppose we observe from which we compute
σ ). X
over a xed time interval
1/h
[0, 1]
intraday returns at frequency
Z
ih
Z
yi ≡ Xih − X(i−1)h =
µt dt + (i−1)h
where we will let
yki
at regular time points
to denote the
i-th
ih,
for
i = 0, . . . , 1/h,
h,
ih
1 σt dWt , i = 1, . . . , , h (i−1)h k , k = 1, . . . , d.
intraday return on asset
(3)
The parameters of
interest in this paper are functions of the elements of the integrated covariance matrix of
X,
i.e., the
process
Z
t
Γt ≡
Σs ds, t ∈ [0, 1] . 0
Without loss of generality, we let over the period
[0, 1] ,
we let
Γkl
t=1
and de ne
denote the
Γ = Γ1 =
(k, l)-th
As equation (3) shows, the intraday returns
yi
R1 0
element of
Σs ds
as the integrated covariance of
Γ.
depend on the drift
out inference for observations in a xed time interval the process For most purposes it is only a nuisance parameter.
X
µt
µ,
unfortunately when carrying
cannot be consistently estimated.
To deal with this, Mykland and Zhang (2009)
propose to work with a new probability measure which is measure theoretically equivalent to
P
and
under which there is no drift (a statistical risk neutral measure). They pursue the analysis further and
4
propose an approximation measure
Qh,M
de ned on the discretized observations
1 M h non overlapping blocks of size 1 of high frequency returns within a block, we have that M ≤ . h the volatility is constant on each of the
Speci cally, under the approximate measure
yi+(j−1)M h = C(j) · where
σt
ηi+(j−1)M ∼ i.i.d.N (0, Id ), Id
at the beginning of the
j -th
is a
√
d×d
Qh,M ,
in each block
hηi+(j−1)M ,
for
M.
Xih
M
Since
j = 1, . . . , M1h ,
only, for which is the number
we have,
1 ≤ i ≤ M,
identity matrix and
C(j) = σ(j−1)M h ,
(4) i.e., the value of
block (see Mykland and Zhang (2009), p.1417 for a formal de nition of
Qh,M ). The true distribution is
P,
but we may prefer to work with
simpler. Afterwards we adjust results back to
P
Qh,M
since then calculations are much
using the likelihood ratio (Radon-Nikodym derivative)
dQh,M /dP .
Remark 1. As pointed out in Mykland and Zhang's (2009) Theorem 3 and, in Mykland and Zhang's (2011) Theorem 1, the measure P and its approximation Qh,M are contiguous on the observables. This is to say that for any sequence Ah of sets, P (Ah ) → 0 if and only if Qh,M (Ah ) → 0. In particular, if an estimator is consistent under Qh,M , it is also consistent under P . Rates of convergence (typically h−1/2 ) are also preserved, but the asymptotic distribution may change. More speci cally, when adjusting from Qh,M to P , the asymptotic variance of the estimator is unchanged (due to the preservation of quadratic variation under limit operations), while the asymptotic bias may change. It appears that a given sequence Zh of martingales will have exactly the same asymptotic distribution under Qh,M and P , when the Qh,M martingale part of the log likelihood ratio log(dP /dQh,M ) has zero asymptotic covariation with Zh . In this case, we do not need to adjust the distributional result from Qh,M to P . Two important examples where this is true are the realized volatility and realized beta which we will study in details in Sections 3 and 4. Remark 2. In the particular case where the window length M increases with the sample size h−1 such that M = o(h−1/2 ), there is also no contiguity adjustment (see Remark 2 of Mykland and Zhang (2011)). However, it is important to highlight that all results using contiguity arguments in Mykland and Zhang (2009) apply only to the case with a bounded M. For the case M → ∞, Mykland and Zhang (2011) use a dierent representation than (4). It should be noted that throughout this paper, we will focus only on the representation given in (4). As a consequence, we would not assume that the approximate measure Qh,M is contiguous to the measure P when M grows to in nity. This means that, we would neither use results in Mykland and Zhang (2009), nor those in Mykland and Zhang (2011) when M → ∞. Next we introduce a new bootstrap method that exploits the structure of (4). In particular, we
C(j) by 0 i=1 yi+(j−1)M yi+(j−1)M . That is, we follow
mimic the original observed vector of returns, and we use the normality of the data and replace
ˆ(j) , where Cˆ(j) is such that Cˆ(j) Cˆ 0 its estimate C (j)
= 5
1 Mh
PM
the main idea of Mykland and Zhang (2009), and assume constant volatility within blocks. inside each block
where
j
M (j = 1, . . . , M1h ), we generate the M vector of returns √ ∗ ∗ = Cˆ(j) · hηi+(j−1)M , for i = 1, . . . , M, yi+(j−1)M
of size
∗ ηi+(j−1)M ∼ i.i.d.N (0, Id )
across
(i, j),
and
Id
is a
d×d
Then,
as follows, (5)
identity matrix.
Although we motivate the local Gaussian bootstrap data-generating process (DGP) by relying on
Qh,M
the approximate measure
with representation given by (4), note that to generate the bootstrap
vector of returns, we only need an estimator of the spot covariance matrix and a normal external random vector. The existence of the approximate measure validity of the local Gaussian bootstrap approach. bootstrap remains asymptotically valid when the approximate measure
Qh,M
Qh,M
is not a necessary condition to the
As we show in this paper, the local Gaussian
M → ∞,
i.e., a setting where we do not know whether
(with representation given by (4)) is contiguous to
P. In such situation,
for the proof of the validity of the bootstrap the key aspect is that we proceed directly under the true
P,
probability measure
without using any contiguity arguments.
In this paper, and as usual in the bootstrap literature,
P ∗ (E ∗
and
V ar∗ )
denotes the proba-
bility measure (expected value and variance) induced by the bootstrap resampling, conditional on a realization of the original time series. probability distribution statistics
∗ , Zh,M
ability under
P
we write
υ,
In the following additional notations,
or the approximate probability measure
For a sequence of bootstrap ∗
in probability under
h
∗ = OP ∗ (1) as h → 0, in probability under υ Zh,M i h ∗ limh→0 υ P ∗ Zh,M > Mε > ε = 0. Finally, we
υ,
denotes the true
∗ →P 0, as h → 0, in υ , or Zh,M i ∗ ε > 0, δ > 0, limh→0 υ P ∗ Zh,M > δ > ε = 0. Similarly, we
∗ Zh,M = oP ∗ (1)
if for any
Qh,M .
υ
if conditional on the sample,
∗ Zh,M
∗ write Zh,M
weakly converges to
set with probability converging to one under
ε>0
if for all
Z
→d∗
under
there exists a
Z
as
P ∗,
h → 0,
Mε < ∞
probwrite
such that
in probability under
for all samples contained in a
υ.
The following result is crucial in obtaining our bootstrap results only in the context where those
Qh,M .
results are derived under the approximate measure
∗ Theorem 2.1. Let Zh,M be a sequence of bootstrap statistics. Given the probability measure P and its approximation Qh,M , we have that ∗ ∗ ∗ ∗ Zh,M →P 0, as h → 0, in probability under P , if and only if Zh,M →P 0, as h → 0, in probability under Qh,M .
Proof of Theorem 2.1 ∗ Zh,M
∗ →P
0,
as
For any
h → 0,
ε > 0, δ > 0,
in probability under
is equivalent to
limh→0 Qh,M (Ah,M ) = 0,
∗ then that Zh,M
∗ →P
0,
as
letting
h → 0,
since
P,
P
o n ∗ Ah,M ≡ P ∗ Zh,M >δ >ε ,
if for any
and
in probability under
Qh,M
we have that
ε > 0, δ > 0, limh→0 P (Ah,M ) = 0.
This
are contiguous (see Remark 1). It follows
Qh,M .
The inverse follows similarly.
Theorem 2.1 provides a theoretical justi cation to derive bootstrap consistency results under the approximate measure
Qh,M
as well as under
P.
This may simplify the bootstrap inference. We will
6
subsequently rely on this theorem to establish the bootstrap consistency results, when necessary (see e.g., Theorem 4.2 and Theorem 4.3 in Section 4.2).
3
Results for realized volatility
3.1 Existing asymptotic theory To describe the asymptotic properties of realized volatility, we need to introduce some notation. In the following we assume that
1/M h
is an integer. For any
q > 0,
de ne the realized
q -th
order power
variation as
1/M h
Rq ≡ M h
X j=1
RVj,M Mh
q/2 .
(6)
PM
2 i=1 yi+(j−1)M is the realized volatility over the period
where
RVj,M =
1, . . . ,
1 M h . Note that when
q = 2, R2 = RV
[(j − 1) M h, jM h]
(realized volatility). Similarly, for any
q > 0,
for
j =
de ne the
integrated power variation by
σq ≡
Z1
σuq du.
0 Mykland and Zhang (2009) show that standard
χ2
distribution with
M
1
cM,q Rq
cM,q = Γ is the Gamma function.
(CLT) result for
Rq
with
M
→
σ q , where
cM,q ≡ E
with
χ2M
the
(7)
Similarly, Mykland and Zhang (2009) provide a central limit theorem
bounded, whereas Mykland and Zhang (2011) based also on a contiguity
−1 , provided the sample size h
Rq ,
q/2
q/2 Γ q+M 2 2 , M M Γ 2
argument, but using a dierent representation than (4) allow the block size
M h → 0,
χ2M M
degrees of freedom and
where
P
M
−1/2 ). Note that when is of order O(h
without using any contiguity argument between
Qh,M
and
M
to go to in nity with
M →∞
P,
as
h→0
such that
a CLT result still holds for
although this may involves the presence of de-biasing term, see e.g., Jacod and Protter (2012),
Jacod and Rosembaum (2013) (cf. equations (3.8) and (3.11)) and the recent work of Li, Todorov and Tauchen (2016) (cf. Theorems 2 and 3). We would like to highlight that for the special case
M → ∞, Rq
q = 2 (i.e.,
the case of realized volatility), when
enjoys a CLT without any bias correction term. Speci cally, for
q = 2,
as the number of
intraday observations increases to in nity
√
h−1 R2 − σ 2 d p → N (0, 1), Vσ¯2
7
(8)
where
1 M cM,4 − c2M,2 Z
Vσ¯2 =
Note that for all that for xed
c2M,2
0
M (cM,4 −c2M,2 ) M +2 , it follows that M c2M,2
M > 0, cM,2 = 1, cM,4 =
M,
σu4 du.
(8) holds true under both
P
and
Qh,M
= 2.
In addition, note
(see Mykland and Zhang (2009)). Whereas
M → ∞, (8) holds directly under the true measure P, as soon as h → 0 such that M 3 h → ∞ and
when
M 2h → 0
(see e.g., Jacod and Rosembaum (2013) (cf. equations (3.6), (3.8) and (3.11))). In practice,
Vσ¯2 ,M
result in (8) is infeasible since the asymptotic variance integrated quarticity
R1
σu4 du.
For, xed
M,
depends on an unobserved quantity, the
Mykland and Zhang (2009) (cf.
Remark 8) propose a
0 consistent estimator of have
P
Rq → σ q ,
M (cM,4 −c2M,2 ) 1 cM,4 R4 ). When c2M,2
Vσ¯2 (Vˆσ¯2 ,h,M =
M → ∞, cM,q → 1
and we still
see e.g., Jacod and Rosembaum (2013) (cf. equation (3.13) and Corollary 3.7), and
Theorem 3 of Li, Todorov and Tauchen (2016), see also Theorem 9.4.1 of Jacod and Protter (2012) for similar result.
M → ∞.
Hence together with (8), we have the feasible CLT, for both cases: xed
It follows that as
M
and
h→0 √ TσMZ ¯2 ,h,M ≡
h−1 R2 − σ 2 d q → N (0, 1), Vˆ ¯2
(9)
σ ,h,M
where we added the superscripts MZ to precise that the statistic is related to Mykland and Zhang (2009) blocking method to avoid confusion when necessary. When the block size
M
is xed such that
M = 1,
this result is equivalent to the CLT for realized
volatility derived by Barndor-Nielsen and Shephard (2002).
c1,4 = E
2 χ21
= 3.
We de ne for
In particular,
c1,2 = E χ21 = 1,
and
M = 1, Tσ¯2 ,h = Tσ¯2 ,h,1 . Here, when M > 1, the realized volatility RV
using the blocking approach is the same realized volatility studied by Barndor-Nielsen and Shephard (2002), but the
t-statistic
is dierent because
Vˆσ¯2 ,h,M
changes with
M.
One advantage of the block-
based estimator is to improve e ciency by varying the size of the block (see for e.g. Mykland, Shephard and Sheppard (2012)).
3.2 Bootstrap consistency Here we show that the new bootstrap method we proposed in Section 2 is consistent when applied to realized volatility. Speci cally, given (5) with
r ∗ yi+(j−1)M = where
∗ ηi+(j−1)M ∼
i.i.d.N (0, 1) across
d = 1,
for
j = 1, . . . , 1/M h,
we let
RVj,M ∗ ηi+(j−1)M , i = 1, . . . , M, M
(i, j).
Note that this bootstrap method is related to the wild
bootstrap approach proposed by Gonçalves and Meddahi (2009). In particular, when
RVj,M M
= yj2
(10)
M = 1 and d = 1,
this amounts to the wild bootstrap based on a standard normal external random variable.
8
As the results of Gonçalves and Meddahi (2009) show, this choice of on
σ2
because it does not replicate the higher order cumulants of
RV
η∗
is not optimal for inference
up to
o h1/2
propose a two-point distribution that matches the rst three cumulants up to provides a second-order re nement. We show that by replacing
RV
an asymptotically valid bootstrap method for through order
o (h)
h1/2
. Instead, they
with the local average
and therefore
RVj,M M , yields
and can matches the rst four cumulants of
even when volatility is stochastic.
RV
To see the gain of using the local Gaussian
dXt = σdWt
bootstrap method. Let consider a simpli ed model without drift), where
yj2
o
d
(i.e. the constant volatility model
d
yi = σh1/2 νi , with νi ∼ i.i.d. N (0, 1) , and `=' denotes equivalence in distribution.
We have that
2
RVj,M = σ h
M X
! 2 νi+(j−1)M
≡ σ 2 h · χ2M,j ,
i=1 where
σ 2 h,
χ2M,j
is i.i.d.∼
χ2M .
Thus,
E (RVj,M ) = σ 2 hE χ2M,j = σ 2 hM ,
the integrated volatility over the
2 of σ h, it is clear that choosing RVj,M is variance of M
2σ 4 h2 /M ,
h-horizon.
M > 1
Although
yi2
implying that
RVj,M estimates M
can also be viewed as an estimator
improves the e ciency of the estimator (for given
which decreases with increasing
M ).
h,
the
The local Gaussian bootstrap
exploited this source of e ciency while at the same time preserving the local Gaussianity inherent in the semimartingale continuous time model driving
Xt .
Although (10) is motivated by this simpli ed
model, as we will prove below, this does not prevent the local Gaussian bootstrap method to be valid more generally.
Its rst-order validity extends to the case where volatility is stochastic, there is a
leverage eect and the drift is non-zero.
In particular, it remains asymptotically valid for general
stochastic volatility models described by (1) and (2). We de ne the bootstrap realized volatility estimator as follows
R2∗ =
1/h X
1/M h
yi∗2 =
i=1 where
∗ = RVj,M
X
∗ RVj,M ,
j=1
PM
∗2 i=1 yi+(j−1)M . Letting M χ2j,M 1 X ∗2 ηi+(j−1)M ≡ , M M i=1
it follows that
∗ = RVj,M
χ2j,M M
RVj,M .
E ∗ (R2∗ ) = cM,2 R2 ,
We can easily show that
and
Vσ∗¯2 ,h,M ≡ V ar∗ h−1/2 R2∗ = M cM,4 −µ22 R4 .
Hence, we propose the following consistent estimator of
Vˆσ∗¯2MZ =M ,h,M
Vσ∗¯2 ,h,M :
cM,4 − c2M,2 cM,4
9
R4∗ .
The bootstrap analogue of
TσMZ ¯2 ,h,M
is given by
√ Tσ∗¯2MZ ,h,M
≡
h−1 (R2∗ − cM,2 R2 ) q . Vˆ ∗¯MZ
(11)
σ 2 ,h,M
Theorem 3.1. Suppose (1), (2) and (5) hold. We have that −c2
M,2 ) 1 a) For xed M, Vσ∗¯2 ,h,M = MM+2 Vˆσ¯2 ,h,M , where Vˆσ¯2 ,h,M = ( M,4 cM,4 R4 is a consistent estimator c2M,2 of Vσ¯2 . If M is xed or M → ∞ as h → 0 such that M = o h−1/2 , then as h → 0,
M c
M V ∗¯ − Vσ¯2 = oP (1) , M + 2 σ2 ,h,M
˜MZ and sup P ∗ T˜σ∗¯2MZ ≤ x − P T ≤ x → 0, ¯ 2 ,h,M σ ,h,M
in probability under P , where T˜σ∗¯2MZ ≡ ,h,M
b) If M is xed or M
→∞
x∈<
q
M M +2
√
h−1 (R2∗ − cM,2 R2 ) and T˜σMZ ¯2 ,h,M ≡
√
h−1 R2 − σ 2 .
as h → 0 such that M = o h−1/2 , then as h → 0,
MZ sup P ∗ Tσ∗¯2MZ ≤ x − P T ≤ x → 0, ,h,M σ¯2 ,h,M
x∈<
in probability under P. Part a) of Theorem 3.1 justi es using the local Gaussian bootstrap to estimate the entire distribution of
T˜σ¯2 ,h,M ,
thus to construct bootstrap unstudentized (percentile) intervals for integrated volatility.
Whereas part b) of Theorem 3.1 provides a theoretical justi cation for using the bootstrap distribution of
Tσ∗¯2 ,h,M
to estimate the entire distribution of
Tσ¯2 ,h,M . This result also justi es the use of the bootstrap
for constructing the studentized bootstrap (percentile-t) intervals. Once again, it is worth emphasising that these results hold under the general context studied by Mykland and Zhang (2009) which allow for the presence of drifts and leverage eects under
P.
As the proof of Theorem 3.1 shows, here we
derive directly bootstrap results under the true probability measure measure
Qh,M ;
P , without using any approximate
and /or the machinery of contiguity. Then we do not need to rely on Theorem 2.1 to
establish the bootstrap consistency results. The asymptotic validity of the local Gaussian bootstrap depends on the availability of a CLT result of
R2
and a law of large numbers for
under the assumptions of Mykland and Zhang (2009) when Jacod and Rosembaum (2013) when
M
Rq , which hold directly
is xed, and under the assumptions of
M → ∞.
To see the gain from the new local Gaussian bootstrap procedure, one should compare these results with those of Gonçalves and Meddahi (2009). Results in part a) of Theorem 3.1 is in contrast to the bootstrap methods studied by Gonçalves and Meddahi (2009), which are only valid for percentile-t intervals i.e., when the statistic is normalized by its (bootstrap) standard deviation.
In particular,
the i.i.d. bootstrap variance estimator for the asymptotic variance of the realized volatility studied by Gonçalves and Meddahi (2009) (cf. page 287) is given by
h−1
1/h X i=1
2 Z 1/h X yi4 − yi2 →P 3 0
i=1 10
1
σs4 ds −
Z 0
1
2 σs2 ds ,
which is equal to
2
R1
σs4 ds
(i.e.
the asymptotic conditional variance of the realized volatility) only
when the volatility is constant.
Similarly, the wild bootstrap variance estimator of Gonçalves and
0
Meddahi (2009) (cf. page 288) for the asymptotic variance of the realized volatility is given by
µ∗4
−
µ∗2 2
h−1
1/h X
yi4
→
µ∗4
P
i=1
where
µ∗q ≡ E ∗ |ν ∗ |q ,
with
ν∗
−
µ∗2 2
|
Z
·3 {z
≡VGM
0
1
σs4 ds, }
(12)
the external random variable used in Gonçalves and Meddahi (2009) to
generate the wild bootstrap observations. In particular, using the best existing choice of
ν∗
(i.e. the
optimal two-point distribution) suggested in Proposition 4.5 of Gonçalves and Meddahi (2009) yield:
µ∗q where
p=
1 2
−
q q q q √ √ 1 1 31 + 186 · p + − 31 − 186 · (1 − p) = 5 5
√ 3 . Thus, we have 186
µ∗2 ≈ 0.991, It follows that
VGM
and
µ∗4 ≈ 1.217.
becomes
1
Z
σs4 ds
VGM ≈ 0.679 ·
Z
1
σs4 ds.
6= 2
0
0
This means that we would neither use the i.i.d.
bootstrap nor the best existing bootstrap method
of realized volatility (i.e. the optimal two-point wild bootstrap of Gonçalves and Meddahi (2009)) to estimate
Vσ¯2 .
However, note that although the bootstrap methods in Gonçalves and Meddahi (2009)
do not consistently estimate (percentile-t) intervals.
Vσ¯2 , their bootstrap methods are still asymptotically valid for studentized
It is also possible to consistently estimate the asymptotic variance of
R2 ,
by using the wild bootstrap of Gonçalves and Meddahi (2009) with certain external random variable. Speci cally, given (12), we can show that a necessary condition for the consistency of the wild bootstrap variance in Gonçalves and Meddahi (2009) is that this choice of
ν∗
µ∗4 − µ∗2 2 =
2 3 . Unfortunately, it is easy to see that
does not deliver an asymptotic re nement.
According to part a) of Theorem 3.1, the local Gaussian bootstrap variance estimator out adjustment is not consistent for (small) as
h → 0, Vσ∗¯2 ,h,M →P
Vσ¯2
M +2 M Vσ¯2
when the block size
6= Vσ¯2 ,
for large
mately). This explains the presence of the scale factor of the local Gaussian bootstrap variance even when this section, the estimator of
Vσ¯2
q
M
M
M
with-
is nite. In particular, for xed
(but bounded)
M M +2 in
Vσ∗¯2 ,h,M
Vσ∗¯2 ,h,M → Vσ¯2
M
(approxi-
T˜σMZ ¯2 ,h,M , in order to restore consistency
is nite. In the univariate context considered in
is rather simple (it is given by a (scaled) version of
R4 ), but this is not
necessarily the case for other applications. For instance, for realized regression coecients de ned by the Mykland and Zhang's (2009) blocking approach the bootstrap percentile method could be useful in that context. As for all blocking methods, for the local Gaussian bootstrap there is an inherent trade-o at stake
11
by increasing the block size
M.
To gain further insight, let's consider once again the simpli ed model
h Vσ¯2 = 2σ 4 , Vσ∗¯2 ,h,M = 2σ 4 M
dXt = σdWt .
It is easy to see that for this model
χ2M,j is i.i.d.∼
χ2M . A straightforward calculation shows that
Bias
Vσ∗¯2 ,h,M
4σ 2 =O = M
1 M
and
V ar
Vσ∗¯2 ,h,M
whereas the variance increases with increasing
j=1
χ2M,j
2
where
(M + 2) M 2 + 9M + 24 4 σ = O (M h) , = 4M h M3
see Lemma B.3 in the Appendix for further details. Then the bias of
M,
P1/M h
M.
Vσ∗¯2 ,h,M
decreases with increasing
Hence, in order to pick up any biases due to
the blocking in our local Gaussian bootstrap procedure, one should choose
M
very large. Of course,
∗ ∗ this comes with a cost: large variance of V ¯2 . It follows that the mean squared error of V ¯2 σ ,h,M σ ,h,M 2 + O (M h) . To minimize it, one should pick M proportional to h−1/3 , in which is of order O 1/M case
M SE Vσ∗¯2 ,h,M = O h−2/3 .
However, the optimal block size for mean squared error may be
suboptimal for the purpose of distribution estimation; see Hall, Horowitz and Jing (1995). We will not pursue this approach further here. We leave this analysis to future work. To summarize, Theorem 3.1 provides a theoretical justi cation for using the local Gaussian bootstrap approach as alternative to the existing bootstrap methods proposed by Gonçalves and Meddahi (2009). Compared with the latter, there is no gain to use the local Gaussian bootstrap method when
M = 1.
However, when
M > 1,
M
with
large, in addition to consistently estimate the (conditional)
asymptotic variance of realized volatility, as we will see shortly (see Section 5.1 below), one can also reduce the local Gaussian bootstrap error in estimating cally, if the block length
M → ∞,
as
h→0
such that
P Tσ¯2 ,h ≤ x
M h → 0,
up to
oP (h).
More speci -
then the local Gaussian bootstrap is
an improvement of the best existing bootstrap methods of Gonçalves and Meddahi (2009). However, in contrast to Gonçalves and Meddahi (2009) our new bootstrap method requires the choice of an additional tuning parameter i.e., the block size
4
M.
Results for realized regression
Our aim in this section is to study the rst-order asymptotic properties of the local Gaussian bootstrap method in multivariate setting. In particular, we investigate the validity of our approach for two dierent estimators of the slope coe cient in a linear-time regression model for the continuous martingale parts of two semimartingales observed over the xed time interval
[0, 1].
The rst is the estimator in
Barndor-Nielsen and Shephard (2004). The second is an estimator based on the results in Mykland and Zhang (2009). To simplify the discussion, in the following, we consider the bivariate case where
d=2 and
and look at results for assets
yli ,
Let
k
and
l,
whose
ith
high frequency returns will be written as
yki
respectively.
Cˆ(j) ≡
Cˆkk(j) 0 ˆ ˆ Clk(j) Cll(j)
!
q ˆ kk(j) Γ 0 r = Γˆ kl(j) ˆ ll(j) − q Γ ˆ kk(j) Γ
12
ˆ2 be the Cholesky decomposition of Γ kl(j) ˆ Γkk(j)
1 Mh
PM
0 i=1 yi+(j−1)M yi+(j−1)M . Given (5) with ∗ yk,i+(j−1)M ∗ yl,i+(j−1)M
! = Cˆ(j) ·
√ h
for
j = 1, . . . , 1/M h, !
and
i = 1, . . . , M,
∗ ηk,i+(j−1)M , ∗ ηl,i+(j−1)M q ˆ kk(j) η ∗ Γ rk,i+(j−1)M ˆ2 ˆ ll(j) − Γkl(j) η ∗ η∗ + Γ
√ h
=
d = 2,
ˆ Γ q kl(j) ˆ kk(j) k,i+(j−1)M Γ
,
(13)
ˆ kk(j) l,i+(j−1)M Γ
where
M M M X X X 2 2 ˆ ll(j) = 1 ˆ kl(j) = 1 ˆ kk(j) = 1 , Γ , Γ Γ yk,i+(j−1)M yl,i+(j−1)M yk,i+(j−1)M yl,i+(j−1)M , Mh Mh Mh i=1 i=1 i=1 ! ∗ ηk,i+(j−1)M ∼ i.i.d.N (0, I2 ), with I2 a 2 × 2 identity matrix. and ∗ ηl,i+(j−1)M
4.1 Barndor-Nielsen and Shephard's (2004) type estimator Under (1) and given our assumptions on
µ and σ
(Barndor-Nielsen and Shephard (2004) cf. equations
(25) and (26) in conjunction with, e.g., Jacod and Protter (2012)), we have that
1/h P
2 yk,i
R1
− 0 Σkk,s ds i=1 1/h 1/h X √ √ R1 P Sh = h−1 vech yi yi0 − vech (Γ) = h−1 yk,i yl,i − 0 Σkl,s ds i=1 i=1 1/h P R 2 − 1Σ yl,i ll,s ds 0
st −→ N (0, Π) ,
(14)
i=1
where
st
−→
denotes stable convergence and
R1 R1 R1 2 0 Σ2kl,s ds 2 0 Σkk,s Σkl,s ds 2 0 Σ2kk,s ds R R R R Π = 2 01 Σkk,s Σkl,s ds 01 Σkk,s Σll,s ds + 01 Σ2kl,s ds 2 01 Σll,s Σkl,s ds . R1 R1 R1 2 0 Σll,s Σkl,s ds 2 0 Σ2ll,s ds 2 0 Σ2kl,s ds
Barndor-Nielsen and Shephard (2004) propose the following consistent estimator of
ˆ h = h−1 Π
1/h X
xi x0i
i=1
(15)
Π:
1/h−1 1 −1 X − h xi x0i+1 + xi+1 x0i , 2 i=1
where
xi =
2 2 yk,i yk,i yl,i yl,i
0
.
Thus
−1/2
ˆ Th = Π h where here
d(d+1) 2
= 3,
since we let
d = 2.
st Sh −→ N 0, I d(d+1) ,
(16)
2
We consider a general class of nonlinear transformations
that satisfy the following assumption. Throughout we let
13
∇f, (d × 1
vector-valued function) denote
the gradient of
f.
Assumption F:
The function
f : R
d(d+1) 2
→ R
is continuously dierentiable with
∇f (vech (Γ))
is
Γ.
non-zero for any sample path of
The corresponding statistics are de ned as
√ Sf,h =
h−1 f vech
1/h X
yi yi0 − f (vech (Γ))
and
−1/2 ˆ f,h Tf,h = Π Sf,h ,
i=1 where
ˆ f,h ≡ ∇0 f vech Π
1/h X
1/h X P ˆ h ∇f vech yi yi0 −→ Πf . yi yi0 Π
(17)
i=1
i=1
In particular, we consider the question of estimating the integrated beta, i.e., the parameter
R1 βlk = R 01 0
Σkl,s ds Σkk,s ds
.
(18)
Given (16) and by using the delta method, the asymptotic distribution of the realized regression coe cient
βˆlk =
1/h X
−1 2 yk,i
i=1 obtained from regressing
yl,i
on
yk,i
1/h X
yk,i yl,i ,
(19)
i=1
is
Sβ,h st Tβ,h = q −→ N (0, 1) , Vˆβ
(20)
where
√ Sβ,h =
h−1 βˆlk − βlk ,
Vˆβ =
1/h X
−2 h−1 gˆβlk
2 yk,i
such that
i=1
gˆβlk =
1/h X
1/h−1
x2βi
−
i=1
X
xβi xβ,i+1 ,
and
2 xβi = yl,i yk,i − βˆlk yk,i .
i=1
Sf,h is given by 1/h 1/h X X √ = h−1 f vech yi∗ yi∗0 − f vech yi yi0 .
The local Gaussian bootstrap version of
∗ Sf,h,M
i=1
i=1
For the raw statistic
∗ Sf,h,M
=
∗ Sh,M
√ =
h−1
1/h X
(x∗i − xi ) ,
i=1 where
x∗i =
∗2 y ∗ y ∗ ∗2 yk,i k,i l,i yl,i 14
0
.
It is elementary (though somewhat tedious) to show that
∗ E ∗ Sh,M = 0, and
ˆ2 ˆ kk(j) Γ ˆ kl(j) ˆ2 2Γ 2Γ 2Γ kk(j) kl(j) X ∗ ˆ kk(j) Γ ˆ kl(j) Γ ˆ kk(j) Γ ˆ ll(j) + Γ ˆ2 ˆ ˆ Sh,M = Mh 2Γ kl(j) 2Γkl(j) Γll(j) ˆ2 ˆ kl(j) Γ ˆ ll(j) ˆ2 j=1 2Γ 2Γ 2Γ kl(j) ll(j) 1/M h
Π∗h,M ≡ V ar∗
(see Lemma B.6 in the Appendix). Similarly, we denote the bootstrap analogue of
∗ Tβ,h by Tf,h,M ,
∗ , S∗ βˆlk β,h,M ,
Tf,h , βˆlk , Sβ,h ,
(21)
and
∗ and Tβ,h,M , respectively. In particular, we de ne
−1/2 ∗ ∗ ˆ∗ Sf,h.M , Tf,h,M ≡ Π f,h,M where
1/h 1/h X X ˆ ∗ ∇f vech yi∗ yi∗0 , ≡ ∇0 f vech yi∗ yi∗0 Π h,M
ˆ∗ Π f,h,M
i=1
i=1 with
ˆ ∗2 ˆ∗ Γ ˆ∗ ˆ ∗2 2Γ 2Γ 2Γ kk(j) kk(j) kl(j) kl(j) X ˆ∗ Γ ˆ∗ ˆ∗ ˆ∗ ˆ ∗2 ˆ∗ ˆ∗ = Mh 2Γ kk(j) kl(j) Γkk(j) Γll(j) + Γkl(j) 2Γkl(j) Γll(j) ˆ ∗2 ˆ∗ Γ ˆ∗ ˆ ∗2 j=1 2Γ 2Γ 2Γ kl(j) kl(j) ll(j) ll(j) ˆ2 ˆ kk(j) Γ ˆ kl(j) ˆ2 ˆ kk(j) Γ ˆ ll(j) 4 Γ 4 Γ 2 Γ + Γ 1/M h kk(j) kl(j) X 1 2 ˆ ˆ ˆ ˆ ˆ kl(j) Γ ˆ ll(j) ˆ 4 Γ Γ Γ Γ + 3 Γ 4Γ · Mh + kk(j) kl(j) kk(j) ll(j) kl(j) M j=1 4 ˆ2 ˆ2 ˆ ˆ ˆ kl(j) Γ ˆ ll(j) 2 Γ 4Γ M Γll(j) kl(j) + Γkk(j) Γll(j) 1/M h
ˆ∗ Π h,M
,
such that
M
M
M
i=1
i=1
i=1
X 1 X ∗2 1 X ∗ ∗2 ∗ ˆ∗ ˆ∗ = 1 ˆ∗ Γ yk,i+(j−1)M , Γ yl,i+(j−1)M , Γ yk,i+(j−1)M yl,i+(j−1)M . kk(j) = ll(j) kl(j) = Mh Mh Mh We also let
∗ βˆlk =
1/h X
−1 ∗2 yk,i
∗ Sβ,h,M =
∗ h−1 βˆlk − βˆlk ,
and
∗ Tβ,h,M
∗ ∗ yk,i yl,i ,
i=1
i=1
√
1/h X
∗ Sβ,h,M =q Vˆ ∗
such that
β,h,M
∗ Vˆβ,h,M
−2 1/h X ∗2 = yk,i h−1 gˆβ∗lk , i=1
with
gˆβ∗lk =
1/h X i=1
1/h−1
x∗2 βi −
X
x∗βi x∗β,i+1 ,
and
∗ ∗ ∗ ∗2 x∗βi = yl,i yk,i − βˆlk yk,i .
i=1
Theorem 4.1. Suppose (1), (2) and (5) hold. Under Assumption F and the true probability distribution P , if M → ∞ as h → 0 such that M = o h−1/2 , then as h → 0, the following hold 15
a)
Π∗f,h,M ≡ ∇0 f vech
1/h X
yi yi0 Π∗h,M ∇f vech
1/h X
P
yi yi0 −→ Πf ,
i=1
i=1
where Πf = p lim V ar (Sf,h ) , in particular h→0
where
P ∗ V ar∗ Sβ,h,M −→ Vβ , R −2 1 Vβ = p lim V ar (Sβ,h ) ≡ 0 Σkk,s ds B, with h→0
Z B=
1
2 2 Σ2kl,s + Σll,s Σkk,s − 4βlk Σkl,s Σkk,s + 2βlk Σkk,s ds.
0
b) P ∗ sup P ∗ Sf,h,M ≤ x − P (Sf,h ≤ x) −→ 0, x∈R
in particular P ∗ ≤ x − P (Sβ,h ≤ x) −→ 0. sup P ∗ Sβ,h,M x∈R
c) P ∗ ≤ x − P (Tf,h ≤ x) −→ 0, sup P ∗ Tf,h,M x∈R
in particular P ∗ ≤ x − P (Tβ,h ≤ x) −→ 0. sup P ∗ Tβ,h,M x∈R
Theorem 4.1 justi es using the local Gaussian bootstrap to estimate the distribution (and functionals of it such as the variance) of smooth function of realized covariance matrix
P1/h
0 i=1 yi yi , in particular
the realized beta. This contrasts with results in Dovonon et al. (2013) (cf. Theorem 5.1) where the traditional pairs bootstrap is not able to mimic the score heterogeneity.
4.2 New variance estimator of Mykland and Zhang's (2009) realized beta type estimator and bootstrap consistency The goal of this subsection is to describe the realized beta in the context of Mykland and Zhang's (2009) blocking approach. In order to obtain a feasible CLT, we propose a consistent estimator of the variance of the realized beta, which is a new estimator in this literature. To derive this result, we use the approach of Dovonon et al. (2013) and suppose that
σ
is independent of
Dovonon et al. (2013), we do not need here to suppose that directly proceed under
1
Qh,M
µt = 0.
W.1
Note that contrary to
The main reason is because we can
(we will see shortly how). This is in fact one gain to revisit Dovonon et al.
We make the assumption of no leverage for notational simplicity and because this allows us to easily compute the
moments of the intraday returns conditionally on the volatility path. It would cleary be desirable to have a formal proof of this result by allowing for leverage eect.
16
(2013) analysis with Mykland and Zhang's (2009) techniques of contiguity. In Dovonon et al. (2013), letting
µt = 0,
is mathematically convenient in order to easily compute the moments of cross product
of high frequency returns. One can see that when
µt = 0,
we can simply write a high frequency return
of a given asset as the product of its volatility (the spot volatility) multiplied by a normal standard distribution under the true probability measure
dXt = σt dWt ,
suppose that
where
equation (5)), conditionally on
σ,
σ
P.
Speci cally, Dovonon et al. (2013) (cf. Session 5.1)
is independent of
W.
Then, following Dovonon et al. (2013) (cf.
for a given frequency of the observations, we can write
yli = βlki yki + ui , where independently across with
Γlki =
R ih
(i−1)h Σlk
i = 1, . . . , 1/h, ui |yki ∼ N (0, Vi ) ,
(u) du.
(22) with
Vi ≡ Γli −
Γ2lki Γki , and
βlki ≡
Γlki Γki ,
2
Notice that (22) only makes sense in discrete time.
As Dovonon et al. (2013) argue, the conditional mean parameters of realized regression models are heterogeneous under stochastic volatility. This heterogeneity justi es why the pairs bootstrap method that they studied is not second-order accurate. Here we consider the general stochastic volatility model described by (1), but we rule out leverage eects.
Given (4) (without assume that
but under
√
Qh,M .
It follows that under
√
µt = 0)
high frequency returns have similar representation
√ Qh,M , yk,i+(j−1)M = Ckk(j) hηk,i+(j−1)M
and
yl,i+(j−1)M =
1 M h where
Clk(j) hηk,i+(j−1)M + Cll(j) hηl,i+(j−1)M , for i = 1, . . . , M and j = 1, . . . , Ckk(j) 0 ηk,i+(j−1)M C(j) ≡ , ηi+(j−1)M ≡ ∼ i.i.d.N (0, I2 ), Clk(j) Cll(j) ηl,i+(j−1)M 0 with C I2 is a 2×2 identity matrix, C(j) is the Cholesky decomposition of Σ(j) ≡ C(j) C(j) (j) = σ(j−1)M h , i.e., the value of
σt
observables in the
at the beginning of the
j th
block (j
j -th
= 1, . . . , M1h ),
block. Under the approximate measure
Qh,M
for the
the regression (22) becomes
yl,i+(j−1)M = βlk(j) yk,i+(j−1)M + ui+(j−1)M ,
(23)
Clk(j) 2 , and β ui+(j−1)M |yk,i+(j−1)M ∼i.i.d.N 0, V(j) , for i = 1, . . . , M , with V(j) = hCll(j) lk(j) ≡ Ckk(j) . ˇlk(j) the ordinary least squares (OLS) estimator of βlk(j) . To estimate the parameter Let us denote by β R1 ˇ given by βlk = 0 βlk,s ds, Mykland and Zhang (2009) proposed to use βlk de ned as follows, !−1 M ! 1/M h 1/M h M X X X X 2 βˇlk = M h βˇlk(j) = M h y yk,i+(j−1)M yl,i+(j−1)M . (24)
where
k,i+(j−1)M
j=1
j=1
i=1
i=1
Note that the realized beta estimator discussed in Section 4.1.1 (see equation (18) and (19)) and also studied among others by Dovonon et al. (2013) is a dierent statistic than the term given by (24). Here, the realized beta estimator average of
is not directly a least squares estimator, but is the result of the
βˇlk(j) , the OLS estimators for each block.
in each block
2
βˇlk
j,
Since under
Qh,M , the volatility matrix is constant
implying consequently that the score is not heterogeneous and has mean zero. This
The underlying data generating process is in continuous time while one observes a discrete time sample of the process.
17
simpli es the asymptotic inference on
βlk(j) , and on βlk .
It is worth emphasising that contrary to what 3
M = 1,
we have observed in the case of realized volatility estimator in Section 3.1 , here when
the
realized beta estimator using the blocking approach becomes
βˇlk = h
1/h X yl,i , yk,i i=1
which is a dierent statistic than the statistic studied by Barndor-Nielsen and Shephard (2004). But when
M = h−1 ,
when
M → ∞ with the sample size h−1 ,
both estimators are equivalent. However, as Mykland and Zhang (2011) pointed out, the local approximation is good only when
M = O(h−1/2 ).
It
follows then that we are not comfortable to contrast Mykland and Zhang (2009) block-based "realized
M =
beta" estimator asymptotic results with those of Barndor-Nielsen and Shephard (2004a) when
h−1 .
M = h−1 )
The rst reason is that in this case (i.e.
M → ∞,
enough. The second and main reason is that when know whether the approximate measure
Qh,M
the local approximation is not accurate to the best of our knowledge we do not
with representation given by (4) is contiguous to
Mykland and Zhang (2009) provide a CLT result for
βlk .
In particular, we have under
as the number of intraday observations increases to in nity (i.e. Mykland and Zhang (2009), as
if
h → 0),
h → 0, for any δ > 0 such that M > 2 (1+δ) √ h−1 βˇlk − βlk d q → N (0, 1), Vβˇ
P
and
P. Qh,M ,
by using Section 4.2 of
with
M = O(1), (25)
where
M Vβˇ = M −2
Z
1
0
Σll,s 2 − βlk,s ds. Σkk,s
In practice, this result is infeasible since the asymptotic variance
Vβˇ
depends on unobserved quantities.
Mykland and Zhang (2009) did not provide any consistent estimator of to propose a consistent estimator of
Vβˇ.
Vβˇ.
To this end, we exploit the special structure of the regression
model. To nd the asymptotic variance of realized regression estimator
√
One of our contributions is
βˇlk ,
we can write
Xh √ 1/M βˇlk(j) − βlk(j) . h−1 βˇlk − βlk = M h j=1
Since
βˇlk(j)
are independent across
Vβ,h,M ≡ V ar ˇ
√
j,
it follows that
1/M h
h−1
βˇlk − βlk
2
=M h
X
V ar βˇlk(j) − βlk(j) .
(26)
j=1 3
Where the realized volatility estimator
R2
using Mykland and Zhang's (2009) blocking approach with
same realized volatility studied by Barndor-Nielsen and Shephard (2002).
18
M = 1,
is the
To compute (26), note that from standard regression theory, we have that under
M X
V ar βˇlk(j) − βlk(j) = E
Qh,M ,
!−1 2 yk,i+(j−1)M
V(j) ,
i=1 which implies that
1/M h
X
= M 2h Vβ,h,M ˇ
!−1
M X
E
j=1
2 yk,i+(j−1)M
V(j) .
(27)
i=1
Vβˇ with equation (72) of Mykland and Zhang (2009). In fact, we can PM 2 d d 2 2 2 2 i=1 yk,i+(j−1)M = hCkk(j) i=1 vi+(j−1)M = hCkk(j) χj,M , where vi+(j−1)M ∼ 2 follow the standard χ distribution with M degrees of freedom. Then for any
Note that we can contrast write under
Qh,M ,
i.i.d.N (0, 1)
and
integer
M >2
PM
χ2j,M
and conditionally on the volatility path, by using the expectation of the inverse of a
Chi square distribution we have,
E
M X
!−1 2 yk,i+(j−1)M
=E
i=1
!
1
1 −2 h−1 Ckk(j) . M −2
−2 h−1 Ckk(j) =
χ2j,M
(28)
It follows then that
Vβ,h,M ˇ
1/M h X Cll(j) 2 M = Mh . M −2 Ckk(j) j=1
Vβ,h,M ˇ
By using the structure of (27), a natural consistent estimator of
1/M h
Vˆβ,h,M ≡ M 2h ˇ where
X
M X
j=1
i=1
!−1
! ,
(see Lemma B.13 and Lemma B.14 in the Appendix).
P
and
Qh,M
the feasible result
h−1 (βˇlk − βlk ) d q → N (0, 1) . Vˆβ,h,M ˇ
Next we show that the local Gaussian bootstrap method is consistent when applied to realized beta estimator given by (24). Recall (13), and let
∗ from the regression of y l,i+(j−1)M on
∗ βˇlk(j)
j.
X
βˇ∗lk(j) .
j=1 ∗ βˇlk
converges in probability (under
1/M h
βˇlk = M h
X j=1
E∗
M X
!−1 2∗ yk,i+(j−1)M
i=1
P ∗)
M X i=1
19
i.e., the
The bootstrap realized beta
1/M h ∗ βˇlk = Mh
βˇlk
denote the OLS bootstrap estimator
∗ yk,i+(j−1)M inside the block
estimator is
It is easy to check that
(29)
i=1
Together with the CLT result (25), we have under
≡ Tβ,h,M ˇ
1 X 2 u ˆi+(j−1)M M −1
2 yk,i+(j−1)M
u ˆi+(j−1)M = yl,i+(j−1)M − βˇlk(j) yk,i+(j−1)M √
M
is
to
! ∗ ∗ . yk,i+(j−1)M yl,i+(j−1)M
The bootstrap analogue of the regression error
∗ βˇlk(j) yk,i+(j−1)M , ∗ ∗ βˇ y .
ui+(j−1)M
in model (23) is thus
whereas the bootstrap OLS residuals are de ned as
∗ u∗i+(j−1)M = yl,i+(j−1)M −
∗ u ˆ∗i+(j−1)M = yl,i+(j−1)M −
Thus, conditionally on the observed vector of returns lk(j) k,i+(j−1)M ∗ u∗i+(j−1)M |yk,i+(j−1)M ∼ i.i.d.N 0, Vˆ(j) , for i = 1, . . . , M , where
yi+(j−1)M ,
it follows that
2 . Vˆ(j) ≡ hCˆll(j) We can show that for xed
M, V ar∗
Hence, for any xed
Vβˇ.
M,
√ M −1 ∗ h−1 (βˇlk − βˇlk ) = Vˆ ˇ . M − 2 β,h,M
we would not use the local Gaussian bootstrap variance estimator to estimate
However, since we know the scaled factor
M −1 M −2 , this does not create a problem if we adjust the
bootstrap statistic accordingly. Our next theorem summarizes these results.
Theorem 4.2. Consider DGP (1), (2) and suppose (5) holds. Then conditionally on σ, under Qh,M and P for xed M , the following hold, as h → 0, for any δ > 0 such that M > 2 (1+δ) with M = O(1), a) ∗ Vβ,h,M ˇ
≡ V ar∗ P
→
√
∗ − βˇlk ) h−1 (βˇlk
M −1 V ˇ, M −2 β
b) sup P ∗ x∈R
r
M − 2 √ −1 ˇ∗ h βlk − βˇlk ≤ x M −1
! −P
√
h−1
P βˇlk − βlk → 0.
Part (a) of Theorem 4.2 shows that the bootstrap variance estimator is not consistent for the block size any
M
is nite. Note that for large
δ > 0 such that M > 2 (1+δ),
M −1 M −2
−1
M,
∗ but bounded V ˇ β,h,M
∗ Vβ,h,M → Vβˇ. ˇ
→ Vβˇ,
Vβˇ
when
(approximately) and for
Results in part (b) imply that the bootstrap
realized beta estimator has a rst-order asymptotic normal distribution with mean zero and covariance matrix
Vβˇ.
This is in line with the existing results in the cross section regression context, where the
wild bootstrap and the pairs bootstrap variance estimator of the least squares estimator are robust to heteroskedasticity in the error term. Bootstrap percentile intervals do not promise asymptotic re nements. Next, we propose a consistent bootstrap variance estimator that allows us to form bootstrap percentile-t intervals. More speci cally, we can show that the following bootstrap variance estimator consistently estimates
1/M h ∗ Vˆβ,h,M ˇ
2
≡M h
X
M X
j=1
i=1
!−1 ∗2 yk,i+(j−1)M
! M 1 X ∗2 u ˆi+(j−1)M . M −1 i=1
20
V ∗β,h,M : ˇ (30)
Our proposal is to use this estimator to construct the bootstrap t-statistic, associated with the bootstrap realized regression coe cient
∗. βˇlk
Let
√ ∗ ≡ Tβ,h,M ˇ
∗ −β ˇlk h−1 βˇlk q , Vˆ ∗
(31)
ˇ β,h,M
be the bootstrap analogue of
Tβ,h,M . ˇ
Theorem 4.3. Consider DGP (1), (2) and suppose (5) holds. Let that M is bounded, conditionally on σ, as h → 0, the following hold ∗
∗ Tβ,h,M →d N (0, 1) , ˇ
Note that when the block size
M
M > 4 (2+δ)
for any δ > 0 such
in probability, under Qh,M and P.
is nite the bootstrap is also rst-order asymptotically valid
∗ when applied to the t-statistic T ˇ (de ned in (31) without any scaled factor), as our Theorem 4.3 β,h,M ∗ proves. This rst-order asymptotic validity occurs despite the fact that Vβ,h,M does not consistently estimate
∗ Vˆβ,h,M ˇ
Vβˇ
M
is xed. The key aspect is that we studentize the bootstrap OLS estimator with
(de ned in (30)), a consistent estimator of
bootstrap
5
when
t-statistic
∗ Vβ,h,M , ˇ
implying that the asymptotic variance of the
is one.
Higher-order properties
In this section, we investigate the asymptotic higher order properties of the bootstrap cumulants. Section 5.1 considers the case of realized volatility whereas Section 5.2 considers realized beta estimator as studied by Barndor-Nielsen and Shephard (2004). The ability of the bootstrap to accurately match the cumulants of the statistic of interest is a rst step to showing that the bootstrap oers an asymptotic re nement. The results in this section are derived under the assumption of zero drift and no leverage (i.e. is assumed independent of
σ ).
As in Dovonon et al. (2013), a nonzero drift changes the expressions
of the cumulants derived here. For instance, for the realized volatility, the eect of the drift on
√ OP h .
W
Th
is
While this eect is asymptotically negligible at rst-order, it is not at higher orders. See
also the recent work of Hounyo and Veliyev (2016) (cf. equation (3.8)) where an additional term shows up in the Edgeworth expansions for realized volatility estimator when the drift
µt
is non zero. The no
leverage assumption is mathematically convenient as it allows us to condition on the path of volatility when computing the cumulants of our statistics.
Allowing for leverage is a di cult but promising
extension of the results derived here. We introduce some notation. For any statistics
Th
and
Th∗ ,
we write
κj (Th ) to denote the j th
∗ ∗ cumulant of Th and κj (Th ) to denote the corresponding bootstrap cumulant. For
denotes the coe cient of the terms of order
O
√
h
21
of the asymptotic expansion of
j = 1
and
κj (Th ),
order
3, κj
whereas
for
j = 2
κ∗j,h,M
and
4 , κj
O (h).
denotes the coe cients of the terms of order
The bootstrap coe cients
are de ned similarly.
5.1 Higher order cumulants of realized volatility Here, we focus on the
t-statistic Tσ¯2 ,h
t-statistic Tσ∗¯2 ,h,M √ h−1 R2 − µ2 σ 2 q , = Vˆ ¯2
and the bootstrap
Tσ¯2 ,h
de ned by
(32)
σ ,h
where
(µ4 −µ22 ) −1 P1/h 4 h Vˆσ¯2 ,h = i=1 yi , µ4
and
√ Tσ∗¯2 ,h,M
=
h−1 (R2∗ − µ2 R2 ) q , Vˆ ∗¯
(33)
σ 2 ,h,M
where
Vˆσ∗¯2 ,h,M
1/M h h−1 M X RVj,M 2 1 X µ4 − µ22 −1 X ∗4 µ4 − µ22 ∗4 = h Mh ηi+(j−1)M − µ4 , yi = µ4 µ4 Mh M i=1
i=1
such that
Rq,p ≡
∗ ηi+(j−1)M ∼ i.i.d.N (0, 1), µq = E |η ∗ |q
for
i=1
q > 0.
Let
σq,p ≡
σq
q/p , σp
( )
for any
q, p > 0,
and
Rq . We make the following assumption. (Rp )q/p
Assumption H. volatility
σ
The log price process follows (1) with
µt = 0 and σt
is independent of
Wt , where the
is a càdlàg process, bounded away from zero, and satis es the following regularity
condition:
lim h
1/2
h→0 for some
r>0
and for any
ηi
and
ξi
1/h X r ση − σ r = 0, ξ i i i=1
such that
0 ≤ ξ1 ≤ η1 ≤ h ≤ ξ2 ≤ η2 ≤ 2h ≤ . . . ≤ ξ1/h ≤
η1/h ≤ 1. Assumption H is stronger than required to prove the CLT for
R2 ,
and the rst-order validity of
the local Gaussian bootstrap, but it is a convenient assumption to derive the cumulants expansions of
Tσ¯2 ,h
and
Tσ∗¯2 ,h,M .
that for any
q>
Speci cally, under Assumption H, Barndor-Nielsen and Shephard (2004b) show
0, σhq
− σ q = o(
√
h), where σhq
=
h1−q/2
1/h P
Rsh
s=1
(s−1)h
subsequently rely on to establish the cumulants expansion of
22
Tσ¯2 ,h .
!q/2 σu2 du
, a result on which we
The following result gives the expressions of
κj
and
κ∗j,h,M
for
j = 1, 2, 3
and
4.
We need to
introduce some notation. Let
A1 = B1 =
√ µ6 − µ2 µ4 = 2 2, 1/2 µ4 µ4 − µ22 √ µ6 − 3µ2 µ4 + 2µ32 = 2 2, 3/2 µ4 − µ22
A2 =
µ8 − µ24 − 2µ2 µ6 + 2µ22 µ4 = 12, µ4 µ4 − µ22
B2 =
µ8 − 4µ2 µ6 + 12µ22 µ4 − 6µ42 − 3µ24 = 12, 2 µ4 − µ22
C1 =
µ8 − µ24 32 = . 2 3 µ4
Theorem 5.1. Suppose (1) and (2) hold with µ = 0 and W independent of σ. Furthermore assume (5). Under Assumption H, conditionally on σ and under P, it follows that, as h → 0 a) κ1 Tσ¯2 ,h κ2 Tσ¯2 ,h κ3 Tσ¯2 ,h κ4 Tσ¯2 ,h
=
√ hκ1 + o (h) ,
with κ1 = −
A1 σ6,4 , 2
7 2 = 1 + hκ2 + o (h) , with κ2 = (C1 − A2 ) σ8,4 + A21 σ6,4 , 4 √ hκ3 + o (h) , with κ3 = (B1 − 3A1 ) σ6,4 , = = hκ4 + o (h) ,
2 with κ4 = (B2 + 3C 1 − 6A2 ) σ8,4 + 18A21 − 6A1 B1 σ6,4 .
b) κ∗1 Tσ∗¯2 ,h,M κ∗2 Tσ∗¯2 ,h,M κ∗3 Tσ∗¯2 ,h,M κ∗4 Tσ∗¯2 ,h,M
√ =
hκ∗1,h,M + oP (h) ,
with κ∗1,h,M = −
A1 R6,4 , 2
7 2 = 1 + hκ∗2,h,M + oP (h) , with κ∗2,h,M = (C1 − A2 ) R8,4 + A21 R6,4 , 4 √ ∗ = hκ3,h,M + oP (h) , with κ∗3,h,M = (B1 − 3A1 ) R6,4 , = hκ∗4,h,M + oP (h) ,
2 . with κ∗4,h,M = (B2 + 3C 1 − 6A2 ) R8,4 + 18A21 − 6A1 B1 R6,4
c) For j = 1, 2, 3 and 4, p limκ∗j,h,M − κj is nonzero if M is nite and it is zero if M such that M = o
h→0 h−1/2 .
→∞
as h → 0
Theorem 5.1 states our main ndings for realized volatility. Part (a) of Theorem 5.1 are well known in the literature (see e.g., Gonçalves and Meddahi (2009) (cf. Theorem A.1) and Hounyo and Veliyev (2016)). These results are only given here for completeness. The remaining results in parts (b) and (c) are new. Part (b) gives the corresponding results for the local Gaussian bootstrap. Part (c) shows that the cumulants of
Tσ¯2 ,h
and
Tσ∗¯2 ,h,M
do not agree up to
o h1/2
when the block size
M
is xed (although
they are consistent), implying that the bootstrap does not provide a higher-order asymptotic re nement
23
for nite values of up to
o h1/2
M.
when
M
is nite is that
as
h→0
p limRq,p − σq,p
does not always equal to zero. Nevertheless,
h→0
M h → 0, p limRq,p − σq,p = 0, then the bootstrap matches the h→0 1/2 , which implies that it provides a second-order rst and third order cumulants through order O h ∗ T∗ re nement, i.e. the bootstrap distribution P ≤ x consistently estimates P Tσ¯2 ,h ≤ x with ¯ 2 σ ,h,M 1/2 (assuming the corresponding Edgeworth expansions exist).4 This is an error that vanishes as o h −1/2 . in contrast with the rst-order asymptotic Gaussian distribution whose error converges as O h
when
M →∞
The main reason why the local Gaussian bootstrap is not able to match cumulants
such that
Note that Gonçalves and Meddahi (2009) also proposed a choice of the external random variable for their wild bootstrap method which delivers second-order re nements. Our results for the bootstrap method based on the local Gaussianity are new. We will compare the two methods in the simulation section. Theorem 5.1 also shows that the new bootstrap method we propose is able to match the second and
Tσ¯2 ,h imply that through order O (h) when M → ∞ as h → 0. These results ∗ distribution of T ¯2 consistently estimate the distribution of Tσ¯2 ,h through order σ ,h,M
fourth order cumulants of the bootstrap
o (h),
in which case the bootstrap oers a third-order asymptotic re nement (this again assumes that
the corresponding Edgeworth expansions of
Tσ∗¯2 ,h,M
exist, something we have not attempted to prove
in this paper). If this is the case, then the local Gaussian bootstrap will deliver symmetric percentile-t intervals for integrated volatility with coverage probabilities that converge to zero at the rate
o (h) .
In contrast, the coverage probability implied by the asymptotic theory-based intervals converge to the desired nominal level at the rate
O (h) .
Remark 3. We would like to highlight that results in Theorem 5.1 are stated for the t-statistics Tσ¯2 ,h ∗MZ and Tσ∗¯2 ,h,M (see (32) and (33)) but not for TσMZ ¯2 ,h,M and Tσ¯2 ,h,M (see (9) and (11)). However, under the same conditions as in Theorem 5.1 but strengthened by piecewise constant assumption on the volatility ∗MZ process σt , it is easy to see that analogue results as in Theorem 5.1 hold true for TσMZ ¯2 ,h,M and Tσ¯2 ,h,M directly under P (without assuming that Qh,M is contiguous to P ). More speci cally, if the volatility process σt is such that for s = 1, . . . , M1h , σt = σ(s−1)M h > 0,
for t ∈ ((s − 1) M h, sM h],
(34)
for any sample path of σ, then we can deduce that, when M → ∞ as h → 0, we also have ∗ ∗MZ −1/2 P TσMZ ≤ x − P T ≤ x = o h , P ¯2 ,h,M σ¯2 ,h,M ∗ ∗MZ P TσMZ = oP (h) Tσ¯2 ,h,M ≤ x ¯2 ,h,M ≤ x − P 4
Recently, Hounyo and Veliyev (2016) rigorously justify the Edgeworth expansions for realized volatility derived by
As a consequence the cumulants expansions of Tσ¯2 ,h exist under our assumptions. Our focus in parts (b) and (c) of Theorem 5.1 is on using on formal expansions to explain the superior nite sample Gonçalves and Meddahi (2009).
properties of the new local Gaussian bootstrap theoretically (see e.g., Mammen (1993), Davidson and Flachaire (2001) and Gonçalves and Meddahi (2009) for a similar approach). This approach does not seek to determine the conditions under which the relevant expansions are valid.
24
under P. The proof follows exactly the same line as the proof of Theorem 5.1. For reasons of space we leave the details for the reader. When M → ∞ condition (34) amounts almost to say that the volatility is constant (σt = σ). The potential for the local Gaussian bootstrap intervals to yield third-order asymptotic re nements is particularly interesting because Gonçalves and Meddahi (2009) show that their wild bootstrap method is not able to deliver such re nements. Thus, our method is an improvement not only of the Gaussian asymptotic distribution but also of the best existing bootstrap methods for realized volatility in the context of no microstructure eects (where prices are observed without any error).
5.2 Higher order cumulants of realized beta In this section, we provide the rst and third order cumulants of realized beta estimator given in (19). These cumulants enter the Edgeworth expansions of the one-sided distribution functions of
∗ Tβ,h,M ,
i.e.,
∗ P ∗ Tβ,h,M ≤x
and
P (Tβ,h ≤ x),
Tβ,h
and
respectively. To describe the Edgeworth expansions,
we need to introduce additional notation. Let
A˜0 =
Z
A˜1 =
Z
˜ = B
Z
1
Σkk,s Σkl,s + βlk Σ2kk,s ds,
0 1
3 3 2 2 Σkk,s ds, Σkk,s Σkl,s − 8βlk 2Σ3kl,s + 6Σkk,s Σkl,s Σll,s − 18βlk Σkk,s Σ2kl,s − 6βlk Σ2kk,s Σll,s + 24βlk
0 1
2 2 Σkk,s ds, Σ2kl,s + Σkk,s Σll,s − 4βlk Σkk,s Σkl,s + 2βlk
0
˜1 = H
4A˜0 p ˜ Γkk B
and
˜ ˜ 2 = A1 . H ˜ 3/2 B
Similarly we let
1/M h
A˜∗0,h,M
= Mh
X
1/M h
ˆ kk(j) Γ ˆ kl(j) − βˆlk M h Γ
j=1
= Mh
ˆ ˆ ˆ3 ˆ ˆ ˆ ˆ2 2Γ kl(j) + 6Γkk(j) Γkl(j) Γll(j) − 18βlk Γkk(j) Γkl(j) ˆ2 ˆ 2 ˆ3 ˆ 3 ˆ2 Γ ˆ ˆ −6βˆlk Γ kk(j) ll(j) + 24βlk Γkk(j) Γkl(j) − 8βlk Γkk(j)
X j=1
˜∗ B h,M
= Mh
ˆ2 , Γ kk(j)
j=1
1/M h
A˜∗1,h,M
X
1/M h
X
! ,
ˆ2 ˆ kk(j) Γ ˆ ll(j) − 4βˆlk Γ ˆ kk(j) Γ ˆ kl(j) + 2βˆ2 Γ ˆ2 Γ + Γ lk kk(j) , kl(j)
j=1
˜∗ R 1,h,M
= −
˜∗ H 1,h,M
=
1 ˜ ∗3/2 B h,M
P1/M h ˆ 3 P1/M h ˆ ˆ ˆ 3M h j=1 Γ j=1 Γkk(j) Γkl(j) Γll(j) kl(j) + 5M h P P 1/M h ˆ 2 −19βˆlk M h 1/M h Γ ˆ ˆ kk(j) Γ ˆ2 ˆ j=1 j=1 Γkk(j) Γll(j) kl(j) − 5βlk M h P P 1/M h 1/M h 2 Mh ˆ3 ˆ2 ˆ ˆ3 +24βˆlk j=1 Γkk(j) Γkl(j) − 8βlk M h j=1 Γkl(j)
4A˜∗0,h,M q ˆ kk B ˜∗ Γ
h,M
and
˜ 2,h,M = H
A˜∗1,h,M ˜ ∗3/2 B h,M
,
where
ˆ kk = Γ
1/h X
,
2 yk,i .
i=1
Theorem 5.2. Suppose (1) and (2) hold with µ = 0 and W independent of σ. Furthermore assume (5). Under Assumption H, conditionally on σ and under P, it follows that, as h → 0 25
a) √ h , √ √ κ3 (Tβ,h ) = hκ3 + o h , √
κ1 (Tβ,h ) =
hκ1 + o
1˜ ˜2 , H1 − H 2 ˜ 1 − 2H ˜ 2, κ3 = 3H
with κ1 = with
b) √
∗ κ∗1 Tβ,h,M
=
∗ κ∗3 Tβ,h,M
=
√
√ h , √ + oP h ,
R ˜∗ 1 ˜∗ 1,h,M ˜∗ H1,h,M − H + , 2,h,M 2 M 1 ˜∗ ˜∗ = 3H , 1,h,M − 2H2,h,M + oP M
hκ∗1,h,M + oP
with κ∗1,h,M =
hκ∗3,h,M
with κ∗3,h,M
c) For j = 1 and 3, p limκ∗j,h,M − κj is nonzero if M is nite and it is zero if M that M = o
Theorem 5.2 shows that the cumulants of
Tβ,h,M
implies that the error of the bootstrap approximation of order
√ o h .
→∞
h→0 h−1/2 .
T∗ agree β,h,M ∗ ≤x P ∗ Tβ,h,M
and
through order
as h → 0 such
O
√ h ,
to the distribution of
Since the normal approximation has an error of the order
√ O h ,
which
Tβ,h,M
is
this implies that
the local Gaussian bootstrap is second-order correct. This result is an improvement over the bootstrap results in Dovonon, Gonçalves and Meddahi (2013), who showed that the pairs bootstrap is not secondorder correct in the general case of stochastic volatility. Thus, for realized beta, the local Gaussian bootstrap is able to replicate the rst and third cumulants of
βˆlk
through order
o
√ h when M → ∞.
We conjecture that similar analysis at higher-order (possibly third-order asymptotic re nement) holds for the realized beta estimator, but a full exploration of this is left for future research.
6
Monte Carlo results
In this section we assess by Monte Carlo simulation the accuracy of the feasible asymptotic theory approach of Mykland and Zhang (2009).
We nd that this approach leads to important coverage
probability distortions when returns are not sampled too frequently. We also compare the nite sample performance of the new local Gaussian bootstrap method with the existing bootstrap method for realized volatility proposed by Gonçalves and Meddahi (2009). For integrated volatility, we consider two data generating processes in our simulations.
First,
following Zhang, Mykland and Aït-Sahalia (2005), we use the one-factor stochastic volatility (SV1F) model of Heston (1993) as our data-generating process, i.e.
dXt = (µ − νt /2) dt + σt dBt , and
dνt = κ (α − νt ) dt + γ (νt )1/2 dWt ,
26
where
νt = σt2 , B
and
W
are two Brownian motions, and we assume
values are all annualized. In particular, we let
Corr(B, W ) = ρ.
The parameter
µ = 0.05/252, κ = 5/252, α = 0.04/252, γ = 0.05/252,
ρ = −0.5. We also consider the two-factor stochastic volatility (SV2F) model analyzed by Barndor-Nielsen 5
et al. (2008) and also by Gonçalves and Meddahi (2009), where
dXt = µdt + σt dBt , σt = s-exp (β0 , +β1 τ1t + β2 τ2t ) , dτ1t = α1 τ1t dt + dB1t , dτ2t = α2 τ2t dt + (1 + φτ2t ) dB2t , corr (dWt , dB1t ) = ϕ1 , corr (dWt , dB2t ) = ϕ2 . We follow Huang and Tauchen (2005) and set
α2 = −1.386, φ = 0.25, ϕ1 = ϕ2 = −0.3.
µ = 0.03, β0 = −1.2, β1 = 0.04, β2 = 1.5, α1 = −0.00137,
We initialize the two factors at the start of each interval by
drawing the persistent factor from its unconditional distribution,
−1 and by starting the τ10 ∼ N 0, 2α 1
strongly mean-reverting factor at zero. For integrated beta, the design of our Monte Carlo study is roughly identical to that used by Barndor-Nielsen and Shephard (2004a), and Dovonon Gonçalves and Meddahi (2013) with a minor dierence. In particular, we add a constant drift component to the design of Barndor-Nielsen and Shephard (2004a). Here we brie y describe the Monte Carlo design we use. We assume that
σ (t) dW (t) +
Σ (t) = and
= Σ (t), where 2 Σ11 (t) Σ12 (t) σ1 (t) σ12 (t) , = σ21 (t) σ22 (t) Σ21 (t) Σ22 (t)
σ12 (t) = σ1 (t) σ2 (t) ρ (t) .
2(2) σ1 (t), where for
and
µ=
µ1 µ2
As Barndor-Nielsen and Shephard (2004a), we let
√ 2(s) 2(s) (s) s = 1, 2, dσ1 (t) = −λs (σ1 (t) − ξs )dt + ωs σ1 (t) λs dbs (t),
component of a vector of standard Brownian motions, independent from
ξ1 = 0.110,
dX (t) =
µd (t), with σ (t) σ 0 (t)
ω1 = 1.346,
λ2 = 3.74,
ξ2 = 0.398,
and
ω2 = 1.346.
GARCH(1,1) diusion studied by Andersen and Bollerslev (1998):
W.
2(1)
σ12 (t) = σ1 bi
where
We let
(t) +
is the
i-th
λ1 = 0.0429,
Our model for
σ22 (t)
is the
dσ22 (t) = −0.035(σ22 (t) − 0.636)dt +
0.236σ22 (t)db3 (t). We follow Barndor-Nielsen and Shephard (2004), and let ρ(t) = (e2x(t) − 1)/(e2x(t) + 1),
where
x
let
µ1 = µ2 = 0.03.
follows the GARCH diusion:
We simulate data for the unit interval an Euler scheme. We then construct the
dx(t) = −0.03(x(t) − 0.64)dt + 0.118x(t)db4 (t). [0, 1].
The observed log-price process
h-horizon
returns
yi ≡ Xih − X(i−1)h
X
Finally, we
is generated using
based on samples of size
1/h. Tables 1 and 2 give the actual rates of 95% con dence intervals of integrated volatility and integrated
5
The function s-exp is the usual exponential function with a linear growth function splined in at high values of its s-exp(x) = exp(x) if x ≤ x0 and s-exp(x) = √ exp(x20 ) 2 if x > xo , with x0 = log(1.5). x0 −x0 +x
argument:
27
beta, computed over 10,000 replications. Results are presented for six dierent samples sizes:
1/h 1152,
576, 288, 96, 48, and 12, corresponding to 1.25-minute, 2.5-minute, 5-minute, 15-minute, halfhour and 2-hour returns. In Table 1, for each sample size we have computed the coverage rate by varying the block size, whereas in Table 2 we summarize results by selecting the optimal block size. We also report results for con dence intervals based on a logarithmic version of the statistic
Th,M
and
its bootstrap version. In our simulations, bootstrap intervals use 999 bootstrap replications for each of the 10,000 Monte Carlo replications. We consider the studentized (percentile-t) symmetric bootstrap con dence interval method computed at the 95% level. As for all blocking methods, to implement our bootstrap methods, we need to choose the block
M.
size
We follow Politis and Romano (1999) and Hounyo, Gonçalves and Meddahi (2013) and use
the Minimum Volatility Method. Although the block length
M
that result for this approach may be
suboptimal for the purpose of distribution estimation (in the context of realized volatility) as we do in our application, it is readily accessible to practitioners, typically perform well, and allows meaningful comparisons of the local Gaussian bootstrap and asymptotic normal theory-based methods. Here we describe the algorithm we employ for a two-sided con dence interval.
Algorithm: Choice of the block size M by minimizing con dence interval volatility (i)
For
M = Msmall to M = Mbig compute a bootstrap interval for the parameter of interest (integrated
volatility or integrated beta) at the desired con dence level, this resulting in endpoints and
(ii)
ICM,low
ICM,up .
For each
M
compute the volatility index
in a neighborhood of
M.
V IM
as the standard deviation of the interval endpoints
More speci cally, for a smaller integer
dard deviation of the endpoints
{ICM −l,low , . . . , ICM +l,low }
l,
let
V IM
equal to the stan-
plus the standard deviation of the
{ICM −l,up , . . . , ICM +l,up }, i.e. v v u u l l u 1 X 2 u 1 X t ¯ ¯ up 2 , V IM ≡ ICM +i,low − IC low + t ICM +i,up − IC 2l + 1 2l + 1
endpoints
i=−l
where
(iii)
¯ low = IC
Pick the value
1 2l+1
M∗
Pl
i=−l
i=−l
ICM +i,low
and
¯ up = IC
1 2l+1
Pl
i=−l
ICM +i,up .
corresponding to the smallest volatility index and report
{ICM ∗ ,low , ICM ∗ ,up }
as the nal con dence interval.
One might ask what is a selection of reasonable size
1/h = 1152,
the choices
Msmall = 1
and
1152, 576, 288, 96, and 48, we have used used
l = 2
in our simulations.
Msmall
Mbig = 12
Msmall = 1
and
Mbig ?
In our experience, for a sample
usually su ce, for the samples sizes : and
Mbig = 12.
1/h =
For results in Table 2, we
Some initial simulations (not recorded here) showed that the actual
28
coverage rate of the con dence intervals using the bootstrap is not sensitive to reasonable choice of l, in particular, for
l = 1, 2, 3.
Starting with integrated volatility, the Monte Carlo results in Tables 1 and 2 show that for both models (SV1F and SV2F), the asymptotic intervals tend to undercover. The degree of undercoverage is especially large, when sampling is not too frequent. for the log-based statistics.
It is also larger for the raw statistics than
The SV2F model exhibits overall larger coverage distortions than the
SV1F model, for all sample sizes.
When
M = 1,
the Gaussian bootstrap method is equivalent to
the wild bootstrap of Gonçalves and Meddahi (2009) that uses the normal distribution as external random variable. One can see that the bootstrap replicates their simulations results. In particular, the Gaussian bootstrap intervals tend to overcover across all models. The actual coverage probabilities of the con dence intervals using the Gaussian bootstrap are typically monotonically decreasing in and does not tend to decrease very fast in
M
M,
for larger values of sample size.
A comparison of the local Gaussian bootstrap with the best existing bootstrap methods for realized 6
volatility
shows that, for smaller samples sizes, the con dence intervals based on Gaussian bootstrap
are conservative, yielding coverage rates larger than
95% for the SV1F model.
The con dence intervals
tend to be closer to the desired nominal level for the SV2F than the best bootstrap proposed by Gonçalves and Meddahi (2009). For instance, for SV1F model, the Gaussian bootstrap covers of the time when
87.42%.
h−1 = 12
96.51%
whereas the best bootstrap of Gonçalves and Meddahi (2009) does only
These rates decrease to
93.21%
and
80.42%
for the SV2F model, respectively.
We also consider intervals based on the i.i.d. bootstrap studied by Gonçalves and Meddahi (2009). Despite the fact that the i.i.d. bootstrap does not theoretically provide an asymptotic re nement for two-sided symmetric con dence intervals, it performs well. While none of the intervals discussed here (bootstrap or asymptotic theory-based) allow for
h−1 ,
we have also studied this setup which is nevertheless an obvious interest in practice.
M =
For the
SV1F model, results are not very sensitive to the choice of the block size, whereas for the SV2F model coverage rates for intervals using a very large value of block size (M lower than
95%
even for the largest sample sizes. When
M=
= h−1 )
are systematically much
h−1 , the realized volatility
R2
using the
blocking approach is the same realized volatility studied by Barndor-Nielsen and Shephard (2002), but the estimator of integrated quarticity using the blocking approach is
h−1 +2 2 R2 . This means that h−1
2 1 2 , which is only valid under constant volatility. By σ dt 0 0 t R 2 R1 1 2 Cauchy-Schwarz inequality, we have ≤ 0 σt4 dt, it follows then that we underestimated 0 σt dt
asymptotically we replace
R1
σt4 dt
by
R
the asymptotic variance of the realized volatility estimator.
This explains the poor performance of
the theory based on the blocking approach when the block size is too large. This also con rms the theoretical prediction, which require measure
6
√ M = O( h−1 )
for a good approximation for the probability
P.
The wild bootstrap based on Proposition 4.5 of Gonçalves and Meddahi (2009).
29
For realized beta, we see that intervals based on the feasible asymptotic procedure using Mykland and Zhang's (2009) blocking approach and the bootstrap tend to be similar for larger sample sizes whereas, at the smaller sample sizes, intervals based on the asymptotic normal distribution are quite severely distorted. For instance, the coverage rate for the feasible asymptotic theory of Mykland and Zhang (2009) when to
95.17% (94.84%),
h−1 = 12
(cf.
h−1 = 48)
is only equal to
88.49% (92.86%),
whereas it is equal
for the Gaussian bootstrap (the corresponding symmetric interval based on the
pairs bootstrap of Dovonon Gonçalves and Meddahi (2013) yields a coverage rate of
93.59% (93.96%),
better than Mykland and Zhang (2009) but worse than the Gaussian bootstrap interval). Our Monte Carlo results also con rm that for a good approximation, a very large block size is not recommended. Overall, all methods behave similarly for larger sample sizes, in particular the coverage rate tends to be closer to the desired nominal level. The Gaussian bootstrap performance is quite remarkable and
−1
outperforms the existing methods, especially for smaller samples sizes (h
7
= 12
and
48).
Empirical results
As a brief illustration, in this section we implement the local Gaussian bootstrap method with real high-frequency nancial intraday data, and compare it to the existing feasible asymptotic procedure of Mykland and Zhang (2009). The data consists of transaction log prices of General Electric (GE) shares carried out on the New York Stock Exchange (NYSE) in August 2011. Before analyzing the data we have cleaned the data. For each day, we consider data from the regular exchange opening hours from time stamped between 9:30 a.m. till 4 p.m. Our procedure for cleaning data is exactly identical to that used by Barndor-Nielsen et al. (2008). We detail in Appendix A the cleaning we carried out on the data. We implemented the realized volatility estimator of Mykland and Zhang (2009) on returns recorded every
S
transactions, where
S
is selected each day so that there are 96 observations a day. This means
that on average these returns are recorded roughly every 15 minutes. Table 3 in the Appendix provides the number of transactions per day, and the sample size used. Typically each interval corresponds to about 131 transactions. This choice is motivated by the empirical study of Hansen and Lunde (2006), who investigate 30 stocks of the Dow Jones Industrial Average, in particular they have presented detailed work for the GE shares. They suggest to use 10 to 15 minutes horizon for liquid assets to avoid the market microstructure noise eect. Hence the main assumptions underlying the validity of the Mykland and Zhang (2009) block-based method and our new bootstrap method are roughly satis ed and we feel comfortable to implement them on this data. To implement the realized volatility estimator, we need to choose the block size Minimum Volatility Method described above to choose
30
M.
M.
We use the
We consider bootstrap percentile-t intervals, computed at the 95% level. The results are displayed in Figure 1 in the appendix in terms of daily 95% con dence intervals (CIs) for integrated volatility. Two types of intervals are presented: our proposed new local Gaussian bootstrap method , and the the feasible asymptotic theory using Mykland and Zhang (2009) blocking approach. The realized volatility estimate
R2
is in the center of both con dence intervals by construction. A comparison of the local
Gaussian bootstrap intervals with the intervals based on the feasible asymptotic theory using Mykland and Zhang (2009) block-based approach suggests that the both types of intervals tend to be similar. The width of these intervals varies through time.
However there are instances where the bootstrap
intervals are wider than the asymptotic theory-based interval. These days often correspond to days with large estimate of volatility.
We have asked whether it will be due to jumps.
At this end we
have implemented the jumps test using blocked bipower variation of Mykland, Shephard and Sheppard (2012). We have found no evidence of jumps at 5% signi cance level for these two days. The gures also show a lot of variability in the daily estimate of integrated volatility.
8
Concluding remarks
This paper proposes a new bootstrap method for statistics that are smooth functions of the realized multivariate volatility matrix based on Mykland and Zhang's (2009) blocking approach.
We show
how and to what extent the local Gaussianity assumption can be explored to generate a bootstrap approximation.
We use Monte Carlo simulations and derive higher order expansions for cumulants
to compare the accuracy of the bootstrap and the normal approximations at estimating con dence intervals for integrated volatility and integrated beta. Based on these expansions, we show that the local Gaussian bootstrap provides second-order re nements for the realized beta, whereas it provides a third-order asymptotic re nement for realized volatility.
This is an improvement of the existing
bootstrap results. Our new bootstrap method also generalizes the wild bootstrap of Gonçalves and Meddahi (2009). Monte Carlo simulations suggest that the Gaussian bootstrap improves upon the rstorder asymptotic theory in nite samples and outperform the existing bootstrap methods for realized volatility and realized betas. An important assumption we make throughout this paper is that asset prices are observed at regular time intervals and without any error (so that markets are frictionless).
If prices are nonequidistant
but non noisy, results in Mykland and Zhang (2009) show that the contiguity argument holds and the same variance estimators of the realized covariation measures remain consistent. same
t-statistics
Consequently, the
as those considered here can be used for inference purposes. In this case, we can rely
on the same local Gaussian bootstrap
t-statistics
to approximate their distributions.
The case of noisy data is much more challenging because in this case the realized covariation measures studied in this paper are not consistent estimators of their integrated volatility measures. Dierent microstructure noise robust measures are required. We conjecture that for the noise robust
31
measures based on pre-averaging approach (a dierent kind of blocking method) of Podolskij and Vetter (2009) and Jacod et al. (2009), the contiguity results can be found under common types of noise. Since, in these papers, the latent semimartingale is itself given a locally constant approximation. However, for the bootstrap, it is worth emphasising that we have to distinguish two cases. The non-overlapping pre-averaging based estimators as in Podolskij and Vetter (2009), and the overlapping pre-averaging estimators as in Jacod et al.
(2009).
For the latter, the local Gaussian bootstrap would not be
valid, since the pre-averaged returns are very strongly dependent because they rely on many common high frequency returns.
A dierent bootstrap method such as the wild blocks of blocks bootstrap
studied by Hounyo, Gonçalves and Meddahi (2013) will be appropriate. But we conjecture that, when the pre-averaging method is apply on non-overlapping intervals (as in Podolskij and Vetter (2009) and Gonçalves, Hounyo and Meddahi (2014)), the local Gaussian bootstrap method remains valid. However, it is important to highlight that in this case, we should apply the local Gaussian bootstrap method to the pre-averaged returns, instead of the raw returns.
Since in the presence of noise, the
non-overlapping pre-averaged returns remains asymptotically Gaussian and conditionally independent, which are not the case for noisy high frequency raw returns. Another promising extension would be to allow for jumps in the log-price process. Following the initial draft of this paper, Dovonon et al.
(2014) generalize the local Gaussian bootstrap method
and use it for jump tests. In particular, instead of the local Gaussian bootstrap observations within a given block will have a normal distribution with variance equal to the realized volatility over the corresponding block. They propose to generate the bootstrap observations with normal distribution, but now with a variance given by a local jumps-robust realized measure of volatility, for instance the local block multipower variation estimator, similar to that of Mykland, Shephard and Sheppard (2012). We also have supposed in multivariate framework that prices on dierent assets are observed synchronously, when prices are non-synchronous, dierent robust measures (eg.
Hayashi and Yoshida
(2005) covariance estimator) are required. Another important extension is to prove the validity of the Edgeworth expansions derived here and to provide a theoretical optimal choice of the block size
M
for
con dence interval construction. The extension of the local Gaussian bootstrap to these alternative estimators and test statistics is left for future research.
Appendix A This appendix is organized as follows. First, we details the cleaning we carried out on the data. Second, we report simulation results. Finally we report empirical results.
Data Cleaning In line with Barndor-Nielsen et al. (2009) we perform the following data cleaning steps:
(i)
Delete entries outside the 9:30pm and 4pm time window.
32
(ii)
Delete entries with a quote or transaction price equal to be zero.
(iii)
Delete all entries with negative prices or quotes.
(iv)
Delete all entries with negative spreads.
(v)
Delete entries whenever the price is outside the interval [bid
(vi)
− 2 ∗ spread ; ask + 2 ∗ spread].
Delete all entries with the spread greater or equal than 50 times the median spread of that day.
(vii)
Delete all entries with the price greater or equal than 5 times the median mid-quote of that day.
(viii)
Delete all entries with the mid-quote greater or equal than 10 times the mean absolute deviation from the local median mid-quote.
(ix)
Delete all entries with the price greater or equal than 10 times the mean absolute deviation from the local median mid-quote.
We report in Table 1 below, the actual coverage rates for the feasible asymptotic theory approach and for our bootstrap methods.
In Table 2 we summarize results using the optimal block size by
minimizing con dence interval volatility.
Table 3 provides some statistics of GE shares in August
2011.
33
Table 1. Coverage rates of nominal 95% CI for integrated volatility and integrated beta Integrated volatility
Integrated beta
SV1F Raw
M
CLT
Boot
SV2F Log
CLT
Boot
Raw CLT
Boot
Log
Raw
CLT
Boot
M
CLT
Boot
1/h = 12 1
85.44
98.49
90.08
97.86
80.38
96.62
86.17
96.24
2
83.58
95.67
2
85.56
97.31
90.31
96.80
80.43
94.70
86.27
94.73
3
87.57
94.98
3
85.71
96.46
90.84
96.08
80.34
93.77
85.89
93.70
4
89.15
94.81
4
85.88
96.20
90.97
95.93
80.34
92.88
85.52
92.89
6
90.65
94.47
12
86.11
94.84
91.27
94.87
77.66
88.89
81.65
86.97
12
90.46
93.62
1/h = 48 1
92.04
98.55
93.51
97.71
88.28
97.09
90.93
96.67
3
92.36
95.65
2
92.10
97.28
93.59
96.50
88.13
95.63
91.08
95.48
4
92.69
95.29
4
92.20
96.40
93.80
95.80
88.16
94.55
91.10
94.53
8
92.91
94.71
8
92.33
95.60
93.88
95.18
87.89
93.32
90.33
93.20
12
92.62
93.75
48
92.74
95.06
94.22
95.04
81.83
86.63
82.92
84.57
48
91.61
92.45
1/h = 96 1
93.35
97.94
94.09
97.10
90.20
97.06
92.10
96.66
3
92.60
95.51
2
93.43
96.78
93.99
96.06
90.37
95.84
92.24
95.67
4
93.12
95.01
4
93.47
95.78
94.03
95.61
90.46
94.70
92.09
94.83
8
93.78
94.81
8
93.50
95.26
94.09
95.32
90.07
93.81
91.75
94.01
12
93.79
94.55
96
93.42
94.80
94.35
94.87
81.93
84.61
82.79
83.60
96
91.97
92.36
1/h = 288 1
94.57
97.09
94.61
96.25
93.39
97.44
93.96
96.76
3
93.81
95.72
2
94.56
96.00
94.61
95.67
93.51
96.35
93.95
95.95
4
94.73
95.61
4
94.62
95.48
94.67
95.36
93.50
95.57
93.98
95.28
8
94.92
95.45
8
94.55
95.26
94.81
95.19
93.43
95.06
93.82
94.75
12
94.65
94.99
288
94.46
94.78
94.84
94.99
82.43
83.86
83.34
83.53
288
90.08
90.33
1/h = 576 1
94.53
96.12
94.75
95.84
94.19
96.96
94.49
96.52
3
93.93
95.58
2
94.57
95.53
94.68
95.41
94.17
96.23
94.52
95.78
4
94.49
95.36
4
94.74
95.15
94.70
95.16
94.32
95.59
94.56
95.45
8
94.51
94.88
8
94.67
95.08
94.72
94.96
94.22
95.38
94.46
95.16
12
94.48
94.86
576
94.58
94.85
94.76
94.92
82.01
82.37
82.05
82.32
576
87.09
87.09
1/h = 1152 1
95.06
96.06
95.16
95.70
94.51
96.52
94.47
95.95
3
94.75
95.91
2
95.13
95.68
95.20
95.65
94.53
95.79
94.47
95.42
4
94.90
95.45
4
95.05
95.49
95.20
95.31
94.42
95.21
94.50
95.11
8
94.83
95.13
8
95.15
95.47
95.18
95.20
94.39
95.03
94.47
94.85
12
94.95
94.85
1152
94.86
94.97
94.83
94.91
82.60
82.73
82.85
82.89
1152
81.69
81.60
Notes: CLT-intervals based on the Normal; Boot-intervals based on our proposed new local Gaussian bootstrap;
M
is the block size used to compute con dence intervals. 10,000 Monte Carlo trials with 999 bootstrap
replications each.
34
35
94.49
94.50
5.37
5.86
5.66
5.82
6.01
48
96
288
576
1152
94.55
94.25
4.96
5.60
5.76
5.79
5.94
48
96
288
576
1152
94.68
94.72
94.75
94.98
93.96
93.60
PairsB
94.87
94.54
94.70
94.67
94.62
93.66
iidB
94.76
94.64
94.98
94.62
94.85
95.16
Boot
95.13
94.51
94.98
94.38
93.98
87.42
WB
95.04
94.95
95.05
95.29
95.76
96.51
Boot
95.15
94.58
94.60
93.68
92.31
90.79
CLT
Log
94.85
94.65
94.86
95.48
95.45
96.11
iidB
95.14
94.66
94.81
94.85
94.71
88.35
WB
95.02
94.98
94.96
95.11
95.35
96.07
Boot
6.25
6.06
5.94
5.81
5.75
3.84
M∗
SV2F
94.41
94.08
93.32
88.94
85.40
80.42
CLT
Raw
94.83
94.89
94.85
94.05
92.77
90.82
iidB
94.56
94.47
94.18
92.01
89.98
79.41
WB
94.92
95.19
95.02
94.28
93.63
93.21
Boot
94.38
94.38
93.62
90.48
87.59
86.70
CLT
Log
94.92
94.90
94.99
94.28
94.15
93.43
iidB
94.76
94.81
94.36
93.11
90.82
80.41
WB
94.83
95.03
94.90
94.21
93.80
93.35
Boot
Monte Carlo trials with 999 bootstrap replications each.
the pairs bootstrap of Dovonon Gonçalves and Meddahi (2013);
M∗
is the optimal block size selected by using the Minimum Volatility method. 10,000
on Proposition 4.5 of Gonçalves and Meddahi (2009); Boot-intervals based on our proposed new local Gaussian bootstrap; PairsB-intervals based on
Notes: CLT-intervals based on the Normal; iidB-intervals based on the i.i.d. bootstrap of Gonçalves and Meddahi (2009); WB-wild bootstrap based
94.75
93.70
92.87
88.49
3.62
12
CLT
M∗
n
Integrated beta
95.05
93.01
90.89
86.44
3.75
12
CLT
M∗
Raw
n
SV1F
Integrated volatility
Table 2. Coverage rates of nominal 95% intervals for integrated volatility and integrated beta using the optimal block size
Table 3. Summary statistics Days
Trans
n
S
1 Aug
11303
96
118
2 Aug
13873
96
145
3 Aug
13205
96
138
4 Aug
16443
96
172
5 Aug
16212
96
169
8 Aug
18107
96
189
9 Aug
18184
96
190
10 Aug
15826
96
165
11 Aug
15148
96
158
12 Aug
12432
96
130
15 Aug
12042
96
126
16 Aug
10128
96
106
17 Aug
9104
96
95
18 Aug
15102
96
158
19 Aug
11468
96
120
22 Aug
10236
96
107
23 Aug
11518
96
120
24 Aug
10429
96
109
25 Aug
9794
96
102
26 Aug
9007
96
94
29 Aug
10721
96
112
30 Aug
9131
96
96
31 Aug
10724
96
112
Trans denotes the number of transactions, every
S 'th
n
the sample size used to compute the realized volatility, and sampling of
transaction price, so the period over which returns are calculated is roughly 15 minutes.
Figure 1: 95% Con dence Intervals (CI's) for the daily
σ2 ,
for each regular exchange opening days in August 2011,
calculated using the asymptotic theory of Mykland and Zhang (CI's with bars), and the new wild bootstrap method (CI's with lines). The realized volatility estimator is the middle of all CI's by construction. Days on the
x-axis.
36
Appendix B This appendix concerns only the case where
d = 1 (i.e.
when the parameter of interest is the integrated
volatility). We organized this appendix as follows. First, we introduce some notation. Second, we state Lemmas B.1 and B.2 and their proofs useful for proofs of theorem Theorem 5.1 presented in the main text. These results are used to obtain the formal Edgeworth expansions through order
O(h) for realized
volatility. Finally, we prove Theorem 3.1, Lemma B.3 and Theorem 5.1.
Notation To make for greater comparability, and in order to use some existing results, we have kept the notation from Gonçalves and Meddahi (2009) whenever possible. We introduce some notation, recall that, for any
q > 0, σ q ≡
R1
σuq du,
and let
RVj,M
σi2 h
q/2
σq,p,h ,
and
i=1
0
σq,p ≡
1/h
P σ¯hq ≡ h
q when σ q is replaced with σh ( ) M P 2 = yi+(j−1)M . We also let Rq,p ≡ σq
q/p , σp
we write
Rq . Let (Rp )q/p
i=1
and note that
µ2 = 1, µ4 = 3, µ6 = 15,
and
Rq ≡
µq = E |η|q
µ8 = 105.
σi2 ≡
Rih
σu2 du. We let (i−1)h 1/M Ph RVj,M q/2 Mh , where Mh j=1
, where
η ∼ N (0, 1), with q > 0, χ2M q/2 cM,q ≡ E with M
where
Recall that
cM,2 = 1, cM,4 = MM+2 , It follows by using the de nition of cM,q gives in equation (7) and this property of the Gamma function, for all x > 0, Γ (x + 1) = xΓ (x). Recall the −1 (µ4 −µ22 ) −1 hP 4 ˆ de nition of Tσ¯2 ,h given in (32) and Vσ¯2 ,h = h yi . µ4 χ2M the standard χ2 distribution with M degrees +4) (M +2)(M +4)(M +6) cM,6 = (M +2)(M and cM,8 = . M2 M3
of freedom.
Note that
i=1
We follow Gonçalves and Meddahi (2009) and we write
√ Tσ∗¯2 ,h,M
=
h−1 (R2∗ − µ2 R2 ) q , Vˆ ∗¯ σ 2 ,h
where
Vˆσ∗¯2 ,h,M
1/M h h−1 M X RVj,M 2 1 X µ4 − µ22 −1 X ∗4 µ4 − µ22 ∗4 h yi = Mh ηi+(j−1)M − µ4 . = µ4 µ4 Mh M i=1
i=1
i=1
We can write
Tσ∗¯2 ,h,M = Sσ∗¯2 ,h,M
Vˆσ∗¯2 ,h,M
−1/2
Vσ∗¯2 ,h,M
−1/2 √ = Sσ∗¯2 ,h,M 1 + hUσ∗¯2 ,h,M ,
where
√ Sσ∗¯2 ,h,M =
and
√ h−1 (R2∗ − µ2 R2 ) q , Vσ∗¯2 ,h,M
Uσ∗¯2 ,h,M ≡
h−1 Vˆσ∗¯2 ,h − Vσ∗¯2 ,h,M
1/M Ph RVj,M 2 Vσ∗¯2 ,h,M = V ar∗ n1/2 R2∗ = µ4 − µ22 · M h . Mh i=1
37
Vσ∗¯2 ,h,M
,
RV q/2 q > 0, Mj,M h
Note that for any with zero mean since
hq/2 M
q M P ∗ ηi+(j−1)M − µq
∗ ηi+(j−1)M ∼ i.i.d. N (0, 1).
R2∗
−
P
i6=j6=k
=
σ
independent
and
Vˆσ∗¯2 ,h,M −Vσ∗¯2 ,h,M
as follows
1/M h
M 2 X RVj,M 1 X ∗ − µ2 R2 = M h ηi+(j−1)M − µ2 , Mh M
Vσ∗¯2 ,h,M
i=1
1/M h M 4 X RVj,M 2 1 X µ4 − µ22 ∗ = − µ Mh η i+(j−1)M 4 . Mh M µ4 j=1 i=1 P
Finally, note that throughout we will use example
R2∗ − µ2 R2
We rewrite
j=1
Vˆσ∗¯2 ,h,M
are conditionally on
i=1
i6=j6=...6=k to denote a sum where all indices dier, for
P
i6=j,i6=k,j6=k .
Lemma B.1. Suppose (1), (2) and (5) hold. Let M q > 0, we have
≥1
such that M ≈ ch−α with α ∈ [0, 1) for any
a2)
q ∗ RV q/2 q/2 E ∗ yi+(j−1)M h , for i = 1, . . . , M = µq Mj,M h √ Vσ∗¯2 ,h,M ≡ V ar∗ h−1 R2∗ = µ4 − µ22 R4 ,
a3)
E∗
a4)
i h 2 E ∗ (R2∗ − µ2 R2 )4 = 3h2 µ4 −µ22 (R4 )2 + h3 µ8 − 4µ2 µ6 + 12µ22 µ4 − 6µ42 − 3µ24 R8 ,
a5)
h i µ −µ2 E ∗ (R2∗ − µ2 R2 ) Vˆσ∗¯2 ,h,M − Vσ∗¯2 ,h,M = 4µ4 2 (µ6 − µ2 µ4 ) hR6 ,
a6)
i h µ −µ2 E ∗ (R2∗ − µ2 R2 )2 Vˆσ∗¯2 ,h,M − Vσ∗¯2 ,h,M = 4µ4 2 h2 µ8 − µ24 −2µ2 µ6 + µ22 µ4 R8 ,
a7)
i h 2 (µ4 −µ22 ) (µ6 −µ2 µ4 ) E (R2∗ − µ2 R2 )3 Vˆσ∗¯2 ,h,M − Vσ∗¯2 ,h,M = 3h2 R4 R6 + OP h3 as h → 0, µ4
a8)
h i E ∗ (R2∗ − µ2 R2 )4 Vˆσ∗¯2 ,h,M − Vσ∗¯2 ,h,M " # 2 3 2 4 µ6 − 3µ2 µ4 + 2µ2 (µ6 − µ2 µ4 ) R µ −µ 6 = h3 4µ4 2 + OP h4 as h → 0, 2 2 2 +6 µ8 − µ4 − 2µ2 µ6 + 2µ2 µ4 µ4 −µ2 R4 R8 2 ∗ ∗ ∗ ∗ ˆ E (R2 − µ2 R2 ) Vσ¯2 ,h,M − Vσ¯2 ,h,M = OP h2 as h → 0,
a1)
a9)
a10)
h
(R2∗
− µ2 R2 )
3
i
and j = 1, . . . , M1h
= h2 µ6 − 3µ2 µ4 + 2µ32 R6 ,
2 2 ˆ∗ ∗ ∗ (R2 − µ2 R2 ) Vσ¯2 ,h,M − Vσ¯2 ,h,M 2 (µ4 −µ2 ) = h2 µ2 2 µ4 − µ22 µ8 − µ24 R4 R8 + 2 (µ6 − µ2 µ4 )2 R62 + OP h3 E∗
4
2 3 ˆ∗ ∗ ∗ (R2 − µ2 R2 ) Vσ¯2 ,h,M − Vσ¯2 ,h,M = OP h3
as h → 0,
a11)
E∗
a12)
2 4 ˆ∗ ∗ ∗ (R2 − µ2 R2 ) Vσ¯2 ,h,M − Vσ¯2 ,h,M i 2 h 2 2 2 (µ4 −µ2 ) = h3 µ2 2 3 µ4 − µ22 µ8 − µ24 R4 R8 +12 µ4 − µ22 (µ6 − µ2 µ4 ) R6 R4 +OP h4 as h → 0. E∗
4
38
as h → 0,
Lemma B.2. Suppose (1), (2) and (5) hold. Let M q > 0, we have E ∗ Sσ∗¯2 ,h,M E ∗ Sσ∗2 ¯2 ,h,M E ∗ Sσ∗3 ¯2 ,h,M E ∗ Sσ∗4 ¯2 ,h,M E ∗ Sσ∗¯2 ,h,M Uσ∗¯2 ,h,M ∗ U E ∗ Sσ∗2 ¯2 ,h,M σ¯2 ,h,M
≥1
such that M ≈ ch−α with α ∈ [0, 1) for any
= 0, = 1, √ = hB1 R6,4 , = 3 + hB2 R8,4 , = A1 R6,4 , √ hA2 R8,4 , =
and as h → 0 we have, 3 E ∗ Sh,M Uσ∗¯2 ,h,M 4 E ∗ Sh,M Uσ∗¯2 ,h,M E ∗ Sσ∗¯2 ,h,M Uσ∗2 ¯2 ,h,M ∗2 E ∗ Sσ∗3 ¯2 ,h,M Uσ¯2 ,h,M ∗2 E ∗ Sσ∗2 ¯2 ,h,M Uσ¯2 ,h,M ∗2 E ∗ Sσ∗4 ¯2 ,h,M Uσ¯2 ,h,M
= A3 R6,4 + OP (h) , √ 2 h D1 R8,4 + D2 R6,4 + OP h3/2 , = = OP h1/2 , = OP h1/2 , 2 = C1 R8,4 + C2 R6,4 + OP (h) , 2 = E1 R8,4 + E2 R6,4 + OP (h) .
The constant A1 , B1 , A2 , B2 and C1 are de ned in the text, and we have A3 = 3A1 , C2 = 2A21 , D1 = 6A2 , D2 = 4A1 B1 , E1 = 3C1 and E2 = 12A21 .
Proof of Lemma B.1 (a1) follows from
q q ∗ RVj,M q/2 q/2 ∗ yi+(j−1)M = M h h ηi+(j−1)M ,
for
∗ ηi+(j−1)M ∼ i.i.d. N (0, 1). For (a2) recall the de nition of and j = 1, . . . , ∗ ∗ that given result in (a1) we have E (R2 ) = µ2 R2 , then note that we can write 1 M h , where
R2∗
i = 1, . . . , M
R2∗ and remark
1/M h
M 2 X RVj,M 1 X ∗ − µ2 R2 = M h ηi+(j−1)M − µ2 . Mh M j=1
i=1
It follows that,
V ar∗
√
h−1 R2∗
= h−1 E ∗ [R2∗ − µ2 R2 ]2 −1
= h
1/M h
2 M 2 X RVj,M 2 1 X ∗ ∗ (M h) E ηi+(j−1)M − µ2 Mh M2 2
j=1
= =
i=1
1/M h X RVj,M 2 2 µ4 −µ2 M h Mh j=1 2 µ4 −µ2 R4 .
39
For (a3), we have
3 M 2 X RVj,M h X ∗ = E ∗ M ηi+(j−1)M − µ2 Mh M
E ∗ [R2∗ − µ2 R2 ]3
1/M h
i=1
j=1
1/M h 1/M h 1/M h
M X M X M X X X RVj ,M RVj ,M RVj ,M X 1 2 3 = h Ii∗1 ,j1 ,i2 ,j2 ,i3 ,j3 , Mh Mh Mh 3
j1 =1 j2 =1 j3 =1
where
i1 =1 i2 =1 i3 =1
2 2 2 ∗ ∗ ∗ ∗ = E ηi1 +(j1 −1)M − µ2 ηi2 +(j2 −1)M − µ2 ηi3 +(j3 −1)M − µ2 .
Ii∗1 ,j1 ,i2 ,j2 ,i3 ,j3
easy to see that the only nonzero contribution to
Ii∗1 ,j1 ,i2 ,j2 ,i3 ,j3
is when
It is
(i1 , j1 ) = (i2 , j2 ) = (i3 , j3 ) ,
in
which case we obtain
Ii∗1 ,j1 ,i2 ,j2 ,i3 ,j3
3 M 2 X ∗ = ηi+(j−1)M − µ2 = M µ6 − 3µ2 µ4 + 2µ32 . i=1
Hence,
E
∗
[R2∗
3
− µ2 R2 ]
1/M h
X RVj,M 3 3 = h M h M µ6 − 3µ2 µ4 + 2µ2 j=1 2 = h µ6 − 3µ2 µ4 + 2µ32 R6 . 3
To prove the remaining results we follow the same structure of proofs as Gonçalves and Meddahi (2009). Here
RVj,M q/2 Mh
hq/2 M
q M P ∗ ηi+(j−1)M − µq
plays the role of
i=1
|ri∗ |q − µq |ri |q ,
where
ri∗
denotes
the wild bootstrap returns in Gonçalves and Meddahi (2009).
Proof of Lemma B.2 Results follow immediately by using Lemma B.1 given the de nitions of Sσ∗¯2 ,h,M and
Uσ∗¯2 ,h,M .
M (cM,4 −c2M,2 ) 1 ∗ cM,4 R4 and Vσ¯2 ,h,M = c2M,2 M cM,4 −µ22 R4 the rst part of the result follows immediately, i.e. for xed M, Vσ∗¯2 ,h,M = MM+2 Vˆσ¯2 ,h,M . ∗ For the second part, given that V ¯2 = MM+2 Vˆσ¯2 ,h,M , we can write σ ,h,M
Proof of Theorem 3.1 part a).
Given the de nitions of
M V ∗¯ − Vσ¯2 M + 2 σ2 ,h,M
Vˆσ¯2 ,h,M =
= Vˆσ¯2 ,h,M − Vσ¯2 = oP (1) ,
Vˆσ¯2 ,h,M is a consitent estimator of Vσ¯2 . Next h → 0 such that M 2 h → 0, then as h → 0, ˜MZ sup P ∗ T˜σ∗¯2MZ ≤ x − P T ≤ x → 0. ¯ 2 ,h,M σ ,h,M
where consistency follows since xed or
M →∞
as
x∈<
in probability under
P.
To this end, let
1/M Ph ∗ T˜σ∗¯2MZ = zj , ,h,M
where
j=1
zj∗
r =
M √ −1 ∗ ∗ h RVj,M − E ∗ RVj,M . M +2
40
we show that, if
M
is
1/M Ph
∗ Note that E
j=1
! zj∗
= 0,
and
1/M Ph
ar∗
V
j=1
! P = Vˆσ¯2 ,h,M → Vσ¯2 .
zj∗
Moreover, since
are conditionally independent, by the Berry-Esseen bound, for some small
C>0
∗ z1∗ , . . . , z1/M h
δ > 0 and for some constant
(which changes from line to line),
1/M h X 2+δ p ∗ ˜∗MZ sup P Tσ¯2 ,h,M ≤ x −Φ x/ Vσ¯2 ≤ C , E ∗ zj∗
x∈<
j=1
which converges to zero in probability under as
h → 0.
X
1/M h
2+δ = E ∗ zj∗
X
j=1
j=1
≤ 2
= 2
M ≥1
such that
M ≈ Ch−α
with
α ∈ [0, 1/2),
r 2+δ √ M ∗ ∗ h−1 RVj,M − E ∗ RVj,M E∗ M +2
M M +2
M M +2
where the inequality follows from the and
R2(2+δ) , X
2+δ E ∗ zj∗
≤ 2
j=1
2
h
−(2+δ) 2
X
≤ C
2+δ 2
2+δ P M ∗2 η 1/M h (j−1)M +i X −(2+δ) i=1 ∗ h 2 E |RVj,M |2+δ , M j=1
and the Jensen inequalities. Then, given the de nitions of
M M +2
2+δ 2
M M +2
2+δ 2
δ
cM,2(2+δ) M 1+δ h 2 R2(2+δ)
P 1 cM,2(2+δ) R2(2+δ) →
1
δ
c2M,2(2+δ) h 2 −α(1+δ)
δ > 0, M ≥ 1 such that M ≈ Ch−α
− α(1 + δ) > 0,
∗ 2+δ E ∗ RVj,M
j=1
Cr
Note that for any
1/M h
2+δ
we can write
1/M h
δ 2
for any
Indeed, we have that
1/M h
cM,2(2+δ)
P
with
σ 2(2+δ) = OP (1),
cM,2(2+δ)
R2(2+δ) .
α ∈ [0, 1/2), as h → 0,
and
cM,2(2+δ) → 1.
M M +2
2+δ 2
= O (1) ,
Thus, results follow, in
particular, we have
1/M h
X
2+δ E ∗ zj∗
= OP h
δ −α(1+δ) 2
c2M,2(2+δ)
j=1
= oP (1) . It is worthwhile to precise that for a bounded
M
(i.e.,
α = 0), the consistency result
P 1 cM,2(2+δ) R2(2+δ) →
σ 2(2+δ) = OP (1) follows from Mykland and Zhang (2009). Whereas when M → ∞, in particular for α ∈ (0, 1/2) , the consistency result still holds since as M → ∞, cM,2(2+δ) → 1 and using e.g., Jacod and Rosembaum (2013) (cf. equations (3.8) and (3.11)) or the recent work of Li, Todorov and Tauchen (2016) (cf.
Theorems 2 and 3), we have
P
R2(2+δ) → σ 2(2+δ) .
Protter (2012) for similar result.
41
See also Theorem 9.4.1 of Jacod and
Proof of Theorem 3.1 part b). probability under
P.
Given that
d
TσMZ ¯2 ,h,M → N (0, 1),
it su ces that
d∗
Tσ∗¯2MZ → N (0, 1) ,h,M
in
Let
∗MZ T˜h,M Hσ∗¯2 ,h,M = q Vˆ ¯2
,
σ ,h,M
and note that
v u u M + 2 Vˆσ¯2 ,h,M ∗ = Hσ¯2 ,h,M t , M Vˆ ∗¯2MZ
Tσ∗¯2MZ ,h,M
σ ,h,M
where
Tσ∗¯2 ,h,M , Vˆσ¯2 ,h,M
and
Vˆσ∗¯2MZ ,h,M
are de ned in the main text. Part a) of Theorem 3.1 proved that
d∗
Hσ∗¯2 ,h,M → N (0, 1) in probability under P . under
P.
In particular, we show that (1)
in probability under
Thus, it su ces to show that
Bias∗
M ˆ ∗MZ M +2 Vσ¯2 ,h,M
= 0,
Vˆσ¯2 ,h,M M ˆ ∗MZ V M +2 σ¯2 ,h,M
and (2)
P∗
→ 1 in probability V ar∗ MM+2 Vˆσ∗¯2MZ → 0, ,h,M
P.
We have that
Bias
∗
M ˆ ∗MZ V¯ M + 2 σ2 ,h,M
M ˆ∗ M = E Vσ¯2 ,h,M − V ∗¯ M +2 M + 2 σ2 ,h,M = E ∗ Vˆσ¯2 ,h,M − Vˆσ¯2 ,h,M ∗
= Vˆσ¯2 ,h,M − Vˆσ¯2 ,h,M = 0, we also have
V ar∗
M ˆ ∗MZ V¯ M + 2 σ2 ,h,M
=
M M +2
2
V ar∗ Vˆσ∗¯2 ,h,M ,
(35)
where
2 2 = E ∗ Vˆσ∗¯2 ,h,M −Vσ∗¯2 ,h,M − E ∗ Vˆσ∗¯2 ,h,M −Vσ∗¯2 ,h,M V ar∗ Vˆσ∗¯2 ,h,M 2 !2 1/M h 2 X c − c M,4 M,2 2∗ 2 = M2 (M h)−2 E ∗ RVj,M − cM,4 RVj,M cM,4 j=1 2 !2 !2 1/M h 2 2 X c − c χ M,4 j,M M,2 4 (M h)−2 RVj,M E∗ = M2 − cM,4 , cM,4 M j=1
then given (35), the de nitions of
V ar
∗
M ˆ ∗MZ V¯ M + 2 σ2 ,h,M
= =
cM,2 , cM,4 , cM,8 M M +2
2
M M +2
2
M
2
M2
and
=
we can write
cM,4 − c2M,2
!2 −2
(M h)
cM,4 cM,4 − c2M,2
cM,8 −
c2M,4
1/M h X
4 RVj,M
j=1
!2 (M h) cM,8 − c2M,4 R8
cM,4 2 2 M 2M (M + 2) (M + 4) (M + 6) − M (M + 2)2 h R8 M +2 M +2 M2 OP (M h)
=
R8 ,
→ 0 42
in probability under
P,
as long as
Mh → 0
as
Lemma B.3. Consider the simpli ed model drift), then we have
h → 0. dXt = σdWt
a1)
Bias
a2)
(M +2)(M 2 +9M +24) 4 V ar Vσ∗¯2 ,h,M = 4M h σ M3
Vσ∗¯2 ,h,M
=
4σ 2 M ,
Proof of Lemma B.3 part a1). we can write
Given that
yi = Xih − X(i−1)h =d σh1/2 νi , RVj,M
(i.e. the constant volatility model without
2
= σ h
M X
dXt = σdWt , for a given frequency h of the observations, νi ∼ i.i.d. N (0, 1) . It follows that !
with
2 νi+(j−1)M
= σ 2 h · χ2M,j ,
i=1 1/M h
R4 = (M h)
−1
X
1/M h 2 RVj,M
−1
4 2
= σ h (M h)
j=1
Vσ∗¯2 ,h,M
= M cM,4 −
X
χ2M,j
2
,
j=1
c2M,2
1/M h 2 h X R4 = 2σ χ2M,j , M 4
and
Vσ¯2 = 2σ 4 .
j=1
where
χ2M,j
χ2M . Thus, Bias Vσ∗¯2 ,h,M = E Vσ∗¯2 ,h,M − Vσ¯2
is i.i.d.∼
= 2σ 4
1/M h 2 h X E χ2M,j − 2σ 4 M j=1
4σ 2 h (M h)−1 2M + M 2 − 2σ 4 = . M M 1/M P h 2 2 ∗ 4 h that V ¯2 = 2σ χM,j we can M σ ,h,M
= 2σ 4
Proof of Lemma B.3 part a2). V ar
Vσ∗¯2 ,h,M
= =
= = = =
Proof of Theorem 5.1
Given
write
j=1
1/M h 2 h2 X V ar χ2M,j 4σ 2 M j=1 2 4 2 2 −1 8 h 2 2 4σ (M h) E χM,j − E χM,j M2 !4 !2 2 2 2 2 χ χ h M,j M,j − E 4σ 8 2 (M h)−1 M 4 E M M M 4σ 8 M h cM,8 − c2M,4 " # 2 (M + 2) (M + 4) (M + 6) (M + 2) 4σ 8 M h − M3 M2 (M + 2) M 2 + 9M + 24 4 4M h σ . M3 8
Part (a) follows under our assumptions by using Theorem A.1 of Gonçalves
and Meddahi (2009). The proof of part (b) use the same technique as in the proof of Theorem A.3 in
43
Tσ∗¯2 ,h,M
Gonçalves and Meddahi (2009). In particular, the rst four cumulants of
are given by (e.g.,
Hall, 1992, p.42):
κ∗1 Tσ∗¯2 ,h,M κ∗2 Tσ∗¯2 ,h,M κ∗3 Tσ∗¯2 ,h,M κ∗4 Tσ∗¯2 ,h,M
= E ∗ Tσ∗¯2 ,h,M , 2 ∗ = E ∗ Tσ∗2 Tσ∗¯2 ,h,M , ¯2 ,h,M − E 3 ∗ ∗2 ∗ ∗ ∗ = E ∗ Tσ∗3 − 3E T E T + 2 E T , ¯2 ,h,M σ¯2 ,h,M σ¯2 ,h,M σ¯2 ,h,M 2 ∗ ∗ = E ∗ Tσ∗4 Tσ∗3 Tσ∗¯2 ,h,M − 3 E ∗ Tσ∗2 ¯2 ,h,M − 4E ¯2 ,h,M E ¯2 ,h,M 4 ∗ +12E ∗ Tσ∗2 Tσ∗¯2 ,h,M )2 − 6 E ∗ Tσ∗¯2 ,h,M . ¯2 ,h,M (E
Our goal is to identify the terms of order up to
OP (h)
in the asymptotic expansions of these four
cumulants. We will rst provide asymptotic expansions through order of
Tσ∗¯2 ,h,M
by using a Taylor expansion.
f (x) = (1 + x)−k/2 integer k ,
k = 1, · · · , 4,
0
yields
k,
O(h)
for the rst four moments
a second-order Taylor expansion of
f (x) = 1 − k2 x + k4 ( k2 + 1)x2 + O(x3 ).
We have that for any xed
−k/2 √ ∗ = Sσ∗k + OP ∗ (h3/2 ), ¯2 ,h,M 1 + hUσ¯2 ,h,M k √ ∗k k k 3/2 = Sσ∗k hSσ¯2 ,h,M Uσ∗¯2 ,h,M + ( + 1)hSσ∗¯2 ,h,M Uσ∗2 ) ¯2 ,h,M − ¯2 ,h,M + OP ∗ (h 2 4 2 3/2 ≡ Tˆσ∗k ). ¯2 ,h,M + OP ∗ (h
Tσ∗k ¯2 ,h,M
For
around
For a xed value
the moments of
Tσ∗k ¯2 ,h,M
up to order
O(h3/2 )
7
are given by
√
3 h ∗ ∗ E Sσ¯2 ,h,M Uσ∗¯2 ,h,M + hE ∗ Sσ∗¯2 ,h,M Uσ∗2 ¯2 ,h,M 2 8 √ ∗ ∗2 ∗2 ∗2 ∗ ∗ = 1 − hE Sσ¯2 ,h,M Uσ¯2 ,h,M + hE ∗ Sσ∗2 E Tσ¯2 ,h,M ¯2 ,h,M Uσ¯2 ,h,M √ 3 15 ∗ ∗3 ∗3 ∗ ∗3 ∗2 ∗ ∗ E ∗ Tσ∗3 = E S − + S U S U h E hE ¯2 ,h,M ¯ ¯ σ¯2 ,h,M σ¯2 ,h,M σ¯2 ,h,M 2 σ2 ,h,M σ2 ,h,M 8 √ ∗ ∗4 ∗4 ∗2 ∗ ∗ ∗4 ∗ + 3hE S U . U E ∗ Tσ∗4 = E S − 2 hE S ¯2 ,h,M σ¯2 ,h,M σ¯2 ,h,M σ¯2 ,h,M σ¯2 ,h,M σ¯2 ,h,M E
∗
Tσ∗¯2 ,h,M
where we used
= 0−
E ∗ Sσ∗¯2 ,h,M = 0,
E ∗ Tσ∗¯2 ,h,M E ∗ Tσ∗2 ¯2 ,h,M E ∗ Tσ∗3 ¯2 ,h,M E ∗ Tσ∗4 ¯2 ,h,M Thus
7
and
E ∗ Sσ∗2 ¯2 ,h,M = 1.
By Lemma B.2 in Appendix B, we have that
√
A1 h − R6,4 + OP (h3/2 ), 2 √ 2 = 1 + h (C1 − A2 ) R8,4 + C2 R6,4 + OP (h2 ) √ 3 h B1 − A3 R6,4 + OP (h3/2 ) = 2
=
2 = 3 + h (B2 − 2D1 + 3E1 ) R8,4 + (3E2 − 2D2 ) R6,4 + OP (h2 ).
√ κ∗1 Tσ∗¯2 ,h,M = − h A21 R6,4 , this proves the rst result.
To be strictly rigorous, we should be taking expected value of
Tˆσ∗k ¯2 ,h,M
The remaining results follow similarly. rather than of
(cf. p. 72) or Gonçalves and Meddahi (2009) (cf. p. 302) for similar approach.
44
Tσ∗k ¯2 ,h,M .
See e.g. Hall (1992)
For part (c), results follows by noting that
1 cM,q Rq
→ σq
use results in Section 4.1 of Myklang and Zhang (2009),
P . For xed M, M → ∞ use Jacod and
in probability under
whereas when
Rosembaum (2013) (cf. equations (3.8) and (3.11)).
Appendix C This appendix concerns the multivariate case. First, we show results introduced in Sections 3.1 and 4.1. Then, we show results for Sections 3.2 and 4.2.
Proof of Bootstrap results in Section 3.1 and 4.1: Barndor-Nielsen and Shephard's (2004) type estimator Notation ∗ j = 1, . . . , 1/M h, and i = 1, . . . , M, let ∗i+(j−1)M = yl,i+(j−1)M − = y∗ − βˆ∗ y ∗ be the bootstrap OLS residuals. We
We introduce some notation. For
∗ βˆlk yk,i+(j−1)M ,
and let
ˆ∗i+(j−1)M
lk k,i+(j−1)M
l,i+(j−1)M
can write
√
∗ Tβ,h,M
√
∗ −β ˆlk h−1 βˆlk ≡ r P1/h ∗2 −2 ˆ ∗ B y i=1 k,i
where
=
P
1/h ∗ ∗ i=1 yk,i i
h−1
P
q ˆ∗ B
−1/2 √ ∗ ∗ 1 + hUβ,h,M = Sβ,h,M ,
(36)
h,M
h,M
√ ∗ = Sβ,h,M
h−1
1/h ∗ ∗ i=1 yk,i i
√
q ∗ Bh,M
and
∗ ≡ Uβ,h,M
ˆ ∗ − B∗ h−1 B h,M h,M ∗ Bh,M
,
such that
∗ Bh,M
1/M h 1/h X X ∗ ∗ ˆ2 Γ = V ar∗ h−1 i = Mh yk,i √
kl(j)
ˆ kk(j) Γ ˆ ll(j) − 4βˆlk Γ ˆ kk(j) Γ ˆ2 ˆ kl(j) + 2βˆ2 Γ +Γ lk kk(j) ,
j=1
i=1 and
ˆ∗ B h,M
Recall that Note that
ˆ ∗2 + Γ ˆ∗ Γ ˆ∗ + 1 Γ ˆ kk(j) Γ ˆ ll(j) + 3Γ ˆ2 Γ X M kl(j) kk(j) ll(j) kl(j) ˆ2 = Mh . Γ kk(j) 2 ˆ ∗2 ∗2 ∗ ∗ ∗ ˆ ˆ ˆ ˆ ˆ ˆ Γ Γ + 2 β Γ −4 β Γ + 2 Γ + kk(j) kl(j) j=1 lk lk M M kk(j) kl(j) kk(j) 1/M h
cM,q ≡ E
χ2M M
cM,2 = 1, cM,4 =
q/2 M +2 M ,
with
χ2M
cM,6 =
the standard
(M +2)(M +4) and M2
χ2
distribution with
cM,8 =
M
degrees of freedom.
(M +2)(M +4)(M +6) . M3
Lemma B.4. Suppose (1), (2) and (5) hold. We have that, for any q1 , q2 ≥ 0 such that q1 + q2 > 0, and for any k, l, k0 , l0 = 1, . . . , d, 1/M h
q1 q2 X ˆ ˆ Mh Γkl(j) Γk0 l0 (j) = OP (1) , j=1
where Γˆ kl(j) =
1 Mh
PM
i=1 yk,i+(j−1)M yl,i+(j−1)M ,
Proof of Lemma B.4.
for j = 1, . . . , 1/M h.
Apply Theorem 3 of Li, Todorov and Tauchen (2016).
45
Lemma B.5. Suppose (1), (2) and (5) hold. We have that, for any q1 , q2 ≥ 0 such that q1 + q2 > 0, and for any k, l, k0 , l0 = 1, . . . , d, as soon as M → ∞ such that M h → 0, as h → 0. a1)
Mh
a2)
Mh
1/M Ph j=1
1/M h P∗ ˆ ∗ −M h P Γ ˆ kl(j) −→ Γ 0, kl(j) j=1
1/M Ph j=1
1/M Ph ˆ P∗ ˆ∗ Γ ˆ∗ ˆ k0 l0 (j) −→ Γ Γkl(j) Γ 0, kl(j) k0 l0 (j) −M h j=1
Proof of Lemma B.5.
in probability-P.
We show that the results hold in quadratic mean with respect to
probability approaching one. under
in probability-P.
P ∗,
with
This ensures that the bootstrap convergence also holds in probability
P.
For part a1). Recall (13), and notice that we can write
x∗i+(j−1)M =
∗2 yk,i+(j−1)M
∗ ∗ yk,i+(j−1)M yl,i+(j−1)M
∗2 yl,i+(j−1)M
0
,
such that
∗2 yk,i+(j−1)M
ˆ kk(j) η ∗2 = Γ k,i+(j−1)M ,
∗ ∗ yk,i+(j−1)M yl,i+(j−1)M ∗2 yl,i+(j−1)M
ˆ2 Γ kl(j)
=
ˆ kk(j) Γ
2∗ ηk,i+(j−1)M +2
ˆ ll(j) − + Γ
where
∗ ηk,i+(j−1)M ∗ ηl,i+(j−1)M
1/M h
E ∗ M h
X
q ∗ ˆ kk(j) Γ ˆ ll(j) − Γ ˆ 2 η∗ Γ kl(j) k,i+(j−1)M ηl,i+(j−1)M ,
ˆ kl(j) η 2∗ = Γ k,i+(j−1)M +
ˆ2 Γ kl(j)
ˆ kl(j) q Γ ∗ ˆ kk(j) Γ ˆ ll(j) − Γ ˆ 2 η∗ Γ kl(j) k,i+(j−1)M ηl,i+(j−1)M ˆ Γkk(j)
! 2∗ ηl,i+(j−1)M ,
ˆ kk(j) Γ
! ∼ i.i.d.N (0, I2 ).
It follows that
! M X 1 ∗ ∗ yk,i+(j−1)M yl,i+(j−1)M Mh
1/M h
X
ˆ ∗ = E ∗ M h Γ kl(j)
j=1
j=1
=
1/M h M X X
i=1
∗ ∗ E ∗ yk,i+(j−1)M yl,i+(j−1)M
j=1 i=1
= h
1/M h M X X
E
∗
ˆ kl(j) η 2∗ Γ k,i+(j−1)M +
j=1 i=1 1/M h
= Mh
X j=1
P ˆ kl(j) −→ Γ
Z
1
Σkk,s ds. 0
46
q ∗ ∗ 2 ˆ ˆ ˆ Γkk(j) Γll(j) − Γkl(j) ηk,i+(j−1)M ηl,i+(j−1)M
Similarly, we have
V ar∗ M h
1/M h
X
1/M h M X X
ˆ ∗ = V ar∗ Γ kl(j)
∗ ∗ yl,i+(j−1)M E ∗ yk,i+(j−1)M
j=1 i=1
j=1
= h2
1/M h M X X
q ∗ ˆ kl(j) η 2∗ ˆ kk(j) Γ ˆ ll(j) − Γ ˆ 2 η∗ + η Γ V ar∗ Γ k,i+(j−1)M kl(j) k,i+(j−1)M l,i+(j−1)M
j=1 i=1
= h2
1/M h M X X
q 2 ∗ ˆ kl(j) η 2∗ ˆ kk(j) Γ ˆ ll(j) − Γ ˆ 2 η∗ + E∗ Γ η Γ k,i+(j−1)M kl(j) k,i+(j−1)M l,i+(j−1)M
j=1 i=1 1/M h M h X X
q i2 ∗ ˆ kl(j) η 2∗ ˆ kk(j) Γ ˆ ll(j) − Γ ˆ 2 η∗ E∗ Γ + Γ η k,i+(j−1)M kl(j) k,i+(j−1)M l,i+(j−1)M
−h2
j=1 i=1
= Mh
2
1/M h
X
1/M h M X X 2 2 ˆ ˆ ˆ ˆ2 2Γkl(j) + Γkk(j) Γll(j) − M h Γ kl(j)
j=1
= hM h
j=1 i=1
1/M h
X
ˆ2 ˆ kk(j) Γ ˆ ll(j) = OP (h) = oP (1) . Γ + Γ kl(j)
j=1
{z
|
}
=OP (1)
The proof of part a2) follows similarly and therefore we omit the details.
Lemma B.6. Suppose (1), (2) and (5) hold. We have that, a1)
∗ = 0, E ∗ Sh,M
a2)
ˆ2 ˆ kk(j) Γ ˆ kl(j) ˆ2 2Γ 2Γ 2Γ kk(j) kl(j) ˆ ˆ kl(j) Γ ˆ kk(j) Γ ˆ ll(j) + Γ ˆ2 ˆ ˆ 2Γkk(j) Γ kl(j) 2Γkl(j) Γll(j) . ˆ2 ˆ kl(j) Γ ˆ ll(j) ˆ2 2Γ 2Γ 2Γ kl(j) ll(j)
∗ = Mh Π∗h,M ≡ V ar∗ Sh,M
1/M Ph j=1
Proof of Lemma B.6 part a1). ∗ E ∗ Sh,M
√ =
h−1
Given the de nitions of
1/M h M X X
∗ , x∗ , Sh,M i
and
xi
we have
E ∗ x∗i+(j−1)M − xi+(j−1)M
j=1 i=1
√ =
h−1
1/M h M X X
E
∗
x∗i+(j−1)M
j=1 i=1
where results follow, since we have
1/M M Ph P j=1 i=1
−
√
h−1
1/M h M X X
xi+(j−1)M = 0,
j=1 i=1
E∗
∗ ∗ yk,i+(j−1)M yl,i+(j−1)M
47
=
−1 hP
i=1
yk,i yl,i .
Proof of Lemma B.6 part a2).
Given the de nition of
Π∗M,h ,
we have
1/h X √ = V ar∗ h−1 (x∗i − xi )
Π∗h,M
i=1 1/h
= h−1
X
V ar∗ (x∗i )
i=1
= h
−1
1/M h M X X
E
∗
x∗i+(j−1)M x∗0 i+(j−1)M
−E
∗
x∗i+(j−1)M
E
∗
x∗i+(j−1)M
0
j=1 i=1
= h
1/M h M X X
0 ∗ ∗ ∗ ∗ h−2 E ∗ x∗i+(j−1)M x∗0 − E x E x . i+(j−1)M i+(j−1)M i+(j−1)M
j=1 i=1 Let
∗ h−2 x∗i+(j−1)M x∗0 i+(j−1)M ≡ ai1 ,i2
1≤i1 ,i2 ≤3
.
It follows that
ˆ 2 η ∗4 a∗1,1 = Γ kk(j) k,i+(j−1)M , ˆ kk(j) Γ ˆ kl(j) η ∗4 a∗2,1 = Γ k,i+(j−1)M + a∗3,1
q
∗3 ∗ ˆ3 Γ ˆ ˆ2 ˆ2 Γ kk(j) ll(j) − Γkk(j) Γkl(j) ηk,i+(j−1)M ηl,i+(j−1)M , q ∗ ˆ kk(j) Γ ˆ ll(j) − Γ ˆ 2 η ∗3 ˆ 2 η 4∗ ˆ kl(j) Γ = Γ + 2 Γ kl(j) k,i+(j−1)M kl(j) k,i+(j−1)M ηl,i+(j−1)M ∗2 2∗ ˆ kk(j) Γ ˆ ll(j) − Γ ˆ2 + Γ kl(j) ηk,i+(j−1)M ηl,i+(j−1)M ,
a∗1,2 = a∗2,1 , q h i2 ∗ ˆ kl(j) η 2∗ ˆ kk(j) Γ ˆ ll(j) − Γ ˆ 2 η∗ a∗2,2 = Γ + Γ η , k,i+(j−1)M kl(j) k,i+(j−1)M l,i+(j−1)M v u ˆ2 ˆ2 ˆ3 u Γ Γ Γ kl(j) t ˆ kl(j) 3∗ kl(j) 4∗ ∗ ηk,i+(j−1)M + 2 q Γll(j) − η η∗ a3,2 = ˆ kk(j) ˆ kk(j) k,i+(j−1)M l,i+(j−1)M Γ Γ ˆ Γkk(j) ! ˆ3 Γ kl(j) 2∗ 2∗ ˆ ˆ + Γkl(j) Γll(j) − ηk,i+(j−1)M ηl,i+(j−1)M ˆ Γkk(j) ˆ2 q Γ kl(j) ∗ ˆ kk(j) Γ ˆ ll(j) − Γ ˆ 2 η 3∗ + Γ kl(j) k,i+(j−1)M ηl,i+(j−1)M ˆ Γkk(j) ! ˆ2 Γ kl(j) 2∗ 2∗ ˆ kl(j) Γ ˆ ll(j) − +2Γ ηk,i+(j−1)M ηl,i+(j−1)M ˆ kk(j) Γ ! ˆ2 q Γ kl(j) 3∗ ˆ ˆ kk(j) Γ ˆ ll(j) − Γ ˆ 2 η∗ + Γll(j) − Γ kl(j) k,i+(j−1)M ηl,i+(j−1)M , ˆ kk(j) Γ
48
(37)
a∗1,3 = a∗3,1 , a∗2,3 = a∗3,2 , ˆ2 ˆ kl(j) Γ Γkl(j) 2∗ ∗ a3,3 = ηk,i+(j−1)M + 2 q ˆ kk(j) Γ ˆ Γ
kk(j)
v u ˆ2 u Γ tΓ ˆ ll(j) − kl(j) η ∗ η∗ + ˆ kk(j) k,i+(j−1)M l,i+(j−1)M Γ
ˆ ll(j) − Γ
ˆ2 Γ kl(j)
2
!
2∗ ηl,i+(j−1)M .
ˆ kk(j) Γ
Hence, we have
ˆ2 ˆ kk(j) Γ ˆ kl(j) ˆ2 ˆ ˆ 3Γ 3Γ 2Γ kk(j) kl(j) + Γkk(j) Γll(j) ˆ kk(j) Γ ˆ kl(j) ˆ2 ˆ ˆ ˆ kl(j) Γ ˆ ll(j) 3Γ 2Γ 3Γ = . kl(j) + Γkk(j) Γll(j) 2 2 ˆ ˆ ˆ ˆ kl(j) Γ ˆ ll(j) ˆ 2Γ 3Γ 3Γ kl(j) + Γkk(j) Γll(j) ll(j)
h−2 E ∗ x∗i+(j−1)M x∗0 i+(j−1)M
(38) Next, remark that given the de nition of
x∗i+(j−1)M ,
and by using the linearization property of
E ∗ (·) ,
we have
0 ˆ kk(j) Γ ˆ kl(j) Γ ˆ ll(j) . E ∗ x∗i+(j−1)M = h Γ Thus, we can write
ˆ2 ˆ kk(j) Γ ˆ kl(j) Γ ˆ kk(j) Γ ˆ ll(j) Γ Γ kk(j) ˆ ˆ ˆ2 ˆ kl(j) Γ ˆ ll(j) Γ Γ = Γ . kk(j) Γkl(j) kl(j) 2 ˆ kk(j) Γ ˆ ll(j) Γ ˆ kl(j) Γ ˆ ll(j) ˆ Γ Γ ll(j)
h−2 E ∗ x∗i+(j−1)M E ∗ x∗i+(j−1)M
0
(39)
Given (38) and (39), we have
−2
h
E
∗
x∗i+(j−1)M x∗0 i+(j−1)M
−E
∗
x∗i+(j−1)M
∗
E
x∗i+(j−1)M
0
ˆ2 ˆ kk(j) Γ ˆ kl(j) ˆ2 2Γ 2Γ 2Γ kk(j) kl(j) ˆ 2 ˆ ˆ ˆ ˆ ˆ ˆ Γ Γ Γ + Γ 2 Γ = 2Γ kk(j) kl(j) kk(j) ll(j) kl(j) Γll(j) . kl(j) ˆ2 ˆ kl(j) Γ ˆ ll(j) ˆ2 2Γ 2Γ 2Γ kl(j) ll(j) Result follows by using (37).
Proof of Theorem 4.1. nonlinear
We prove results for
f (z) = z;
the delta method implies the result for
f.
Part (a). Result follows from Theorem 3 of Li, Todorov and Tauchen (2016) by noting the following: First the elements of
Π∗h,M
are all of the form of
Mh
1/M Ph
ˆ kl(j) Γ ˆ k0 l0 (j) . Γ
Second, since
X
is continuous
j=1
in our framework, the truncation in equation (3.1) of Li, Todorov and Tauchen (2016) is useless: one may use a treshold
vn = ∞
which reduce to our de nition of
ˆ kl(j) . Γ
∗ Part (b). Given the de nition of Sh,M , we have
∗ Sh,M =
√
h−1
1/M h M X X
x∗i+(j−1)M − xi+(j−1)M =
√
j=1 i=1
1/M h
h−1
X
M X
j=1
i=1
"
x∗i+(j−1)M − E ∗
M X
!# x∗i+(j−1)M
.
i=1
d(d+1) P1/M h ∗ λ ∈ R 2 such that λ0 λ = 1, supx∈R |P ∗ ( j=1 x ˜j ≤ x)− M M P √ P P P 1/M h ∗ x∗i+(j−1)M − E ∗ x∗i+(j−1)M . Note that, E ∗ x ˜ Φ(x/ (λ0 Πλ))| → 0, where x ˜∗j = h−1 λ0 j = j=1 i=1 i=1 P P 1/M h ∗ 0 and V ar∗ ˜j = λΠ∗h,M λ → λ0 Πλ by part a). Thus, by Katz's (1963) Berry-Essen Bound, j=1 x
The proof follows from showing that for any
49
for some small
ε>0
C > 0 which changes from line to line, 1/M h Xh X ∗ 1/M ∗ 0 x ˜j ≤ x − Φ(x/ λ Πλ ) ≤ C sup P E ∗ |˜ x∗j |2+ . x∈R j=1 j=1
Next, we show that
and some constant
P1/M h j=1
1/M h
X
E ∗ |˜ x∗j |2+ε = op (1).
We have that
"M !# 2+ε M √ X X 0 ∗ ∗ ∗ ∗ −1 xi+(j−1)M − E xi+(j−1)M = E h λ i=1 i=1 j=1 2+ε 1/M h M X −(2+ε) X ≤ 22+ h 2 E ∗ λ0 x∗i+(j−1)M j=1 i=1 2+ε 1/M h M X −(2+ε) X E∗ ≤ 22+ h 2 x∗i+(j−1)M 1/M h
E ∗ |˜ x∗j |2+ε
j=1
X
j=1
≤ 22+ Ch
−(2+ε) 2
i=1
M 1+ε
1/M h M X X
2+ε E ∗ x∗i+(j−1)M ,
j=1 i=1 where the rst inequality follows from the
Cr
and the Jensen inequalities; the second inequality uses the
Cauchy-Schwarz inequality and the fact that We let
2
|z| =
(z 0 z) for any vector
z.
E ∗ |˜ x∗j |2+ε ≤ Ch
−(2+ε) 2
1/M h M X X
M 1+ε
j=1 i=1
j=1
≤ Ch
and the third inequality uses the
Cr
inequality.
Then, we have
1/M h
X
λ0 λ = 1;
−(2+ε) 2
1/M h M X X
M 1+ε
d X d X 2 1+ε/2 ∗ ∗ E∗ yk,i+(j−1)M yl,i+(j−1)M k=1 l=1
2+ε ∗ ∗ ∗ E yk,i+(j−1)M yl,i+(j−1)M
j=1 i=1
≤ Ch
2+ε 2
∗ 2(2+ε) E ∗ ηk,1 M 2+ε
{z
| +Ch
2+ε 2
∗ ∗ 2+ε E ∗ ηk,1 ηl,1 M 2+ε
|
{z
1+ε ≤ Ch | M {z }M h =o(1) |
1/M h q
2+ε X ˆ kk(j) Γ ˆ ll(j) − Γ ˆ 2 Γ kl(j) j=1
}
=O(1) 1/M h
ε 2
2+ε X ˆ kl(j) Γ j=1
}
=O(1)
1/M h
"
X
2+ε 2+ε ˆ ˆ 2 ˆ + Γkk(j) Γll(j) Γkl(j) {z
}
=OP (1)
∗ ∗ yk,i+(j−1)M yl,i+(j−1)M .
Cr
inequality, the third inequality uses in
Finally, the last inequality and the consistency follow
from Minkowski inequality, Lemma B.4 and the fact that for any
α ∈ 0,
ε 2(1+ε)
Part (c).
,h
ε 2
M 1+ε
Given that
→ 0,
= oP (1) .
j=1
where the second and third inequalities follow from the addition the de nition of
#
h → 0. d Th → N 0, I d(d+1) ,
M
such that
M ≈ Ch−α
with
as
it su ces that
2
d∗ ∗ Th,M → N 0, I d(d+1) 2
50
in probability
under
P.
Next, note that from Lemma B.5 ∗
P ˆ ∗ − Π∗ −→ 0, Π h,M h,M
P.
in probability under
In addition, both
probability approaching one, as
h → 0.
ˆ∗ Π h,M
and
Π∗h,M
are non singular in large samples with
Then, using results in parts (a) and (b) of Theorem 4.1, it
follows that
−1/2 1/2 ∗ −1/2 ∗ d∗ ∗ ∗ ˆ∗ Π Th,M = Π Π → N 0, I , S d(d+1) h,M h,M h,M h,M 2 {z } | {z }| d∗
P∗
→N 0,I d(d+1)
→I d(d+1)
2
2
in probability-P.
Lemma B.7. Suppose (1), (2) and (5) hold. We have that, a1)
ˆ ∗3 E∗ Γ kl(j) =
a2)
ˆ∗ Γ ˆ∗ Γ ˆ∗ E∗ Γ kk(j) kl(j) ll(j) =
a3)
ˆ∗ Γ ˆ ∗2 E∗ Γ kk(j) kl(j) =
(M +2)(M +3) ˆ ˆ2 Γkk(j) Γ kl(j) M2
a4)
ˆ ∗2 Γ ˆ∗ E∗ Γ kk(j) ll(j) =
M (M +2) ˆ 2 ˆ ll(j) Γkk(j) Γ M2
a5)
ˆ ∗2 Γ ˆ∗ E∗ Γ kk(j) kl(j) =
M (M +2)(M +4) ˆ 2 ˆ kl(j) ; Γkk(j) Γ M3
a6)
ˆ ∗2 Γ ˆ ∗2 E∗ Γ kk(j) kl(j) =
M 3 +11M 2 +42M +36 ˆ 2 ˆ2 Γkk(j) Γ kl(j) M3
a7)
ˆ ∗3 Γ ˆ∗ E∗ Γ kk(j) kl(j) =
M (M +2)(M +4)(M +6) ˆ 3 ˆ kl(j) ; Γkk(j) Γ M3
a8)
ˆ∗ Γ ˆ∗ E∗ Γ kk(j) kl(j) =
M (M +2) ˆ ˆ kl(j) ; Γkk(j) Γ M2
a9)
ˆ ∗2 E∗ Γ kl(j) =
(M +2)(M +1) ˆ 3 Γkl(j) M2
M +1 ˆ 2 M Γkl(j)
a10)
ˆ ∗2 E∗ Γ ll(j) =
a11)
ˆ ∗2 E∗ Γ kk(j) =
+
3(M +2) ˆ ˆ kl(j) Γ ˆ ll(j) ; Γkk(j) Γ M2
(M +2)2 ˆ ˆ kl(j) Γ ˆ ll(j) Γkk(j) Γ M2
+
+
ˆ3 + 2 (MM+2) 2 Γkl(j) ;
(M +2) ˆ 2 ˆ ll(j) ; Γkk(j) Γ M2
+
4(M +2) ˆ ˆ2 ; Γkk(j) Γ kl(j) M2
+
M 2 +2M +12 ˆ 3 ˆ ll(j) ; Γkk(j) Γ M3
1 ˆ ˆ M Γkk(j) Γll(j) ;
M +2 ˆ 2 M Γll(j) ; M +2 ˆ 2 M Γkk(j) .
Proof of Lemma B.7.
ˆ∗ , Γ ˆ∗ ˆ∗ Γ kk(j) kl(j) Γll(j) and " # M X 1 ∗2 ˆ kk(j) ηk,i+(j−1)M =Γ , M
Note that given the de nitions of
M
1 X ∗2 ˆ∗ yk,i+(j−1)M Γ = kk(j) Mh i=1
(13), we have
i=1
M
1 X ∗ ∗ yk,i+(j−1)M yl,i+(j−1)M Mh i=1 ! M q X 1 2∗ ˆ kl(j) ˆ kk(j) Γ ˆ ll(j) − Γ ˆ2 = Γ ηk,i+(j−1)M + Γ kl(j) M
ˆ∗ Γ kl(j) =
i=1
51
M 1 X ∗ ∗ ηk,i+(j−1)M ηl,i+(j−1)M M i=1
!
and
M
ˆ∗ Γ ll(j) =
1 X ∗2 yl,i+(j−1)M Mh i=1
ˆ2 Γ kl(j)
! ! M M ˆ kl(j) q X Γ 1 X 2∗ 1 ∗ ∗ 2 ˆ ll(j) − Γ ˆ ˆ kk(j) Γ = ηk,i+(j−1)M + 2 Γ ηk,i+(j−1)M ηl,i+(j−1)M kl(j) ˆ kk(j) M ˆ kk(j) M Γ Γ i=1 i=1 ! ! M ˆ2 X Γ 1 2∗ ˆ ll(j) − kl(j) + Γ , ηl,i+(j−1)M ˆ kk(j) M Γ i=1 ! ∗ ηk,i+(j−1)M ∼ i.i.d.N (0, I2 ). By tedious but simple algebra, all results follow from the norwhere ∗ ηl,i+(j−1)M ∗ ∗ mality and i.i.d properties of η k,i+(j−1)M , ηl,i+(j−1)M across (i, j) . For instance, for (a1) E
∗
ˆ ∗3 Γ kl(j)
= E
∗
= E∗
M X 2∗ ˆ kl(j) 1 Γ ηk,i+(j−1)M M i=1 ! M X 1 2∗ ˆ3 ηk,i+(j−1)M Γ kl(j) M
!
q ˆ kk(j) Γ ˆ ll(j) − Γ ˆ2 + Γ
kl(j)
M 1 X ∗ ∗ ηk,i+(j−1)M ηl,i+(j−1)M M
!!3
i=1
i=1
M 1 X 2∗ ηk,i+(j−1)M M
q ∗ ˆ ˆ ˆ2 ˆ2 +3Γ kl(j) Γkk(j) Γll(j) − Γkl(j) E
!2
i=1
ˆ kl(j) Γ ˆ kk(j) Γ ˆ ll(j) − +3Γ
ˆ2 Γ kl(j)
E
∗
!
M 1 X ∗ ∗ ηk,i+(j−1)M ηl,i+(j−1)M M i=1 !3
!2
i=1
M 1 X 2∗ ηk,i+(j−1)M M
!
i=1
3/2 ˆ kk(j) Γ ˆ ll(j) − Γ ˆ2 + Γ E∗ kl(j)
M 1 X ∗ ∗ ηk,i+(j−1)M ηl,i+(j−1)M M
M 1 X ∗ ∗ ηk,i+(j−1)M ηl,i+(j−1)M M i=1
ˆ3 = cM,6 Γ kl(j) + 0 3 ˆ 2 ∗ ˆ ˆ ˆ + 3Γ Γ Γ − Γ kl(j) E M kl(j) kk(j) ll(j) = =
M X i=1
! 2∗ ηk,i+(j−1)M
M X
!2 ∗ ∗ ηk,i+(j−1)M ηl,i+(j−1)M
i=1
M (M + 2) (M + 4) ˆ 3 3M (M + 2) ˆ ˆ kk(j) Γ ˆ ll(j) − Γ ˆ2 Γ + Γ Γ kl(j) kl(j) kl(j) M3 M3 (M + 2) (M + 1) ˆ 3 3 (M + 2) ˆ ˆ kl(j) Γ ˆ ll(j) . Γkl(j) + Γkk(j) Γ M2 M2
52
+0
For (a2), note that
ˆ∗ ˆ∗ ˆ∗ Γ Γ kk(j) kl(j) Γll(j) ˆ3 = Γ kl(j)
M 1 X 2∗ ηk,i+(j−1)M M
!3
i=1
!2 ! M M 1 X ∗ 1 X 2∗ ∗ ηk,i+(j−1)M ηk,i+(j−1)M ηl,i+(j−1)M M M i=1 i=1 ! ! 2 M M 1 X X 1 2∗ 2∗ ˆ kk(j) Γ ˆ kl(j) Γ ˆ ll(j) − Γ ˆ3 + Γ ηk,i+(j−1)M ηl,i+(j−1)M kl(j) M M i=1 i=1 ! ! 2 M M q X X 1 1 ∗ 2∗ ∗ ˆ2 ˆ ˆ ˆ2 +Γ ηl,i+(j−1)M ηk,i+(j−1)M ηk,i+(j−1)M kl(j) Γkk(j) Γll(j) − Γkl(j) M M i=1 i=1 ! !2 M M 1 X X 1 2∗ ∗ ∗ ˆ kk(j) Γ ˆ kl(j) Γ ˆ ll(j) − Γ ˆ3 +2 Γ ηk,i+(j−1)M ηk,i+(j−1)M ηl,i+(j−1)M kl(j) M M i=1 i=1 ! ! ! M M M 3/2 1 X X X 1 1 2∗ 2∗ ∗ ∗ ˆ kk(j) Γ ˆ ll(j) − Γ ˆ2 + Γ . ηk,i+(j−1)M ηl,i+(j−1)M ηk,i+(j−1)M ηl,i+(j−1)M kl(j) M M M q ˆ2 ˆ ˆ ˆ2 +2Γ kl(j) Γkk(j) Γll(j) − Γkl(j)
i=1
i=1
i=1
Then, we have
M (M + 2) (M + 4) ˆ 3 M 2 (M + 2) ˆ 3 ˆ ˆ ˆ Γ + 0 + Γ Γ Γ − Γ kk(j) kl(j) ll(j) kl(j) kl(j) M3 M3 M (M + 2) ˆ ˆ kl(j) Γ ˆ ll(j) − Γ ˆ3 +0 + 2 Γkk(j) Γ kl(j) + 0 M3 (M + 2)2 ˆ ˆ kl(j) Γ ˆ ll(j) + 2 (M + 2) Γ ˆ3 . = Γkk(j) Γ kl(j) 2 M M2
ˆ∗ Γ ˆ∗ Γ ˆ∗ = E∗ Γ kk(j) kl(j) ll(j)
For (a3), note that
ˆ∗ Γ ˆ ∗2 ˆ ˆ2 Γ kk(j) kl(j) = Γkk(j) Γkl(j)
M 1 X 2∗ ηk,i+(j−1)M M
!3
i=1
!2 ! M M 1 X ∗ 1 X 2∗ ∗ ηk,i+(j−1)M ηk,i+(j−1)M ηl,i+(j−1)M M M i=1 i=1 ! !2 M M 1 X 2∗ 1 X ∗ ∗ ηk,i+(j−1)M ηk,i+(j−1)M ηl,i+(j−1)M , M M
q ˆ2 ˆ kk(j) Γ ˆ kl(j) Γ ˆ kk(j) Γ ˆ ll(j) − Γ +2Γ kl(j) ˆ2 Γ ˆ ll(j) − Γ ˆ kk(j) Γ ˆ2 + Γ kk(j) kl(j)
i=1
i=1
then we have
ˆ∗ Γ ˆ ∗2 E∗ Γ = kk(j) kl(j) =
M (M + 2) ˆ 2 M (M + 2) (M + 4) ˆ ˆ2 ˆ ll(j) − Γ ˆ kk(j) Γ ˆ2 Γ Γ Γ + 0 + Γ kk(j) kl(j) kk(j) kl(j) M3 M3 (M + 2) ˆ 2 (M + 2) (M + 3) ˆ ˆ2 ˆ ll(j) . Γkk(j) Γ Γkk(j) Γ kl(j) + M2 M2
53
For (a4), note that
ˆ ∗2 Γ ˆ∗ ˆ ˆ2 Γ kk(j) ll(j) = Γkk(j) Γkl(j)
M 1 X 2∗ ηk,i+(j−1)M M
!3
i=1
!2 ! M M 1 X ∗ 1 X 2∗ ∗ ηk,i+(j−1)M ηk,i+(j−1)M ηl,i+(j−1)M M M i=1 i=1 ! !2 M M 1 X 2∗ 1 X 2∗ , ηl,i+(j−1)M ηk,i+(j−1)M M M
ˆ kl(j) q Γ ˆ2 ˆ kk(j) Γ ˆ ll(j) − Γ ˆ2 +2Γ Γ kk(j) ˆ kl(j) Γkk(j) ˆ2 Γ ˆ ll(j) − Γ ˆ kk(j) Γ ˆ2 + Γ kk(j) kl(j)
i=1
i=1
then we can write
ˆ∗ ˆ ∗2 Γ = E∗ Γ kk(j) ll(j) =
M (M + 2) (M + 4) ˆ M 2 (M + 2) ˆ 2 2 2 ˆ ˆ ˆ ˆ + 0 + Γ − Γ Γ Γ Γ Γ kk(j) kl(j) kk(j) kl(j) kk(j) ll(j) M3 M3 M (M + 2) ˆ 2 ˆ ll(j) + 4 (M + 2) Γ ˆ kk(j) Γ ˆ2 . Γkk(j) Γ kl(j) 2 M M2
For (a5), note that
ˆ ∗2 Γ ˆ ∗2 Γ kk(j) kl(j)
=
ˆ2 Γ ˆ2 Γ kk(j) kl(j)
M 1 X 2∗ ηk,i+(j−1)M M
!4
i=1
!3 ! M M 1 X ∗ 1 X 2∗ ∗ ηk,i+(j−1)M ηk,i+(j−1)M ηl,i+(j−1)M M M i=1 i=1 !2 !2 M M 1 X 2∗ 1 X ∗ ∗ ηk,i+(j−1)M ηk,i+(j−1)M ηl,i+(j−1)M , M M
q 2 ˆ kk(j) Γ ˆ ll(j) − Γ ˆ2 ˆ ˆ +2Γkk(j) Γkl(j) Γ kl(j) ˆ3 Γ ˆ ˆ2 ˆ2 + Γ kk(j) ll(j) − Γkk(j) Γkl(j)
i=1
i=1
then we have
M (M + 2) (M + 4) (M + 6) ˆ 2 ˆ2 Γkk(j) Γ kl(j) + 0 M4 15M + 3 M 2 − M + M 3 − M 2 ˆ 3 ˆ ll(j) − Γ ˆ2 Γ ˆ2 + Γ Γ kk(j) kk(j) kl(j) M4 3 2 M 2 + 2M + 12 ˆ 3 M + 11M + 42M + 36 ˆ 2 ˆ2 ˆ ll(j) . = Γ Γ + Γkk(j) Γ kk(j) kl(j) M3 M3
ˆ ∗2 Γ ˆ ∗2 = E∗ Γ kk(j) kl(j)
Result follows for (a6), by noting that
ˆ ∗3 Γ ˆ∗ Γ kk(j) kl(j)
=
ˆ3 Γ ˆ Γ kk(j) kl(j)
M 1 X 2∗ ηk,i+(j−1)M M
!4
i=1
ˆ3 +Γ kk(j)
q ˆ kk(j) Γ ˆ ll(j) − Γ ˆ2 Γ
kl(j)
M 1 X 2∗ ηk,i+(j−1)M M i=1
54
!3
M 1 X ∗ ∗ ηk,i+(j−1)M ηl,i+(j−1)M M i=1
! ,
and
M 1 X 2∗ E∗ ηk,i+(j−1)M M
!4
= cM,8 = M (M + 2) (M + 4) (M + 6) M4
i=1
E∗
M 1 X 2∗ ηk,i+(j−1)M M
!3
M 1 X ∗ ∗ ηk,i+(j−1)M ηl,i+(j−1)M M
i=1
! = 0.
i=1
The remaining results follow similarly and therefore we omit the details.
Lemma B.8. Suppose (1), (2) and (10) hold. We have that, a1)
E∗
P
a2)
E∗
P
a3)
E∗
P
a4)
E∗
hP 1/h
h
hP
h
= −
a5)
E∗ =
1/h ∗ ∗ i=1 yk,i i
= 0;
1/h ∗ ∗ 2 y i=1 k,i i
˜∗ ; = hB h,M
1/h ∗ ∗ 3 y i=1 k,i i
= h2 A˜∗1,h,M ; ii
ii
1/h ∗ ∗ P1/M h ˆ ∗ ∗ Γ ˆ∗ Γ ˆ∗ ˆ∗ Γkk(j) Γ j=1 i=1 yk,i i kk(j) kl(j) kl(j) − E h P1/M h ˆ P1/M h ˆ 2 (M +2) ˆ2 ˆ 3M h j=1 Γ kk(j) Γkl(j) + M h j=1 Γkk(j) Γll(j) M2
a6)
E∗
hP
a7)
E∗
P
a8)
2 P 1/h ∗ ∗ E∗ i=1 yk,i i
1/h ∗ ∗ P1/M h i=1 yk,i i j=1
1/h ∗ ∗ 4 y i=1 k,i i
h
ii ˆ ∗2 − E ∗ Γ ˆ ∗2 Γ = kk(j) kk(j)
˜ ∗2 + OP (h) , = 3h2 B h,M
as h → 0;
a9)
P1/M h ˆ ∗2 ∗ Γ ∗ ∗ ˆ∗ Γ ˆ∗ ˆ ∗2 + Γ ˆ∗ Γ ˆ∗ Γkl(j) + Γ j=1 i=1 yk,i i kk(j) ll(j) − E kl(j) kk(j) ll(j) P1/M h ˆ 3 P1/M h ˆ 2(M +3) 6M +10 ˆ ˆ M h j=1 Γ j=1 Γkk(j) Γkl(j) Γll(j) kl(j) + M 2 M h M2 P P 1/M h ˆ 1/M h ˆ 2 2(M +1) ˆ (6M +14) ˆ ˆ2 ˆ βlk M h j=1 Γ βlk M h j=1 Γ kk(j) Γkl(j) − kk(j) Γll(j) ; M2 M2
P 3 1/h ∗ ∗ E∗ i=1 yk,i i
i P1/M h ˆ 2 ˆ kl(j) ; Γ Γ j=1 kk(j)
i h P1/M h ˆ 2 ˆ kl(j) − βˆlk M h P1/M h Γ ˆ3 M h j=1 Γ Γ j=1 kk(j) kl(j) ;
as h → 0;
i P1/M h h ˆ ∗2 ˆ ∗2 + Γ ˆ∗ Γ ˆ∗ ˆ∗ Γ ˆ∗ − E∗ Γ Γ + Γ j=1 kl(j) kk(j) ll(j) kk(j) ll(j) i = OP h2 , kl(j) P1/M h h ˆ ∗ ∗ Γ ∗ Γ ˆ∗ Γ ˆ∗ ˆ ∗2 ˆ ∗2 ˆ∗ + j=1 Γkk(j) Γ kk(j) kk(j) kl(j) + Γkk(j) − E kl(j) − E i P1/M h h ˆ ∗2 ˆ∗ Γ ˆ∗ − E∗ Γ ˆ ∗2 + Γ ˆ∗ Γ ˆ∗ Γ + Γ j=1 kl(j) kk(j) ll(j) kk(j) ll(j) kl(j) i P1/M h h ˆ ∗ ∗ ∗ ∗ ∗ ∗2 ˆ ˆ ˆ ˆ ˆ ∗2 + j=1 Γkk(j) Γkl(j) − E Γkk(j) Γkl(j) + Γkk(j) − E ∗ Γ kk(j)
˜ ∗ A˜∗2 = 3h2 B h,M 1,h,M + OP (h),
Proof of Lemma B.8.
4(M +2) M2
− 4βˆlk M h
as h → 0.
In the following, recall that by de nition for
55
j = 1, . . . , 1/M h,
and
i =
∗ ∗ 1, . . . , M, ∗i+(j−1)M = yl,i+(j−1)M − βˆlk yk,i+(j−1)M , 1/h 1/h X X ∗ ∗ ∗ ∗ E∗ yk,i i = E∗ yk,i i i=1
and (13). For (a1), we can write
i=1 1/M h M h X X
=
i ∗ 2∗ ∗ yl,i+(j−1)M − βˆlk E ∗ yk,i+(j−1)M E ∗ yk,i+(j−1)M
j=1 i=1 1/M h
X
= Mh
1/M h
ˆ kl(j) − βˆlk M h Γ
j=1
ˆ kk(j) Γ
j=1
1/M h
X
= Mh
X
M
1 X yk,i+(j−1)M yl,i+(j−1)M Mh i=1
j=1
!
1/M h X i=1 yk,i yl,i − P1/h Mh 2 j=1 i=1 yk,i
P1/h
M
1 X 2 yk,i+(j−1)M Mh i=1
= 0. For (a2), note that by de nition
E∗
1/h X
2
∗ ∗ i = V ar∗ yk,i
i=1
1/h X i=1
1/h X ∗ ∗ ∗ ∗ i + E ∗ i . yk,i yk,i i=1
˜ ∗ , and the fact that by (a1) E ∗ P1/h y ∗ ∗ = 0. For the B i=1 k,i i h,M ∗ ∗ ∗ ∗ ∗ ∗ yk,i i and note that by de nition, the zi∗0 s are conditionally remaining results, write zi = yk,i i − E P P1/h ∗ P1/h ∗ ∗ 1/h ∗ ∗ ∗ ∗ ∗ , since E independent with E (zi ) = 0. Next, note also that z = y y i=1 i i=1 k,i i i=1 k,i i = 0.
Then, result follows given the de nition of
For (a3), note that
E∗
1/h X
3 1/h 1/h 1/M h M X X X X ∗3 ∗ ∗ . i = E ∗ zi∗ = E ∗ zi∗3 = E ∗ zi+(j−1)M yk,i
i=1
3
i=1
i=1
j=1 i=1
Thus, we can write
E∗
1/h X
3 ∗ ∗ i yk,i
i=1
=
1/M h M X X
h i3 ∗ ∗ ∗i+(j−1)M E ∗ yk,i+(j−1)M ∗i+(j−1)M − E ∗ yk,i+(j−1)M
j=1 i=1
=
∗3 ∗ y ∗2 ∗2 ∗ y∗ ∗ E ∗ yk,i+(j−1)M ∗3 − 3E E i+(j−1)M k,i+(j−1)M i+(j−1)M k,i+(j−1)M i+(j−1)M . h i3 ∗ ∗ ∗ +2 E yk,i+(j−1)M i+(j−1)M i=1
1/M h M X X j=1
56
!
Next, to arrive at the desired expression for
1/M h M X X
E∗
1/h ∗ ∗ 3 y , note that i=1 k,i i
P
∗3 E ∗ yk,i+(j−1)M ∗3 i+(j−1)M
j=1 i=1 1/M h M X X
=
3 ∗ ∗3 ∗ ˆ E yk,i+(j−1)M yl,i+(j−1)M − βlk yk,i+(j−1)M ∗
j=1 i=1 3
= h
1/M h M X X
ˆ3 ˆ ˆ ˆ ˆ ˆ2 ˆ 6Γ kl(j) + 9Γkk(j) Γkl(j) Γll(j) − 36Γkk(j) Γkl(j) βlk ˆ ˆ2 ˆ 2 ˆ3 ˆ 3 ˆ2 Γ ˆ ˆ −9Γ kk(j) ll(j) βlk + 45βlk Γkk(j) Γkl(j) − 15βlk Γkk(j)
j=1 i=1
−3
1/M h M X X
! ,
∗2 ∗ ∗ ∗ E ∗ yk,i+(j−1)M ∗2 E y i+(j−1)M k,i+(j−1)M i+(j−1)M
j=1 i=1
= −3h
1/M h M X X
ˆ ˆ ˆ2 ˆ 2 ˆ2 ˆ ˆ ˆ 2Γ kl(j) + Γkk(j) Γll(j) − 6βlk Γkk(j) Γkl(j) + 3βlk Γkk(j)
ˆ kk(j) ˆ kl(j) − βˆlk Γ Γ
j=1 i=1
= h
1/M h M X X
ˆ2 ˆ 2 ˆ ˆ ˆ ˆ2 ˆ3 ˆ ˆ ˆ −6Γ kl(j) − 3Γkk(j) Γkl(j) Γll(j) + 18βlk Γkk(j) Γkl(j) − 9βlk Γkk(j) Γkl(j) ˆ3 ˆ 3 ˆ2 ˆ 2 ˆ ˆ2 ˆ ˆ ˆ kk(j) Γ ˆ2 +6βˆlk Γ kl(j) + 3βlk Γkk(j) Γll(j) − 186βlk Γkk(j) Γkl(j) + 9βlk Γkk(j)
j=1 i=1
! ,
and
2
1/M h M h X X
i3 ∗ ∗ ∗ yl,i+(j−1)M − βˆlk yk,i+(j−1)M E ∗ yk,i+(j−1)M
j=1 i=1 3
= 2h
1/M h M X X
ˆ kk(j) ˆ kl(j) − βˆlk Γ Γ
3
j=1 i=1
= h3
1/M h M X X
ˆlk Γ ˆ2 Γ ˆ3 ˆ3 ˆ kk(j) Γ ˆ2 Γ ˆ kl(j) − 2βˆ3 Γ ˆ2 2Γ − 6 β + 6 β lk kk(j) . lk kk(j) kl(j) kl(j)
j=1 i=1 For (a4), note that
1/h 1/M h h X X ∗ ∗ ∗ ˆ ∗2 yk,i i Γ E
kl(j)
i=1
∗ ˆ ∗2 ˆ∗ Γ ˆ∗ ˆ∗ Γ ˆ∗ +Γ Γkl(j) + Γ kk(j) ll(j) − E kk(j) ll(j)
i
j=1
1/h 1/M h h i X X ∗ ˆ ∗2 ˆ ∗2 + Γ ˆ∗ Γ ˆ∗ ˆ∗ Γ ˆ∗ = E ∗ zi∗ Γ Γkl(j) + Γ kl(j) kk(j) ll(j) − E kk(j) ll(j) i=1
j=1
ˆlk Γ ˆ∗ Γ ˆ∗ ˆ∗ ˆ∗ ˆ ∗2 + Γ E∗ Γ − β Γ kl(j) kk(j) kl(j) kk(j) ll(j) = Mh 1 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ − Γ − β Γ Γ + Γ Γ + Γ Γ + 3 Γ lk kk(j) kl(j) kk(j) ll(j) kk(j) ll(j) j=1 M kl(j) kl(j) 1/M h ˆ ∗3 + Γ ˆ∗ Γ ˆ∗ Γ ˆ ∗ − βˆlk Γ ˆ∗ Γ ˆ ∗2 − βˆlk Γ ˆ ∗2 Γ ˆ∗ X E∗ Γ kl(j) kk(j) kk(j) kl(j) kk(j) ll(j) kl(j) ll(j) = Mh 1 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ2 ˆ ˆ − Γ + Γ Γ + Γ Γ + 3 Γ Γ − β Γ lk kl(j) kk(j) kk(j) ll(j) kk(j) ll(j) j=1 M kl(j) kl(j) 1/M h
X
57
.
Using Lemma B.7 we deduce that
ˆ∗ Γ ˆ∗ Γ ˆ ∗ − βˆlk Γ ˆ∗ Γ ˆ ∗2 − βˆlk Γ ˆ ∗2 Γ ˆ∗ ˆ ∗3 + Γ E∗ Γ kk(j) kl(j) ll(j) kk(j) kl(j) kk(j) ll(j) kl(j) ∗ ˆ∗ ˆlk E ∗ Γ ˆlk E ∗ Γ ˆ∗ ˆ ∗2 Γ ˆ ∗2 ˆ ∗3 ˆ∗ Γ ˆ∗ ˆ∗ Γ − β − β = E∗ Γ + E Γ Γ kk(j) ll(j) kk(j) kl(j) kl(j) kk(j) kl(j) ll(j) (M + 2) (M + 1) ˆ 3 3 (M + 2) ˆ ˆ kl(j) Γ ˆ ll(j) Γkl(j) + Γkk(j) Γ 2 M M2 (M + 2)2 ˆ ˆ ll(j) + 2 (M + 2) Γ ˆ kl(j) Γ ˆ3 Γkk(j) Γ + kl(j) 2 2 M M (M + 2) ˆ 2 (M + 2) (M + 3) ˆ 2 ˆ ˆ ˆ −βlk Γkk(j) Γkl(j) + Γkk(j) Γll(j) M2 M2 M (M + 2) ˆ 2 4 (M + 2) ˆ 2 ˆ ˆ ˆ −βlk Γkk(j) Γll(j) + Γkk(j) Γkl(j) M2 M2 (M + 2) (M + 5) ˆ (M + 2) (M + 3) ˆ 3 ˆ kl(j) Γ ˆ ll(j) Γkl(j) + Γkk(j) Γ = 2 2 M M (M + 2) (M + 7) ˆ (M + 2) (M + 1) ˆ 2 2 ˆ ˆ ˆ −βlk Γkk(j) Γkl(j) + Γkk(j) Γll(j) . M2 M2 =
Adding and substracting appropriately gives result for (a4). For (a5), note that
1/M h h 1/h X X ∗ ∗ ∗ ˆ∗ Γ yk,i i E
ˆ∗ kk(j) Γkl(j)
ˆ∗ Γ ˆ∗ − E∗ Γ kk(j) kl(j)
i
j=1
i=1
1/h 1/M h h i X X ∗ ˆ∗ ˆ∗ ˆ∗ Γ ˆ∗ Γkk(j) Γ Γ = E ∗ zi∗ kl(j) kk(j) kl(j) − E i=1
j=1
1/M h X 2 ˆ ∗2 ∗2 ∗ ∗ ˆ∗ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Γkk(j) Γkl(j) + = Mh E Γkk(j) Γkl(j) − βlk Γkk(j) Γkl(j) − Γkl(j) − βlk Γkk(j) Γ Γ , M kk(j) kl(j) j=1
then use Lemma B.7 and remark that
ˆ ∗2 Γ ˆ∗ ˆ∗ Γ ˆ ∗2 − βˆlk Γ E∗ Γ kk(j) kl(j) kk(j) kl(j) =
(M + 2) (M + 3) ˆ (M + 2) ˆ 2 ˆ2 ˆ ll(j) − βˆlk (M + 2) (M + 4) Γ ˆ2 Γ ˆ Γkk(j) Γ Γkk(j) Γ kl(j) + kk(j) kl(j) . M2 M2 M2
Adding and substracting appropriately gives result for (a5). For (a6), note that
1/h 1/M h h X X ∗ ∗ ∗ ˆ ∗2 E yk,i i Γ
kk(j)
i=1
ˆ ∗2 − E∗ Γ kk(j)
i
j=1
1/M h X 2 ˆ2 ∗ ˆ ∗2 ∗ ∗3 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ = Mh E Γkk(j) Γkl(j) − βlk Γkk(j) − Γkl(j) − βlk Γkk(j) Γkk(j) + Γ . M kk(j) j=1
Next use Lemma B.7 and compute
M (M + 2) (M + 4) ˆlk Γ ˆ ∗2 Γ ˆ∗ ˆ ∗3 ˆ2 Γ ˆ kl(j) − βˆlk Γ ˆ3 Γ E∗ Γ − β = kk(j) kl(j) kk(j) kk(j) kk(j) . M3
58
Adding and substracting appropriately gives result for (a6). The remaining results follows similarly and therefore we omit the details.
Lemma B.9. Suppose (1), (2) and (10) hold. We have that, a1)
∗ E ∗ Sβ,h,M = 0;
a2)
2 ∗ E ∗ Sβ,h,M = 1;
a3)
3 √ A˜∗ ∗ E ∗ Sβ,h,M = h ˜1,h,M ∗3/2 ;
a4)
Bh,M
# P1/M h ˆ 3 P1/M h ˆ ˆ kl(j) Γ ˆ ll(j) 2M h Γ + 6M h Γ Γ kk(j) j=1 j=1 kl(j) 1 ∗ ∗ E ∗ Sβ,h,M U1,β,h,M = ˜ ∗3/2 P1/M h ˆ 2 P1/M h ˆ ˆ ˆ ll(j) ˆ2 Bh,M Γ Γ −6βˆlk M h j=1 Γ kk(j) Γkl(j) − 2βlk M h " #j=1 kk(j) P1/M h ˆ P1/M h ˆ 3 ˆ kl(j) Γ ˆ ll(j) 6M h j=1 Γkl(j) + 10M h j=1 Γkk(j) Γ 1 ; + ˜ ∗3/2 P P 1/M h 1/M h ˆ2 Γ ˆ ll(j) ˆ kk(j) Γ ˆ2 Bh,M M Γ Γ − 2βˆlk M h −14βˆlk M h
"
j=1
a5)
∗ ∗ U2,β,h,M E ∗ Sβ,h,M ˆ
lk = −4 ˜β∗3/2 1+
Bh,M
2 M
h
3M h
2 βˆlk ∗ ∗ = 8 ˜ ∗3/2 U3,β,h,M E ∗ Sβ,h,M 1+
a7)
˜∗ A ∗ ∗ = −4 ˜ ∗1/2 0,h,M U4,β,h,M E ∗ Sβ,h,M P1/h
a9)
kk(j)
i P1/M h ˆ P1/M h ˆ 2 P1/M h ˆ 2 ˆ ˆ ˆ2 ˆ j=1 Γkk(j) Γkl(j) + M h j=1 Γkk(j) Γll(j) − 4βlk M h j=1 Γkk(j) Γkl(j) ;
a6)
a8)
j=1
kl(j)
Bh,M
Bh,M
2 M
i=1
h
2 yk,i
Mh
i P1/M h ˆ 2 ˆ kl(j) − βˆlk M h P1/M h Γ ˆ3 Γ Γ j=1 j=1 kk(j) kl(j) ;
;
4 ∗ E ∗ Sβ,h,M = 3 + OP (h) , as h → 0; √ i h ˇ∗ = OP E ∗ S ∗2 U h , as h → 0;
a10)
β,h,M
β,h,M
h i ˜∗ A 1,h,M ∗3 ˇ∗ E ∗ Sβ,h,M U = 3 + OP (h) , β,h,M ˜ ∗3/2 Bh,M
Proof of Lemma B.9.
We apply Lemma
∗ ∗ ∗ ∗ , U4,β,h,M , U3,β,h,M U1,β,h,M , U2,β,h,M
and
as h → 0.
∗ , B.8, results follow directly given the de nitions of Sβ,h,M
ˇ∗ U β,h,M .
Lemma B.10. Suppose (1), (2) and (10) hold. We have that, ∗ Uβ,h,M
∗ ˇ∗ =U β,h,M + U4,β,h,M + OP ∗
√ h ,
in probability, where ˇ∗ U β,h,M
=
√ 1/M h i M h X h ˆ ∗2 ˆ∗ Γ ˆ∗ − E∗ Γ ˆ ∗2 + Γ ˆ∗ Γ ˆ∗ Γ + Γ kl(j) kk(j) ll(j) kl(j) kk(j) ll(j) ˜∗ B h,M j=1 i Xh h √ 1/M βˆlk ∗ ˆ∗ ˆ∗ Γ ˆ∗ ˆ∗ Γ − E Γ Γ −4 ∗ M h kk(j) kl(j) kk(j) kl(j) ˜ B h,M j=1 +2
≡
i 2 Xh h √ 1/M βˆlk ˆ ∗2 ˆ ∗2 − E ∗ Γ M h Γ kk(j) kk(j) ˜∗ B
h,M ∗ U1,β,h,M +
j=1
∗ ∗ U2,β,h,M + U3,β,h,M , 59
and A˜∗0,h,M ∗ ∗ U4,β,h,M = −4 q Sβ,h,M . P 1/h ˜∗ B y2
Proof of Lemma B.10.
Using the de nition of
h,M
i=1 k,i
ˆ∗ , B h,M
by adding and substracting appropriately, we
can write
ˆ∗ B h,M
Since
X = Mh j=1 1/M h
ˆ kk(j) Γ ˆ ll(j) + 3Γ ˆ2 Γ kl(j) ˆ2 Γ kk(j) 2 ˆ ∗ ∗ ∗ ∗2 ∗ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ −4 βlk − βlk Γkk(j) Γkl(j) − M Γkk(j) Γkl(j) + 4βlk βlk − βlk Γkk(j) − 2 M 2 ˆ ˆ∗ Γ ˆ∗ ˆ kl(j) −4βˆlk Γ kk(j) kl(j) − M Γkk(j) Γ ˆ2 ˆ2 2 Γ Γ kk(j) kk(j) ∗ ∗2 2 ∗2 ˆ ˆ + 2 βˆlk − βˆlk Γ +2βˆlk Γ kk(j) − 2 M kk(j) − 2 M ˆ ∗2 + Γ ˆ∗ Γ ˆ∗ Γ kl(j) kk(j) ll(j) −
∗ −β ˆlk = OP ∗ h1/2 βˆlk
∗ βˆlk
and
Mh
− βˆlk
2
P1/M h j=1
ˆ2 Γ kk(j) ∗2 ˆ Γkk(j) − 2 M = OP ∗ (1) ,
1/M h
Mh
1 M
X
ˆ ∗2 Γ kk(j)
−2
ˆ2 Γ kk(j)
j=1
M
in probability, then
! = OP ∗ (h) ,
in probability. Next, note that
1/M h
Mh
X j=1
2 ˆ ˆ ˆ∗ Γ ˆ∗ Γ Γ Γ kk(j) kl(j) − M kk(j) kl(j)
in probability, which together with
1/M h
∗ −β ˆlk = OP ∗ βˆlk
= Mh
X
ˆ kk(j) Γ ˆ kl(j) + OP ∗ h1/2 , Γ
j=1
h1/2 ,
and
βˆlk = OP ∗ (1)
implies that
1/M h 1/M h X X 2 ˆ ∗ ∗ ∗ ∗ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ kk(j) Γ ˆ kl(j) + OP ∗ (h) Γkk(j) Γkl(j) − βlk − βlk · M h Γkk(j) Γkl(j) Γ = βlk − βlk · M h M j=1 j=1 ! 1/M h 1/M h ˆ2 X X Γ ∗ ˆlk βˆ∗ − βˆlk · M h ˆ ∗2 − 2 kk(j) ˆ2 Γ βˆlk βˆlk − βˆlk · M h Γ = β lk kk(j) kk(j) + OP ∗ (h) . M j=1
j=1
Note also that
ˆ ∗2 + Γ ˆ kk(j) Γ ˆ ll(j) + 3Γ ˆ2 ˆ∗ Γ ˆ∗ ˆ2 ˆ kk(j) Γ ˆ ll(j) + 1 Γ E∗ Γ = Γ + Γ kl(j) , kl(j) kk(j) ll(j) kl(j) M ˆ ˆ ˆ∗ Γ ˆ∗ ˆ kk(j) Γ ˆ kl(j) + 2 Γ E∗ Γ Γ , = Γ kk(j) kl(j) M kk(j) kl(j) ˆ2 Γ kk(j) ∗ ˆ ∗2 2 ˆ E Γkk(j) , = Γkk(j) + 2 M and recall that by de nition
˜∗ = M h B h,M
1/M h
X
ˆ2 ˆ kk(j) Γ ˆ ll(j) − 4βˆlk Γ ˆ kk(j) Γ ˆ kl(j) + 2βˆ2 Γ ˆ2 Γ + Γ lk kk(j) . kl(j)
j=1
60
.
Hence, we obtain
ˆ∗ B h,M
˜∗ + M h = B h,M
1/M h h
X
i ∗ ˆ ∗2 ˆ∗ ˆ∗ Γ ˆ∗ ˆ∗ Γ ˆ ∗2 + Γ Γkl(j) + Γ Γ kk(j) ll(j) kk(j) ll(j) − E kl(j)
j=1
−4βˆlk M h
1/M h h
X
1/M h h i i X ∗ ˆ∗ ∗ 2 ∗ ∗ ˆ ˆ ∗2 ˆ ˆ ∗2 − E ∗ Γ ˆ ˆ Γ Γkk(j) Γkl(j) − E Γkk(j) Γkl(j) + 2βlk M h kk(j) kk(j) j=1
j=1
∗ −4A˜∗0,h,M βˆlk − βˆlk + OP ∗ (h) . ∗ −β ˆlk = βˆlk
Next, we can use
∗ A˜∗0,h,M βˆlk − βˆlk
P1/h
∗ ∗ i=1 yk,i i P1/h ∗2 i=1 yk,i
√ ∗ and the de nition of Sβ,h,M
=
h−1
P1/h
√
i=1
∗ ∗ yk,i i
∗ Bh,M
to write
q P −1 1/h ∗2 ˜∗ B √ ∗ y h,M i=1 k,i ∗ = hA˜0,h,M Sβ,h,M P1/h 2 P1/h 2 i=1 yk,i i=1 yk,i q P1/h ∗2 P1/h 2 ˜∗ B √ ∗ y − y h,M i=1 k,i k,i ∗ 1 + i=1 P hA˜0,h,M P1/h = Sβ,h,M + OP ∗ (h) 1/h 2 2 y y i=1 k,i i=1 k,i q q P P1/h 2 1/h ∗2 ˜∗ ˜∗ ˜∗ A B √ √ A˜∗0,h,M B − y 0,h,M h,M ∗ h,M ∗ i=1 yk,i i=1 k,i Sβ,h,M + h P1/h Sβ,h,M h P1/h + OP ∗ (h) , = P 1/h 2 2 2 i=1 yk,i i=1 yk,i i=1 yk,i | {z } =OP ∗ (h)
where we have used the fact that
P.
∗ = OP ∗ (1) and Sβ,h,M
P1/h
∗2 i=1 yk,i −
P1/h
2 i=1 yk,i =OP ∗
√ h , in probability-
It follows that
∗ Uβ,h,M
√ 1/M h i M h X h ˆ ∗2 ˆ ∗2 + Γ ˆ∗ Γ ˆ∗ ˆ∗ Γ ˆ∗ − E∗ Γ Γ + Γ kl(j) kk(j) ll(j) kl(j) kk(j) ll(j) ˜∗ B h,M j=1
=
i Xh h √ 1/M βˆlk ∗ ˆ∗ ˆ∗ ˆ∗ Γ ˆ∗ −4 ∗ M h Γ Γ Γ − E kk(j) kl(j) kk(j) kl(j) ˜ B h,M j=1 i 2 Xh h √ 1/M A˜∗0,h,M βˆlk ∗ ˆ ∗2 − E ∗ Γ ˆ ∗2 q +2 ∗ M h Γ − 4 kk(j) kk(j) P1/h 2 Sβ,h,M + OP ∗ (h) ˜ ∗ B ˜ Bh,M i=1 yk,i h,M j=1 √ ∗ ∗ ∗ ∗ ≡ U1,β,h,M + U2,β,h,M + U3,β,h,M + U4,β,h,M + OP ∗ h . | {z } ˇ∗ ≡U β,h,M
Proof of proposition 5.2.
Part (a) follows under our assume conditions by using Theorem 5.2 of
Dovonon et al. (2013). The proof of part (b) use the same technique as in the proof of Theorem A.3 in Gonçalves and Meddahi (2009). Given (36) and Lemma B.10, we may decompose
√ −1/2 √ ∗ ∗ ∗ ∗ ˇ ∗ Tβ,h,M = Sβ,h,M 1+ h U + U + O h . P β,h,M 4,β,h,M k, we have that √ k ∗k ∗ ∗ ∗ ˇ = Sβ,h,M 1 − h Uβ,h,M + U4,β,h,M + OP ∗ (h) ≡ T˜β,h,M + OP ∗ (h) . 2
Hence, for any xed integer
∗k Tβ,h,M
61
For
k = 1, 2, 3,
The
the moments of
∗ T˜β,h,M
are given by
√ k √ k ∗k ∗k ∗k ∗ ∗k ∗ ˇ∗ E ∗ T˜β,h,M = E ∗ Sβ,h,M − h E ∗ Sβ,h,M U Sβ,h,M U4,β,h,M . β,h,M − h E 2 2 ˜∗ rst and third cumulants of T are given by β,h,M
∗ ∗ κ∗1 T˜β,h,M = E ∗ T˜β,h,M h i3 ∗ ∗3 ∗2 ∗ ∗ . κ∗3 T˜β,h,M = E ∗ T˜β,h,M − 3E ∗ T˜β,h,M E ∗ T˜β,h,M + 2 E ∗ T˜β,h,M By Lemma B.9, we deduce
√
∗ E ∗ T˜β,h,M
1/M h
1/M h
X
X
h ˆ kk(j) Γ ˆ kl(j) Γ ˆ ll(j) ˆ3 Γ Mh Γ kl(j) + 3M h ∗3/2 ˜ Bh,M j=1 j=1 √ 1/M h 1/M h X X h ˆ ˆ ˆ kk(j) Γ ˆ2 ˆ2 Γ ˆ − ∗3/2 −3βlk M h Γ Γ kl(j) − βlk M h kk(j) ll(j) ˜ Bh,M j=1 j=1 √ 1/M h 1/M h 1/M h X X X h ˆ ˆ2 ˆ ˆ2 Γ ˆ ˆ2 Γ ˆ ˆ kk(j) Γ ˆ2 Γ Γ Γ − ∗3/2 −6βlk M h kk(j) kl(j) kk(j) ll(j) + 8βlk M h kl(j) − 2βlk M h ˜ Bh,M j=1 j=1 j=1 √ 1/M h 1/M h X X √ 2A˜∗0,h,M h 2 ˆ2 Γ ˆ kl(j) − 4βˆ3 M h ˆ3 + h q − ∗3/2 4βˆlk Mh Γ Γ lk kk(j) kl(j) ˜ ∗ P1/h y 2 Bh,M B j=1 j=1 i=1 k,i h,M √ 1/M h 1/M h 1/M h X X X h ˆ kk(j) Γ ˆ2 ˆ kk(j) Γ ˆ kl(j) Γ ˆ ll(j) − 19βˆlk M h ˆ3 − ∗3/2 Γ Γ Γ 3M h kl(j) kl(j) + 5M h ˜ Bh,M M j=1 j=1 j=1 √ 1/M h 1/M h 1/M h X X X h ˆ ˆ3 ˆ2 ˆ3 . ˆ2 Γ ˆ ˆ2 Γ ˆ − ∗3/2 Γ Γ Γ −5βlk M h kl(j) kk(j) kl(j) − 8βlk M h kk(j) ll(j) + 24βlk M h ˜ Bh,M M j=1 j=1 j=1
= −
Result follows since, we can write
∗ = E ∗ T˜β,h,M
where
˜∗ A˜∗0,h,M , A˜∗1,h,M , B h,M
and
√
˜∗ 2A˜∗0,h,M A˜∗1,h,M R 1,h,M h q + − , P ∗3/2 1/h M ˜ 2 ˜∗ 2 B B y h,M i=1 k,i h,M
˜∗ R 1,h,M
are de ned in the main text.
The remaining results follow
similarly. For part (c), apply Theorem 3 of Li, Todorov and Tauchen (2016).
62
Proof of Bootstrap results in Section 3.2 and 4.2: Mykland and Zhang's (2009) type estimator Notation We introduce some notation. We let
V1,(j) =
V2,(j) = We denote by
M 2h M −1
M X
M 2h M −1
M X
!−1
M X
2 yk,i+(j−1)M
i=1
! u2i+(j−1)M
,
i=1
!−2
M X
2 yk,i+(j−1)M
i=1
!2 yk,i+(j−1)M ui+(j−1)M
.
i=1
yk(j) = yk,1+(j−1)M , · · · , yk,M j
0
, the
M
returns of asset
k
observed within the block
j.
Similarly for the bootstrap, we let
∗ V1,(j)
≡
∗ V2,(j)
and
M X
M 2h M −1
M X
1/M Ph j=1
M M −1
q
2
Cˆkk(j) 0 ˆ ˆ Clk(j) Cll(j)
1 bM,q cM −1,q
bll(j) C b Ckk(j)
M X
! u∗2 i+(j−1)M
,
i=1
!−2
M X
∗2 yk,i+(j−1)M
i=1
∗ ∗ ∗ yk(j) = yk,1+(j−1)M , · · · , yk,M j
Cˆ(j) ≡
!−1 ∗2 yk,i+(j−1)M
i=1
Recall that
Mh
≡
M 2h M −1
!2 ∗ yk,i+(j−1)M u∗i+(j−1)M
,
i=1
0
.
q
!
=
ˆ kk(j) Γ
ˆ Γ q kl(j) ˆ kk(j) Γ
0 r
ˆ ll(j) − Γ
ˆ2 . For any Γ kl(j) ˆ kk(j) Γ
q > M,
let
Rβ,q ≡
q , where the de nition of
q
cM,q
is given in equation (7), and for
2q Γ( M2 − 2q ) 2 2 , where χM is the standard χ distribution Γ( M ) 2 M M2 M3 with M degrees of freedom. Note that bM,2 = , bM,4 = M −2 (M −2)(M −4) , and bM,6 = (M −2)(M −4)(M −6) . It follows by using the de nition of bM,q and this property of the Gamma function, for all x > 0, Γ (x + 1) = xΓ (x). any
q > M,
we have
bM,q ≡ E
M χ2M
2
M 2
=
Auxiliary Lemmas Lemma B.11. Suppose (1) and (2) hold. Then, we have that 1/M h
Vˆβ,h,M = ˇ
X
1/M h
V1,(j) −
j=1
X
V2,(j) .
j=1
Lemma B.12. Suppose (1) and (2) hold with W independent of σ. Assume that conditionally on σ and under Qh,M , the following hold a1)
E V1,(j) =
a2)
2 E V1,(j) =
M 3h (M −1)(M −2)
Cll(j) Ckk(j)
2
M 5 (M +2) h2 (M −1)2 (M −2)(M −4)
, for M > 2;
Cll(j) Ckk(j)
4
, for M > 4;
63
M = O (1) ,
then
2
a3)
E V2,(j) =
a4)
2 E V2,(j) =
a5)
E V1,(j) V2,(j) =
a6)
V ar V1,(j) =
4M 5 h2 (M −1)(M −2)2 (M −4)
Cll(j) Ckk(j)
4
, for M > 4;
a7)
V ar V2,(j) =
2M 4 h2 (M −1)(M −2)2 (M −4)
Cll(j) Ckk(j)
4
, for M > 4;
a8)
Cov V1,(j) , V2,(j) =
a9)
V ar V1,(j) − V2,(j) =
M 2h (M −1)(M −2)
Cll(j) Ckk(j)
3M 4 h2 (M −1) (M −2)(M −4)
, for M > 2;
2
Cll(j) Ckk(j)
M 4 (M +2) h2 (M −1)2 (M −2)(M −4)
4
, for M > 4;
Cll(j) Ckk(j)
4M 4 h2 (M −1)(M −2)2 (M −4)
4
, for M > 4;
Cll(j) Ckk(j)
2M 5 (2M −3) h2 (M −1)(M −2)2 (M −4)
4
Cll(j) Ckk(j)
, for M > 4;
4
, for M > 4.
Lemma B.13. Suppose (1) and (2) hold with W independent of σ. Assume that conditionally on σ and under Qh,M , for any M > 4, the following hold a1)
= Vβ,h,M ; E Vˆβ,h,M ˇ ˇ
a2)
= V ar Vˆβ,h,M ˇ
a3)
Vˆβ,h,M − Vβ,h,M →0 ˇ ˇ
a4)
Vβ,h,M → Vβˇ. ˇ
2M 4 (2M −3) h (M −1)(M −2)2 (M −4)
· Mh
1/M Ph j=1
Cll(j) Ckk(j)
4
ˆ kk(j) )−1 (Γ
ˆ2i+(j−1)M = i=1 u
q
E
a3)
Rβ,q − M h
M −1 M
1/M Ph j=1
δ > 0;
a4)
=
2q
Cll(j) Ckk(j)
Vˆβ,h,M − Vβ,h,M →0 ˇ ˇ
1/M h
Vˆβ,h,M ˇ
Cll(j) Ckk(j)
bM,q cM −1,q q
→0
2
;
Cll(j) Ckk(j)
q
, for M > q;
in probability under Qh,M and P , for any M > q (1+δ) , for some
Given the de nition of
Vˆβ,h,M
u ˆi+(j−1)M = yl,i+(j−1)M − βˇlk(j) yk,i+(j−1)M , 2
= M h
X
M X
j=1
i=1
1/M h
=
then
in probability under Qh,M and P , for any M > 2 (1+δ) , for some δ > 0.
Proof of Lemma B.11. de nition of
M = O (1) ,
in probability;
PM
a2)
bll(j) C b Ckk(j)
then
;
Lemma B.14. Suppose (1) and (2) hold with W independent of σ. Assume that conditionally on σ and under Qh,M , the following hold a1)
M = O (1) ,
M 2h X M −1 j=1
!−1 2 yk,i+(j−1)M
M X i=1
in the text (see Equation (29)), and the
we can write
M
2 1 X yl,i+(j−1)M − βˇlk(j) yk,i+(j−1)M M −1
!
i=1
!−1 2 yk,i+(j−1)M
M X i=1 64
ui+(j−1)M − βˇlk(j) − βlk(j) yk,i+(j−1)M
2
! ,
where we used the de nition of
yl,i+(j−1)M
see equation (29). Adding and subtracting appropriately,
it follows that
Vˆβ,h,M ˇ
=
1/M h M M 2h X X 2 yk,i+(j−1)M M −1 j=1
!−1
i=1
M X
! + βˇlk(j) − βlk(j)
u2i+(j−1)M
2
!
i=1
!−1 M ! 1/M h M X X M 2h X ˇ 2 βlk(j) − βlk(j) yk,i+(j−1)M yk,i+(j−1)M ui+(j−1)M −2 M −1 j=1 i=1 i=1 !−1 M ! 1/M h M X M 2h X X 2 yk,i+(j−1)M u2i+(j−1)M = M −1 j=1 i=1 i=1 ! !2 −2 1/M h M M X M 2h X X 2 − yk,i+(j−1)M yk,i+(j−1)M ui+(j−1)M M −1 j=1
1/M h
=
X j=1
i=1
i=1
1/M h
V1,(j) −
X
V2,(j) ,
j=1
−1 P PM 2 M βˇlk(j) − βlk(j) = y y u i=1 k,i+(j−1)M i=1 k,i+(j−1)M i+(j−1)M . Proof of Lemma B.12 part a1). Given the de nition of V1,(j) , the law of iterated expectations the fact that ui+(j−1)M |yk(j) ∼ i.i.d.N 0, V(j) , we can write E V1,(j) = E E V1,(j) |yk(j) !−1 M ! M 2 X X M h 2 E E yk,i+(j−1)M u2i+(j−1)M |yk(j) = M −1 i=1 i=1 ! ! −1 M M 2 X X M h 2 E yk,i+(j−1)M E u2i+(j−1)M |yk(j) = M −1 i=1 i=1 ! −1 M 3 X M h 2 , = V E yk,i+(j−1)M M − 1 (j) where we used
and
i=1
then given equation (28) in the text and by replacing
E V1,(j) =
Proof of Lemma B.12 part a2).
V(j)
by
M 3h (M − 1) (M − 2)
Given the de nition of
2 , hCll(j)
Cll(j) Ckk(j)
V1,(j)
we deduce
2 .
and the law of iterated expectations,
we can write
2 2 E V1,(j) = E E V1,(j) |yk(j) !−2 M !2 M X X M 4 h2 2 E E yk,i+(j−1)M u2i+(j−1)M |yk(j) = (M − 1)2 i=1 i=1 2 !−2 !2 M M X X u M 4h i+(j−1)M 2 p = yk,i+(j−1)M E |yk(j) . V2 E (M − 1)2 (j) V(j) i=1
i=1
65
Note that since
M (M + 2)
ui+(j−1)M |yk(j) ∼ i.i.d.N 0, V(j)
where
χ2j,M
follow the standard
2 E V1,(j) =
then given the fact that
χ2
,
PM
E
√
i=1
+ 2) h 2 V(j) E (M − 1)2
M X
M
!2
2 |yk(j)
V(j)
2 = E χ2j,M =
degrees of freedom. Then we have
!−2 2 yk,i+(j−1)M
,
i=1
d 2 i=1 yk,i+(j−1)M =
by using the second moment of an inverse
ui+(j−1)M
distribution with
M 5 (M
PM
d
2 hCkk(j) χ2j,M , where `=' denotes equivalence in distribution, 2 1 1 2 = (M −2)(M of χ distribution, we have E −4) , and χ2 j,M
by replacing
V(j)
by
2 hCll(j)
E
it follows that
2 V1,(j)
M 5 (M + 2) = h2 (M − 1)2 (M − 2) (M − 4)
Proof of Lemma B.12 part a3).
Given the de nition of
V1,(j) ,
Cll(j) Ckk(j)
4 .
the law of iterated expectations and
ui+(j−1)M |yk(j) ∼ i.i.d.N 0, V(j) , we can write E V2,(j) = E E V2,(j) |yk(j) !−2 M !2 M 2 X X M h 2 E E = yk,i+(j−1)M yk,i+(j−1)M ui+(j−1)M |yk(j) M −1 i=1 i=1 ! ! −2 M M 2 X X M h 2 2 E = yk,i+(j−1)M yk,i+(j−1)M E u2i+(j−1)M |yk(j) M −1 i=1 i=1 ! −1 M 2 X M h 2 , V E yk,i+(j−1)M = M − 1 (j)
the fact that
i=1
then using equation (28) in the text and replacing
E V2,(j) =
Proof of Lemma B.12 part a4).
V(j)
by
2 hCll(j)
M 2h (M − 1) (M − 2)
Given the de nition of
yields
Cll(j) Ckk(j)
V2,(j)
2 .
and the law of iterated expectations,
we can write
2 2 = E E V2,(j) |yk(j) E V2,(j) !−4 !4 M M 4 2 X X M h 2 E yk,i+(j−1)M E yk,i+(j−1)M ui+(j−1)M |yk(j) = (M − 1)2 i=1 i=1 ! −4 M X M 4 h2 2 ≡ E yk,i+(j−1)M Ξ . (M − 1)2 i=1
66
Then using the conditional independence and mean zero property of
Ξ ≡ E
M X
!4
yk,i+(j−1)M ui+(j−1)M
we have that
|yk(j)
yk,i+(j−1)M ui+(j−1)M
i=1
=
M X
4 u4i+(j−1)M |yk(j) E yk,i+(j−1)M
i=1
+3
X
2 2 E yk,i+(j−1)M u2i+(j−1)M |yk(j) E yk,s+(j−1)M u2s+(j−1)M |yk(j)
i6=s 2 = 3V(j)
M X y4
k,i+(j−1)M
i=1
+
X
2 2 = 3V 2 yk,s+(j−1)M yk,i+(j−1)M (j)
M X
!2 4 yk,i+(j−1)M
,
i=1
i6=s
thus we can write
2 E V2,(j)
!−2 M X M 4 h2 2 , = 3V 2 E yk,i+(j−1)M (M − 1)2 (j) i=1
result follows similarly where we use the same arguments as in the proof of Lemma B.12 part a2).
Proof of Lemma B.12 part a5).
The proof follows similarly as parts a2) and a4) of Lemma B.12
and therefore we omit the details.
Proof of Lemma B.13 part a1).
Given the de nitions of
Vˆβ,h,M , V1,(j) , V2,(j)
and by using Lemma
B.11 and part 1 of Lemma B.12, we can write
1/M h
E Vˆβ,h,M ˇ
= E
X
V1,(j) − E
j=1 1/M h
=
X
1/M h
X
V2,(j)
j=1
1/M h X E V1,(j) − E V1,(j)
j=1
j=1 1/M h
2
1/M h X Cll(j) 2 M 2h − (M − 1) (M − 2) Ckk(j)
=
X M 3h (M − 1) (M − 2)
=
1 M Vβ,h,M − Vˇ = Vβ,h,M . ˇ ˇ M −1 M − 1 β,h,M
j=1
Proof of Lemma B.13 part a2).
Cll(j) Ckk(j)
j=1
Given the de nitions of
Vˆβ,h,M , V1,(j) , V2,(j) ˇ
and Lemma B.11, we
can write
V ar Vˆβ,h,M = V ar ˇ
1/M h
X j=1
V1,(j) + V ar
1/M h
X j=1
V2,(j) − 2Cov
1/M h
1/M h
X
X
j=1
V1,(j) ,
V2,(j) ,
j=1
ui+(j−1)M |yk(j) ∼ i.i.d.N 0, V(j) , we have V1,(j) and V2,(j) are conditionally indeyk(j) , V1,(j) and V2,(t) are conditionally independent for all t 6= j given yk(j) . It follows
given the fact that pendent given
67
that
V ar Vˆβ,h,M ˇ
1/M h
X
=
2 2 2 2 E V1,(j) − E V1,(j) + E V2,(j) − E V2,(j)
j=1 1/M h
−2
X
E V2,(j) V2,(j) − E V1,(j) E V2,(j) ,
j=1 nally results follow given Lemma B.12.
Proof of Lemma B.13 part a3). Results follow directly given Lemma B.12 parts a1) and a2) since → 0 as h → 0. − Vβ,h,M V ar Vˆβ,h,M ˇ ˇ Proof of Lemma B.13 part a4). This result follows from the boundedness of Σkk,s , Σll,s Reimann integrable of Σkl,s for any k, l = 1, · · · , d. Proof of Lemma B.14 part a1). Given the de nition of uˆi+(j−1)M , we can write =0 − Vβ,h,M E Vˆβ,h,M ˇ ˇ
ˆ kk(j) )−1 (Γ
M X
u ˆ2i+(j−1)M
and
M X
1
=
ˆ kk(j) Γ
i=1
ˆ kk(j) Γ 1
=
2 yl,i+(j−1)M − 2βˇlk(j)
i=1
βˇlk(j) =
Proof of Lemma B.14 part a2).
2 2 2 yl,i+(j−1)M − 2βˇlk(j) yl,i+(j−1)M yk,i+(j−1)M + βˇlk(j) yk,i+(j−1)M
i=1 M X
ˆ kk(j) Γ
thus results follow by replacing
2
i=1 M X
1
=
yl,i+(j−1)M − βˇlk(j) yk,i+(j−1)M
and the
M X
2 yl,i+(j−1)M yk,i+(j−1)M + βˇlk(j)
i=1
M X
! 2 yk,i+(j−1)M
i=1
ˆ lk(j) Γ ˆ kk(j) . Γ
Given the de nitions of
ˆ ll(j) , Γ ˆ kl(j) Γ
and
ˆ lk(j) Γ
and using part a1)
of Lemma B.14, we can write
E
Cll(j) Ckk(j)
PM
ˆ2i+(j−1)M i=1 u
q = E
! 2q
ˆ kk(j) Γ
= E
M X
!− 2q 2 yk,i+(j−1)M
M X
E
i=1
= E
M X
! 2q u ˆ2i+(j−1)M
2 yk,i+(j−1)M
V(j) E
E
cM,q ,
V(j)
! 2q
|yk(j) = E
χ2j,M q
q 2
= (M − 1) 2 cM −1,q ,
68
! 2q
|yk(j) ,
ui+(j−1)M |yk(j) ∼ i.i.d.N 0, V(j)
we can write
M u X ˆ2i+(j−1)M i=1
V(j)
i=1
where we use the law of iterated expectations and the fact that
|yk(j)
M u X ˆ2i+(j−1)M
q 2
i=1
Then given the de nition of
i=1
!− 2q
.
,
it follows that
E
bll(j) C bkk(j) C
!q
PM
ˆ2i+(j−1)M i=1 u
= E
! 2q
ˆ k(j) Γ
M X
q 2
q
= (M − 1) 2 cM −1,q V(j) E
!− 2q 2 yk,i+(j−1)M
i=1
!q 2 M = (M − 1) cM −1,q V(j) Γk(j) E χ2j,M
q 2
q 2
= where
bM,q = E
M χ2j,M
M −1 M
− 2q
q
2
bM,q cM −1,q
q ;
q ! 2
, for
M > q.
Proof of Lemma B.14 part a3).
We verify the moments conditions of the Weak Law of Large q
Numbers for independent and nonidentically distributed on
1, . . . , M1h .
Cll(j) Ckk(j)
By using part a2) of Lemma B.14, for any
1+δ
E |zj |
=
M −1 M
δq 2
δ > 0,
zj ≡
M2 q (M −1) 2 bM,q cM −1,q
and conditionally on
bM,(1+δ)q cM −1,(1+δ)q bM,q cM −1,q
Cll(j) Ckk(j)
σ,
bll(j) C b Ckk(j)
q
,
j =
we can write
(1+δ)q <∞
Σ is an adapted càdlàg spot covolatility matrix and locally bounded and invertible (in particular, 2 Ckk(j) > 0). and Proof of Lemma B.14 part a4). Result follows directly given the de nition of Vˆˇ , V ˇ since
part a3) of Lemma B.14, where we let
β,h,M
q = 2.
β,h,M
∗ , V ∗ Remark 4. The bootstrap analogue of Lemma B.11 and B.12 replace V1(j) with V1(j) 2(j) with V2(j) . ll(j) ∗ ∗ The bootstrap analogue of Lemma B.13 replaces Vˆβ,h,M with Vˆβ,h,M , Vβ,h,M with Vβ,h,M and CCkk(j) with ˇ ˇ ˇ ˇ bll(j) C b Ckk(j)
.
Lemma B.15. Suppose (1) and (2) hold with W independent of σ. Assume that conditionally on σ and under Qh,M , for some small δ > 0, the following hold −2(2+δ)
a1)
E∗
P
a2)
E∗
2(2+δ) PM ∗ ∗ ˆ 2+δ Vˆ 2+δ ; ≤ µ22(2+δ) M 2+δ Γ i=1 yk,i+(j−1)M ui+(j−1)M k(j) (j)
M 2∗ i=1 yk,i+(j−1)M
Proof of Lemma B.15 part a1).
−2(2+δ)
ˆ = bM,4(2+δ) Γ k(j)
M = O (1) ,
then
, for M > 4 (2 + δ);
Given the de nition of
∗ yk,i+(j−1)M , we can write
d 2∗ i=1 yk,i+(j−1)M =
PM
PM 2 2 2 b2 b2 hC i=1 vi+(j−1)M = hCkk(j) χj,M , where vi+(j−1)M ∼ i.i.d.N (0, 1), and χj,M follow the standard kk(j) χ2 distribution with M degrees of freedom. Then for any integer M > 4 (2 + δ), we have that, !−2(2+δ) !2(2+δ) M X M 2 ˆ −2(2+δ) = bM,4(2+δ) Γ ˆ −2(2+δ) . E yk,i+(j−1)M =E Γ 2 kk(j) kk(j) χ j,M i=1 69
Proof of Lemma B.15 part a2). and the fact that
Cr
Indeed by using the
∗ u∗i+(j−1)M |yk(j) ∼ i.i.d.N 0, Vˆ(j)
inequality, the law of iterated expectations
, we can write for any
δ > 0,
2(2+δ) M M 2(2+δ) X X ∗ ∗ ≤ M 3+2δ u∗i+(j−1)M E ∗ u∗i+(j−1)M yk,i+(j−1)M E ∗ yk,i+(j−1)M i=1
i=1
= M = where the last equality follows since and
2(2+δ)
µ2(2+δ) = E |v|
3+2δ
M X
∗2(2+δ) ∗2(2+δ) ∗ E ∗ yk,i+(j−1)M E ∗ ui+(j−1)M |yk(j)
i=1 2 ˆ 2+δ Vˆ 2+δ , µ2(2+δ) M 2+δ Γ kk(j) (j) d
2∗ b2 v2 yk,i+(j−1)M = hC kk(j) i+(j−1)M ,
where
vi+(j−1)M ∼ i.i.d.N (0, 1)
.
Proof of Theorem 4.2
For part a), the proof follows the same steps as the proof of
we explain in the main text, in particular, given the de nition of
∗ Vβ,h,M ˇ
= V ar∗
√
∗, βˆlk
Vβ,h,M ˇ
which
we have that
∗ − βˇlk ) h−1 (βˇlk
1/M h
X
2
= M h
∗ V ar∗ βˇlk(j) − βˇlk(j)
j=1
1/M h
X
= M 2h
M X
E∗
j=1
=
M 2h M −2
!−1 2∗ yk,i+(j−1)M
Vˆ(j)
i=1
1/M h X j=1
Cll(j) Ckk(j)
2 =
M −1ˆ , Vˇ M − 2 β,h,M
.
then results follows, given Lemma B.13 or part a4) of Lemma B.14 For part b), we have
zj,∗ βˇ =
r
q
M −2 M −1
√
∗ −β ˇlk = h−1 βˇlk
1/M Ph j=1
M X M −2 √ ∗2 M h yk,i+(j−1)M M −1
zj,∗ βˇ,
!−1
i=1
Note that
E ∗ zj,∗ βˇ = 0,
where
M X
! ∗ yk,i+(j−1)M u∗i+(j−1)M
.
i=1
and that
1/M h
X
zj∗ =
V ar∗
j=1 by part a) moreover, since
∗ , . . . , z∗ z1, βˇ 1/M h,βˇ
M −2 ∗ P Vˇ = Vˆβ,h,M → Vβˇ, ˇ M − 1 β,h,M
are conditionally independent, by the Berry-Esseen bound,
δ > 0 and for some constant C > 0, ! r M − 2 √ −1 ˇ∗ ∗ sup P h (βlk − βˇlk ) ≤ x − Φ M −1 x∈<
for some small
70
x Vβˇ
! 1/M h 2+δ X E ∗ zj,∗ βˇ , ≤C j=1
Next, we show that
1/M Ph j=1
1/M h
X j=1
2+δ E ∗ zj,∗ βˇ = op (1).
We have that
2+δ !−(2+δ) M 2+δ 1/M h M 2+δ X X X √ 2 2+δ M −2 ∗ ∗2 ∗ = u M h E∗ yk,i+(j−1)M y E ∗ zj,∗ βˇ k,i+(j−1)M i+(j−1)M M −1 j=1
≡
i=1
i=1
2+δ 1/M h M − 2 2 √ 2+δ X ∗ ∗ ∗ M h E Aj B j , M −1 j=1
it follows then by using Cauchy-Schwarz inequality that
E∗
v u u A∗j Bj∗ ≤ tE ∗
M X i=1
v 2(2+δ) M !−2(2+δ) u u X u ∗ ∗2 ∗ tE yk,i+(j−1)M u∗i+(j−1)M yk,i+(j−1)M i=1
− 2+δ 2
1
δ 2 ˆ ≤ µ2(2+δ) bM,4(2+δ) M 1+ 2 Γ k(j) ˆ 1 Γ 2 l(j) − = µ2(2+δ) bM,4(2+δ) ˆ k(j) Γ
2+δ 2
Vˆ(j)
ˆ lk(j) Γ ˆ k(j) Γ
!2 2+δ 2 ,
where the second inequatily used part a1) and a2) of Lemma B.15 and
v ∼ N (0, 1).
Finally, given the de nition of
1/M h
X
2+δ
E ∗ zj,∗ βˇ
≤
j=1
=
M −2 M −1
2+δ
M −2 M −1
2+δ
= OP
δ 2
Rβ,2+δ ,
1/M h
µ2(2+δ) bM,4(2+δ) M
2+δ 1+ 2δ
h
X j=1
1
2
2 µ2(2+δ) bM,4(2+δ)
1 2
h bM,4(2+δ)
M −2 M
with
we can write
1 2
2
µ2(2+δ) = E |v|2(2+δ)
M −1 M
2+δ 2
2+δ 2
bll(j) C bkk(j) C
!2+δ
δ
bM,2+δ cM −1,2+δ M 1+δ h 2 Rβ,2+δ !
bM,2+δ cM −1,2+δ
= oP (1) . δ > 0, µ2(2+δ) = E |v|2(2+δ) ≤ ∆ < ∞ where v ∼ N (0, 1), moreover as h → 0, cM −1,2+δ = O (1), bM,4(2+δ) = O (1), bM,2+δ = O (1) and by using Lemma B.14 we have Rβ,2+δ = OP (1).
Since for any
Proof of Theorem 4.3 Let
q ∗ Hβ,h,M ˇ
=
M −2 M −1
√
√ ∗ −β ˇlk ) ∗ −β h−1 (βˇlk ˇlk ) h−1 (βˇlk q q = , ∗ Vβ,h,M Vˆ ˇ ˇ β,h,M
and note that
∗ Tβ,h,M ˇ
v uV ∗ u β,h,M ˇ ∗ t = Hβ,h,M , ˇ Vˆ ∗ ˇ β,h,M
where
∗ Vˆβ,h,M ˇ
is de ned in the main text. Theorem 4.2 proved that
show that (1)
Bias∗
in probability.
∗ Vˆ ˇ∗ − Vβ,h,M → 0 in probability under Qh,M and P . In particular, we ˇ β,h,M P ∗ ∗ V ˆ∗ Vˆβ,h,M = 0, and (2) V ar → 0. Results follows directly by using the ˇ ˇ β,h,M
Thus, it su ces to show that
P∗
d∗
∗ Hβ,h,M → N (0, 1) ˇ
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bootstrap analogue of parts a1), a2) and a3) of Lemma B.13.
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