Cross-sectional Dependence in Idiosyncratic Volatility

Cross-sectional Dependence in Idiosyncratic Volatility∗
Ilze Kalnina†
Kokouvi Tewou‡
First version: February 25, 2015
This version: January 10, 2016
Abstract
This paper introduces a framework for analysis of cross-sectional dependence in the idiosyncratic volatilities of assets using high frequency data. We first consider the estimation
of standard measures of dependence in the idiosyncratic volatilities such as covariances and
correlations. Next, we study an idiosyncratic volatility factor model, in which we decompose
the co-movements in idiosyncratic volatilities into two parts: those related to factors such as
the market volatility, and the residual co-movements. When using high frequency data, naive
estimators of all of the above measures are biased due to the estimation errors in idiosyncratic
volatility. We provide bias-corrected estimators and establish their asymptotic properties. We
apply our estimators to high-frequency data on 27 individual stocks from nine sectors, and
document strong cross-sectional dependence in their idiosyncratic volatilities. We also find that
on average, 49% of this dependence can be explained by the market volatility.
Keywords: high frequency data; idiosyncratic volatility; errors-in-variables; cross-sectional returns.
JEL Codes: C22, C14.
∗
We benefited from discussions with Marine Carrasco, Sı́lvia Gonçalves, Jean Jacod, Benoit Perron, and Dacheng
Xiu. We thank conference participants at the 49th Annual Conference of the Canadian Economics Association and
the 11th World Congress of the Econometric Society.
†
Département de sciences économiques and CIREQ, Université de Montréal. Address: C.P.6128, succ. CentreVille, Montréal, QC, H3C 3J7, Canada. E-mail address: [email protected]. Kalnina’s research was
supported by Institut de finance mathématique de Montréal, SSHRC, and FQRSC. She is grateful to the Economics
Department, the Gregory C. Chow Econometrics Research Program, and the Bendheim Center for Finance at
Princeton University for their hospitality and support in the Spring of 2015.
‡
Département de sciences économiques, Université de Montréal. Address: C.P.6128, succ. Centre-Ville, Montréal,
QC, H3C 3J7, Canada.E-mail address: [email protected].
1
1
Introduction
Idiosyncratic Volatility (IV) of returns of an asset or a portfolio is the subject of many recent
papers in empirical finance. The IV is usually defined with respect to some popular empirical
asset pricing model such as the Fama and French (1993) model, so that IV is the volatility of the
risk-adjusted returns. Even if the idiosyncratic returns are not correlated in the cross-section, their
volatilities may well be. In fact, cross-sectional correlation of IVs has emerged as a stylized fact,
see, e.g., Herskovic, Kelly, Lustig, and Nieuwerburgh (2014) and Duarte, Kamara, Siegel, and Sun
(2014). The current paper develops the tools to formally study this empirical phenomenon.
We provide a flexible framework for studying the cross-sectional dependencies in IVs using
high-frequency data. Our framework incorporates important stylized facts in asset returns and
volatilities and offers a solution to the measurement error problem in estimated IVs.
First, we study the behavior of standard measures of cross-sectional dependence in IVs using
high-frequency data. We show that the naive estimators of these measures are biased, and provide
bias-corrected estimators. We then obtain the relevant asymptotic distributions, which allow us
to perform statistical tests.
Second, we study an idiosyncratic volatility factor model (IV-FM).1 The IV-FM decomposes
the cross-sectional dependence in IVs into two components. The first component is the crosssectional dependence due to popular factors. The IV factors can include the volatility of the
return factors, non-linear transforms of the spot covariance matrices such as correlations, as well
as the average variance factor of Chen and Petkova (2012). The second component in the IV-FM is
residual dependence in IVs not explained by the IV factors. Again, the standard estimators of this
decomposition are biased due to the latency in volatility. We provide bias-corrected estimators,
and derive their asymptotic distributions. We build a test for whether the IV-FM can fully account
for the dependence between the IVs.
We apply our estimators to high-frequency data on 27 individual US stocks from nine different
sectors. We study idiosyncratic volatilities with respect to two models for asset prices, CAPM
and the three-factor Fama-French model. In both cases, the average correlation between the
idiosyncratic volatilities is above 0.55. Moreover, the average correlation between the IVs is on
average the same among those pairs of stocks, which have close to zero correlations between their
idiosyncratic returns. In other words, the dependencies in IVs cannot be explained by a missing
return factor. This is in line with the recent findings of Herskovic, Kelly, Lustig, and Nieuwerburgh
(2014) who use daily and monthly return data. We then consider the IV-FM with market volatility
as the IV factor. We find that on average, the systematic component of IV that arises due to IV
exposure to market volatility, accounts for 74% of the cross-sectional dependence in the IVs. We
find that in 110 out of 351 pairs of stocks analyzed, this idiosyncratic volatility factor model fully
1
Throughout the paper, we use the term “factor model” to denote a regression model, e.g., we call the Fama and
French (1993) model a factor model.
2
accounts for the cross-sectional dependence in IVs, so that their non-systematic components are
no longer significantly correlated.
The importance of accounting for estimation errors in volatilities has been demonstrated in
other contexts. Recently, Aı̈t-Sahalia, Fan, and Li (2013) show that failure to account for the
latency of volatility drives the leverage effect puzzle.2 An important aspect of our methods is that
we fully account for the latency of IV.
Our paper is related to several strands of the literature. Our inference theory extends the results
on estimation of the integrated one-dimensional (total) volatility of volatility (Vetter (2012), Aı̈tSahalia and Jacod (2014)). The (total) leverage effect is also a quantity, for which the naive
nonparametric estimators are inconsistent due to the measurement errors in volatilities, see Wang
and Mykland (2014), Kalnina and Xiu (2015), Aı̈t-Sahalia, Fan, Laeven, Wang, and Yang (2013)
and Aı̈t-Sahalia, Fan, and Li (2013) for one-dimensional results. Due to the decomposition of
total returns into systematic and idiosyncratic part, our estimators involve aggregation of nonlinear functionals of the return volatility matrix, hence our bias-correction terms are related to
the general theory developed in Jacod and Rosenbaum (2012) and Jacod and Rosenbaum (2013).
We define the IV with respect to a continuous-time factor model for asset returns with observable
return factors. This framework was originally studied in Mykland and Zhang (2006) in the case
of one factor and in the absence of jumps. It was extended to multiple factors and jumps in
Aı̈t-Sahalia, Kalnina, and Xiu (2014).
The remainder of the paper is organized as follows. Section 2 introduces the model and describes
quantities of interest. Section 3 describes the identification and estimation of these quantities of
interest. Section 4 presents the asymptotic properties of our estimators. Section 5 investigates
their finite sample properties. Section 6 uses high-frequency stock return data to study the crosssectional dependence in IVs using our framework. Section 7 concludes. The Appendix contains
the proofs.
2
Model and Quantities of Interest
We first describe a general factor model for the returns, in which the idiosyncratic volatility is
defined. We then introduce the idiosyncratic volatility factor model (IV-FM).
Suppose we have (log) prices on dS assets such as stocks and on dF observable factors. These
factors serve as the observable factors in the model P-FM below. We stack them into the ddimensional process Yt = (S1,t , . . . , SdS ,t , F1,t , . . . , FdF ,t )> where d = dS + dF . We assume Yt
follows an Itô semimartingale,
Z
Yt = Y0 +
t
Z
bs ds +
0
t
σs dWs + Jt ,
0
2
See Wang and Mykland (2014), Kalnina and Xiu (2015), and Aı̈t-Sahalia, Fan, Laeven, Wang, and Yang (2013)
for related results on the leverage effect.
3
where W is a d0 -dimensional Brownian motion (d0 ≥ d), σs is a d × d0 stochastic volatility process,
and Jt denotes a finite variation jump process. We assume also that the spot variance matrix
process Ct = σt σt> of Yt is a continuous Itô semimartingale,3
Z t
Z t
e
σ
es dWs ,
Ct = C0 +
bs ds +
0
(1)
0
see Section 4 for the full list of assumptions. We denote Ct = (Cab,t )1≤a,b≤d . For convenience, we
also employ the alternative notation CU V,t to refer to the spot covariance between two elements U
and V of Y .
We assume a standard continuous-time factor model for the (log) prices of the assets:
Definition (Factor Model for Prices, P-FM). For for all 0 ≤ t ≤ T and j = 1, . . . , dS ,4
>
c
= βj,t
dFtc + dZj,t
c
dSj,t
with
[Zjc , F c ]t = 0,
(2)
We do not need the return factors Ft to be the same across assets to identify the model, but
without loss of generality, we keep this structure because it is standard in empirical finance. These
return factors are assumed to be observable. For example, in the empirical application, we use two
sets of return factors: the market portfolio and the three Fama-French factors.5
The process Zj,t in the P-FM is the idiosyncratic component of the price of the j th stock with
respect to the return factors. We use the superscript c to emphasize that the P-FM only involves
the continuous martingale parts of the two observable processes Yj,t and Ft . The jump parts of
these processes are left unrestricted. For j = 1, . . . , dS , the factor loadings βj,t is a RdF -valued
process which represents the continuous beta.6 The k-th component of βj,t corresponds to the
time-varying loading of the continuous part of the return on stock j to the continuous part of
the return on the k-th factor. We set βt = (β1,t , . . . , βdS ,t )> and Zt = (Z1,t , . . . , ZdS ,t )> . This
framework was originally studied in Mykland and Zhang (2006) in the case of one factor and in
the absence of jumps. It was extended to multiple factors and jumps in Aı̈t-Sahalia, Kalnina,
and Xiu (2014). See also Li, Todorov, and Tauchen (2013), Fan, Furger, and Xiu (2015), and
3
Note that assuming that Y and C are driven by the same d0 -dimensional Brownian motion W is without loss of
generality provided that d0 is large enough, see, e.g., equation (8.12) of Aı̈t-Sahalia and Jacod (2014).
4
If X and Y are two vector-valued Itô semimartingales, their quadratic covariation over the time span [0, T ] is
defined
[X, Y ]T = p − lim
M →∞
M
−1
X
(Xtj+1 − Xtj )(Ytj+1 − Ytj )> ,
j=0
for any sequence t0 < t1 < . . . < tM = T with sup {tj+1 −tj } → 0 as M → ∞, where p-lim stands for the probability
j
limit. Barndorff-Nielsen and Shephard (2004) discuss its estimation when both X and Y are observed.
5
High-frequency observations on the Fama-French factors are constructed in Aı̈t-Sahalia, Kalnina, and Xiu (2014).
6
Interestingly, it is possible to define a discontinuous beta, see, e.g., Bollerslev and Todorov (2010) and Li,
Todorov, and Tauchen (2014).
4
Reiß, Todorov, and Tauchen (2015). Our framework can be potentially extended to use principal
components instead of observable return factors as in Ait-Sahalia and Xiu (2015).
Idiosyncratic Volatility of stock j is the spot volatility of the residual process Zj , and we denote
it by CZjZj . Notice that the factor loadings, as well as IV, in (2) are functions of the total spot
variance matrix Ct . First, the vector of factor loadings satisfies
βjt = (CF F,t )−1 CF Sj ,t ,
(3)
for j = 1, . . . , dS , where CF F,t denotes the spot covariance of the factors F , which is the lower
dF × dF sub-matrix of Ct , and CF Sj,t denotes the covariance of the factors and the j th stock, which
are the last dF elements of the j th column of Ct . Second, the IV of stock j satisfies
CZjZj,t = CY jY j,t − (CF Sj,t )> (CF F,t )−1 CF Sj,t .
(4)
Since factor loadings and the IV are functions of the total spot variance matrix, by the Itô lemma,
they are also Itô semimartingales with their characteristics related to those of Ct .
We take the following quadratic-covariation based quantity as the natural measure of dependence between the IV shocks of stocks i and j,
[CZiZi , CZjZj ]T
p
ρZi,Zj = p
.
[CZiZi , CZiZi ]T [CZjZj , CZjZj ]T
(5)
Alternatively, one can consider the quadratic covariation [CZiZi , CZjZj ]T . In Section 4.4, we use
the estimator of the latter quantity to test for the presence of cross-sectional dependence in IVs.
We now introduce the Idiosyncratic Volatility Factor model (IV-FM). We assume that the
cross-sectional dependence in the IVs can be potentially explained by the IV factors. We assume
the IV factors are known functions of the matrix Ct . In the empirical application, we use the
market volatility as the IV factor; we discuss other possibilities below. We allow the IV factors to
be any known function of Ct as long as it satisfies a certain polynomial growth condition in the
sense of being in the class G(p) below,
G(p) = {H : H is three-times continuously differentiable and for some K > 0,
k∂ j H(x)k ≤ K(1 + kxk)p−j , j = 0, 1, 2, 3}, for some p ≥ 3.
Definition (Idiosyncratic Volatility Factor Model, IV-FM). For all 0 ≤ t ≤ T and j =
1, . . . , dS , the idiosyncratic volatility CZjZj follows,
dCZjZj,t
NS
= b>
Zj dΠt + dCZjZj,t with
N S , Π]
[CZjZj
t = 0.
(6)
(7)
where Πt = (Π1t , . . . , ΠdΠ t ) is a RdΠ -valued vector of IV factors, which satisfy Πkt = Πk (Ct ) with
the function Πk (.) belonging to G(p) for k = 1, . . . , dΠ .
5
NS
We call the residual term CZjZj,t
the non-systematic idiosyncratic volatility of asset j, and
we abbreviate it as NS-IV. We refer to bZj as the IV beta. The IV beta is time-invariant. Our
NS
assumptions imply that the components of the IV-FM, CZjZj,t , Πt and CZjZj,t
, are continuous Itô
semimartingales. We remark that both the regressand and the regressors in our IV-FM are not
directly observable and have to be estimated. As will see in Section 3, this preliminary estimation
implies that the naive estimators of all the quantities of interest in the IV-FM are biased. One of
the contributions of this paper is to quantify this bias and propose bias-corrected estimators for
all the quantities of interest.
The class of IV factors permitted by our theory is rather wide as it includes general non-linear
transforms of the spot variance process Ct . For example, IV factors can be linear combinations
of the total variances of stocks, see, e.g., the average variance factor of Chen and Petkova (2012).
Other examples of IV factors are linear combinations of the IVs, such as the equally-weighted
average of the IVs, the “CIV”, of Herskovic, Kelly, Lustig, and Nieuwerburgh (2014). The IV
factors can also be the volatilities of any other observable processes.
To measure the residual cross-sectional dependence between two IVs after accounting for the
effect of the IV factors, we use a quadratic-covariation based correlation measure between NS-IVs,
NS , CNS ]
[CZiZi
ZjZj T
S
q
.
ρN
=
q
Zi,Zj
N
S
N
S
N
S
N
S
[CZiZi , CZiZi ]T [CZjZj , CZjZj ]T
(8)
When testing for the presence of residual correlation between NS-IVs, we use the quadratic covariN S , C N S ] without normalization.
ation [CZiZi
ZjZj T
We want to capture how well the IV factors explain the time variation of j th IV. For this
purpose, we use the quadratic-covariation based analog of the coefficient of determination. For
j = 1, . . . , dS ,
2,IV -F M
RZj
=
b>
Zj [Π, Π]T bZj
[CZjZj , CZjZj ]T
.
(9)
It is interesting to compare the correlation measure between IVs in equation (5) with the
correlation between the non-systematic parts of IVs in (8). We consider their difference,
S
ρZi,Zj − ρN
Zi,Zj ,
(10)
to see how much of the dependence between IVs can be attributed to the IV factors. In practice,
if we compare assets that are known to have positive covolatilities (typically, stocks have that
property), another useful measure of the systematic part in the overall covariation between IVs is
the following quantity,
-F M =
QIV
Zi,Zj
b>
Zi [Π, Π]T bZj
.
[CZiZi , CZjZj ]T
(11)
This measure is bounded by 1 if the covariations between NS-IVs are nonnegative and smaller than
the covariations between IVs, which is what we find for every pair in our empirical application with
6
high-frequency observations on stock returns.
The next section outlines identification and estimation of the above key quantities. It also
presents the asymptotic distributions, which can be used to conduct statistical tests. We conduct
three tests. First, we test whether the total cross-correlation in IVs is nonzero for a given pair of
assets, which corresponds to the hypothesis [CZiZi , CZjZj ]T = 0. Second, we test whether the IV
factors contribute to the cross-correlation in IVs by considering the null hypothesis [CZjZj , Π]T = 0.
Third, we test the hypothesis of whether IV-FM can explain all the cross-sectional IV dependence,
N S , C N S ] = 0.
i.e., [CZiZi
ZjZj T
It is interesting to compare our framework with the following null hypothesis studied in Li,
Todorov, and Tauchen (2013), H0 : CZjZj,t = aZj + b>
Zj Πt , 0 ≤ t ≤ T. This H0 implies that the
IV is a deterministic function of the factors, which does not allow for a non-systematic error term.
In particular, this null hypothesis implies R2,IV -F M = 1.
Zj
3
Estimation
We now discuss the estimation of our main quantities of interest introduced in Section 2,
2,IV -F M
NS
NS
S
IV -F M
,
[CZiZi , CZjZj ]T , ρZi,Zj , [CZjZj
, CZjZj
]T , ρN
Zi,Zj , QZi,Zj , and RZi
(12)
for i, j = 1, . . . , dS . We first show that each of them can be written as
ϕ ([H1 (C), G1 (C)]T , . . . , [Hκ (C), Gκ (C)]T ) ,
where ϕ as well as Hr and Gr , for r = 1, . . . , κ, are known real-valued functions. Each element in
this expression is of the form [H(C), G(C)]T , i.e., it is a quadratic covariation between functions
of Ct . We then show how to estimate [H(C), G(C)]T .
First, consider the quadratic covariation between ith and j th IV, [CZiZi , CZjZj ]T . It can be
written as [H(C), G(C)]T if we choose H(Ct ) = CZiZi,t and G(Ct ) = CZjZj,t . By (4), both CZiZi,t
and CZjZj,t are smooth functions of Ct . Next, consider the correlation ρZi,Zj defined in (5). By
the argument above, its numerator and each of the two components in the denominator can be
written as [H(C), G(C)]T for different functions H and G. Therefore, ρZi,Zj is itself a known
smooth function of three objects of the form [H(C), G(C)]T .
To show that the remaining quantities in (12) can also be expressed in terms of objects of the
form [H(C), G(C)]T , note that the IV-FM implies
NS
NS
bZj = ([Π, Π]T )−1 [Π, CZjZj ]T and [CZiZi
, CZjZj
]T = [CZiZi , CZjZj ]T − b>
Zi [Π, Π]T bZj ,
for i, j = 1, . . . , dS . Since CZiZi,t , CZjZj,t and every element in Πt are real-valued functions of
Ct , the above equalities imply that all quantities of interest in (12) can be written as real-valued,
known smooth functions of a finite number of quantities of the form [H(C), G(C)]T .
To estimate [H(C), G(C)]T , suppose we have discrete observations on Yt over an interval [0, T ].
7
Denote by ∆n the distance between observations. Note that we can estimate the spot covariance
matrix Ct at time (i − 1)∆n with a local truncated realized volatility estimator (Mancini (2001)),
bi∆n =
C
kn −1
1 X
(∆ni+j Y )(∆ni+j Y )> 1{k∆ni+j Y k≤χ∆$
,
n}
kn ∆n
(13)
j=0
where ∆ni Y = Yi∆n − Y(i−1)∆n and where kn is the number of observations in a local window.7
bi∆n = (C
bab,i∆ )1≤a,b≤d .
Throughout the paper we set C
n
We propose two estimators for the general quantity [H(C), G(C)]T .8 The first is based on the
analog of the definition of quadratic covariation between two Itô processes,
AN
\
[H(C),
G(C)]T
=
2
−
kn
3
2kn
d
X
[T /∆n ]−2kn +1
X
b(i+k )∆ ) − H(C
bi∆n )
H(C
n
n
b(i+k )∆ ) − G(C
bi∆n )
G(C
n
n
i=1
!
bgb,i∆ + C
bgb,i∆ C
b
bga,i∆n C
bi∆n ) C
,
(∂gh H∂ab G)(C
n
n ha,i∆n
(14)
g,h,a,b=1
where the factor 3/2 and last term correct for the biases arising due to the estimation of volatility
Ct . The increments used in the above expression are computed over overlapping blocks, which
results in a smaller asymptotic variance compared to the version using non-overlapping blocks.
Our second estimator is based on the following equality, which follows by the Itô lemma,
d
X
[H(C), G(C)]T =
Z
T
gh,ab
∂gh H∂ab G (Ct )C t
dt,
(15)
g,h,a,b=1 0
gh,ab
where C t
denotes the covariation between the volatility processes Cgh,t and Cab,t . The quantity
is thus a non-linear functional of the spot covariance and spot volatility of volatility matrices. Our
second estimator is based on this “linearized” expression,
LIN
\
[H(C),
G(C)]T
=
3
2kn
d
X
[T /∆n ]−2kn +1
g,h,a,b=1
i=1
X
bgh,(i+k )∆ − C
bgh,i∆ )(C
bab,(i+k )∆
(C
n
n
n
n
n
bi∆n )×
(∂gh H∂ab G)(C
(16)
bha,i∆ ) .
bgb,i∆ C
bab,i∆ ) − 2 (C
bga,i∆n C
bgb,i∆ + C
−C
n
n
n
n
kn
Consistency for a similar estimator has been established by Jacod and Rosenbaum (2012).9 We go
beyond their result by deriving the asymptotic distribution and proposing a consistent estimator
7
It is also possible to define kernel-based estimators as in Kristensen (2010).
As evident from their formulas, the computation time required for the calculation of the two estimators is
increasing with the number of stocks and factors d. To ease the implementation of the procedure, we compute all
the quantities of interest for pairs of stocks which means practically one needs only to set dS = 2 so that d = dF + 2.
9
Jacod and Rosenbaum (2012) derive the probability limit of the following estimator:
8
3
2kn
d
X
[T /∆n ]−2kn +1
g,h,a,b=1
X
i=1
2
bi∆n ) (C
bga,i∆n C
bgb,i∆n +C
b(i+k )∆ −C
bi∆n )(C
b(i+k )∆ −C
bgb,i∆n C
bha,i∆n ) .
bi∆n )− 2 (C
(∂gh,ab
H)(C
n
n
n
n
kn
8
of its asymptotic variance.
Note that the same additive bias-correcting term,
3
− 2
kn
[T /∆n ]−2kn +1
X
d
X
i=1
g,h,a,b=1
!
bi∆n ) C
bga,i∆n C
bgb,i∆ + C
bgb,i∆ C
b
(∂gh H∂ab G)(C
,
n
n ha,i∆n
(17)
is used for the two estimators. This term is (up to a scale factor) an estimator of the asymptotic
RT
RT
covariance between the sampling errors embedded in estimators of 0 H(Ct )dt and 0 G(Ct )dt
defined in Jacod and Rosenbaum (2013).
The two estimators are identical when H and G are linear, for example, when estimating the
covariation between two volatility processes. In the univariate case d = 1, when H(C) = G(C) = C,
our estimator coincides with the volatility of volatility estimator of Vetter (2012) , which was
extended to allow for jumps in Jacod and Rosenbaum (2012). Our contribution is the extension of
this theory to the multivariate d > 1 case with nonlinear functionals.
4
Asymptotic Properties
We start by outlining the full list of assumptions for our asymptotic results. We then state the
asymptotic distribution for the general functionals introduced in the previous section, and develop
estimators for the asymptotic variance. Finally, we outline three statistical tests of interest that
follow from our theoretical results.
4.1
Assumptions
Recall that the d-dimensional process Yt represents the (log) prices of stocks and factors.
Assumption 1. Suppose Y is an Itô semimartingale on a filtered space (Ω, F, (Ft )t≥0 , P),
Z t
Z t
Z tZ
Yt = Y0 +
bs ds +
σs dWs +
δ(s, z)µ(ds, dz),
0
0
0
E
where W is a d0 -dimensional Brownian motion (d0 ≥ d) and µ is a Poisson random measure on
R+ × E, with E an auxiliary Polish space with intensity measure ν(dt, dz) = dt ⊗ λ(dz) for some σ0
finite measure λ on E. The process bt is Rd -valued optional, σt is Rd ×Rd -valued, and δ = δ(w, t, z)
is a predictable Rd -valued function on Ω × R+ × E. Moreover, kδ(w, t ∧ τm (w), z)k ∧ 1 ≤ Γm (z),
for all (w,t,z), where (τm ) is a localizing sequence of stopping times and, for some r ∈ [0, 1], the
R
function Γm on E satisfies E Γm (z)r λ(dz) < ∞. The spot volatility matrix of Y is then defined
as Ct = σt σt> . We assume that Ct is a continuous Itô semimartingale,10
Z t
Z t
e
Ct = C0 +
bs ds +
σ
es dWs .
0
(18)
0
Note that σ
es = (e
σsgh,m ) is (d × d × d0 )-dimensional and σ
es dWs is (d × d)-dimensional with (e
σs dWs )gh =
gh,m
m
es
dWs .
m=1 σ
10
Pd0
9
where eb is Rd × Rd -valued optional.
With the above notation, the elements of the spot volatility of volatility matrix and spot
covariation of the continuous martingale parts of X and c are defined as follows,
0
gh,ab
Ct
=
d
X
0
σ
etgh,m σ
etab,m ,
0g,ab
Ct
m=1
=
d
X
σtgm σ
etab,m .
(19)
m=1
The process σ
et is restricted as follows:
Assumption 2. σ
et is a continuous Itô semimartingale with its characteristics satisfying the same
requirements as that of Ct .
Assumption 1 is very general and nests most of the multivariate continuous-time models used in
economics and finance. It allows for potential stochastic volatility and jumps in prices. Assumption
2 is required to obtain the asymptotic distribution of estimators of the quadratic covariation
between functionals of the spot covariance matrix Ct . It is not needed to prove consistency. This
restriction also appears in Vetter (2012), Kalnina and Xiu (2015) and Wang and Mykland (2014).
4.2
Asymptotic Distribution
We have seen in Section 3 that all quantities of interest in (12) are functions of multiple objects of
the form [H(C), G(C)]T . Therefore, if we can obtain a multivariate asymptotic distribution for a
vector with elements of the form [H(C), G(C)]T , the asymptotic distributions for all our estimators
follow by the delta method. Presenting this asymptotic distribution is the purpose of the current
section.
Let H1 , G1 , . . . , Hκ , Gκ be some arbitrary elements of G(p). We are interested in the asymptotic
behavior of vectors
AN
AN >
LIN
LIN >
\
\
\
\
[H1 (C),
G1 (C)]T , . . . , [Hκ (C),
Gκ (C)]T
and [H1 (C),
G1 (C)]T , . . . , [Hκ (C),
Gκ (C)]T
.
The smoothness requirement on the different functions Hj and Gj is useful for obtaining the asymptotic distribution of the bias correcting terms (see for example Jacod and Rosenbaum (2012) and
Jacod and Rosenbaum (2013)).The following theorem summarizes the joint asymptotic behavior
of the estimators.
AN
\
\
Theorem 1. Let [Hr (C),
Gr (C)]T be either [Hr (C),
Gr (C)]T
LIN
\
or [Hr (C),
Gr (C)]T
defined in
−1/2
(14) and (16), respectively. Suppose Assumption 1 and Assumption 2 hold. Fix kn = θ∆n
some θ ∈ (0, ∞) and set (8p − 1)/4(4p − r) ≤ $ <
1
2.
for
Then, as ∆n −→ 0,


\
[H1 (C),
G1 (C)]T − [H1 (C), G1 (C)]T

 L−s
..

 −→ M N (0, ΣT ),
∆−1/4
.
n


\
[Hκ (C), Gκ (C)]T − [Hκ (C), Gκ (C)]T
10
(20)
where ΣT = Σr,s
T
1≤r,s≤κ
\
denotes the asymptotic covariance between the estimators [Hr (C),
Gr (C)]T
\
and [Hs (C),
Gs (C)]T . The elements of the matrix ΣT are
r,s,(1)
Σr,s
T = ΣT
r,s,(1)
ΣT
6
= 3
θ
r,s,(2)
r,s,(3)
+ ΣT
d
X
+ ΣT
d
X
,
T
Z
h
∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs (Cs ) Ct (gh, jk)Ct (ab, lm)
g,h,a,b=1 j,k,l,m=1 0
i
+ Ct (ab, jk)Ct (gh, lm) dt,
r,s,(2)
ΣT
151θ
=
140
r,s,(3)
ΣT
3
=
2θ
d
X
d
X
T
Z
i
h gh,jk ab,lm
ab,jk gh,lm
Ct
+ Ct
Ct
dt,
∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs (Ct ) C t
g,h,a,b=1 j,k,l,m=1 0
d
X
d
X
T
Z
h
ab,lm
gh,jk
∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs (Ct ) Ct (gh, jk)C t
+ Ct (ab, lm)C t
g,h,a,b=1 j,k,l,m=1 0
ab,jk
+ Ct (gh, lm)C t
gh,lm
+ Ct (ab, jk)C t
i
dt,
with
Ct (gh, jk) = Cgj,t Chk,t + Cgk,t Chj,t .
The convergence in Theorem 1 is stable in law (denoted L-s, see for example Aldous and
Eagleson (1978) and Jacod and Protter (2012)). The limit is mixed gaussian and the precision of the
estimators depends on the paths of the spot covariance and the volatility of volatility process. The
−1/4
rate of convergence ∆n
has been shown to be the optimal for volatility of volatility estimation
(under the assumption of no volatility jumps).
The asymptotic variance of the estimators depends on the tuning parameter θ whose choice
may be crucial for the reliability of the inference. We document the sensitivity of the inference
theory to the choice of the parameter θ in a Monte Carlo experiment (see Section 5).
4.3
Estimation of the Asymptotic Covariance Matrix
To provide a consistent estimator for the element Σr,s
T of the asymptotic covariance matrix in
Theorem 1, we introduce the following quantities:
d
X
b r,s,(1) = ∆n
Ω
T
d
X
[T /∆n ]−4kn +1
X
h
bn ) C
bi∆n (gh, jk)C
bi∆n (ab, lm)
∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs (C
i
i=1
g,h,a,b=1 j,k,l,m=1
i
bi∆n (ab, jk)C
bi∆n (gh, lm) ,
+C
b r,s,(2) =
Ω
T
d
X
d
X
[T /∆n ]−4kn +1
h n,gh n,jk n,ab n,lm
bn ) 1 γ
∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs (C
b
γ
bi γ
bi+2kn γ
bi+2kn +
i
2 i
i=1
g,h,a,b=1 j,k,l,m=1
1 n,ab n,lm n,gh n,jk
1 n,ab n,jk n,gh n,lm
1 n,gh n,lm n,ab n,jk i
γ
bi γ
bi γ
bi+2kn γ
bi+2kn + γ
bi γ
bi γ
bi+2kn γ
bi+2kn + γ
b
γ
bi γ
bi+2kn γ
bi+2kn ,
2
2
2 i
X
11
[T /∆n ]−4kn +1
d
d
X
X
X
3
r,s,(3)
b
bn ) ×
ΩT
=
∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs (C
i
2kn
i=1
g,h,a,b=1 j,k,l,m=1
h
i
bi∆n (gh, jk)b
bi∆n (ab, lm)b
bi∆n (gh, lm)b
bi∆n (ab, jk)b
C
γin,ab γ
bin,lm + C
γin,gh γ
bin,jk + C
γin,ab γ
bin,jk + (C
γin,gh γ
bin,lm ,
b n,jk − C
b n,jk and C
bi∆n (gh, jk) = (C
bgj,i∆n C
bhk,i∆ + C
bgk,i∆ C
b
with γ
bin,jk = C
n
n hj,i∆n ).
i
i+kn
The following result holds,
Theorem 2. Suppose the assumptions of Theorem 1 hold, then, as ∆n −→ 0
6 b r,s,(1) P
r,s,(1)
−→ ΣT
Ω
θ3 T
3 b r,s,(3) 6 b r,s,(1) P
r,s,(3)
[ΩT
− ΩT
] −→ ΣT
2θ
θ
151θ 9 b r,s,(2)
4 b r,s,(1) 4 b r,s,(3) P
r,s,(2)
[ΩT
+ 2Ω
− ΩT
] −→ ΣT
.
2
140 4θ
θ T
3
b T is symmetric but is not guaranteed to be positive semi-definite. By
The estimated matrix Σ
b T is positive semi-definite in large samples. The form of the asymptotic variance is
Theorem 1, Σ
relatively complicated. Estimating it using subsampling or bootstrap techniques is an interesting
research question that is beyond the scope of this paper.
Remark 1: Using results in Jacod and Rosenbaum (2012), it can be shown that the first con−1/2
vergence in Theorem 2 holds at a rate of ∆n
−1/4
while the last convergence rate is ∆n
straightforward extension of Theorem 1. Our estimator of
convergence
r,s,(2)
ΣT
by a
can be shown to have a rate of
−1/4
∆n .
r,s,(2)
Remark 2: In the one-dimensional case (d = 1), much simpler estimators of ΣT
constructed using the quantities
n,gh n,xy
γ
b
γ
bin,jk γ
bin,lm γ
bi+k
n i+kn
or
γ
bin,jk γ
bin,lm γ
bin,gh γ
bin,xy
can be
as in Vetter (2012).
However, in the multidimensional case, the latter quantities do not identify separately the quantity
Ct
jk,lm
Ct
gh,xy
since the combination Ct
jk,lm
Ct
gh,xy
+ Ct
jk,gh
Ct
lm,xy
+ Ct
jk,xy
Ct
gh,lm
shows up in a
non-trivial way in the limit of the estimator.
AN
\
\
Corollary 3. For 1 ≤ r ≤ κ, let [Hr (C),
Gr (C)]T be either [Hr (C),
Gr (C)]T
LIN
\
or [Hr (C),
Gr (C)]T
defined in (16) and (14), respectively. Suppose the assumptions of theorem 1 hold, then we have:


\
[H1 (C),
G1 (C)]T − [H1 (C), G1 (C)]T

 L
..
b −1/2 
 −→ N (0, Iκ ),
∆−1/4
Σ
(21)
.
n
T


\
[Hκ (C),
Gκ (C)]T − [Hκ (C), Gκ (C)]T
In the above, we use the notation L to denote the convergence in distribution and Iκ the
identity matrix of order κ. Corollary 3 states the standardized asymptotic distribution, which
follows directly from the properties of stable-in-law convergence. Similarly, by the delta method,
standardized asymptotic distribution can also be derived for the estimators of the quantities in
(12). These standardized distributions allow the construction of confidence intervals for all the
latent quantities of the form [Hr (C), Gr (C)]T and, more generally, functions of these quantities.
12
4.4
Testing procedure
We now describe the three statistical tests that we are interested in. The test of absence of
dependence between the IV of the returns on asset i and j can be formulated as:
H01 : [CZiZi , CZjZj ]T = 0 vs H11 : [CZiZi , CZjZj ]T 6= 0.
The null hypothesis H01 is rejected whenever,
\
[C
,
C
]
ZiZi ZjZj T −1/4
∆n r
> Zα .
\
AV
AR CZiZi , CZjZj
The test of absence of dependence between the IV of stock j and all IV factors Π takes the following
form:
H02 : [CZjZj , Π]T = 0 vs H12 : [CZjZj , Π]T 6= 0.
Recalling that dΠ denotes the number of IV factors, we reject the above null hypothesis H02 when,
> −1
\
\, Π] > X 2
\
∆−1/4
[C
,
Π]
AV
AR
C
,
Π
[CZjZj
ZjZj
ZjZj
n
dΠ ,1−α .
T
T
The test of absence of dependence between the NS-IVs can be stated as:
NS
NS
NS
NS
H03 : [CZiZi
, CZjZj
]T = 0 vs H13 : [CZiZi
, CZjZj
]T 6= 0,
with the null rejected if
N S\N S
[CZiZi , CZjZj ]T −1/4
∆n r
> Zα .
N
S
N
S
\
AV AR CZiZi , CZjZj
Our inference theory also allows to test more general hypotheses, which are joint across any
AN
\
\
subset of the panel. In the above statements, [H(C),
G(C)]T can be either [H(C),
G(C)]T or
LIN
\
\
\
[H(C),
G(C)]T , AV
AR H(C), G(C) is an estimate of the asymptotic variance of [H(C),
G(C)]T ,
Zα stands for the (1 − α) quantile of the N (0, 1), and Xd2q ,1−α stands for (1 − α) quantile of the Xd2q
distribution. For the first two tests, the expression for the true asymptotic variance is obtained
using Theorem 1 and its estimation follows from Theorem 2. The asymptotic variance of the third
test is obtained by an application of the delta method to the convergence result in Theorem 1. The
expression of the AVAR for the third test involves some of the latent quantities defined in (12),
which can be estimated using either AN- or LIN-type estimators. Therefore in general, we have
two tests for each null hypothesis, corresponding to the two type of estimators for [H(C), G(C)]T .
Under (2) and the assumptions of Theorem 1, Corollary 3 implies that the asymptotic size of the
two types of tests for the null hypotheses H01 and H02 is α, and their power approaches 1. The
same properties apply for the tests of the null hypotheses H03 as long as (2) and our IV-FM rep13
resentation (6) hold.
Theoretically, it is possible to test for absence of dependence in the IVs at the spot level. In
this case the null hypothesis is H001 : [CZiZi , CZjZj ]t = 0 for all 0 ≤ t ≤ T , which is, in theory,
stronger than our H001 . In particular, Theorem 1 can be used to set up Kolmogorov-Smirnov type
of tests for H001 in the same spirit as Vetter (2012). However, we do not pursue this direction in
the current paper for two reasons. First, the testing procedure would be more involved. Second,
empirical evidence suggests nonnegative dependence between IVs, which means that in practice,
it is not too restrictive to assume [CZiZi , CZjZj ]t ≥ 0 ∀t, under which H01 and H001 are equivalent.
5
Monte Carlo
This section investigates the finite sample properties of our estimators and tests. The data generating process (DGP) is constructed as follows. Denote by (Yj )1≤j≤27 the log-prices of 27 individual
stocks, and by X the log-price of the market portfolio. The number of stocks matches the one we
use in the empirical application. Recall that the superscript c indicates the continuous part of a
process. We assume
dXt = dXtc + dJt , dXtc =
p
CXX,t dWt ,
and, for j = 1, . . . , 27,
c
c
dYj,t = βj,t dXtc + dYej,t
+ dJj,t , dYej,t
=
p
fj,t .
CZjZj,t dW
fj )1≤j≤27 , are Brownian motions
In the above, CXX is the spot volatility of the market portfolio, (W
fj,t , dW
f ) = 0.2ωj where (ωj )1≤j≤27 are independent variables uniformly distributed
with Corr(dW
f are two independent Brownian motions; J and (Jj )1≤j≤27 are independent
over [-1,1]; W and W
compound Poisson processes with intensity equal to 2 jumps per year and jump size distribution
N(0,0.022 ). The beta process is time-varying and is specified as βj,t = πj βt with the πj ’s being
independent variables uniformly distributed over [0,1] and βt = 0.5 + 0.1 sin(100t).
We next specify the volatility processes. The market volatility process CXX,t is generated as
follows,
p
p
dCXX,t = 5(0.09 − CXX,t )dt + 0.35 CXX,t (−0.8dWt + 1 − 0.82 dBt ),
where B is a standard Brownian Motion, which is independent from the Brownian Motions of the
return Factor Model. The common Brownian Motion Wt in the market portfolio price process Xt
and its volatility process CXX,t generates a leverage effect for the market portfolio.11 The value
of the leverage effect is -0.8, which is standard in the literature, see Kalnina and Xiu (2015), Aı̈tSahalia, Fan, and Li (2013) and Aı̈t-Sahalia, Fan, Laeven, Wang, and Yang (2013). To create a
11
Due to the factor structure of the stocks returns, the leverage effect of the market portfolio translates into a
leverage effect for the individual stocks.
14
factor structure in the IV processes CZjZj,t , we generate the following processes
p
dfj,t = 5(0.09 − fj,t )dt + 0.35 fj,t dBj,t , for j = 0, . . . , 27,
where (Bj )0≤j≤27 are independent Browian motions which are independent from existing ones, see
Table 1 for the different specifications for the IV processes.12
Model 1
Model 2
Model 3
CZjZj,t
0.67uj + 0.8fj,t
0.67uj + (0.32 + 0.8vj )CXX,t + 0.8fj,t
0.67uj + (0.32 + 0.8vj )CXX,t + (0.28 + 0.8wj )f0,t + 0.8fj,t
Table 1: Different specifications for the Idiosyncratic Volatility processes CZjZj,t for j = 1, . . . , 27. The
random variables uj , vj and wj and independent and uniformly distributed over [0,1].
We set the time span T equal 1260 or 2520 days, which correspond approximately to 5 and 10
business years. These values are close to those typically used in the nonparametric leverage effect
estimation literature (see Aı̈t-Sahalia, Fan, and Li (2013) and Kalnina and Xiu (2015)), which
is related to the problem of volatility of volatility estimation. Each day consists of 6.5 trading
hours. We consider two different values for the sampling frequency, ∆n = 1 minute and ∆n = 5
minutes. We follow Li, Todorov, and Tauchen (2013) and set the truncation threshold un in day t
at 3b
σt ∆n0.49 , where σ
bt is the squared root of the annualized bipower variation of Barndorff-Nielsen
and Shephard (2004). We use 10 000 Monte Carlo replications in all the experiments.
We first investigate the finite sample properties of the estimators under Model 3. The considered
estimators include:
• the IV beta of the first stock bZ1 ,
2,IV -F M • the contribution of the market volatility to the variation of the IV of the first stock RZ1
,
• the correlation between the idiosyncratic volatilities of stocks 1 and 2 ρZ1,Z2 ,
S
• the correlation between non-systematic idiosyncratic volatilities ρN
Z1,Z2 ,
The interpretation of simulation results is much simpler when the quantities of interest do not
change across simulations. To achieve that, we generate once and keep fixed the paths of the
processes CXX,t and (fj,t )0≤j≤27 and replicate several times the other parts of the DGP. In Table
2, we report the bias and the interquartile range (IQR) of the two type of estimators for each
quantity using 5 minutes data sampled over 10 years. We choose four different values for the width
of the subsamples, which corresponds to θ = 1.5, 2, 2.5 and 3 (recall that the number of observations
√
in a window is kn = θ/ ∆n ). It seems that larger values of the parameters produce better results.
12
The (fj,t )0≤j≤27 are independent Cox Ingersoll Ross processes which fulfill the Feller property. Therefore they
are ensured to take always positive values.
15
Next, we investigate how these results change when we increase the sampling frequency. In Table
3, we report the results with ∆n = 1 minute in the same setting. We note a reduction of the bias
and IQR at all levels of significance. However, the magnitude of the decrease of the IQR is very
small. Finally, we conduct the same experiment using data sampled at one minute over 5 years.
Despite using more than twice as many observations than in the first experiment, the precision
is not as good. In other words, increasing the time span is more effective for precision gain than
1/4
increasing the sampling frequency. This result is typical for ∆n -convergent estimators, see, e.g.,
Kalnina and Xiu (2015).
AN
θ
1.5
2
LIN
2.5
bbZ1
b2,IV -F M
R
Z1
ρbZ1,Z2
S
ρbN
Z1,Z2
-0.047
0.176
-0.288
-0.189
-0.025
0.130
-0.212
-0.113
-0.011
0.103
-0.163
-0.064
bbZ1
b2,IV -F M
R
Z1
ρbZ1,Z2
S
ρbN
Z1,Z2
0.222
0.210
0.404
0.456
0.166
0.188
0.325
0.384
0.138
0.172
0.263
0.315
3
1.5
Median Bias
-0.003
-0.006
0.085
0.181
-0.133
-0.249
-0.034
-0.150
IQR
0.121
0.226
0.152
0.181
0.223
0.338
0.272
0.388
2
2.5
3
0.001
0.140
-0.190
-0.091
0.009
0.112
-0.146
-0.047
0.015
0.092
-0.120
-0.021
0.168
0.166
0.283
0.337
0.139
0.152
0.237
0.285
0.122
0.140
0.205
0.250
Table 2: Finite sample properties of our estimators using 10 years of data sampled at 5 minutes. The true
S
IV -F M
= 0.342, ρZ1,Z2 = 0.523, ρN
values are bZ1 = 0.450, RZ1
Z1,Z2 = 0.424.
AN
θ
1.5
2
LIN
2.5
bbZ1
bIV -F M
R
Z1
ρbZ1,Z2
S
ρbN
Z1,Z2
-0.022
0.107
-0.147
-0.135
-0.012
0.091
-0.104
-0.086
-0.003
0.073
-0.073
-0.058
bbZ1
bIV -F M
R
Z1
ρbZ1,Z2
S
ρbN
Z1,Z2
0.156
0.201
0.340
0.417
0.112
0.146
0.238
0.291
0.088
0.118
0.184
0.228
3
1.5
Median Bias
0.004
-0.003
0.056
0.113
-0.048
-0.133
-0.039
-0.119
IQR
0.075
0.157
0.100
0.184
0.150
0.309
0.184
0.378
2
2.5
3
-0.000
0.095
-0.097
-0.078
0.006
0.075
-0.067
-0.052
0.012
0.058
-0.042
-0.032
0.112
0.138
0.226
0.274
0.088
0.113
0.177
0.217
0.075
0.096
0.145
0.177
Table 3: Finite sample properties of our estimators using 10 years of data sampled at 1 minute. The true
2,IV -F M
S
values are bZ1 = 0.450, RZ1
= 0.336, ρZ1,Z2 = 0.514, ρN
Z1,Z2 = 0.408.
Next, we study the size and power of the three statistical tests as outlined in Section 4.4. We
16
AN
θ
1.5
2
LIN
2.5
bbZ1
b2,IV -F M
R
Z1
ρbZ1,Z2
S
ρbN
Z1,Z2
-0.019
0.115
-0.168
-0.141
-0.011
0.096
-0.101
-0.079
-0.007
0.081
-0.064
-0.035
bbZ1
b2,IV -F M
R
Z1
ρbZ1,Z2
S
ρbN
Z1,Z2
0.215
0.282
0.472
0.541
0.159
0.204
0.337
0.412
0.128
0.168
0.263
0.324
3
1.5
Median Bias
0.000
-0.001
0.069
0.119
-0.038
-0.149
-0.007
-0.127
IQR
0.110
0.216
0.144
0.260
0.213
0.436
0.266
0.510
2
2.5
3
-0.001
0.100
-0.092
-0.067
0.002
0.084
-0.057
-0.029
0.008
0.071
-0.033
-0.001
0.158
0.194
0.319
0.391
0.129
0.161
0.252
0.311
0.110
0.139
0.206
0.256
Table 4: Finite sample properties of our estimators using 5 years of data sampled at 1 minute. The true
2,IV -F M
S
values are bZ1 = 0.450, RZ1
= 0.35, ρZ1,Z2 = 0.517, ρN
Z1,Z2 = 0.417.
use Model 1 to study the size properties of the first two tests: the test of the absence of dependence
between the IVs (H01 : [CZ1Z1 , CZ2Z2 ]T = 0), and the absence of dependence between the IV of the
first stock and the market volatility (H02 : [CZ1Z1 , CXX ]T = 0). We use Model 2 to study the size
N S , C N S ] = 0). Finally, we use Model 3 to study power
properties of the third test (H03 : [CZ1Z1
Z2Z2 T
properties of all three tests.
The upper panel Tables 5, 6, and 7 reports the size results while the lower panels shows the
results for the power. We present the results for the two sampling frequencies (∆n = 1 minute
and ∆n = 5 minutes) and the two type of tests (AN and LIN). We observe that the size of three
tests are reasonably close to their nominal levels. The rejection probabilities under the alternatives
are rather high, except when the data is sampled at 5 minutes frequency and the nominal level
at 1%.13 We note that the tests based on LIN estimators have better testing power compared to
those that build on AN estimators. Increasing the window length induces some size distortions
but is very effective for power gain. Consistent with the asymptotic theory, the size of the three
tests are closer to the nominal levels and the power is higher at the one minute sampling frequency.
Clearly, the test of absence of dependence between IV and the market volatility has the best power,
followed by the test of absence of dependence between the two IVs. This ranking is compatible
with the notion that the finite sample properties of the tests deteriorate with the degree of latency
embedded in each null hypothesis.
13
We set the nominal level at 5% in the empirical application.
17
Type of the test
θ = 1.5
AN LIN
∆n = 5 minutes
θ = 2.0
θ = 2.5
AN
LIN
AN LIN
θ = 1.5
AN LIN
∆n = 1 minute
θ = 2.0
θ = 2.5
AN LIN AN LIN
α = 20%
α = 10%
α = 5%
α = 1%
19.5
9.7
4.7
0.9
21.0
10.6
5.1
1.1
19.4
10.6
4.5
0.9
Panel A : Size Analysis-Model 1
20.4 19.4 20.7
20.2 19.6 19.7
12.6
9.7 10.3
10.2 9.7 10.0
5.3
4.8
5.6
5.3
5.3
5.2
1.2
0.9
1.1
1.1
1.1
1.2
19.9
10.2
5.3
1.1
19.8
9.8
4.9
1.0
20.1
10.2
5.1
1.0
α = 20%
α = 10%
α = 5%
α = 1%
33.5
20.5
11.9
3.3
45.7
31.5
21.0
6.9
50.4
35.7
23.9
8.7
Panel
63.1
48.3
35.76
15.6
82.3
71.6
59.8
34.5
94.1
88.0
79.6
57.4
95.8
91.2
84.4
64.1
B : Power Analysis-Model 3
67.8 78.1
48.5 56.2 77.7
53.3 65.8
33.9 41.0 65.6
40.6 53.4
22.3 29.5 52.8
18.4 28.6
8.9 12.4 28.6
Table 5: Size and Power of the test of absence of dependence between idiosyncratic volatilities for T =
10 years.
6
Empirical Analysis
We apply our methods to study the cross-sectional dependence in IV using high frequency data.
We find that stocks’ idiosyncratic volatilities co-move strongly with the market volatility. This is
a quite surprising finding. It is of course well known that the total volatility of stocks moves with
the market volatility. However, we stress that we find that the strong effect is still present when
considering the idiosyncratic volatilities.
We use full record transaction prices from NYSE TAQ database for 27 stocks over the time
period 2003-2012. After removing the non-trading days, our sample contains 2517 days. The
selected stocks have been part of S&P 500 index throughout our sample. Our 27 stocks contain
three liquid stocks in each of the nine sectors of the index (Consumer Discretionary, Consumer
Staples, Energy, Financial, Health Care, Industrial, Materials, Technology, and Utilities). For each
day, we consider data from the regular exchange opening hours from time stamped between 9:30
a.m. until 4 p.m. We clean the data following the procedure suggested by Barndorff-Nielsen,
Hansen, Lunde, and Shephard (2008), remove the overnight returns and then sample at 5 minutes.
This sparse sampling has been widely used in the literature because the effect of the microstructure
noise and potential asynchronicity of the data is less important at this frequency, see also Liu,
Patton, and Sheppard (2014).
The parameter choices for the estimators are as follows. Guided by our Monte Carlo results,
we set the length of window to be approximately one week for the estimators in Section 3 (this
−1/2
corresponds to θ = 2.5 where kn = θ∆n
is the number of observations in a window). The
truncation threshold for all estimators is set as in the Monte Carlo study (3b
σt ∆n0.49 where σ
bt2 is the
18
Type of test
∆n = 5 minutes
θ = 1.5
θ = 2.0
θ = 2.5
AN LIN AN LIN AN LIN
∆n = 1 minute
θ = 1.5
θ = 2.0
AN LIN AN LIN
α = 20%
α = 10%
α = 5%
α = 1%
22.6
12.1
6.2
1.5
20.1
10.2
5.0
1.0
20.0
10.0
4.5
0.8
Panel A :
21.0 19.8
10.6 9.8
5.2
4.6
1.0
0.9
α = 20%
α = 10%
α = 5%
α = 1%
73.1
60.0
47.7
24.1
80.7
69.0
57.2
32.3
91.4
84.0
75.0
52.2
Panel B : Power Analysis-Model 3
93.9 97.4 98.3
95.8 97.2 99.7
88.3 94.6 96.1
91.1 93.3 99.2
81.0 89.6 92.6
84.9 88.2 98.2
60.1 73.7 78.9
67.7 72.0 93.0
θ = 2.5
AN LIN
Size Analysis-Model 1
21.5
21.6 20.6 21.6
11.0
11.0 10.4 10.3
5.4
5.5
5.4
5.2
1.2
1.1
1.1
1.0
21.5
10.4
5.1
0.9
21.1
10.4
5.2
0.8
21.5
10.4
5.3
1.0
99.8
99.4
98.6
94.5
100
100
100
99.2
100
100
100
99.4
Table 6: Size and Power of the test of absence of dependence between the idiosyncratic volatility and the
market volatility for T = 10 years.
bipower variation).
We consider two sets of factors in the factor model for the prices: the S&P500 market index
and the three Fama-French factors (FF3 henceforth). All factors are sampled at 5 minutes over
2003-2012.14
Figures 1 and 2 contain plots of the time series of the estimated RY2 j of the return regressions,
i.e., the estimated monthly contribution of the return factors to the total volatility, for each stock in
the two models (CAPM and FF3).15 In Table 8, we report the average of these monthly statistics
over the full sample. As we can see in Table 8, the return factors have a relatively low contribution
to the total variation of the returns in the two models which implies a high contribution of the
idiosyncratic volatility (1 − RY2 j ). Indeed, the minimum value (across all the stocks) for 1 − RY2 j
is 61.5% for the one-factor model and 53.5% in the FF3 model. Figures 1 and 2 show that the
time series of all stocks follow approximately the same trend with a considerable increase in the
contribution around the crisis year 2008, which shows that the systematic risk became relatively
more important during this period. Overall our results suggest that IV contributes to more than
a half of the total variation for each stock. Therefore, studying the source of variation in IVs is
potentially useful. We proceed to investigate the dynamic properties of the panel of idiosyncratic
volatilities.
14
15
The high-frequency data on the Fama-French factors were obtained from Aı̈t-Sahalia, Kalnina, and Xiu (2014).
For theR j th stock, our analog of the coefficient of determination in the return factor model for this stock is
RY2 j = 1 −
T
R 0T
CZjZj,t dt
. We estimate RY2 j using the general method of Jacod and Rosenbaum (2013). The resulting
CY jY j,t dt
RY2 j requires
0
estimator of
a choice of a block size for the spot volatility estimation; we choose two hours in practice
(the number of observations in a block, say ln , has to satisfy ln2 ∆n → 0 and ln3 ∆n → ∞, so it is of smaller order than
the number of observations kn in our estimators of Section 3).
19
Type of test
∆n = 5 minutes
θ = 1.5
θ = 2.0
θ = 2.5
AN LIN AN LIN AN LIN
θ = 1.5
AN LIN
∆n = 1 minute
θ = 2.0
θ = 2.5
AN LIN AN LIN
α = 20%
α = 10%
α = 5%
α = 1%
19.9
10.0
5.0
1.1
23
10.1
6.3
1.5
20.4
12.1
5.1
0.8
Panel A :
23.7 20.2
10.8 9.9
6.3
5.1
1.6
1.1
Size Analysis-Model 2
23.2
19.7 20.5 20.3
12.6
10.1 10.3 10.6
6.7
5.5
5.5
5.3
1.4
1.1
1.2
1.3
21.7
11.3
5.9
1.3
20.0
10.1
5.2
1.3
22.3
11.4
6.0
1.5
α = 20%
α = 10%
α = 5%
α = 1%
25.1
13.7
7.4
1.6
32.1
19.2
11.3
3.1
29.0
16.8
9.3
2.3
Panel B : Power Analysis-Model 3
36.5 42.8 51.7
31.0 35 50.0
23.0 28.1 36.9
19.0 22.2 35.0
14.2 18.3 25.2
11.0 13.7 23.9
3.9
6.0
9.5
2.9
4.0
9.3
54.6
39.4
28.0
11.6
68.0
53.4
40.0
18.8
72.3
58.3
44.9
22.2
Table 7: Size and Power of the test of absence of dependence between NS-IVs for T = 10 years.
We first investigate the (total) dependence in the idiosyncratic volatilities. We have 351 pairs
of stocks available in the panel. For each pair of stocks, we compute the correlation between
the IVs, ρZi,Zj . The upper and lower panels of Table 13 display the correlations estimated using
the LIN-type estimators in the CAPM and FF3 as the factor model for prices.16 The values in
parenthesis correspond to the p-values of the test of dependence in the IVs (see Section 4.4 for
an expression of the test statistic). The reader should be careful when interpreting these p-values
because they are not adjusted for multiple testing. Clearly, there is evidence for strong dependence
between the IVs. Indeed, the absolute values of the t-statistics are bigger than 1.96 for 350 pairs
over 351. Only the dependence between the IV of the Goldman Sachs (GS) and IBM gives rise to
the absolute value of the t-statistic smaller than 1.96. Using the Bonferroni correction, the p-value
of the test of absence of dependence in all pairs is less than 0.0001. The estimated correlation is
positive for each pair of stocks. We also observe substantial heterogeneity in the correlation with a
maximum value of 94.4% (Exxon Mobil (XOM) and Chevron Corp (CVX)) and a minimum value
of 24.8% (Duke Energy (DUK) and Avery Dennison Corporation (AVY)). Table 9 is a summary of
the results of Table 13; it shows the number of pairs with the estimated correlation greater than a
set of thresholds. For example, it shows that the correlation is greater than 50% for more than two
thirds of the pairs (265). Interestingly, the results of the test are unchanged for the FF3 model,
and the estimated correlations are very close to those obtained in the CAPM. This result is not
surprising given the relatively small difference between the values of RY2 j in the two models.
We next ask the question of whether potential missing factors in the factor model for returns
might be responsible for the strong dependence in IVs. Omitted factors in the factor model for
16
This choice is motivated by our simulation results where LIN type of estimators and tests appear to have better
finite sample properties than AN type estimators.
20
returns induce correlation between the estimated idiosyncratic returns, Corr(Zi , Zj ).17 We report
in Table 12 the estimated correlations Corr(Zi , Zj ). Table 10 presents a summary of how estimates
of Corr(Zi , Zj ) in Table 12 are related to the estimates of correlation in IVs, ρZi,Zj , in Table 13. In
particular, different rows in Table 10 display average values of ρZi,Zj among those pairs, for which
Corr(Zi , Zj ) is below some threshold. For example, the last-but-one row in Table 10 indicates
that there are 56 pairs of stocks with estimated Corr(Zi , Zj ) < 0.01, and among those stocks, the
average correlation between IVs, ρZi,Zj , is estimated to be 0.579. This estimate ρZi,Zj is virtually
the same among pairs of stocks with high Corr(Zi , Zj ). Therefore, we know that among 56 pairs of
stocks, a missing return factor cannot explain dependence in IVs. Moreover, these results suggest
that missing return factor cannot explain dependence in IVs for all considered stocks. These results
are in line with recent findings of Herskovic, Kelly, Lustig, and Nieuwerburgh (2014) with daily
and monthly returns.
To understand the source of the strong dependence in the IVs, we consider the Idiosyncratic
Volatility Factor Model (IV-FM) of Section 2. We use the market volatility as the single IV factor.18
We start by considering individual stocks separately. In Table 11, we report the estimates of the
idiosyncratic volatility beta (bbZi ) and the contribution of the market volatility to the aggregate
variation in IV (R2,IV -F M ). The absolute values of the t-statistics based on the covariation between
Zi
IV and the market volatility are bigger than 1.96 for each stock. For every stock, the estimated
IV beta is positive, suggesting that the idiosyncratic volatility co-moves with the market volatility.
For 10 stocks out of 27, the NS-IV contributes to more than 50% of the variation in their IV, with
the average being 44%.
Next, we turn to the implications of the IV-FM for the cross-section. We conduct inference
on dependence in the NS-IVs. Table 14 displays the estimated correlation between the NS-IVs.
The residual correlations are smaller than the total IV correlations. There are only 26 pairs of
stocks with this correlation higher than 50% in the CAPM model and 27 pairs in the FF3 model.
Interestingly, they all remain positive. The t-statistics based on covariation between NS-IVs are
larger than 1.96 for 241 pairs in the CAPM and 244 pairs in the FF3 model (see the values in
parenthesis of both Tables 14 and 13). From Table 11, each stock has at least eight other stocks
with whom it produces a t-statistic bigger than 1.96. Using the Bonferroni correction, the p-value
of the test of absence of dependence in all pairs is less than 0.0001. In Table 14 we report, for
S
each pair of stocks, the correlation between the NS-IVs, ρN
Zi,Zj . The values are much smaller than
17
Our measure of correlation between the idiosyncratic returns Zi and Zj is
RT
CZiZj,t dt
0
qR
Corr(Zi , Zj ) = qR
, i, j = 1, . . . , dS ,
T
T
C
dt
C
dt
ZiZi,t
ZjZj,t
0
0
(22)
where CZiZj,t is the spot covariation between the idiosyncratic returns Zi and Zj . Similarly to RY2 j , we estimate
Corr(Zi , Zj ) using the method of Jacod and Rosenbaum (2013).
18
We also considered the volatility of size and value Fama-French factors. However, both these factors turned out
to have very low volatility of volatility and therefore did not significantly change the results.
21
the correlations between the total IVs (ρZi,Zj ) in Table 13. We conclude that despite the market
volatility explaining most of the cross-sectional dependence in IVs, it does not explain all of it.
Additional IV factors may help to explain all the dependence in the idiosyncratic volatilities.
Sector
Financial
Energy
Consumer Staples
Industrials
Technology
Health Care
Consumer Discretionary
Utilities
Material
Stock
American Express Co.
Goldman Sachs Group
JPMorgan Chase & Co.
Chevron Corp.
Schlumberger Ltd.
Exxon Mobil Corp.
Coca Cola Company
Procter & Gamble
Wal-Mart Stores
Caterpillar Inc.
3M Company
United Technologies
Cisco Systems
International Bus. Machines
Intel Corp.
Johnson & Johnson
Merck & Co.
Pfizer Inc.
Home Depot
McDonald’s Corp.
Nike
Air Products & Chemicals Inc
Allegheny Technologies Inc
Avery Dennison Corp
Duke Energy
CenterPoint Energy
Exelon Corp.
Ticker
AXP
GS
JPM
CVX
SLB
XOM
KO
PG
WMT
CAT
MMM
UTX
CSCO
IBM
INTC
JNJ
MRK
PFE
HD
MCD
NKE
APD
ATI
AVY
DUK
CNP
EXC
CAPM
34.8
34.5
37.0
35.8
26.0
38.5
24.8
25.2
26.3
36.8
36.6
36.2
34.5
367
37.0
24.4
21.4
25.5
30.9
24.2
25.5
33.2
28.7
29.3
19.1
17.2
19.2
FF3 Model
43.5
43.1
45.2
44.2
35.7
46.5
34.6
34.9
36.0
45.2
45.0
44.5
43.1
44.8
45.2
34.3
31.5
33.9
39.9
34.1
35.3
42.0
38
38.5
29.7
28.0
29.8
Table 8: Average of the monthly contribution of the market volatility to stocks total volatility(RY2 j ) over
the period 2003:2012 in percentages. The first column provides information on the sectors, the second the
names of the companies and the third their tickers. The fourth and and fifth columns show RY2 j for the
CAPM and FF3 return model.
22
ρbZi,Zj
> 0.9
> 0.8
> 0.7
> 0.6
> 0.5
> 0.4
> 0.3
> 0.2
> 0.1
6= 0
CAPM
2
13
60
158
265
323
350
350
350
350
FF3 Model
2
13
58
163
265
323
350
350
350
350
Table 9: Number of pairs of stocks with significant dependence between their IVs and the estimated correlation greater than the threshold given in the first column. The second column shows the results for the
CAPM model. The results for the FF3 model are reported in the third column.
[ i , Zj )|
|Corr(Z
< 0.6
< 0.4
< 0.3
< 0.2
< 0.1
< 0.075
< 0.05
< 0.025
< 0.01
< 0.005
Pairs
351
350
348
343
323
303
265
152
56
29
CAPM
[
Avg |Corr(Zi , Zj )|
0.043
0.042
0.040
0.037
0.031
0.028
0.023
0.013
0.005
0.003
Avg ρbZi,Zj
0.585
0.584
0.583
0.583
0.580
0.579
0.570
0.568
0.579
0.580
Pairs
351
350
348
343
323
304
266
152
56
27
FF3 Model
[
Avg |Corr(Zi , Zj )|
0.043
0.042
0.041
0.038
0.031
0.028
0.023
0.013
0.005
0.003
Avg ρbZi,Zj
0.586
0.585
0.584
0.584
0.581
0.581
0.571
0.566
0.574
0.580
Table 10: We report the number of pairs of stocks with the absolute value of the correlation between their
idiosyncratic returns smaller than the threshold given in the first column, the average of the absolute value
of the idiosyncratic returns correlation for those pairs as well as the average of the IVs correlation for the
same pairs. The results are presented both for the CAPM and the FF3 models.
23
Stock
AXP
GS
JPM
CVX
SLB
XOM
KO
PG
WMT
CAT
MMM
UTX
CSCO
IBM
INTC
JNJ
MRK
PFE
HD
MCD
NKE
APD
ATI
AVY
DUK
CNP
EXC
bbz
1.600
2.313
1.899
0.611
1.064
0.576
0.327
0.427
0.445
0.589
0.389
0.501
0.571
0.339
0.454
0.404
0.535
0.434
0.500
0.516
0.528
0.535
1.698
0.315
0.415
0.740
0.927
CAPM
2,IV -F M
b
RZ
(%)
46.9
24.8
29.1
51.0
52.2
58.0
56.8
61.5
56.5
42.1
53.7
48.6
51.4
42.3
44.7
67.6
31.1
34.1
42.5
39.5
44.1
52.4
33.2
18.4
21.3
29.2
60.2
p-val
0.010
0.024
0.004
0.008
0.005
0.004
0.013
0.002
0.007
0.002
0.000
0.004
0.002
0.015
0.003
0.007
0.001
0.002
0.005
0.002
0.000
0.001
0.001
0.001
0.002
0.001
0.001
#
13
13
8
20
16
21
13
18
22
13
16
17
20
9
23
22
24
22
19
15
25
24
20
15
19
20
21
bbz
1.584
2.341
1.894
0.603
1.043
0.575
0.328
0.424
0.444
0.590
0.386
0.501
0.562
0.343
0.451
0.401
0.534
0.425
0.499
0.518
0.526
0.527
1.726
0.312
0.415
0.737
0.926
FF3 Model
2,IV -F M
b
RZ
(%)
46.8
25.5
29.0
51.1
48.4
51.6
56.7
61.5
56.7
42.0
53.1
51.3
44.1
43.2
44.0
67.1
31.1
24.0
42.5
39.7
44.1
51.2
34.1
18.3
21.7
29.3
59.9
p-val
0.011
0.018
0.004
0.008
0.005
0.004
0.012
0.002
0.007
0.003
0.000
0.004
0.002
0.011
0.003
0.007
0.001
0.001
0.004
0.002
0.000
0.001
0.001
0.001
0.002
0.001
0.001
#
13
13
9
20
15
19
14
19
22
14
15
17
21
9
22
21
24
22
16
15
24
24
19
14
20
20
21
Table 11: Estimates of the IV beta (bbZ ) and the contribution of the market volatility to the variation in the
b2,IV -F M ). We use the market volatility as the IV factor. P-val is the p-value of the test of the absence
IV (R
Z
of dependence between the IV and the market volatility for a given individual stock. In the column with
the heading #, we report the number of stocks with their NS-IV having a relatively large covariation with
the NS-IV of the stock listed in the first column (in particular, when the t-statistic based on the covariation
between the NS-IVs is larger than 1.96 in the absolute value).
24
7
Conclusion
This paper provides tools for the analysis of cross-sectional dependencies in idiosyncratic volatilities
using high frequency data. First, using a factor model in prices, we develop inference theory for
covariances and correlations between the idiosyncratic volatilities. Next, we study an idiosyncratic
volatility factor model, in which we decompose the co-movements in idiosyncratic volatilities into
two parts: those related to factors such as the market volatility, and the residual co-movements.
We provide the asymptotic theory for the estimators in the decomposition.
Empirically, we find that our IV Factor Model with market volatility as the only factor explains
a large part of the cross-sectional dependence in IVs. However, it is not able to explain all of it. It
therefore opens the room for the construction of additional IV factors based on economic theory,
for example, along the lines of the heterogeneous agents model of Herskovic, Kelly, Lustig, and
Nieuwerburgh (2014).
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27
Appendix
A
Figures and Tables
AXP
GS
1
1
0.5
0.5
0
2003
2005
2007
JPM
2010
2012
0
2003
1
1
0.5
0.5
0
2003
2005
2007
SLB
2010
2012
0
2003
1
1
0.5
0.5
0
2003
2005
2007
KO
2010
2012
0
2003
1
1
0.5
0.5
0
2003
2005
2007
WMT
2010
2012
0
2003
1
1
0.5
0.5
0
2003
2005
2007
MMM
2010
2012
0
2003
1
1
0.5
0.5
0
2003
2005
2007
CSCO
2010
2012
0
2003
1
1
0.5
0.5
0
2003
2005
2007
2010
2012
0
2003
2005
2007
CVX
2010
2012
2005
2007
XOM
2010
2012
2005
2007
PG
2010
2012
2005
2007
CAT
2010
2012
2005
2007
UTX
2010
2012
2005
2007
IBM
2010
2012
2005
2007
2010
2012
b2 ) over the period 2003:2012.
Figure 1: Monthly contribution of the return factors to the total volatility (R
Yj
The dotted blue line plots this measure calculated in CAPM model. The solid red line plots the same
measure obtained in the FF3 model. We use the ticker of the stocks to label the graphs.
28
INTC
JNJ
1
1
0.5
0.5
0
2003
2005
2007
MRK
2010
2012
0
2003
1
1
0.5
0.5
0
2003
2005
2007
HD
2010
2012
0
2003
1
1
0.5
0.5
0
2003
2005
2007
NKE
2010
2012
0
2003
1
1
0.5
0.5
0
2003
2005
2007
ATI
2010
2012
0
2003
1
1
0.5
0.5
0
2003
2005
2007
DUK
2010
2012
0
2003
1
1
0.5
0.5
0
2003
2005
2007
EXC
2010
2012
2005
2007
2010
2012
0
2003
2005
2007
PFE
2010
2012
2005
2007
MCD
2010
2012
2005
2007
APD
2010
2012
2005
2007
AVY
2010
2012
2005
2007
CNP
2010
2012
2005
2007
2010
2012
1
0.5
0
2003
b2 ) over the period
Figure 2: Monthly contribution of the return factors to stocks total volatility (R
Yj
2003:2012. The dotted blue line plots this measure calculated in CAPM model. The solid red line plots the
same measure obtained in the FF3 model. We use the ticker of the stocks to label the graphs.
29
30
0.148
0.190
-0.032
-0.017
-0.038
-0.004
-0.012
0.028
0.040
0.016
0.024
0.011
0.017
0.015
-0.011
-0.004
-0.000
0.051
0.019
0.032
0.008
0.019
0.022
-0.024
-0.002
-0.013
AXP
0.230
-0.022
-0.002
-0.024
-0.015
-0.010
0.012
0.040
0.026
0.006
0.020
0.017
0.017
-0.027
-0.011
-0.008
0.040
0.014
0.022
0.017
0.043
0.020
-0.020
-0.005
-0.030
GS
0.146
-0.052
-0.031
-0.045
-0.010
-0.014
0.015
0.007
-0.004
-0.009
-0.003
0.007
0.007
-0.014
-0.009
0.002
0.041
0.020
0.008
-0.008
0.014
0.003
-0.013
-0.016
-0.037
JPM
0.189
0.230
0.327
0.514
0.007
-0.004
-0.031
0.024
0.016
0.015
-0.010
-0.004
-0.014
0.003
-0.002
-0.003
-0.045
-0.006
-0.000
0.066
0.090
0.009
0.044
0.049
0.054
CVX
-0.032
-0.022
-0.049
0.295
-0.030
-0.042
-0.045
0.054
0.011
0.010
-0.011
-0.021
-0.012
-0.027
-0.023
-0.028
-0.038
-0.015
-0.000
0.054
0.110
0.012
0.008
0.021
0.021
SLB
-0.018
-0.001
-0.031
0.324
0.016
0.016
-0.027
0.018
0.012
0.011
-0.004
-0.003
-0.017
0.010
-0.004
0.003
-0.041
0.000
-0.011
0.051
0.075
-0.003
0.038
0.036
0.048
XOM
-0.038
-0.025
-0.044
0.515
0.293
0.170
0.078
0.002
0.057
0.061
0.026
0.061
0.018
0.119
0.084
0.082
0.025
0.066
0.043
0.023
-0.023
0.037
0.070
0.048
0.075
KO
-0.003
-0.016
-0.012
0.008
-0.030
0.017
0.079
-0.006
0.056
0.061
0.019
0.050
0.009
0.149
0.094
0.078
0.041
0.074
0.045
0.026
-0.036
0.031
0.077
0.044
0.070
PG
-0.009
-0.009
-0.013
-0.003
-0.041
0.018
0.167
0.018
0.035
0.034
0.038
0.048
0.034
0.055
0.032
0.045
0.192
0.081
0.087
0.028
-0.000
0.034
0.033
0.011
0.020
WMT
0.028
0.013
0.015
-0.031
-0.044
-0.025
0.075
0.080
0.100
0.113
0.041
0.032
0.039
-0.012
-0.012
-0.007
0.041
0.038
0.052
0.098
0.107
0.088
0.007
0.022
0.014
CAT
0.039
0.040
0.006
0.025
0.055
0.019
0.001
-0.005
0.017
0.139
0.028
0.069
0.037
0.054
0.036
0.035
0.045
0.058
0.047
0.116
0.055
0.097
0.042
0.047
0.054
MMM
0.016
0.025
-0.003
0.016
0.011
0.013
0.058
0.056
0.037
0.102
0.043
0.060
0.028
0.048
0.037
0.028
0.035
0.059
0.067
0.089
0.043
0.091
0.040
0.035
0.044
UTX
0.023
0.007
-0.009
0.014
0.011
0.010
0.060
0.063
0.035
0.114
0.139
0.102
0.182
0.020
0.019
0.021
0.035
0.037
0.030
0.024
0.022
0.032
0.010
0.003
0.006
CSCO
0.011
0.019
-0.002
-0.007
-0.012
-0.003
0.025
0.021
0.037
0.041
0.028
0.042
0.102
0.046
0.030
0.039
0.041
0.055
0.045
0.033
0.009
0.036
0.016
0.015
0.022
IBM
0.016
0.016
0.007
-0.002
-0.019
-0.002
0.059
0.053
0.050
0.030
0.068
0.062
0.103
0.008
0.019
0.023
0.046
0.031
0.039
0.027
0.027
0.030
-0.007
-0.007
-0.004
INTC
0.013
0.017
0.007
-0.014
-0.011
-0.017
0.019
0.007
0.034
0.039
0.038
0.027
0.182
0.101
Table 12: The correlation between stocks idiosyncratic returns over 2003-2012, Corr(Zi , Zj ).
The top panel reports the results for the CAPM, the bottom panel presents the same results for the FF3 model.
AXP
GS
JPM
CVX
SLB
XOM
KO
PG
WMT
CAT
MMM
UTX
CSCO
IBM
INTC
JNJ
MRK
PFE
HD
MCD
NKE
APD
ATI
AVY
DUK
CNP
EXC
0.169
0.166
0.017
0.055
0.029
0.018
-0.030
0.014
0.066
0.048
0.067
JNJ
-0.009
-0.025
-0.012
0.004
-0.025
0.012
0.119
0.151
0.057
-0.012
0.054
0.048
0.020
0.046
0.009
0.203
0.016
0.046
0.026
0.020
-0.015
0.021
0.054
0.042
0.059
MRK
-0.003
-0.011
-0.008
-0.001
-0.023
-0.003
0.084
0.095
0.032
-0.013
0.037
0.037
0.018
0.030
0.019
0.170
0.025
0.036
0.022
0.014
-0.020
0.016
0.055
0.047
0.050
PFE
0.002
-0.006
0.003
-0.004
-0.027
0.004
0.081
0.080
0.045
-0.006
0.037
0.027
0.021
0.038
0.022
0.166
0.203
0.093
0.107
0.030
0.016
0.048
0.020
0.009
0.007
HD
0.052
0.040
0.040
-0.047
-0.037
-0.042
0.023
0.040
0.191
0.041
0.044
0.036
0.036
0.041
0.046
0.017
0.018
0.025
0.083
0.044
0.011
0.042
0.036
0.024
0.031
MCD
0.021
0.013
0.020
-0.005
-0.014
0.001
0.066
0.077
0.081
0.038
0.059
0.061
0.037
0.058
0.030
0.056
0.046
0.038
0.094
0.065
0.034
0.071
0.023
0.029
0.032
NKE
0.029
0.020
0.008
-0.001
0.000
-0.013
0.042
0.044
0.086
0.053
0.047
0.067
0.029
0.047
0.038
0.029
0.026
0.022
0.107
0.082
0.144
0.163
0.042
0.050
0.053
APD
0.007
0.014
-0.007
0.066
0.055
0.051
0.023
0.027
0.028
0.099
0.115
0.089
0.024
0.033
0.025
0.019
0.022
0.014
0.030
0.044
0.064
0.086
0.008
0.028
0.012
ATI
0.017
0.043
0.014
0.088
0.108
0.074
-0.023
-0.038
0.000
0.107
0.055
0.043
0.021
0.008
0.028
-0.030
-0.014
-0.020
0.017
0.010
0.035
0.144
0.042
0.055
0.046
AVY
0.022
0.018
0.002
0.011
0.013
-0.002
0.036
0.031
0.035
0.087
0.097
0.091
0.032
0.036
0.030
0.014
0.021
0.016
0.049
0.043
0.071
0.162
0.086
0.230
0.306
DUK
-0.023
-0.018
-0.011
0.045
0.008
0.040
0.070
0.076
0.032
0.005
0.043
0.040
0.010
0.016
-0.007
0.066
0.055
0.056
0.021
0.037
0.023
0.041
0.008
0.043
0.242
CNP
-0.003
-0.003
-0.015
0.048
0.021
0.036
0.048
0.044
0.011
0.022
0.047
0.034
0.003
0.014
-0.008
0.047
0.041
0.046
0.008
0.024
0.030
0.049
0.028
0.056
0.231
EXC
-0.013
-0.027
-0.032
0.053
0.021
0.048
0.076
0.068
0.019
0.014
0.053
0.044
0.006
0.022
-0.005
0.065
0.060
0.050
0.007
0.029
0.031
0.051
0.013
0.046
0.306
0.243
31
0.598
(0.002)
0.532
(0.001)
0.793
(0.001)
0.648
(0.016)
0.800
(0.002)
0.651
(0.021)
0.761
(0.001)
0.782
(0.001)
0.530
(0.008)
0.554
(0.013)
0.647
(0.010)
(0.012)
0.565
0.606
(0.033)
0.667
(0.001)
0.697
(0.004)
0.538
(0.000)
0.563
(0.001)
0.658
(0.000)
0.759
(0.000)
0.550
(0.001)
0.609
(0.003)
0.469
(0.007)
0.331
(0.004)
0.423
(0.000)
0.532
(0.000)
0.686
(0.001)
AXP
0.679
(0.000)
0.587
(0.007)
0.455
(0.025)
0.675
(0.006)
0.539
(0.008)
0.586
(0.007)
0.512
(0.008)
0.455
(0.015)
0.434
(0.034)
0.495
(0.006)
(0.033)
0.455
0.429
(0.095)
0.619
(0.002)
0.492
(0.011)
0.402
(0.004)
0.480
(0.008)
0.621
(0.002)
0.417
(0.016)
0.521
(0.001)
0.661
(0.001)
0.377
(0.038)
0.284
(0.006)
0.393
(0.001)
0.451
(0.004)
0.600
(0.002)
GS
0.600
(0.003)
0.524
(0.000)
0.495
(0.002)
0.592
(0.000)
0.558
(0.001)
0.540
(0.000)
0.492
(0.001)
0.474
(0.002)
0.425
(0.009)
0.690
(0.000)
(0.000)
0.509
0.682
(0.001)
0.581
(0.000)
0.533
(0.001)
0.431
(0.000)
0.407
(0.000)
0.459
(0.003)
0.460
(0.000)
0.570
(0.000)
0.609
(0.000)
0.444
(0.000)
0.320
(0.001)
0.353
(0.000)
0.435
(0.000)
0.563
(0.000)
JPM
0.536
(0.000)
0.678
(0.000)
0.779
(0.006)
0.944
(0.000)
0.743
(0.016)
0.874
(0.000)
0.826
(0.001)
0.573
(0.007)
0.653
(0.001)
0.758
(0.004)
(0.006)
0.640
0.647
(0.034)
0.697
(0.001)
0.791
(0.003)
0.552
(0.000)
0.536
(0.001)
0.700
(0.001)
0.757
(0.000)
0.623
(0.000)
0.677
(0.002)
0.538
(0.002)
0.387
(0.000)
0.476
(0.000)
0.594
(0.000)
0.752
(0.000)
CVX
0.798
(0.001)
0.581
(0.007)
0.529
(0.000)
0.753
(0.007)
0.737
(0.013)
0.700
(0.005)
0.734
(0.005)
0.674
(0.001)
0.601
(0.001)
0.729
(0.006)
(0.005)
0.629
0.681
(0.012)
0.618
(0.004)
0.704
(0.007)
0.511
(0.001)
0.485
(0.004)
0.683
(0.001)
0.639
(0.005)
0.608
(0.000)
0.715
(0.000)
0.578
(0.001)
0.413
(0.000)
0.402
(0.002)
0.492
(0.003)
0.689
(0.001)
SLB
0.652
(0.013)
0.447
(0.029)
0.498
(0.001)
0.779
(0.005)
0.726
(0.020)
0.904
(0.000)
0.853
(0.001)
0.599
(0.006)
0.725
(0.001)
0.762
(0.003)
(0.004)
0.718
0.664
(0.036)
0.717
(0.001)
0.806
(0.003)
0.588
(0.000)
0.621
(0.001)
0.685
(0.001)
0.791
(0.000)
0.673
(0.000)
0.710
(0.001)
0.574
(0.001)
0.427
(0.000)
0.506
(0.000)
0.640
(0.000)
0.811
(0.000)
XOM
0.802
(0.001)
0.666
(0.006)
0.590
(0.000)
0.944
(0.000)
0.756
(0.007)
0.712
(0.014)
0.693
(0.016)
0.663
(0.003)
0.621
(0.003)
0.816
(0.004)
(0.015)
0.587
0.686
(0.025)
0.669
(0.003)
0.751
(0.013)
0.575
(0.001)
0.463
(0.010)
0.654
(0.003)
0.567
(0.016)
0.626
(0.001)
0.669
(0.002)
0.448
(0.013)
0.424
(0.000)
0.481
(0.001)
0.556
(0.002)
0.675
(0.006)
KO
0.651
(0.020)
0.531
(0.008)
0.552
(0.001)
0.739
(0.015)
0.735
(0.013)
0.725
(0.019)
0.847
(0.000)
0.592
(0.004)
0.732
(0.000)
0.756
(0.001)
(0.001)
0.706
0.669
(0.016)
0.698
(0.000)
0.817
(0.002)
0.577
(0.000)
0.604
(0.000)
0.693
(0.000)
0.792
(0.000)
0.685
(0.000)
0.708
(0.001)
0.573
(0.001)
0.398
(0.000)
0.509
(0.000)
0.645
(0.000)
0.790
(0.000)
PG
0.769
(0.000)
0.580
(0.008)
0.547
(0.000)
0.877
(0.000)
0.704
(0.004)
0.904
(0.000)
0.712
(0.014)
0.596
(0.006)
0.705
(0.002)
0.708
(0.002)
(0.003)
0.731
0.656
(0.020)
0.656
(0.001)
0.817
(0.005)
0.574
(0.000)
0.588
(0.001)
0.725
(0.000)
0.805
(0.000)
0.634
(0.000)
0.694
(0.001)
0.544
(0.003)
0.391
(0.001)
0.474
(0.001)
0.585
(0.000)
0.796
(0.000)
WMT
0.785
(0.001)
0.505
(0.010)
0.490
(0.000)
0.831
(0.001)
0.739
(0.004)
0.852
(0.001)
0.694
(0.014)
0.849
(0.000)
0.609
(0.001)
0.609
(0.001)
(0.003)
0.563
0.613
(0.003)
0.623
(0.000)
0.603
(0.007)
0.505
(0.001)
0.504
(0.001)
0.554
(0.001)
0.517
(0.004)
0.601
(0.000)
0.687
(0.000)
0.547
(0.000)
0.406
(0.000)
0.382
(0.002)
0.422
(0.002)
0.593
(0.003)
CAT
0.527
(0.005)
0.452
(0.014)
0.477
(0.002)
0.572
(0.005)
0.671
(0.000)
0.597
(0.006)
0.662
(0.003)
0.587
(0.003)
0.592
(0.004)
0.619
(0.000)
(0.000)
0.644
0.537
(0.026)
0.557
(0.002)
0.735
(0.002)
0.611
(0.000)
0.548
(0.001)
0.559
(0.003)
0.628
(0.001)
0.599
(0.000)
0.700
(0.000)
0.540
(0.000)
0.488
(0.000)
0.451
(0.001)
0.537
(0.000)
0.751
(0.001)
MMM
0.557
(0.012)
0.430
(0.041)
0.421
(0.008)
0.653
(0.002)
0.600
(0.001)
0.727
(0.001)
0.625
(0.003)
0.728
(0.000)
0.702
(0.002)
0.607
(0.000)
(0.000)
0.689
0.742
(0.004)
0.683
(0.001)
0.732
(0.001)
0.565
(0.000)
0.510
(0.000)
0.609
(0.002)
0.668
(0.000)
0.692
(0.000)
0.663
(0.000)
0.549
(0.000)
0.459
(0.000)
0.446
(0.000)
0.581
(0.000)
0.642
(0.000)
UTX
0.647
(0.009)
0.486
(0.007)
0.690
(0.000)
0.756
(0.003)
0.733
(0.005)
0.758
(0.003)
0.813
(0.004)
0.760
(0.001)
0.706
(0.002)
0.608
(0.001)
0.617
(0.000)
0.710
(0.003)
0.665
(0.001)
0.675
(0.008)
0.510
(0.001)
0.537
(0.001)
0.603
(0.001)
0.612
(0.002)
0.643
(0.000)
0.651
(0.001)
0.568
(0.000)
0.337
(0.002)
0.410
(0.002)
0.503
(0.002)
0.684
(0.001)
CSCO
0.568
(0.010)
0.448
(0.032)
0.506
(0.000)
0.643
(0.005)
0.634
(0.004)
0.723
(0.003)
0.584
(0.014)
0.709
(0.001)
0.733
(0.002)
0.558
(0.002)
0.645
(0.000)
0.687
(0.000)
0.673
(0.003)
0.644
(0.032)
0.482
(0.003)
0.464
(0.006)
0.535
(0.012)
0.601
(0.021)
0.606
(0.000)
0.603
(0.002)
0.549
(0.001)
0.345
(0.005)
0.370
(0.009)
0.453
(0.013)
0.619
(0.006)
IBM
0.602
(0.030)
0.421
(0.098)
0.679
(0.001)
0.646
(0.030)
0.680
(0.012)
0.658
(0.036)
0.681
(0.025)
0.670
(0.017)
0.651
(0.018)
0.609
(0.002)
0.527
(0.027)
0.735
(0.006)
(0.002)
0.699
Table 13: The correlation between the idiosyncratic volatilities, ρZi,Zj .
The results for CAPM are in the upper panel; the results for FF3 model are in the bottom panel.
The figures in the parenthesis are the p-values of the LIN test of the absence of dependence in the IVs.
We use the market volatility as IV factor. θ = 2.5, T = 10 years, and ∆n = 5mins.
EXC
CNP
DUK
AVY
ATI
APD
NKE
MCD
HD
PFE
MRK
JNJ
INTC
IBM
CSCO
UTX
MMM
CAT
WMT
PG
KO
XOM
SLB
CVX
JPM
GS
AXP
0.628
(0.003)
0.505
(0.000)
0.495
(0.001)
0.659
(0.000)
0.558
(0.000)
0.655
(0.000)
0.674
(0.000)
0.517
(0.001)
0.382
(0.000)
0.390
(0.001)
0.537
(0.000)
0.652
(0.000)
INTC
0.674
(0.000)
0.612
(0.001)
0.577
(0.000)
0.701
(0.001)
0.622
(0.003)
0.725
(0.001)
0.664
(0.003)
0.709
(0.000)
0.665
(0.001)
0.623
(0.000)
0.564
(0.002)
0.687
(0.001)
(0.001)
0.670
0.673
(0.002)
0.595
(0.001)
0.576
(0.002)
0.633
(0.002)
0.706
(0.001)
0.655
(0.001)
0.695
(0.003)
0.555
(0.003)
0.399
(0.001)
0.582
(0.000)
0.595
(0.001)
0.783
(0.002)
JNJ
0.699
(0.004)
0.483
(0.013)
0.535
(0.000)
0.792
(0.003)
0.710
(0.005)
0.806
(0.003)
0.753
(0.012)
0.814
(0.002)
0.818
(0.004)
0.605
(0.006)
0.730
(0.002)
0.734
(0.001)
(0.006)
0.679
0.643
(0.030)
0.636
(0.002)
0.546
(0.000)
0.522
(0.000)
0.545
(0.000)
0.483
(0.000)
0.585
(0.000)
0.407
(0.001)
0.350
(0.000)
0.373
(0.000)
0.446
(0.000)
0.650
(0.000)
MRK
0.541
(0.000)
0.402
(0.004)
0.433
(0.000)
0.551
(0.000)
0.512
(0.001)
0.587
(0.000)
0.576
(0.001)
0.577
(0.000)
0.574
(0.000)
0.503
(0.000)
0.608
(0.000)
0.565
(0.000)
(0.001)
0.506
0.477
(0.004)
0.509
(0.000)
0.592
(0.001)
0.516
(0.000)
0.552
(0.000)
0.526
(0.000)
0.535
(0.000)
0.457
(0.000)
0.389
(0.000)
0.381
(0.000)
0.446
(0.000)
0.591
(0.000)
PFE
0.577
(0.000)
0.478
(0.008)
0.414
(0.000)
0.549
(0.001)
0.492
(0.003)
0.632
(0.001)
0.460
(0.010)
0.615
(0.000)
0.598
(0.001)
0.498
(0.001)
0.547
(0.001)
0.515
(0.000)
(0.001)
0.546
0.465
(0.005)
0.504
(0.001)
0.582
(0.002)
0.548
(0.000)
0.587
(0.000)
0.572
(0.001)
0.693
(0.000)
0.490
(0.002)
0.366
(0.000)
0.426
(0.000)
0.497
(0.000)
0.669
(0.000)
HD
0.660
(0.000)
0.613
(0.002)
0.454
(0.003)
0.694
(0.001)
0.680
(0.001)
0.678
(0.001)
0.650
(0.003)
0.692
(0.000)
0.724
(0.000)
0.550
(0.001)
0.556
(0.002)
0.605
(0.002)
(0.001)
0.598
0.529
(0.011)
0.657
(0.000)
0.631
(0.002)
0.521
(0.000)
0.518
(0.000)
0.553
(0.000)
0.593
(0.000)
0.537
(0.000)
0.321
(0.001)
0.444
(0.000)
0.526
(0.000)
0.686
(0.000)
MCD
0.759
(0.000)
0.409
(0.020)
0.459
(0.000)
0.759
(0.000)
0.642
(0.004)
0.786
(0.000)
0.566
(0.015)
0.794
(0.000)
0.802
(0.000)
0.512
(0.002)
0.623
(0.001)
0.665
(0.000)
(0.001)
0.613
0.596
(0.019)
0.568
(0.000)
0.702
(0.001)
0.544
(0.000)
0.566
(0.000)
0.583
(0.000)
0.695
(0.000)
0.542
(0.000)
0.436
(0.000)
0.396
(0.000)
0.537
(0.000)
0.685
(0.000)
NKE
0.545
(0.001)
0.513
(0.001)
0.561
(0.000)
0.622
(0.000)
0.609
(0.000)
0.669
(0.000)
0.621
(0.001)
0.684
(0.000)
0.631
(0.000)
0.600
(0.000)
0.593
(0.000)
0.689
(0.000)
(0.000)
0.638
0.598
(0.000)
0.660
(0.000)
0.653
(0.001)
0.479
(0.000)
0.523
(0.000)
0.565
(0.000)
0.549
(0.000)
0.630
(0.000)
0.402
(0.000)
0.467
(0.000)
0.524
(0.000)
0.781
(0.000)
APD
0.607
(0.001)
0.653
(0.001)
0.609
(0.000)
0.673
(0.001)
0.714
(0.000)
0.709
(0.001)
0.671
(0.002)
0.706
(0.000)
0.693
(0.001)
0.688
(0.000)
0.702
(0.000)
0.665
(0.000)
(0.000)
0.652
0.594
(0.002)
0.671
(0.000)
0.699
(0.002)
0.587
(0.000)
0.536
(0.000)
0.689
(0.000)
0.587
(0.000)
0.694
(0.000)
0.343
(0.000)
0.442
(0.001)
0.427
(0.001)
0.578
(0.000)
ATI
0.463
(0.006)
0.372
(0.040)
0.446
(0.000)
0.538
(0.001)
0.576
(0.000)
0.572
(0.001)
0.444
(0.012)
0.568
(0.000)
0.538
(0.002)
0.543
(0.000)
0.534
(0.000)
0.548
(0.000)
(0.000)
0.567
0.543
(0.001)
0.517
(0.001)
0.549
(0.003)
0.402
(0.001)
0.456
(0.000)
0.482
(0.001)
0.532
(0.000)
0.537
(0.000)
0.628
(0.000)
0.251
(0.001)
0.312
(0.000)
0.398
(0.000)
AVY
0.332
(0.003)
0.279
(0.006)
0.317
(0.001)
0.388
(0.000)
0.412
(0.000)
0.427
(0.000)
0.421
(0.000)
0.399
(0.000)
0.390
(0.001)
0.403
(0.000)
0.488
(0.000)
0.458
(0.000)
(0.002)
0.335
0.341
(0.005)
0.381
(0.000)
0.402
(0.001)
0.347
(0.000)
0.387
(0.000)
0.363
(0.000)
0.320
(0.000)
0.432
(0.000)
0.403
(0.000)
0.340
(0.000)
0.540
(0.000)
0.531
(0.000)
DUK
0.419
(0.000)
0.386
(0.002)
0.351
(0.000)
0.472
(0.000)
0.399
(0.002)
0.500
(0.000)
0.477
(0.001)
0.502
(0.000)
0.469
(0.001)
0.376
(0.001)
0.446
(0.001)
0.443
(0.000)
(0.002)
0.406
0.364
(0.010)
0.389
(0.001)
0.576
(0.000)
0.369
(0.000)
0.380
(0.000)
0.422
(0.000)
0.439
(0.000)
0.390
(0.000)
0.462
(0.000)
0.439
(0.001)
0.248
(0.001)
0.615
(0.000)
CNP
0.533
(0.000)
0.444
(0.004)
0.436
(0.000)
0.592
(0.000)
0.494
(0.002)
0.639
(0.000)
0.552
(0.002)
0.644
(0.000)
0.582
(0.000)
0.416
(0.002)
0.533
(0.000)
0.581
(0.000)
(0.001)
0.502
0.450
(0.012)
0.541
(0.000)
0.592
(0.001)
0.447
(0.000)
0.447
(0.000)
0.494
(0.000)
0.524
(0.000)
0.537
(0.000)
0.522
(0.000)
0.422
(0.001)
0.310
(0.000)
0.536
(0.000)
EXC
0.690
(0.001)
0.594
(0.003)
0.562
(0.000)
0.757
(0.000)
0.694
(0.001)
0.812
(0.000)
0.683
(0.005)
0.791
(0.000)
0.796
(0.000)
0.597
(0.002)
0.748
(0.001)
0.648
(0.000)
(0.001)
0.685
0.618
(0.008)
0.662
(0.000)
0.781
(0.002)
0.652
(0.000)
0.595
(0.000)
0.668
(0.000)
0.682
(0.000)
0.687
(0.000)
0.783
(0.000)
0.575
(0.000)
0.398
(0.000)
0.524
(0.000)
0.616
(0.000)
32
0.402
(0.028)
0.266
(0.186)
0.597
(0.025)
0.309
(0.122)
0.592
(0.004)
0.283
(0.196)
0.495
(0.012)
0.555
(0.000)
0.155
(0.233)
0.111
(0.522)
0.309
(0.208)
0.204
(0.110)
0.284
(0.242)
0.390
(0.018)
0.325
(0.000)
0.259
(0.003)
0.276
(0.000)
0.383
(0.050)
0.578
(0.001)
0.175
(0.155)
0.234
(0.074)
0.118
(0.192)
0.058
(0.352)
0.161
(0.003)
0.263
(0.022)
0.340
(0.005)
AXP
0.560
(0.003)
0.375
(0.040)
0.153
(0.306)
0.518
(0.005)
0.279
(0.089)
0.354
(0.067)
0.231
(0.160)
0.193
(0.262)
0.112
(0.339)
0.221
(0.094)
0.186
(0.222)
0.149
(0.634)
0.439
(0.001)
0.158
(0.233)
0.168
(0.007)
0.264
(0.008)
0.446
(0.001)
0.147
(0.369)
0.288
(0.001)
0.498
(0.000)
0.117
(0.307)
0.088
(0.059)
0.206
(0.001)
0.244
(0.019)
0.383
(0.004)
GS
0.410
(0.028)
0.235
(0.277)
0.184
(0.202)
0.334
(0.109)
0.276
(0.086)
0.226
(0.287)
0.157
(0.418)
0.195
(0.175)
0.057
(0.729)
0.518
(0.005)
0.241
(0.147)
0.517
(0.013)
0.355
(0.038)
0.192
(0.187)
0.187
(0.032)
0.136
(0.125)
0.170
(0.333)
0.184
(0.280)
0.338
(0.025)
0.380
(0.003)
0.190
(0.018)
0.118
(0.129)
0.137
(0.057)
0.203
(0.102)
0.276
(0.056)
JPM
0.271
(0.172)
0.561
(0.003)
0.546
(0.022)
0.882
(0.000)
0.445
(0.076)
0.722
(0.001)
0.626
(0.001)
0.205
(0.221)
0.275
(0.047)
0.503
(0.095)
0.316
(0.043)
0.337
(0.267)
0.425
(0.016)
0.512
(0.000)
0.263
(0.000)
0.211
(0.005)
0.441
(0.025)
0.565
(0.007)
0.283
(0.026)
0.337
(0.030)
0.213
(0.034)
0.129
(0.094)
0.231
(0.001)
0.352
(0.008)
0.450
(0.002)
CVX
0.605
(0.022)
0.372
(0.045)
0.244
(0.257)
0.459
(0.049)
0.430
(0.012)
0.316
(0.152)
0.422
(0.014)
0.393
(0.000)
0.163
(0.146)
0.442
(0.035)
0.292
(0.044)
0.399
(0.005)
0.273
(0.039)
0.289
(0.029)
0.191
(0.070)
0.116
(0.201)
0.408
(0.004)
0.345
(0.105)
0.253
(0.008)
0.414
(0.001)
0.282
(0.006)
0.168
(0.023)
0.109
(0.103)
0.177
(0.173)
0.302
(0.009)
SLB
0.312
(0.132)
0.146
(0.340)
0.186
(0.181)
0.544
(0.024)
0.361
(0.171)
0.764
(0.000)
0.657
(0.000)
0.215
(0.140)
0.384
(0.024)
0.480
(0.074)
0.439
(0.003)
0.336
(0.281)
0.438
(0.010)
0.492
(0.000)
0.304
(0.000)
0.337
(0.000)
0.385
(0.044)
0.619
(0.000)
0.345
(0.011)
0.366
(0.008)
0.247
(0.019)
0.174
(0.012)
0.265
(0.000)
0.418
(0.000)
0.542
(0.000)
XOM
0.594
(0.004)
0.510
(0.008)
0.328
(0.119)
0.882
(0.000)
0.460
(0.053)
0.299
(0.153)
0.292
(0.061)
0.349
(0.020)
0.161
(0.352)
0.603
(0.014)
0.177
(0.232)
0.386
(0.018)
0.343
(0.015)
0.355
(0.018)
0.284
(0.002)
0.044
(0.507)
0.327
(0.018)
0.181
(0.358)
0.255
(0.011)
0.284
(0.014)
0.017
(0.983)
0.171
(0.020)
0.224
(0.001)
0.268
(0.019)
0.223
(0.141)
KO
0.282
(0.192)
0.274
(0.110)
0.263
(0.093)
0.437
(0.071)
0.420
(0.012)
0.354
(0.143)
0.628
(0.000)
0.176
(0.261)
0.378
(0.031)
0.449
(0.055)
0.399
(0.003)
0.329
(0.245)
0.383
(0.016)
0.491
(0.001)
0.271
(0.001)
0.290
(0.000)
0.386
(0.040)
0.618
(0.000)
0.354
(0.007)
0.339
(0.032)
0.229
(0.040)
0.113
(0.148)
0.261
(0.000)
0.422
(0.001)
0.467
(0.002)
PG
0.513
(0.012)
0.353
(0.078)
0.237
(0.265)
0.729
(0.001)
0.321
(0.122)
0.764
(0.000)
0.298
(0.146)
0.216
(0.046)
0.347
(0.038)
0.368
(0.047)
0.470
(0.000)
0.326
(0.088)
0.317
(0.016)
0.530
(0.000)
0.282
(0.000)
0.279
(0.000)
0.470
(0.006)
0.647
(0.000)
0.272
(0.010)
0.338
(0.001)
0.197
(0.011)
0.116
(0.028)
0.212
(0.000)
0.321
(0.000)
0.512
(0.000)
WMT
0.562
(0.000)
0.229
(0.166)
0.152
(0.434)
0.637
(0.000)
0.430
(0.014)
0.653
(0.000)
0.295
(0.043)
0.634
(0.000)
0.262
(0.024)
0.272
(0.028)
0.233
(0.062)
0.325
(0.006)
0.338
(0.014)
0.164
(0.152)
0.227
(0.012)
0.203
(0.013)
0.228
(0.099)
0.184
(0.107)
0.300
(0.017)
0.419
(0.001)
0.273
(0.003)
0.187
(0.005)
0.119
(0.073)
0.111
(0.157)
0.189
(0.130)
CAT
0.149
(0.192)
0.196
(0.256)
0.198
(0.167)
0.204
(0.179)
0.384
(0.000)
0.208
(0.160)
0.347
(0.020)
0.166
(0.202)
0.207
(0.028)
0.203
(0.155)
0.312
(0.037)
0.112
(0.577)
0.143
(0.294)
0.352
(0.012)
0.361
(0.000)
0.222
(0.026)
0.162
(0.205)
0.317
(0.077)
0.224
(0.036)
0.374
(0.001)
0.206
(0.040)
0.285
(0.000)
0.184
(0.004)
0.248
(0.009)
0.431
(0.006)
MMM
0.112
(0.505)
0.110
(0.317)
0.044
(0.744)
0.272
(0.045)
0.150
(0.172)
0.382
(0.016)
0.162
(0.248)
0.362
(0.040)
0.337
(0.037)
0.255
(0.019)
0.409
(0.014)
0.516
(0.012)
0.399
(0.021)
0.364
(0.004)
0.286
(0.004)
0.163
(0.018)
0.269
(0.077)
0.401
(0.084)
0.414
(0.004)
0.310
(0.028)
0.231
(0.036)
0.242
(0.002)
0.182
(0.008)
0.330
(0.039)
0.199
(0.127)
UTX
0.308
(0.188)
0.214
(0.106)
0.516
(0.005)
0.501
(0.082)
0.446
(0.034)
0.469
(0.079)
0.594
(0.015)
0.456
(0.043)
0.362
(0.052)
0.268
(0.033)
0.193
(0.153)
0.486
(0.002)
0.401
(0.006)
0.306
(0.005)
0.225
(0.031)
0.247
(0.000)
0.299
(0.031)
0.334
(0.008)
0.361
(0.001)
0.337
(0.006)
0.297
(0.002)
0.078
(0.289)
0.152
(0.030)
0.228
(0.029)
0.359
(0.000)
CSCO
0.206
(0.109)
0.180
(0.215)
0.234
(0.131)
0.319
(0.031)
0.296
(0.036)
0.445
(0.002)
0.166
(0.170)
0.402
(0.002)
0.472
(0.000)
0.221
(0.035)
0.308
(0.036)
0.402
(0.015)
0.421
(0.018)
0.244
(0.309)
0.184
(0.075)
0.132
(0.148)
0.186
(0.390)
0.319
(0.193)
0.301
(0.008)
0.252
(0.106)
0.270
(0.009)
0.095
(0.183)
0.096
(0.281)
0.154
(0.285)
0.232
(0.146)
IBM
0.283
(0.228)
0.148
(0.633)
0.513
(0.014)
0.341
(0.263)
0.401
(0.005)
0.330
(0.288)
0.382
(0.018)
0.340
(0.208)
0.324
(0.103)
0.324
(0.005)
0.098
(0.614)
0.507
(0.022)
0.469
(0.001)
S
Table 14: The correlation between the NS-IVs, ρN
Zi,Zj .
The results for CAPM are in the upper panel; the results for the FF3 model are in the bottom panel.
The figures in the parenthesis are the p-values of the LIN test of the absence of dependence in the NS-IVs.
We use the market volatility as IV factor. θ = 2.5, T = 10 years, and ∆n = 5mins.
EXC
CNP
DUK
AVY
ATI
APD
NKE
MCD
HD
PFE
MRK
JNJ
INTC
IBM
CSCO
UTX
MMM
CAT
WMT
PG
KO
XOM
SLB
CVX
JPM
GS
AXP
0.198
(0.031)
0.218
(0.004)
0.179
(0.010)
0.399
(0.002)
0.240
(0.089)
0.383
(0.004)
0.381
(0.002)
0.213
(0.021)
0.146
(0.041)
0.122
(0.111)
0.282
(0.007)
0.292
(0.021)
INTC
0.399
(0.017)
0.433
(0.001)
0.346
(0.041)
0.430
(0.018)
0.270
(0.050)
0.447
(0.009)
0.328
(0.023)
0.400
(0.014)
0.331
(0.013)
0.334
(0.010)
0.147
(0.181)
0.400
(0.024)
0.404
(0.005)
0.422
(0.016)
0.290
(0.000)
0.211
(0.000)
0.227
(0.050)
0.426
(0.000)
0.258
(0.000)
0.272
(0.007)
0.165
(0.038)
0.095
(0.049)
0.394
(0.000)
0.314
(0.002)
0.410
(0.003)
JNJ
0.327
(0.001)
0.149
(0.290)
0.191
(0.195)
0.515
(0.000)
0.294
(0.014)
0.487
(0.000)
0.358
(0.012)
0.478
(0.002)
0.533
(0.000)
0.166
(0.157)
0.330
(0.017)
0.365
(0.004)
0.310
(0.001)
0.250
(0.296)
0.203
(0.042)
0.327
(0.000)
0.252
(0.010)
0.301
(0.000)
0.181
(0.005)
0.321
(0.000)
0.122
(0.029)
0.149
(0.001)
0.154
(0.000)
0.207
(0.000)
0.417
(0.000)
MRK
0.263
(0.003)
0.173
(0.018)
0.188
(0.035)
0.263
(0.001)
0.189
(0.072)
0.303
(0.000)
0.286
(0.004)
0.272
(0.001)
0.283
(0.000)
0.223
(0.010)
0.353
(0.000)
0.286
(0.005)
0.217
(0.035)
0.181
(0.078)
0.221
(0.003)
0.284
(0.000)
0.220
(0.004)
0.292
(0.000)
0.228
(0.000)
0.208
(0.001)
0.177
(0.001)
0.190
(0.000)
0.153
(0.000)
0.190
(0.000)
0.272
(0.000)
PFE
0.299
(0.000)
0.266
(0.008)
0.145
(0.108)
0.233
(0.002)
0.125
(0.147)
0.357
(0.000)
0.039
(0.545)
0.311
(0.000)
0.297
(0.000)
0.194
(0.010)
0.217
(0.032)
0.170
(0.017)
0.259
(0.000)
0.138
(0.120)
0.188
(0.007)
0.220
(0.000)
0.331
(0.000)
0.299
(0.085)
0.244
(0.027)
0.428
(0.000)
0.178
(0.044)
0.127
(0.020)
0.182
(0.006)
0.226
(0.032)
0.342
(0.008)
HD
0.387
(0.051)
0.439
(0.002)
0.159
(0.312)
0.431
(0.032)
0.399
(0.006)
0.368
(0.055)
0.318
(0.027)
0.384
(0.047)
0.468
(0.007)
0.220
(0.090)
0.152
(0.152)
0.259
(0.088)
0.289
(0.035)
0.181
(0.318)
0.391
(0.005)
0.220
(0.074)
0.250
(0.011)
0.223
(0.007)
0.232
(0.030)
0.261
(0.026)
0.268
(0.007)
0.073
(0.275)
0.220
(0.000)
0.284
(0.005)
0.403
(0.000)
MCD
0.581
(0.001)
0.143
(0.397)
0.183
(0.278)
0.570
(0.005)
0.350
(0.107)
0.610
(0.000)
0.181
(0.347)
0.625
(0.000)
0.642
(0.000)
0.177
(0.103)
0.307
(0.078)
0.395
(0.088)
0.336
(0.005)
0.318
(0.193)
0.256
(0.074)
0.420
(0.000)
0.300
(0.000)
0.315
(0.000)
0.294
(0.094)
0.421
(0.000)
0.253
(0.001)
0.224
(0.003)
0.130
(0.008)
0.283
(0.003)
0.361
(0.000)
NKE
0.166
(0.178)
0.281
(0.002)
0.323
(0.030)
0.282
(0.029)
0.250
(0.011)
0.338
(0.006)
0.247
(0.014)
0.351
(0.008)
0.267
(0.011)
0.298
(0.008)
0.210
(0.052)
0.408
(0.005)
0.350
(0.002)
0.293
(0.009)
0.389
(0.002)
0.252
(0.001)
0.175
(0.006)
0.223
(0.000)
0.233
(0.031)
0.227
(0.035)
0.374
(0.000)
0.153
(0.017)
0.216
(0.001)
0.232
(0.013)
0.514
(0.000)
APD
0.221
(0.086)
0.490
(0.000)
0.376
(0.003)
0.324
(0.025)
0.400
(0.001)
0.353
(0.014)
0.278
(0.017)
0.322
(0.037)
0.326
(0.002)
0.416
(0.001)
0.366
(0.001)
0.303
(0.028)
0.330
(0.007)
0.235
(0.125)
0.364
(0.003)
0.263
(0.008)
0.319
(0.000)
0.202
(0.001)
0.414
(0.001)
0.247
(0.034)
0.414
(0.000)
0.127
(0.086)
0.237
(0.081)
0.163
(0.149)
0.245
(0.000)
ATI
0.115
(0.181)
0.121
(0.276)
0.196
(0.017)
0.221
(0.026)
0.284
(0.005)
0.252
(0.016)
0.019
(0.805)
0.229
(0.033)
0.195
(0.005)
0.272
(0.002)
0.202
(0.028)
0.236
(0.033)
0.300
(0.001)
0.271
(0.007)
0.218
(0.019)
0.161
(0.038)
0.119
(0.029)
0.181
(0.000)
0.172
(0.053)
0.268
(0.004)
0.254
(0.001)
0.373
(0.000)
0.065
(0.212)
0.107
(0.126)
0.117
(0.039)
AVY
0.058
(0.325)
0.084
(0.063)
0.113
(0.153)
0.129
(0.078)
0.163
(0.020)
0.172
(0.014)
0.165
(0.026)
0.111
(0.111)
0.113
(0.027)
0.181
(0.005)
0.282
(0.000)
0.239
(0.002)
0.072
(0.285)
0.090
(0.198)
0.140
(0.053)
0.096
(0.045)
0.144
(0.002)
0.187
(0.000)
0.121
(0.027)
0.072
(0.250)
0.219
(0.002)
0.148
(0.023)
0.126
(0.086)
0.387
(0.022)
0.304
(0.000)
DUK
0.159
(0.003)
0.204
(0.001)
0.136
(0.046)
0.229
(0.000)
0.107
(0.110)
0.259
(0.000)
0.223
(0.001)
0.255
(0.000)
0.209
(0.000)
0.114
(0.059)
0.178
(0.003)
0.182
(0.007)
0.149
(0.024)
0.094
(0.269)
0.123
(0.085)
0.389
(0.000)
0.151
(0.000)
0.154
(0.000)
0.180
(0.005)
0.216
(0.000)
0.127
(0.007)
0.210
(0.001)
0.239
(0.064)
0.062
(0.232)
0.369
(0.000)
CNP
0.266
(0.017)
0.240
(0.021)
0.204
(0.099)
0.351
(0.005)
0.178
(0.127)
0.418
(0.000)
0.263
(0.020)
0.421
(0.000)
0.318
(0.000)
0.103
(0.197)
0.240
(0.012)
0.329
(0.024)
0.226
(0.011)
0.154
(0.280)
0.288
(0.006)
0.308
(0.003)
0.208
(0.000)
0.193
(0.000)
0.222
(0.036)
0.283
(0.003)
0.283
(0.001)
0.225
(0.018)
0.161
(0.145)
0.103
(0.100)
0.385
(0.018)
EXC
0.345
(0.004)
0.381
(0.006)
0.270
(0.064)
0.459
(0.002)
0.306
(0.009)
0.540
(0.000)
0.236
(0.080)
0.465
(0.002)
0.511
(0.000)
0.194
(0.079)
0.418
(0.004)
0.208
(0.097)
0.356
(0.000)
0.236
(0.140)
0.306
(0.008)
0.398
(0.004)
0.419
(0.000)
0.277
(0.000)
0.337
(0.009)
0.397
(0.000)
0.365
(0.000)
0.509
(0.000)
0.249
(0.000)
0.114
(0.043)
0.297
(0.000)
0.371
(0.000)
B
Proofs
Throughout, we denote by K a generic constant, which may change from line to line. When it depends on
Pa0
a parameter p we use the notation Kp instead. We assume by convention i=a = 0 when a > a0 .
B.1
Proof of Theorem 1
We prove this theorem in three steps. For simplicity, in the first two steps we focus on the estimation of
[H(C), G(C)]T with H, G ∈ G(p). The joint estimation is discussed in Step 3.
By a localization argument (See Lemma 4.4.9 of Jacod and Protter (2012)), there exists a λ-integrable
function J on E and a constant such that the stochastic processes in (18) and (19) satisfy
kbk, kebk, kck, ke
ck, J ≤ A, kδ(w, t, z)kr ≤ J(z).
Setting b0t = bt −
R
δ(t, z)1{kδ(t,z)k≤1} λ(dz) and Yt0 =
Rt
0
0
bs ds +
Yt = Y0 + Yt0 +
X
Rt
0
(23)
σs dWs , we have
∆Ys .
s≤t
The local estimator of the spot variance of the unobservable process Y 0 is given by,
bi0n =
C
kX
n −1
1
b 0n,gh )1≤g,h≤d .
(∆n Y 0 )(∆ni+u Y )0> = (C
i
kn ∆n u=0 i+u
(24)
b 0n since the process Y 0 is continuous. ThereNote that no jump truncation in needed in the definition of C
i
LIN 0
\
b 0n rather than C
b n (defined in (13)). Let [H(C),
fore, it is more convenient to work with C
G(C)]T
i
i
AN 0
\
b n by C
b 0 n in the definition of
and [H(C),
G(C)]T
be the infeasible estimators obtained by replacing C
i
i
AN
LIN
\
\
[H(C), G(C)]T
and [H(C), G(C)]T .
Step1: Dealing with price jumps
We prove that, as long as (8p − 1)/4(4p − r) ≤ $ < 21 , we have
LIN
LIN 0 P
AN
AN 0 P
\
\
\
\
∆n−1/4 [H(C),
G(C)]T −[H(C),
G(C)]T
−→ 0 and ∆−1/4
[H(C),
G(C)]
−
[H(C),
G(C)]
−→ 0.
n
T
T
(25)
To show this result, let us define the functions
d
X
R(x, y) =
∂gh H∂ab G (x) y gh − xgh y ab − xab , S(x, y) = H(y) − H(x) G(y) − G(x)
g,h,a,b=1
d
X
U (x) =
∂gh H∂ab G (x) xga xhb + xgb xha ,
g,h,a,b=1
for any Rd × Rd matrices x and y. The following decompositions hold,
AN
\
[H(C),
G(C)]T
AN 0
\
− [H(C),
G(C)]T
=
3
2kn
[T /∆n ]−2kn +1 h
X
i=1
33
i
0
2
n
n
bin , C
bi+k
bi0 n , C
bi+k
bin ) − U (C
bi0 n ) ,
S(C
) − S(C
) −
U (C
n
n
kn
LIN
\
[H(C),
G(C)]T
LIN 0
\
− [H(C),
G(C)]T
3
=
2kn
[T /∆n ]−2kn +1 h
X
i=1
i
b n ) − U (C
b0 n ) .
bn , C
b n ) − R(C
b0 n , C
b 0 n ) − 2 U (C
R(C
i
i
i
i+kn
i
i+kn
kn
By (3.11) in Jacod and Rosenbaum (2012), there exists a sequence of real numbers an converging to zero
such that
bin − C
bi0 n kq ) ≤ Kq an ∆(2q−r)$+1−q
E(kC
, for any q > 0.
(26)
n
Since H and G ∈ G(p), it is easy to see that the functions R and S are continuously differentiable and satisfy
k∂J(x, y)k
k∂U (x)k
≤ K(1 + kxk + kyk)2p−1 for 1 ≤ g, h, a, b ≤ d and J ∈ {S, R},
2p−1
≤ K(1 + kxk)
(27)
,
(28)
where ∂J (respectively, ∂U ) is a vector that collects the first order partial derivatives of the function J
(respectively, U ) with respect to all the elements of (x, y) (resp x). By Taylor expansion, Jensen inequality,
(27) and (28), it can be shown that, for J ∈ {S, R},
0
0
n
n
n
bn , C
b n ) − J(C
b0 n , C
b 0 n )| ≤ K(1 + kC
bi0 n k2p−1 + kC
bi+k
bin − C
bi0 n k + kC
bi+k
bi+k
|J(C
k2p−1 )(kC
−C
k)
i
i+kn
i
i+kn
n
n
n
0
0
n
n 2p
n
n
2p
b
b
b
b
+ KkCi − Ci k + KkCi+k − Ci+k k
and
n
0
0
n
0
0
bin ) − U (C
bi n )| ≤ K(1 + kC
bi n k2p−1 )(kC
bin − C
bi n k) + KkC
bin − C
bi n k2p .
|U (C
b 0n kv ) ≤ Kv , for any v ≥ 0. Hence by Hölder
By (3.20) in Jacod and Rosenbaum (2012), we have E(kC
i
inequality, for > 0 fixed,
/1+
1/1+ b 0 n k(2p−2)(1+)/ )
bn − C
b 0 n k(1+) )
b 0n k2p−2 kC
bn − C
b 0 n k) ≤ E(kC
E(kC
E(kC
i
i
i
i
i
i
1/1+
bin − C
bi0 n k(1+) )
≤ Kp E(kC
1
1
)$+ 1+
−1
(2− 1+
≤ Kp an ∆n
Using the above result and (26), it easy to see that for (25) to hold, the following conditions are sufficient:
(2 −
r
1
3
)$ +
− 1 − ≥ 0,
1+
1+
4
(4p − r)$ + 1 − 2p −
3
≥ 0,
4
and
(2 − r)$ + −
3
≥ 0.
4
Using the fact that 0 < $ < 21 , and taking sufficiently close to zero, we can see that (25) holds if
(8p − 1)/4(4p − r) ≤ $ < 21 , which completes the proof.
Step 2 : First approximation for the estimators
LIN 0
\
Taking advantage of Step 1, it is enough to derive the asymptotic distributions of [H(C),
G(C)]T
AN 0
LIN
0
AN
and
0
\
\
\
[H(C),
G(C)]T . We show that the two estimators [H(C),
G(C)]T
and [H(C),
G(C)]T can be approx−1/4
imated by a certain quantity with an error of approximation of order smaller than ∆n . To see this, we
set
A
3
\
[H(C),
G(C)]T =
2kn
d
X
[T /∆n ]−2kn +1
g,h,a,b=1
i=1
X
h 0
0
0
0
b n,gh − C
b n,gh )(C
b n,ab − C
b n,ab )
∂gh H∂ab G (Cin ) (C
i
i
i+kn
i+kn
!
2 b 0 n,ga b 0 n,hb b 0 n,gb b 0 n,ha i
−
(C
Ci
+ Ci
Ci
) ,
kn i
34
with Cin = C(i−1)∆n and the superscript A being a short for the word ”approximate”. For notational
b 0 n instead C
b n . We aim
simplicity, we do not index the above quantity by a prime although it depends on C
i
i
to prove that
LIN 0
A P
AN 0
A P
\
\
\
\
∆n−1/4 [H(C),
G(C)]T
− [H(C),
G(C)]T −→ 0 and ∆−1/4
[H(C),
G(C)]
−
[H(C),
G(C)]
−→ 0.
n
T
T
(29)
To prove (29), we introduce some new notation. Following Jacod and Rosenbaum (2012), we define
0
n
bi0 n − Cin , and γin = C
bi+k
bi0 n ,
αin = (∆ni Y 0 )(∆ni Y 0 )> − Cin ∆n , βin = C
−C
n
(30)
which satisfy
βin
kX
n −1
1
n
n
(αn + (Ci+j
=
− Cin )∆n ) and γin = βi+kn − βin + ∆n (Ci+k
− Cin ).
n
kn ∆n j=0 i+j
(31)
We have
LIN 0
\
[H(C),
G(C)]T
AN 0
\
[H(C),
G(C)]T
A
3
\
− [H(C),
G(C)]T =
2kn
A
3
\
− [H(C),
G(C)]T =
2kn
d
X
[T /∆n ]−2kn +1
g,h,a,b=1
i=1
X
[T /∆n ]−2kn +1 X
χni −
i=1
ψin (g, h, a, b),
d
X
∂gh H∂ab G (Cin )γin,gh γin,ab ,
g,h,a,b=1
with
0n
b ) − ∂gh H∂ab G (C n ) γ n,gh γ n,ab ,
ψin (g, h, a, b) = ∂gh H∂ab G (C
i
i
i
i
0
0
0
0
n
n
bi n ) .
bi+k
bi n ) G(C
bi+k
) − G(C
) − H(C
χni = H(C
n
n
By Taylor expansion, we have
d X
0n
2
2
bi ) − ∂gh S∂ab G (Cin ) =
∂gh S∂ab G (C
∂xy,gh
S∂ab G + ∂xy,ab
G∂gh S (Cin )βin,xy
x,y=1
+
1
2
d
X
3
2
2
3
2
2
∂jk,xy,gh
S∂ab G + ∂xy,gh
S∂jk,ab
G + ∂jk,xy,ab
G∂gh S + ∂xy,ab
G∂jk,gh
S (e
cni )βin,xy βin,jk
j,k,x,y=1
and
0
0
b n ) − S(C
b n) =
S(C
i+kn
i
X
gh
+
1
2
X
x,y,j,k,g,h
∂gh S(Cin )γin,gh +
X
2
∂jk,gh
S(Cin )γin,gh βin,jk +
x,y,g,h
j,k,g,h
3
∂xy,jk,gh
S(CCin,S )γin,gh βin,xy βin,jk +
1 X 2
∂xy,gh S(Cin )γin,gh γin,xy
2
1
6
X
3
∂jk,xy,gh
S(Cin,S )γin,jk γin,gh γin,xy ,
j,k,x,y,g,h
b 0 n , C n,S = λS C
b 0 n + (1 − λS )C
b 0 n , CC n,S = µS C n + (1 − µS )C
b0 n
for S ∈ {H, G}, e
cni = λCin + (1 − λ)C
i
i
i
i
i
i
i+kn
for λ, λH , µH , λG , µG ∈ [0, 1]. Although e
cni and λ depend on g, h, a, and b, we do not emphasize this in our
notation to simplify the exposition.
35
We remind the reader some well-known results. For any continuous Itô process Zt , we have
q E sup Zt+w − Zt Ft ≤ Kq sq/2 , and E Zt+s − Zt Ft ≤ Ks.
(32)
w∈[0,s]
Set Fin = F(i−1)∆n . By (4.10) in Jacod and Rosenbaum (2013) we have,
n −1
q q kX
n n
E αin Fin ≤ Kq ∆qn for all q ≥ 0 and E αi+j
Fi ≤ Kq ∆qn knq/2 whenever q ≥ 2.
(33)
j=0
Combining (41), (39), (40) with Z = c and the Hölder inequality yields for q ≥ 2,
q q E βin Fin ≤ Kq ∆q/4 , and E γin Fin ≤ Kq ∆q/4 .
(34)
The bound in the first equation of (42) is tighter than that in (4.11) of Jacod and Rosenbaum (2012)
due to the absence of volatility jumps. This tighter bound will be useful later for deriving the asymptotic
distribution for the approximate estimator (Step 3). By the boundedness of Ct and the polynomial growth
assumption, we have
n n,xy n,jk n,gh n,ab 3
2
2
cni k)2(p−2) kβin k2 kγin k2 .
H∂jk,ab
G (e
ci )βi βi γi γi ≤ K(1 + ke
∂jk,xy,ab G∂gh H + ∂xy,gh
0
b n and using the convexity of the function x2(p−2) , we can refine the last
Recalling e
cni = λCin + (1 − λ)C
i
inequality as follows:
n n,xy n,jk n,gh n,ab 3
2
2
H∂jk,ab
G (e
ci )βi βi γi γi ≤ K 1 + kβin k2(p−2) kβin k2 kγin k2 .
(35)
∂jk,xy,ab G∂gh H + ∂xy,gh
By Taylor expansion, the polynomial growth assumption and using similar idea as for (35), we have
χni −
X
(∂gh H∂ab G)(Cin )γin,gh γin,ab =
g,h,a,b
X
X
g,h,a,b,j,k
0n
b ) − ∂gh H∂ab G (C n ) =
∂gh H∂ab G (C
i
i
g,h,a,b
1
2
2
(∂gh H∂jk,xy
G + ∂gh G∂jk,xy
H)(Cin )(γin,gh + βin,gh )γin,ab γin,jk + ϕni
2
X
2
2
(∂gh H∂ab,xy
G + ∂ab G∂gh,xy
G)(Cin )(βin,xy )γin,gh γin,ab + δin
g,h,a,b,x,y
with E(|ϕni |Fin ) ≤ K∆n and E(|δin |Fin ) ≤ K∆n which follow by the Cauchy-Schwartz inequality together
with (42). Given that kn = θ(∆n )−1/2 , a direct implication of the previous inequalities is
]−2kn +1
−1/4 [T /∆nX
3∆n
2kn
P
ϕni −→ 0 and
i=1
]−2kn +1
−1/4 [T /∆nX
3∆n
2kn
P
δin −→ 0.
i=1
Therefore, in order to prove the two claims in (29), it suffices to show
]−2kn +1
−1/4 [T /∆nX
3∆n
2kn
i=1
]−2kn +1
−1/4 [T /∆nX
3∆n
2kn
i=1
X
2
2
(∂gh H∂jk,ab
G + ∂gh H∂jk,ab
G)(Cin )γin,gh γin,ab γin,jk −→ 0,
(36)
2
2
(∂gh H∂jk,ab
G + ∂gh H∂jk,ab
G)(Cin )βin,gh γin,ab γin,jk −→ 0.
(37)
P
g,h,a,b,j,k
X
P
g,h,a,b,j,k
36
For any càdlàg bounded process Z, we set
s ηt,s (Z) = E sup kZt+u − Zt k2 |Fin ,
0<u≤s
s n
ηi,j (Z) = E
sup
0≤u≤j∆n
kZ(i−1)∆n +u − Z(i−1)∆n k2 |Fin .
In order to prove (36) and (37), we introduce the following lemmas.
Lemma 1. For any càdlàg bounded process Z, for all t, s > 0, j, k ≥ 0, set ηt,s = ηt,s (Z). Then,
[t/∆
Xn ]
∆n E
ηi,kn −→ 0,
[t/∆
Xn ]
∆n E
ηi,2kn −→ 0,
i=1
i=1
E ηi+j,k |Fin ≤ ηi,j+k and
[t/∆
Xn ]
∆n E
ηi,4kn −→ 0.
i=1
The first three claims of Lemma 6 are proved in Jacod and Rosenbaum (2012). The last result can be proved
similarly to the first two.
Z
Lemma 2. Let Z be a continuous Itô process with drift bZ
t and spot variance process Ct , and set ηt,s =
ηt,s (bZ , cZ ). Then, the following bounds hold:
|E(Zt |F0 ) − tbZ
0 | ≤ Ktη0,t
p
|E(Ztj Ztk − tC0Z,jk |F0 )| ≤ Kt3/2 ( ∆n + η0,t )
|E (Ztj Ztk − tC0Z,jk )(CtZ,lm − C0Z,lm )|F0 | ≤ Kt2
|E(Ztj Ztk Ztl Ztm |F0 ) − ∆2n (C0Z,jk C0Z,lm + C0Z,jl C0Z,km + C0Z,jm C0Z,kl )| ≤ Kt5/2
|E(Ztj Ztk Ztl |F0 )| ≤ Kt2
6
Y
∆3 X X X Z,jl jl0 Z,jk jk0 Z,jm jm0
C0
C0
C0
| ≤ Kt7/2
|E( Ztjl |F0 ) − n
6
0
0
0
l<l k<k m<m
l=1
The first four claims of Lemma 7 are parts of Lemma 4.1 in Jacod and Rosenbaum (2012). The two remaining
statements can be shown similarly.
n
n
Lemma 3. Let ζin be a r-dimensional Fin measurable process satisfying kE(ζin |Fi−1
)k ≤ L0 and E kζin kq Fi−1
≤
n
Lq . Also, let ϕni be a real-valued Fin -measurable process with E kϕni+j−1 kq Fi−1
≤ Lq for q ≥ 2 and
1 ≤ j ≤ 2kn − 1. Then, we have
!
n −1
2kX
q n
n n
E ϕi+j−1 ζi+j Fi−1 ≤ Kq Lq (Lq knq/2 + L0q knq ).
j=1
Proof of Lemma 5
Set
ξin = ϕni−1 ζin ,
0
00
0
n
n
n
ξi n = E(ξi |Fi−1
) = E(ϕni−1 ζin |Fi−1
) = ϕni−1 E(ζin |Fi−1
), and ξi n = ξin − ξi n .
37
0
n
Given that kE(ζin |Fi−1
)k ≤ L0 , we have kξi n k ≤ L0 |ϕni−1 |. By the convexity of the function xq , which holds
for q ≥ 2, we have
k
2k
n −1
X
2k
n −1
n −1
2kX
X
0
00
n
n q
n q
ξi+j
kq ≤ K k
ξi+j
k +k
ξi+j
k .
j=1
j=1
j=1
Therefore, on the one hand we have
k
2k
n −1
X
2k
n −1
X
0
n q
ξi+j
k ≤ Kknq−1
j=1
0
n q
kξi+j
k ≤ Kknq−1 L0q
j=1
2k
n −1
X
|ϕni+j−1 |q ,
j=1
n
≤ Lq , satisfies
which by E kϕni+j−1 kq Fi−1
E(k
2k
n −1
X
0
n q
n
ξi+j
k |Fi−1
) ≤ KL0q knq−1
j=1
2k
n −1
X
n
E(|ϕni+j−1 |q |Fi−1
) ≤ KL0q knq Lq .
j=1
00
00
n q
n
n
n
n
n
On the other hand, we have E(kξi+j
k |Fi−1
) ≤ E(kξi+j
kq |Fi−1
) ≤ Lq Lq and E(ξi+j
|Fi−1
) = 0, where the
0
n q
n
n
q
n
q
first inequality is a consequence of E(kξi+j k |Fi−1 ) ≤ E(kξi+j k |Fi−1 ) ≤ Lq L , which follows by the Jensen
inequality and the law of iterated expectation. Hence, by Lemma B.2 of Aı̈t-Sahalia and Jacod (2014) we
have
E(k
2k
n −1
X
00
n q
n
ξi+j
k |Fi−1
) ≤ Kq Lq Lq knq/2 .
j=1
n
To see the latter, we first prove that the required condition E(kξin kq |Fi−1
) ≤ Lq Lq ) in the Lemma B.2 of
n
q
n
Aı̈t-Sahalia and Jacod (2014) can be replaced by E(kξi+j k |Fi−1 ) ≤ Lq Lq ) for 1 ≤ j ≤ 2kn − 1 without
altering the result.
Lemma 4. We have:
4
n,jk n,lm n,gh
n,ab
γi+2kn γi+2k
|Fin ) − 2 (Cin,ga Cin,hb + Cin,gb Cin,ha )(Cin,jl Cin,km + Cin,jm Cin,kl )
E(γi γi
n
kn
4∆n n,ga n,hb
4∆n n,jl n,km
n,gh,ab
n,jk,lm
(Ci Ci
+ Cin,jm Cin,kl )C i
−
(Ci Ci
− Cin,gb Cin,ha )C i
−
3
3
4(kn ∆n )2 n,gh,ab n,jk,lm n
Ci
Ci
−
≤ K∆n (∆1/8
n + ηi,4kn ).
9
Throughout, we use the expression “successive conditioning” to refer to the following equalities,
x1 y1 − x0 y0 = x0 (y1 − y0 ) + y0 (x1 − x0 ) + (x1 − x0 )(y1 − y0 ),
x1 y1 z1 − x0 y0 z0 = x0 y0 (z1 − z0 ) + x0 z0 (y1 − y0 ) + y0 z0 (x1 − x0 ) + x0 (y0 − y1 )(z0 − z1 )
+ y0 (x0 − x1 )(z0 − z1 ) + z0 (x0 − x1 )(y0 − y1 ) + (x1 − x0 )(y1 − y0 )(z1 − z0 ),
which hold for any real numbers x0 , y0 , z0 , x1 , y1 , and z1 .
38
Proof of Lemma 4
n
To prove Lemma 4, we first note that γin,jk γin,lm is Fi+2k
-measurable. Then, by the law of iterated
n
expectations, we have
n
n,gh
n,ab
n,gh
n,ab
n,jk n,lm
n
n
|Fi .
E
γ
γ
|F
γ
γ
|F
=
E
γ
E γin,jk γin,lm γi+2k
i+2k
i
i
i
i+2kn i+2kn
n
n i+2kn
By equation (3.27) in Jacod and Rosenbaum (2012), we have
p
2
2kn ∆n n,gh,ab
n,ga
n,hb
n,gb
n,ha
n
(Ci+2k
C
+
C
C
)
−
C
|
≤
K
∆n (∆1/8
i+2k
n + ηi+2kn ,2kn ),
i+2k
i+2k
i+2k
n
n
n
n
n
kn
3
p
2
2kn ∆n n,jk,lm
n
|E(γin,jk γin,lm |Fin ) −
(Cin,jl Cin,km + Cin,jm Cin,kl ) −
Ci
| ≤ K ∆n (∆1/8
n + ηi,2kn ).
kn
3
n,gh
n
|E(γi+2k
γ n,ab |Fi+2k
)−
n
n i+2kn
Also,
h
2
2kn ∆n n,gh,ab i n n,gh
n,ab n
n,jk n,lm
n,ga
n,hb
n,gb
n,ha
E(γi+2kn γi+2kn Fi+2kn ) −
|E γi γi
(C
C
+ Ci+2kn Ci+2kn ) −
C i+2kn Fi |
kn i+2kn i+2kn
3
p
p
n
n,jk
n
≤ ∆n E(|γin,jk ||γin,lm |(∆1/8
∆n ∆1/8
||γin,lm |Fin )
n + ηi+2kn ,2kn )|Fi ) ≤ K
n E(|γi
p
n
n
n
+ K ∆n E(|γin,jk ||γin,lm |ηi+2k
|
Fi ) ≤ K∆n (∆1/8
n + ηi,4kn ),
,2k
n
n
where the last inequality follows from Lemma 6. Using (40) successively with Z = c and Z = C (recall that
the latter holds under Assumption 2), together with the successive conditioning, we have
h 2
2
2kn ∆n n,gh,ab
n,gb
n,ha
|E γin,jk γin,lm
C i+2kn −
(C n,ga C n,hb + Ci+2k
Ci+2k
)+
(C n,ga Cin,hb + Cin,gb Cin,ha )
n
n
kn i+2kn i+2kn
3
kn i
2kn ∆n n,gh,ab i n Ci
−
Fi | ≤ K∆n ∆1/4
n ,
3
h 2
2kn ∆n n,gh,ab i h 2
2kn ∆n n,jk,lm i
Ci
Ci
|E γin,jk γin,lm
−
(Cin,ga Cin,hb + Cin,gb Cin,ha ) +
(Cin,jl Cin,km + Cin,jm Cin,kl ) +
kn
3
kn
3
h 2
2kn ∆n n,gh,ab i n n
(Cin,ga Cin,hb + Cin,gb Cin,ha ) +
×
Ci
Fi | ≤ K∆n (∆1/8
n + ηi,2kn ).
kn
3
The last inequality yields the result.
n
n
Lemma 5. Let ζin be a r-dimensional Fin -measurable process satisfying kE(ζin |Fi−1
)k ≤ L0 and E kζin kq Fi−1
≤
n
n
n
q n
q
Lq . Also, let ϕi be a real-valued Fi -measurable process with E kϕi+j−1 k Fi−1 ≤ L for q ≥ 2 and
1 ≤ j ≤ 2kn − 1. Then,
!
n −1
2kX
q n
n n
E ϕi+j−1 ζi+j Fi−1 ≤ Kq Lq (Lq knq/2 + L0q knq ).
j=1
We introduce some new notation. Following Jacod and Rosenbaum (2012), we define
0
0
0
n
b n − C n , and γ n = C
bi+k
bi n ,
αin = (∆ni Y 0 )(∆ni Y 0 )> − Cin ∆n , βin = C
−C
i
i
i
n
(38)
which satisfy
βin =
kX
n −1
1
n
n
(αn + (Ci+j
− Cin )∆n ) and γin = βi+kn − βin + ∆n (Ci+k
− Cin ).
n
kn ∆n j=0 i+j
39
(39)
We remind some well-known results. For any continuous Itô process Zt , we have
q E sup Zt+w − Zt Ft ≤ Kq sq/2 , and E Zt+s − Zt Ft ≤ Ks.
(40)
w∈[0,s]
Set Fin = F(i−1)∆n . By (4.10) in Jacod and Rosenbaum (2013), we have
n −1
q q kX
n n
E αin Fin ≤ Kq ∆qn for all q ≥ 0 and E αi+j
Fi ≤ Kq ∆qn knq/2 whenever q ≥ 2.
(41)
j=0
Combining (41), (39), (40) with Z = c and the Hölder inequality yields, for q ≥ 2,
q q E βin Fin ≤ Kq ∆q/4 , and E γin Fin ≤ Kq ∆q/4 .
(42)
For any càdlàg bounded process Z, we set
s ηt,s (Z) = E sup kZt+u − Zt k2 |Fin ,
0<u≤s
s n
ηi,j (Z) = E
sup
0≤u≤j∆n
kZ(i−1)∆n +u − Z(i−1)∆n k2 |Fin .
Lemma 6. For any càdlàg bounded process Z, for all t, s > 0, j, k ≥ 0, and set ηt,s = ηt,s (Z). Then,
[t/∆
Xn ]
ηi,kn −→ 0,
∆n E
[t/∆
Xn ]
ηi,2kn −→ 0,
∆n E
i=1
i=1
E ηi+j,k |Fin ≤ ηi,j+k and
[t/∆
Xn ]
∆n E
ηi,4kn −→ 0.
i=1
The first three claims of Lemma 6 are proved in Jacod and Rosenbaum (2012). The last result can be proved
similarly to the first two.
Z
Lemma 7. Let Z be a continuous Itô process with drift term bZ
t and spot variance process Ct , and set
ηt,s = ηt,s (bZ , cZ ). Then, the following bounds hold:
|E(Zt |F0 ) − tbZ
0 | ≤ Ktη0,t
p
|E(Ztj Ztk − tC0Z,jk |F0 )| ≤ Kt3/2 ( ∆n + η0,t )
|E (Ztj Ztk − tC0Z,jk )(CtZ,lm − C0Z,lm )|F0 | ≤ Kt2
|E(Ztj Ztk Ztl Ztm |F0 ) − ∆2n (C0Z,jk C0Z,lm + C0Z,jl C0Z,km + C0Z,jm C0Z,kl )| ≤ Kt5/2
|E(Ztj Ztk Ztl |F0 )| ≤ Kt2
6
Y
∆3 X X X Z,jl jl0 Z,jk jk0 Z,jm jm0
| ≤ Kt7/2
C0
C0
C0
|E( Ztjl |F0 ) − n
6
0
0
0
l=1
l<l k<k m<m
The first four claims of Lemma 7 are parts of Lemma 4.1 in Jacod and Rosenbaum (2012). The two remaining
statements can be shown similarly.
Lemma 8. The following results hold:
1/4
n
|E(βin,jk βin,lm βin,gh |Fin )| ≤ K∆3/4
n (∆n + ηi,kn ),
40
(43)
n,gh
1/4
n
|E(βin,jk βin,lm (cn,gh
)|Fin )| ≤ K∆3/4
n (∆n + ηi,kn ),
i+kn − ci
(44)
n,lm
n,gh
1/4
n
|E(βin,jk (cn,lm
)(cn,gh
)|Fin )| ≤ K∆3/4
n (∆n + ηi,kn ),
i+kn − ci
i+kn − ci
(45)
|E(βin,jk γin,lm γin,gh |Fin )|
|E(γin,jk γin,lm γin,gh |Fin )|
+
n
ηi,2k
),
n
(46)
+
n
ηi,2k
).
n
(47)
≤
1/4
K∆3/4
n (∆n
≤
1/4
K∆3/4
n (∆n
Proof of (43) in Lemma 8
We start by obtaining some useful bounds for some quantities of interest. First, using the second statement
in Lemma 7 applied to Z = Y 0 , we have
p
n
|E(αin,jk |Fin )| ≤ K∆3/2
n ( ∆n + ηi,1 ).
(48)
Second, by repeated application of the Cauchy-Schwartz inequality and making use of the third and last
statements in Lemma 7 as well as (40) with Z = c, it can be shown that
n,jl n,km
n,jk n,lm
(49)
+ Cin,jm Cin,kl ≤ K∆5/2
E(αi αi |Fin ) − ∆2n Ci Ci
n .
Next, by successive conditioning and using the bound in (40) for Z = c as well as (48) and (49) , we have
for 0 ≤ u ≤ kn − 1,
p
n,jk
n
(50)
E(αi+u Fin ) ≤ K∆3/2
n ( ∆n + ηi,u ),
n,jk n,lm
n,jl n,km
+ Cin,jm Cin,kl ≤ K∆5/2
E(αi+u αi+u |Fin ) − ∆2n Ci Ci
n ,
(51)
To show (43), we first observe that βin,jk βin,lm βin,gh can be decomposed as
βin,jk βin,lm βin,gh =
kX
kX
n −1
n −2 kX
n −1 h
1
1
n,jk n,lm n,gh
n,gh n,jk n,lm
ζ
ζ n,jk ζ n,lm ζ n,gh + ζi,u
ζ
ζ
+
ζi,v ζi,v
kn3 ∆3n u=0 i,u i,u i,u
kn3 ∆3n u=0 v=u+1 i,u i,v i,v
i
n,lm n,gh n,jk
+ ζi,u
ζi,v ζi,v +
kX
n −2 kX
n −1
i
1
n,jk n,lm n,gh
n,gh n,jk n,lm
n,lm n,gh n,jk
[ζi,u
ζi,u ζi,v + ζi,u
ζi,u ζi,v + ζi,u
ζi,u ζi,v
3
3
kn ∆n u=0 v=u+1
kX
kX
n −3 kX
n −2
n −1 h
1
n,jk n,gh n,lm
n,lm n,jk n,gh
n,lm n,gh n,jk
ζ n,jk ζ n,lm ζ n,gh + ζi,u
ζi,v ζi,w + ζi,u
ζi,v ζi,w + ζi,u
ζi,v ζi,w
kn3 ∆3n u=0 v=u+1 w=v+1 i,u i,v i,w
i
n,gh n,lm n,jk
n,gh n,jk n,lm
+ ζi,u
ζi,v ζi,w + ζi,u
ζi,v ζi,w ,
+
n
n
n
n q
with ζi,u
= αi+u
+ (Ci+u
− Cin )∆n , which satisfies E(kζi,u
k |Fin ) ≤ K∆qn for q ≥ 2.
Set
ξin (1) =
kX
n −1
1
ζ n,jk ζ n,lm ζ n,gh ,
kn3 ∆3n u=0 i,u i,u i,u
ξin (3) =
kX
kX
kX
n −2 kX
n −1
n −3 kX
n −2
n −1
1
1
n,jk n,lm n,gh
n
ζ
ζ
ζ
and
ξ
(4)
=
ζ n,jk ζ n,lm ζ n,gh .
i
kn3 ∆3n u=0 v=u+1 i,u i,u i,v
kn3 ∆3n u=0 v=u+1 w=v+1 i,u i,v i,w
ξin (2) =
kX
n −2 kX
n −1
1
ζ n,jk ζ n,lm ζ n,gh
kn3 ∆3n u=0 v=u+1 i,u i,v i,v
The following bounds can be established,
|E(ξin (1)|Fin )| ≤ K∆n , |E(ξin (2)|Fin )| ≤ K∆n , |E(ξin (3)|Fin )| ≤ K∆n and
41
1/4
|E(ξin (4)|Fin )| ≤ K∆3/4
n (∆n + ηi,kn ).
Proof of |E(ξin (1)|Fin )| ≤ K∆n
The result readily follows from an application of the Cauchy Schwartz inequality together with the bound
n
E(kζi+u
kq |Fin ) ≤ Kq ∆qn for q ≥ 2.
Proof of |E(ξin (2)|Fin )| ≤ K∆n
Using the law of iterated expectation, we have, for u < v,
n,jk n,lm n,gh
n,jk
n,lm n,gh
n
E(ζi+u
ζi+v ζi+v |Fin ) = E(ζi+u
E(ζi+v
ζi+v |Fi+u+1
)Fin ).
(52)
By successive conditioning, (49), and the Cauchy-Schwartz inequality, we also have
n,lm
n,gh
n,mg
n,lh
n,mh
n,lg
n,lm n,gh
n
− Cin,lm )| ≤ K∆5/2
− Cin,gh )(Ci+u+1
) − ∆2n (Ci+u+1
Ci+u+1
+ Ci+u+1
Ci+u+1
) − ∆2n (Ci+u+1
ζi,v |Fi+u+1
|E(ζi,v
n .
n,jk q n
n,lm n,gh
n
Given that E(|ζi+u
| Fi ) ≤ ∆qn , the approximation error involved in replacing E(ζi+v
ζi+v |Fi+u+1
) by
7/2
n,lg
n,mh
n,lh
n,mg
n,gh
n,gh
n,lm
n,lm
2
2
∆n (Ci+u+1 Ci+u+1 + Ci+u+1 Ci+u+1 ) + ∆n (Ci+u+1 − Ci )(Ci+u+1 − Ci
) in (52) is smaller than ∆n .
From (3.9) in Jacod and Rosenbaum (2012) we have
p
n,jk
n,lm
n,lm
n
|E(αi+u
(Ci+u+1
− Ci+u
)|Fin )| ≤ K∆3/2
n ( ∆n + ηi,kn ).
(53)
n
n
Since (Ci+u
− Cin ) is Fi+u
-measurable, we use the successive conditioning, the Cauchy-Schwartz inequality,
(48), (49), and the fifth statement in Lemma 7 applied to Z = c to obtain
n,lm
n,jk
n,gh
|E(αi+u
(Ci+u
− Cin,lm )(Ci+u
− Cin,jk )|Fin )| ≤ K∆5/2
n
n,jk n,lm
n,gh
|E(αi+u
αi+u (Ci+u
− Cin,gh )|Fin )| ≤ K∆5/2
n
|E
n,lm
(Ci+u
−
n,jk
Cin,lm )(Ci+u
−
n,gh
Cin,jk )(Ci+u
−
Cin,gh ) |Fin )|
(54)
≤ K∆n ,
which can be proved using . The following inequalities can be established easily using (48), the successive
conditioning together with (40) for Z = c,
n,jk
n,lg
n,mh
n,lh
n,mg
E(αi+u (Ci+u+1 Ci+u+1 + Ci+u+1 Ci+u+1 )|Fin ) ≤ K∆3/2
n
n,lg
n,mh
n,lh
n,mg n,jk
n,jk
E (Ci+u − Ci ) Ci+u+1 Ci+u+1 + Ci+u+1 Ci+u+1 |Fin ≤ K∆1/2
n
p
n,jk
n,gh
n,gh
n,lm
n,lm
n
)|Fin ) ≤ K∆3/2
E(αi+u (Ci+u+1 − Ci )(Ci+u+1 − Ci
n ( ∆n + ηi,kn ).
The last three inequalities together yield |E(ξin (2)|Fin )| ≤ K∆n .
Proof of |E(ξin (3)|Fin )| ≤ K∆n
First, note that, for u < v, we have
n,jk n,lm n,gh
n,jk n,lm
n,gh
n
E(ζi+u
ζi+u ζi+v |Fin ) = E(ζi+u
ζi+u E(ζi+v
|Fi+u+1
)Fin ).
(55)
By successive conditioning and (48) , we have
p
n,gh
n
|E(αi+w
|Fi+v+1
)| ≤ K∆3/2
n ( ∆n + ηi+v+1,w−v ).
42
(56)
Using the first statement of Lemma applied to Z = c, it can be shown that
n,gh
n,gh
1/2
|E (Ci+w
− Ci+v+1
))|Fin − ∆n (w − v − 1)ebn,gh
i+v+1 | ≤ K(w − v − 1)∆n ηi+v+1,w−v ≤ K∆n ηi+v+1,w−v .
The last two inequalities together imply
p
n,gh
n,gh
3/2
n
− (Ci+v+1
− Cin,gh )∆n − ∆2n (w − v − 1)ebn,gh
E ζi+w |Fi+v+1
i+v+1 ≤ K∆n ( ∆n + ηi+v+1,w−v ).
(57)
n,gh
n,gh
n,jk q
n
−Cin,gh )∆n +∆2n (w −
) by (Ci+v+1
|Fi+u+1
| |Fin ) ≤ ∆qn , the error induced by replacing E(ζi+v
Since E(|ζi,u
7/2
v − 1)ebn,gh
i+v+1 in (55) is smaller that ∆n .
Using Cauchy Schwartz inequality, successive conditioning, (54), (40) for Z = c and the boundedness of ebt
and Ct we obtain
n,jk n,lm
n,jk
n,gh
n
≤ K∆5/2
E αi+u αi+u (Ci+u+1 − Ci )|Fi+u
n
n,jk n,lmen,gh
n
≤ K∆2n
E αi+u αi+u bi+u+1 |Fi+u
p
n,jk
n,lm
n,lm
n,gh
3/2
n
)(Ci+u+1
− Cin,gh )|Fin ≤ K∆1/4
E αi+u (Ci+u − Ci
n ∆n ( ∆n + ηi,kn )
n,jk
n,lm
n,lm en,gh
)bi+u+1 |Fin ≤ ∆5/4
E αi+u (Ci+u − Ci
n
n,jk
n,gh
n,lm
n,lm en,gh
)bi+u+1 |Fin ≤ K∆1/2
E (Ci+u − Ci )(Ci+u − Ci
n
n,jk
n,jk
n,lm
n,lm
n,gh
n,gh
)(Ci+u+1 − Ci )|Fin ≤ K∆n .
E (Ci+u − Ci )(Ci+u − Ci
The above inequalities together yield |E(ξin (3)|Fin )| ≤ K∆n .
3/4
1/4
n
)
Proof of |E(ξin (4)Fin )| ≤ K∆n (∆n + ηi,k
n
We first observe that ξin (4) can be rewritten as
ξin (4) =
kX
n −1 w−1
X v−1
X n,jk n,lm n,gh
1
ζ
ζ
ζ
,
3
(kn ∆n ) w=2 v=0 u=0 i+u i+v i+w
where
"
n,jk n,lm n,gh
ζi+u
ζi+v ζi+w
n,jk n,lm n,gh
n,jk
n,lm
n,gh
n,jk
n,lm
n,gh
= αi+u
αi+v αi+w + αi+u
∆n αi+v
(Ci+w
− Cin,gh ) + αi+u
∆n (Ci+v
− Cin,lm )αi+w
n,jk
n,lm
n,gh
n,jk
n,lm n,gh
n,jk
n,lm
n,gh
+ ∆2n αi+u
(Ci+v
− Cin,lm )(Ci+w
− Cin,gh ) + ∆n (Ci+u
− Cin,jk )αi+v
αi+w + ∆2n (Ci+u
− Cin,jk )αi+v
(Ci+w
− Cin,gh )
#
n,jk
n,lm
n,gh
n,jk
n,lm
n,gh
+ ∆2n (Ci+u
− Cin,jk )(Ci+v
− Cin,lm )αi+w
+ ∆3n (Ci+u
− Cin,jk )(Ci+v
− Cin,lm )(Ci+w
− Cin,gh ) .
Based on the above decomposition, we set
ξin (4) =
8
X
j=1
43
χ(j),
3/4
1/4
n
with χ(j) defined below. We aim to show that |E(χ(j)Fin )| ≤ K∆n (∆n + ηi,k
), j = 1, . . . , 8.
n
First, set
χ(1) =
kX
n −1 w−1
X v−1
X n,jk n,lm n,gh
1
α
α
α
.
3
(kn ∆n ) w=2 v=0 u=0 i+u i+v i+w
Upon changing the order of the summation, we have
χ(1) =
kX
n −1 w−1
X v−1
X n,jk n,lm n,gh
1
α
α
α
.
(kn ∆n )3 w=2 v=0 u=0 i+u i+v i+w
Define also
χ0 (1) =
kX
n −1 w−1
X v−1
X n,jk n,lm
1
n,gh
n
α
α
E(αi+w
|Fi+v+1
).
(kn ∆n )3 w=2 v=0 u=0 i+u i+v
Note that E(χ(1)|Fin ) = E(χ0 (1)|Fin ).
It is easy to see that by Lemma 5, we have for q ≥ 2,
v−1
X n,jk q n
αi+u Fi ≤ Kq ∆3q/4
E .
n
u=0
The Cauchy-Schwartz inequality yields,
!
n −1 w−1
kX
i h i
h v−1
X v−1
X n,jk n,lm
n
X n,jk 4 n 1/4
n,lm 4 n 1/4
n,gh
n
E αi+u αi+v E(αi+w |Fi+v+1 )Fi ≤ Kkn2 E αi+u Fi
E αi+v
Fi
w=2 v=0 u=0
u=0
2 i1/2
h p
n,gh
n
n
3/2
× E E(αi+w
|Fi+v+1
) Fin
≤ K∆n kn2 ∆3/4
n ∆n ( ∆n + ηi,kn ),
where the last iteration is obtained using (56) as well as the inequality (a + b)1/2 ≤ a1/2 + b1/2 , which holds
for positive real numbers a and b, and the third statement in Lemma 6.
It follows from this result that
p
n
|E χ(1)Fin | ≤ K∆3/4
n ( ∆n + ηi,kn ).
Next, we introduce
χ(2) =
kX
n −1 w−1
X v−1
X
1
n,lm n,gh
n,jk
n,jk
∆
(C
−
C
)
αi+v
αi+w ,
n
i+u
i
(kn ∆n )3 w=2 v=0 u=0
χ(3) =
kX
n −1 w−1
X v−1
X n,jk 1
n,lm
n,gh
α
∆n (Ci+u
− Cin,lm )αi+w
,
3
(kn ∆n ) w=2 v=0 u=0 i+v
χ(4) =
kX
n −1 w−1
X v−1
X
1
n,jk
n,jk
n,lm
n,gh
∆
(C
−
C
)
∆n (Ci+u
− Cin,lm )αi+w
.
n
i+u
i
(kn ∆n )3 w=2 v=0 u=0
Given that for q ≥ 2, we have
q v−1
X
n,jk
n,jk
E ∆n (Ci+u
− Cin,jk ) Fin ≤ Kq ∆3q/4
and E(kCi+u
− Cin,jk kq Fin ) ≤ Kq ∆q/4
n
n ,
u=0
44
one can follow essentially the same steps as for χ(1) to show that
p
p
n
n
n
|E(χ(2)Fin )| ≤ K∆3/4
n ( ∆n + ηi,kn ) and |E(χ(j) Fi )| ≤ K∆n ( ∆n + ηi,kn ) for j = 3, 4.
Define
χ(5) =
kX
n −1 w−1
X v−1
X n,jk n,lm
1
n,gh
α
α
∆n (Ci+w
− Cin,gh )
3
(kn ∆n ) w=2 v=0 u=0 i+u i+v
χ0 (5) =
kX
n −1 w−1
X v−1
X n,jk n,lm
n
1
n,gh
)
αi+u αi+v ∆n E (Ci+w
− Cin,gh )Fi+v+1
3
(kn ∆n ) w=2 v=0 u=0
χ(6) =
kX
n −1 w−1
X v−1
X
1
n,jk
n,jk
n,lm
n,gh
∆
(C
−
C
)
αi+v
∆n (Ci+w
− Cin,gh )
n
i+u
i
(kn ∆n )3 w=2 v=0 u=0
χ(7) =
kX
n −1 w−1
X v−1
X n,jk 1
n,gh
n,lm
− Cin,gh ),
− Cin,lm )∆n (Ci+w
∆n (Ci+v
α
(kn ∆n )3 w=2 v=0 u=0 i+u
where we have E(χ(5)|Fin ) = E(χ0 (5)|Fin ). Recalling (57), we further decompose χ0 (5) as,
χ0 (5) =
5
X
χ(5)[j],
j=1
with
χ0(5)[1] =
kX
n −1 w−1
X v−1
X n,jk n,lm n,gh
1
n,gh
n,gh
n,gh
n
ebn,gh ∆2 (w − v − 1)
α
α
E
C
−
C
|F
−
(C
−
C
)∆
−
n
i+v+1
n
i+u
i+v
i+w
i
i+v+1
i
i+v+1
(kn ∆n )3 w=2 v=0 u=0
χ0(5)[2] =
kX
n −1 w−1
v−1
X n,jk n,lm
X
1
n,gh
n,gh
αi+u αi+v
∆
(C
−
C
)
n
i+v
i
3
(kn ∆n ) w=2 v=0
u=0
χ0(5)[3] =
kX
n −1 w−1
X v−1
X n,jk 1
n,gh
n,gh
n,lm
α
∆n (Ci+v+1
− Ci+v
)αi+v
(kn ∆n )3 w=2 v=0 u=0 i+u
χ0(5)[4] =
kX
n −1 w−1
X v−1
X n,jk 1
en,gh n,lm
α
∆2n (w − v − 1)(ebn,gh
i+v+1 − bi+v )αi+v
(kn ∆n )3 w=2 v=0 u=0 i+u
χ0(5)[5] =
kX
n −1 w−1
v−1
X n,jk n,lm
X
1
2
ebn,gh
αi+u αi+v .
∆
(w
−
v
−
1)
n
i+v
(kn ∆n )3 w=2 v=0
u=0
Using (57), (56), (53) and following the same strategy proof as for χ(1), it can be shown that
p
n
|E χ0(5)[j]Fin | ≤ K∆3/4
n ( ∆n + ηi,kn ), for j = 1, . . . , 5,
which in turn implies
p
n
|E χ(5)Fin | ≤ K∆3/4
n ( ∆n + ηi,kn ), for j = 1, . . . , 5.
The term χ(6) can be handled similarly to χ(5), hence we conclude that
p
n
|E χ(6)Fin | ≤ K∆3/4
n ( ∆n + ηi,kn ).
45
Next, we set
kX
n −1
1
χ(7) =
(kn ∆n )3 w=2
w−1
X v−1
X
v=0
n,jk
αi+u
!
n,lm
∆n (Ci+v
−
n,gh
Cin,lm )∆n (Ci+w
−
Cin,gh )
.
u=0
Define
kX
n −1
1
χ(7)[1] =
(kn ∆n )3 w=2
w−1
X v−1
X
kX
n −1
w−1
X v−1
X
w=2
v=0
1
χ(7)[2] =
(kn ∆n )3
v=0
n,jk
n,lm
n,gh
n,gh
αi+u
∆n (Ci+v
− Cin,lm )∆n (Ci+v+1
− Ci+v
)
u=0
!
n,jk
αi+u
n,lm
∆n (Ci+v
n,jk
αi+u
n,lm
∆n (Ci+v
−
n,gh
Cin,lm )∆n (Ci+v
−
Cin,lm )∆2n (w
−
Cin,gh )
u=0
kX
n −1
1
χ(7)[3] =
(kn ∆n )3 w=2
w−1
X v−1
X
kX
n −1
1
χ(7)[4] =
(kn ∆n )3 w=2
w−1
X
v=0
!
!
−v−
1)(ebn,gh
i+v+1
− ebn,gh
i+v )
u=0
∆2n (w
− v − 1)ebn,gh
i+v
v=0
v−1
X
n,jk
αi+u
!
n,lm
∆n (Ci+v
−
Cin,lm )
,
u=0
so that
χ(7) =
4
X
χ(7)[j].
j=1
Similar to calculations used for χ(1), it can be shown that
1/4
|E(χ(7)[j]Fin )| ≤ K∆1/4
n (∆n + ηi,kn ), for j = 1, . . . , 3.
To handle the remaining term χ(7)[4], we set
χ(7)[4][1] =
kX
n −1 w−1
X n,jk n,lm
X v−1
∆2n
n,lm
n,gh
n,gh
α
(C
− Ci+u
)(Ci+u+1
− Ci+u
)
3
(kn ∆n ) w=2 v=0 u=0 i+u i+u+1
χ(7)[4][2] =
kX
n −1 w−1
X n,gh
X v−1
∆2n
n,jk
n,lm
n,lm
(C
− Cin,gh )αi+u
(Ci+u+1
− Ci+u
)
(kn ∆n )3 w=2 v=0 u=0 i+u
kX
n −1 w−1
X v−1
X n,gh
∆2n
n,jk
n,lm
n,lm
n
(C
− Cin,gh )E(αi+u
χ (7)[4][2] =
(Ci+u+1
− Ci+u
)|Fi+u
)
(kn ∆n )3 w=2 v=0 u=0 i+u
0
χ(7)[4][3] =
kX
n −1 w−1
X v−1
X n,lm
∆2n
n,jk
n,gh
n,gh
(C
− Cin,lm )αi+u
(Ci+u+1
− Ci+u
)
3
(kn ∆n ) w=2 v=0 u=0 i+u
χ(7)[4][4] =
kX
n −1 w−1
X v−1
X n,lm
∆2n
n,gh
n,jk
(C
− Cin,lm )(Ci+u
− Cin,gh )αi+u
3
(kn ∆n ) w=2 v=0 u=0 i+u
χ(7)[4][5] =
kX
n −1 w−1
X v−1
X n,lm
∆2n
n,jk
n,gh
n,gh
(C
− Cin,lm )αi+u
(Ci+v
− Ci+u+1
)
(kn ∆n )3 w=2 v=0 u=0 i+u
kX
n −1 w−1
X v−1
X n,lm
∆2n
n,jk
n,gh
n,gh
n
χ (7)[2][5] =
(C
− Cin,lm )αi+u
E((Ci+v
− Ci+u+1
|Fi+u
)
(kn ∆n )3 w=2 v=0 u=0 i+u
0
χ(7)[4][6] =
kX
n −1 w−1
X v−1
X n,jk n,lm
∆2n
n,gh
n,lm
n,gh
α
(C
− Ci+u
)(Ci+v
− Ci+u+1
)
3
(kn ∆n ) w=2 v=0 u=0 i+u i+u+1
46
χ(7)[4][7] =
kX
n −1 w−1
X v−1
X n,gh
∆2n
n,jk
n,lm
n,lm
(C
− Cin,gh )αi+u
(Ci+v
− Ci+u+1
)
3
(kn ∆n ) w=2 v=0 u=0 i+u
χ(7)[4][8] =
kX
n −1 w−1
X v−1
X n,jk n,gh
∆2n
n,gh
n,lm
n,lm
α
(C
− Ci+u
)(Ci+v
− Ci+u+1
)
3
(kn ∆n ) w=2 v=0 u=0 i+u i+u+1
χ(7)[4][9] =
kX
n −1 w−1
X v−1
X n,jk n,lm
∆2n
n,lm
n,gh
n,gh
α
(C
− Ci+u+1
)(Ci+v
− Ci+u+1
),
(kn ∆n )3 w=2 v=0 u=0 i+u i+v
which satisfy,
χ(7)[4] =
9
X
χ(7)[4][j].
j=1
By using arguments similar to those used for χ(1), it can be shown that
1/4
|E(χ(7)[4][j]Fin )| ≤ K∆1/4
n (∆n + ηi,kn ), for j = 1, . . . , 8,
which yields
1/4
|E(χ(7)Fin )| ≤ K∆1/4
n (∆n + ηi,kn ).
Next, define
kn −1 w−1
X n,jk
X v−1
1 X
n,lm
n,gh
(C
− Cin,jk )(Ci+v
− Cin,lm )(Ci+w
− Cin,gh ).
χ(8) = 3
kn w=2 v=0 u=0 i+u
This term can be further decomposed into 6 components. Successive conditioning and existing bounds give
n,jk
n,lm
n,lm
n,gh
n,gh n
|E (Ci+u
− Cin,jk )(Ci+v
− Ci+u
)(Ci+w
− Ci+v
) Fi | ≤ K∆n
n,jk
n,lm
n,lm
n,gh
n,gh n
1/4
|E (Ci+u
− Cin,jk )(Ci+v
− Ci+u
)(Ci+v
− Ci+u
) Fi | ≤ K∆3/4
n (∆n + ηi,kn )
n,jk
n,lm
n,lm
n,gh
|E (Ci+u
− Cin,jk )(Ci+v
− Ci+u
)(Ci+u
− Cin,gh )Fin | ≤ K∆n
n,jk
n,lm
n,gh
n,gh n
|E (Ci+u
− Cin,jk )(Ci+u
− Cin,lm )(Ci+w
− Ci+v
) Fi | ≤ K∆n
n,jk
n,lm
n,gh
n,gh n
|E (Ci+u
− Cin,jk )(Ci+u
− Cin,lm )(Ci+v
− Ci+u
) Fi | ≤ K∆n
n,jk
n,lm
n,gh
|E (Ci+u
− Cin,jk )(Ci+u
− Cin,lm )(Ci+u
− Cin,gh )Fin | ≤ K∆n
These bounds can be used to deduce
|E(χ(8)Fin )| ≤ K∆n .
This completes the proof.
47
Proof of (44) and (45) in Lemma 8
Observe that
n,lm
n,gh
βin,jk (Ci+k
− Cin,lm )(Ci+k
− Cin,gh ) =
n
n
n,gh
βin,jk βin,lm (Ci+k
− Cin,gh ) =
n
+
kX
n −1
1
n,lm
n,gh
ζ n,jk (Ci+k
− Cin,lm )(Ci+k
− Cin,gh ),
n
n
kn ∆n u=0 i,u
kX
kX
n −1
n −2 kX
n −1
1
1
n,gh
n,gh
n,gh
n,jk n,lm
)
+
(C
−
C
ζ
− Cin,gh )
ζ
ζ n,jk ζ n,lm (Ci+k
i
i+kn
n
kn2 ∆2n u=0 i,u i,u
kn2 ∆2n u=0 v=0 i,u i,v
kX
n −2 kX
n −1
1
n,gh
− Cin,gh ).
ζ n,lm ζ n,jk (Ci+k
n
kn2 ∆2n u=0 v=0 i,u i,v
Hence, (44) and (45) can be proved using the same strategy as for (43).
Proof of (46) and (47) in Lemma 8
Note that we have
n,jk
n,jk
n,jk n,lm
− βin,gh βin,lm βi+k
β
+ βin,gh βin,jk βin,lm − βin,gh βin,lm βi+k
γin,jk γin,lm βin,gh = βin,gh βi+k
n
n
n i+kn
n,jk
n,lm
n,jk
n,lm
n,jk
n,lm
− Cin,jk )
(Ci+k
− Cin,jk ) − βin,gh βin,lm (Ci+k
− Cin,lm ) + βin,gh βi+k
(Ci+k
− Cin,lm ) − βin,gh βin,jk (Ci+k
+ βin,gh βi+k
n
n
n
n
n
n
n,jk
n,lm
+ βin,gh (Ci+k
− Cin,jk )(Ci+k
− Cin,lm ),
n
n
and
n,gh n,jk n,lm
n,gh n,jk n,lm
n,gh n,lm n,jk
n,gh n,lm n,jk
γin,gh γin,jk γin,lm = βi+k
β
β
+ βi+k
β
βi
− βi+k
β
βi+kn − βi+k
β
βi+kn
n i+kn i+kn
n i
n i
n i
n,gh n,jk
n,lm
n,gh n,jk
n,lm
n,gh n,lm
n,jk
n,gh n,lm
n,jk
+ βi+k
β
(Ci+k
− Cin,lm ) − βi+k
β
(Ci+k
− Cin,lm ) + βi+k
β
(Ci+k
− Cin,jk ) − βi+k
β
(Ci+k
− Cin,jk )
n i+kn
n
n i
n
n i+kn
n
n i
n
n,gh
n,jk
n,lm
n,jk n,lm
n,jk
n,jk
+ βi+k
(Ci+k
− Cin,jk )(Ci+k
− Cin,lm ) − βin,gh βi+k
β
− βin,gh βin,jk βin,lm + βin,gh βin,lm βi+k
+ βin,gh βin,lm βi+k
n
n
n
n i+kn
n
n
n,jk
n,lm
n,lm
n,lm
n,jk
n,jk
− βin,gh βi+k
(Ci+k
− Cin,lm ) + βin,gh βin,jk (Ci+k
− Cin,lm ) − βin,gh βi+k
(Ci+k
− Cin,jk ) + βin,gh βin,lm (Ci+k
− Cin,jk )
n
n
n
n
n
n
n,jk
n,lm
n,jk n,lm
n,gh
n,gh
− βin,gh (Ci+k
− Cin,jk )(Ci+k
− Cin,lm ) + βi+k
β
(Ci+k
− Cin,gh ) + βin,jk βin,lm (Ci+k
− Cin,gh )
n
n
n i+kn
n
n
n,jk
n,gh
n,jk
n,gh
n,jk
n,lm
n,gh
− βin,lm βi+k
(Ci+k
− Cin,gh ) − βin,lm βi+k
(Ci+k
− Cin,gh ) + βi+k
(Ci+k
− Cin,lm )(Ci+k
− Cin,gh )
n
n
n
n
n
n
n
n,lm
n,gh
n,lm
n,jk
n,gh
− βin,jk (Ci+k
− Cin,lm )(Ci+k
− Cin,gh ) + βi+k
(Ci+k
− Cin,jk )(Ci+k
− Cin,gh )
n
n
n
n
n
n,jk
n,gh
n,jk
n,lm
n,gh
− βin,lm (Ci+k
− Cin,jk )(Ci+k
− Cin,gh ) + (Ci+k
− Cin,jk )(Ci+k
− Cin,lm )(Ci+k
− Cin,gh ).
n
n
n
n
n
1/2
n
From (39), notice that βin is Fi+k
-measurable and satisfies kE(βin |Fin )k ≤ K∆n .
n
Using the law of iterated expectations and existing bounds, it can be shown that
n,jk
|E(βin,lm βi+k
|Fin )| ≤ K∆3/4
n .
n
n,jk
|E(βin,lm βin,gh βi+k
|Fin )| ≤ K∆n
n
n,gh
n,jk
|E(βin,lm (Ci+k
− Cin,gh )βi+k
|Fin )| ≤ K∆n
n
n
n,lm
n,jk
|E(βi+k
(Ci+k
− Cin,jk )|Fin )| ≤ K∆3/4
n
n
n
n,jk
n,lm
n,gh
|E((Ci+k
− Cin,jk )(Ci+k
− Cin,lm )(Ci+k
− Cin,gh )|Fin )| ≤ K∆n .
n
n
n
(58)
By Lemma 3.3 in Jacod and Rosenbaum (2012), we have
n,gh n,ab
n
|E(βi+k
β
|Fi+k
)−
n
n i+kn
p
1
kn ∆n n,gh,ab
n,ga n,hb
n,gb
n,ha
n
(Ci+k
Ci+kn + Ci+k
Ci+k
)−
C i+kn | ≤ K ∆n (∆1/8
n + ηi+kn ,kn ).
n
n
n
kn
3
48
q/4
n,gh
n,gh
n,gh q n
n,gh
Hence, for ϕn,gh
∈ {βin,gh , Ci+k
−
C
},
which
satisfies
E(|ϕ
|
Fi ) ≤ K∆n and E(ϕi |Fin ) ≤
i
i
i
n
1/2
K∆n , it can be proved that
h
kn ∆n n,jk,lm i n n,jk n,lm
n,gh 1
n,jl
n,km
n,jm n,kl
n
1/4
n
|Fi | ≤ K∆3/4
β
β
|F
)
−
E
ϕ
C i+kn
|E(ϕn,gh
(C
C
+
C
C
)
−
i
n (∆n + ηi,2kn ).
i
i
i+kn i+kn
i+kn i+kn
kn i+kn i+kn
3
Next, successive conditioning and existing bounds give
n,jk,lm
|E(ϕn,gh
C i+kn
i
1/4
n
)| ≤ K∆1/4
n (∆n + ηi,kn )
n,jl
n,km
|E(ϕn,gh
Ci+k
Ci+k
)| ≤ K∆1/2
n ,
i
n
n
which implies
n,jk n,lm
1/4
n
βi+k
β
|Fin )| ≤ K∆3/4
|E(ϕn,gh
n (∆n + ηi,2kn ).
i
n i+kn
(59)
n
n
It is easy to see that (43), (58) and (59) and the inequality ηi,k
≤ ηi,2k
together yield (46) and (47).
n
n
Step 3: Asymptotic Distribution of the approximate estimator
First, we decompose the approximate estimator as
(A)
\
[H(C),
G(C)]T
(A1)
\
= [H(C),
G(C)]T
(A2)
\
− [H(C),
G(C)]T
,
with
(A1)
\
[H(C),
G(C)]T
3
=
2kn
d
X
[T /∆n ]−2kn +1
g,h,a,b=1
i=1
n
0
0
0
0
b n,gh − C
b n,gh )(C
b n,ab − C
b n,ab ),
∂gh H∂ab G (Ci−1
)(C
i
i
i+kn
i+kn
X
and
(A2)
\
[H(C),
G(C)]T
=
3
kn2
d
X
[T /∆n ]−2kn +1
g,h,a,b=1
i=1
X
0n
0
0
0
0
b n,ga C
b n,hb + C
b n,gb C
b )(C
b n,ha ).
∂gh H∂ab G (C
i
i
i
i
i
n
In this section, we use the notation Ci−1
= C(i−1)∆n and Fi = F(i−1)∆n to simplify the exposition. Given
the polynomial growth assumption satisfied by H and G and the fact that kn = θ(∆n )−1/2 , by Theorem 2.2
in Jacod and Rosenbaum (2012) we have
(A2)
1
3
\
√
[H(C),
G(C)]T − 2
θ
∆n
d
X
Z
g,h,a,b=1
!
T
∂gh H∂ab G
hb
(Ct )(cga
t ct
+
ha
cgb
i ct )dt
= Op (1),
0
which yields
1
1/4
∆n
(A2)
\
[H(C),
G(C)]T
3
− 2
θ
d
X
g,h,a,b=1
Z
!
T
∂gh H∂ab G
hb
(Ct )(cga
t ct
+
ha
cgb
i ct )dt
P
−→ 0.
0
(A1)
\
To study the asymptotic behavior of [H(C),
G(C)]T
the following multidimensional quantities
ζ(1)ni =
, we follow Aı̈t-Sahalia and Jacod (2014) and define
1 n 0 n 0 >
n
∆ Y (∆i Y ) − Ci−1
,
∆n i
49
ζ(2)ni = ∆ni c,
n
ζ 0 (u)ni = E(ζ(u)ni |Fi−1
),
ζ 00 (u)ni = ζ(u)ni − ζ 0 (u)ni ,
with
ζ r (u)ni = ζ r (u)n,gh
i
.
1≤g,h≤d
We also define, for m ∈ {0, . . . , 2kn − 1} and j, l ∈ Z,
ε(1)nm =
ε(2)nm =
2k
n −1
X

−1
if 0 ≤ m < kn
+1 if kn ≤ m < 2kn ,
ε(1)nq = (m + 1) ∧ (2kn − m − 1),
q=m+1

1/∆
n
zu,v
=
γ(u, v; m)nj,l =
3
2kn3
if u = v = 1
n
1
otherwise,
(l−m−1)∨(2kn −m−1)
X
ε(u)nq ε(u)nq+m ,
Γ(u, v)nm = γ(u, v; m)n0,2kn ,
q=0∨(j−m)
n
M (u, v; u0 , v 0 )n = zu,v
zun0 ,v0
2k
n −1
X
Γ(u, v)nm Γ(u0 , v 0 )nm .
m=1
The following decompositions hold,
kn −1 X
2
1 X
n
bi0 n = Ci−1
C
+
ε(u)nj ζ(u)ni+j ,
kn j=0 u=1
γin,gh γin,ab =
+
2
2
1 XX
kn2 u=1 v=1
j−1
2k
n −1 X
X
2k
n −1
X
2k
2
n −1 X
X
0
n
bi+k
bi0 n = 1
C
−
C
ε(u)nj ζ(u)ni+j ,
n
kn j=0 u=1
n,ab
ε(u)nj ε(v)nj ζ(u)n,gh
i+j ζ(v)i+j +
j=0
2k
n −2 2k
n −1
X
X
n,ab
ε(u)nj ε(v)nq ζ(u)n,gh
i+j ζ(v)i+q
j=0 q=j+1
!
n,ab
ε(u)nj ε(v)nq ζ(u)n,gh
.
i+j ζ(v)i+q
j=1 q=0
A change of the order of the summation in the last term gives
γin,gh γin,ab =
+
2
2
1 XX
kn2 u=1 v=1
2k
n −2 2k
n −1
X
X
2k
n −1
X
n,ab
ε(u)nj ε(v)nj ζ(u)n,gh
i+j ζ(v)i+j +
j=0
2k
n −2 2k
n −1
X
X
n,ab
ε(u)nj ε(v)nq ζ(u)n,gh
i+j ζ(v)i+q
j=0 q=j+1
!
n,gh
ε(v)nj ε(u)nq ζ(v)n,ab
.
i+j ζ(u)i+q
j=0 q=j+1
(A1)
\
Therefore, we can further rewrite [H(C),
G(C)]T
(A1)
\
[H(C),
G(C)]T
(A11)
\
= [H(C),
G(C)]T
as
(A12)
\
+ [H(C),
G(C)]T
50
(A13)
\
+ [H(C),
G(C)]T
, with
d
X
(A1w)
\
[H(C),
G(C)]T
=
2
X
d
A1w(H,
gh, u; G, ab, v)nT ,
w = 1, 2, 3,
g,h,a,b=1 u,v=1
and,
3
d
A11(H,
gh, u; G, ab, v)nT = 3
2kn
3
d
A12(H,
gh, u; G, ab, v)nT = 3
2kn
3
d
A13(H,
gh, u; G, ab, v)nT = 3
2kn
[T /∆n ]−2kn +1 2kn −1
X
X
i=1
j=0
n,ab
n
(∂gh H∂ab G)(Ci−1
)ε(u)nj ε(v)nj ζ(u)n,gh
i+j ζ(v)i+j ,
[T /∆n ]−2kn +1 2kn −2 2kn −1
X
X
i=1
j=0 q=j+1
X
n,ab
n
(∂gh H∂ab G)(Ci−1
)ε(u)nj ε(v)nq ζ(u)n,gh
i+j ζ(v)i+q ,
[T /∆n ]−2kn +1 2kn −2 2kn −1
X
X
i=1
j=0 q=j+1
X
n,gh
n
(∂gh H∂ab G)(Ci−1
)ε(v)nj ε(u)nq ζ(v)n,ab
i+j ζ(u)i+q ,
d
d
where we clearly have A13(H,
gh, u; G, ab, v)nT = A12(G,
ab, v; H, gh, u)nT . By a change of the order of the
summation,
[T /∆n ]
3 X
n
d
A11(H, gh, u; G, ab, v)T = 3
2kn i=1
(2kn −1)∧(i−1)
X
n
(∂gh H∂ab G)(Ci−j−1
)ε(u)nj ε(v)nj ζ(u)n,gh
ζ(v)n,ab
,
i
i
j=0∨(i+2kn −1−[T /∆n ])
[T /∆n ] (i−1)∧(2kn −1)
3 X
d
A12(H,
gh, u; G, ab, v)nT = 3
2kn i=2
(2kn −m−1)∧(i−m−1)
X
X
m=1
j=0∨(i+2kn −1−m−[T /∆n ])
n
(∂gh H∂ab G)(Ci−1−j−m
)×
ε(u)nj ε(v)nj+m ζgh (u)ni−m ζab (v)ni .
Set
[T /∆n ] 2kn −1
3 X X
n
g
A11(H,
gh, u; G, ab, v)nT = 3
(∂gh H∂ab G)(Ci−j−1
)ε(u)nj ε(v)nj ζ(u)n,gh
ζ(v)n,ab
,
i
i
2kn
j=0
i=2kn
[T /∆n ] (i−1)∧(2kn −1) (2kn −m−1)
X
X
3 X
n
g
(∂gh H∂ab G)(Ci−j−1−m
)ε(u)nj ε(v)nj+m ζgh (u)ni−m ζab (v)ni ,
A12(H,
gh, u; G, ab, v)nT = 3
2kn
m=1
j=0
i=2kn
and
A11(H, gh, u; G, ab, v)nT =
[T /∆n ] 2kn −1
3 X X
n
ε(u)nj ε(v)nj (∂gh H∂ab G)(Ci−2k
)ζ(u)n,gh
ζ(v)n,ab
i
i
n
3
2kn
j=0
i=2kn
[T /∆n ]
= Γ(u, v)n0
X
n
(∂gh H∂ab G)(Ci−2k
)ζ(u)n,gh
ζ(v)n,ab
,
i
i
n
i=2kn
(i−1)∧(2kn −1) (2kn −m−1)
X
X
3 X
n
(∂
H∂
G)(C
)
ε(u)nj ε(v)nj+m ζgh (u)ni−m ζab (v)ni
gh
ab
i−2kn
3
2kn
m=1
j=0
[T /∆n ]
A12(H, gh, u; G, ab, v)nT =
i=2kn
[T /∆n ]
=
X
n
(∂gh H∂ab G)(Ci−2k
)ρgh (u, v)ni ζab (v)ni ,
n
i=2kn
51
with
ρgh (u, v)ni =
2k
n −1
X
Γ(u, v)nm ζgh (u)ni−m .
m=1
The following results hold:
1 d
P
g
A1w(H, gh, u; G, ab, v)nT − A1w(H,
gh, u; G, ab, v)nT −→ 0
1/4
∆n
for all (H, gh, u, G, ab, v) and w = 1, 2.
(60)
1 g
P
A1w(H, gh, u; G, ab, v)nT − A1w(H, gh, u; G, ab, v)nT −→ 0
1/4
∆n
for all (H, gh, u, G, ab, v) and w = 1, 2.
(61)
Proof of (60) for w = 1
The proof is similar to Step 5 on page 548 of Aı̈t-Sahalia and Jacod (2014). Our proof deviates from the
are scaled by random variables
ζ(v)n,ab
latter reference by the fact that, in all the sums, the terms ζ(u)n,gh
i
i
rather that constant real numbers. First, observe that we can write
g
g
g
d − A11
g = A11(1)
d
d
d
A11
+ A11(2)
+ A11(3)
(2kn −1)∧[T /∆n ]
g
d
A11(1)
=
X
i=1
[T /∆n ]
g
d
A11(2)
=
X
i=[T /∆n ]−2kn +2
3
2kn3
3
2kn3
with
(2kn −1)∧(i−1)
!
X
n
(∂gh H∂ab G)(Ci−j−1
)ε(u)nj ε(v)nj
j=0∨(i+2kn −1−[T /∆n ])
(2kn −1)∧(i−1)
X
n
(∂gh H∂ab G)(Ci−j−1
)ε(u)nj ε(v)nj
j=0∨(i+2kn −1−[T /∆n ])
(2kn −1)
−
ζ(u)n,gh
ζ(v)n,ab
,
i
i
X
!
n
(∂gh H∂ab G)(Ci−j−1
)ε(u)nj ε(v)nj
ζ(u)n,gh
ζ(v)n,ab
,
i
i
j=0
[T /∆n ]−2kn +1
g
d
A11(3)
=
X
i=2kn
3
2kn3
(2kn −1)∧(i−1)
X
n
(∂gh H∂ab G)(Ci−j−1
)ε(u)nj ε(v)nj
j=0∨(i+2kn −1−[T /∆n ])
(2kn −1)
−
X
!
n
(∂gh H∂ab G)(Ci−j−1
)ε(u)nj ε(v)nj
ζ(u)n,gh
ζ(v)n,ab
.
i
i
j=0
g
d
It is easy to see that A12(3)
= 0. Using (40) with Z = c and (41), it can be shown that
n
n
E(kζ(1)ni kq |Fi−1
) ≤ Kq , E(kζ(2)ni kq |Fi−1
) ≤ Kq ∆q/2
n .
(62)
n
The polynomial growth assumption on H and G and the boundedness of Ct imply that |(∂gh H∂ab G)(Ci−j−1
)| ≤
P
(2k
−1)∧(i−1)
n
3
n
n
n
K. Hence, the random quantities 2k3 j=0∨(i+2kn −1−[T /∆n ]) (∂gh H∂ab G)(Ci−j−1 )ε(u)j ε(v)j and
n
P(2kn −1)
3
n
n
n
n
n
(∂
H∂
G)(C
)ε(u)
eu,v
defined as
gh
ab
i−j−1
j ε(v)j are Fi−1 − measurable and are bounded by γ
j=0
2k3
n
n
γ
eu,v
=



K



if (u, v) = (2, 2)
K/kn
if (u, v) = (1, 2), (2, 1)
K/kn2
if (u, v) = (1, 1).
52
Similarly, the quantity,
3
2kn3
(2kn −1)∧(i−1)
(2kn −1)
X
X
n
(∂gh H∂ab G)(Ci−j−1
)ε(u)nj ε(v)nj
j=0∨(i+2kn −1−[T /∆n ])
−
!
n
(∂gh H∂ab G)(Ci−j−1
)ε(u)nj ε(v)nj
,
j=0
n
n
is Fi−1
− measurable and bounded by 2e
γu,v
. Note also that, by (62) and the Cauchy Schwartz inequality,
we have,
n
|Fi−1
ζ(v)n,ab
)
E(|ζ(u)n,gh
i
i
≤
n
n
E(kζ(u)ni k2 |Fi−1
)1/2 E(kζ(v)ni k2 |Fi−1
)1/2
≤


K∆n

if (u, v) = (2, 2)
1/2
K∆n
if (u, v) = (1, 2), (2, 1)



K
if (u, v) = (1, 1).
g
g
−1/2
1/2
d
d
The above bounds, together with the fact that kn = θ∆n , give E(|A11(1)|)
≤ K∆n and E(|A11(2)|)
≤
g
g
1/2
−1/4
−1/4
d
d
K∆n for all (u, v). These two results together imply A11(1) = o(∆n ) and A11(2) = o(∆n ), which
yields the result.
Proof of (60) for w = 2
We proceed similarly to Step 6 on page 548 of Aı̈t-Sahalia and Jacod (2014). First, observe that we have
g
g
d − A12
g = A12(1)
d
d
A12
+ A12(2)
(2kn −1)∧[T /∆n ]
(i−1)
X
X
i=2
m=1
g
d
A12(1)
=
with
3 2kn3
(2kn −m−1)∧(i−m−1)
X
n
(∂gh H∂ab G)(Ci−1−j−m
)ε(u)nj ε(v)nj+m
j=0∨(i+2kn −1−m−[T /∆n ])
!
ζgh (u)ni−m ζab (v)ni ,
(i−1)∧(2kn −1) [T /∆n ]
X
g
d
A12(2)
=
X
m=1
i=[T /∆n ]−2kn +2
(2kn −m−1)∧(i−m−1)
3
2kn3
X
n
(∂gh H∂ab G)(Ci−1−j−m
)ε(u)nj ε(v)nj+m
j=0∨(i+2kn −1−m−[T /∆n ])
(2kn −m−1)
X
−
n
(∂gh H∂ab G)(Ci−1−j−m
)ε(u)nj ε(v)nj+m
!
ζgh (u)ni−m
ζab (v)ni .
j=0
It is easy to see that the quantity
κm,n
=
i
3 2kn3
(2kn −m−1)∧(i−m−1)
X
n
(∂gh H∂ab G)(Ci−1−j−m
)ε(u)nj ε(v)nj+m
j=0∨(i+2kn −1−m−[T /∆n ])
n
n
is Fi−m−1
measurable and bounded by γ
eu,v
. Let
(i−1)
κni =
X
m=1
3 2kn3
(2kn −m−1)∧(i−m−1)
X
n
(∂gh H∂ab G)(Ci−1−j−m
)ε(u)nj ε(v)nj+m ζgh (u)ni−m .
j=0∨(i+2kn −1−m−[T /∆n ])
n
It follows that κni is Fi−1
-measurable. We have
n z
E(|κm,n
|z F0 ) ≤ (e
γu,v
)
i
53
|E(ζ(u)ni−m |Fi−m−1 )| ≤

K √ ∆
if u = 1
n
K∆n

K
z
E(kζ(u)ni−m kz |Fi−m−1 ) ≤
Kz ∆z/2
n
,
if u = 2
Using Lemma 5, we deduce that for z ≥ 2,

K (e
n z z/2
z γu,v ) kn
E(|κni |z ) ≤
z/2
Kz (e
γ n )z /kn
u,v

−3z/2
if u = 1 Kz /kn
≤
−z/2
if u = 2 Kz kn
if u = 1
if u = 2
if v = 1
if v = 2
Using the above result, and similarly to step 6 on page 548 of Aı̈t-Sahalia and Jacod (2014), we obtain that
P
P
1 g
1 g
d
d
1/4 A12(1) ⇒ 0. A similar argument yields
1/4 A12(2) ⇒ 0, which completes the proof of (60) for w = 2.
∆n
∆n
Proof of (61) for w = 1
Define
(C),i,n
Θ(u, v)0
=
2kn −1 3 X
n
n
(∂gh H∂ab G)(Ci−j−1
) − (∂gh H∂ab G)(Ci−2k
) ε(u)nj ε(v)nj .
n
3
2kn j=0
By Taylor expansion, the polynomial growth assumption on H and G and using (40) with Z = c, we have
p
n
n
n
) − (∂gh H∂ab G)(Ci−2k
)Fi−2k
E (∂gh H∂ab G)(Ci−j−1
≤ K(kn ∆n ) ≤ K ∆n for j = 0, . . . , 2kn − 1
n
n
n
n
n
E(|(∂gh H∂ab G)(Ci−j−1
) − (∂gh H∂ab G)(Ci−2k
)|q |Fi−2k
)| ≤ K(kn ∆n )q/2 ≤ K∆q/4
for q ≥ 2
n
n
n
(C),i,n
(C),i,n
(C),i,n
n
n
n
, |E Θ(u, v)0
|Fi−2k
|≤
Next, observe that Θ(u, v)0
is Fi−1
-measurable and satisfies |Θ(u, v)0
|≤γ
eu,v
n
(C),i,n q n
q/4 n q
1/2 n
γu,v ) where the latter follows from the Hölder in| Fi−2kn ≤ Kq ∆n (e
eu,v and E |Θ(u, v)0
K∆n γ
equality. We aim to prove that
b=
E
" [T /∆n ]
X
1
1/4
∆n
#
(C),i,n
Θ(u, v)0
ζ(u)n,gh
ζ(v)n,ab
i
i
i=2kn
converges to zero in probability for any H, G, g, h, a, and b with u, v = 1, 2.
To show this result, we first introduce the following quantities:
b
E(1)
=
b
E(2)
=
1
" [T /∆n ]
X
1/4
∆n
i=2kn
" [T /∆n ]
X
1
1/4
∆n
#
(C),i,n
n
Θ(u, v)0
E(ζ(u)n,gh
ζ(v)n,ab
|Fi−1
)
i
i
#
(C),i,n
Θ(u, v)0
ζ(u)n,gh
ζ(v)n,ab
i
i
−
n
)
E(ζ(u)n,gh
ζ(v)n,ab
|Fi−1
i
i
i=2kn
b = E(1)
b
b
with E
+ E(2).
By Cauchy-Schwartz inequality, we have
E(|ζ(u)n,gh
ζ(v)n,ab
|q )
i
i
≤
n q/2
(b
γu,v
) , where
54
n
γ
bu,v
=



K
if (u, v) = (1, 1)
K∆n
if (u, v) = (1, 2), (2, 1)



K∆2n
if (u, v) = (2, 2)
,
n
Since ζ(u)n,gh
ζ(v)n,ab
is Fin -measurable, the martingale property of ζ(u)n,gh
ζ(v)n,ab
−E(ζ(u)n,gh
ζ(v)n,ab
|Fi−1
)
i
i
i
i
i
i
implies, for all (u, v),
2
n 2 n
b
E(|E(2)|
) ≤ K∆−3/2
(∆1/4
eu,v
) γ
bu,v ≤ K∆n .
n
n γ
P
P
b
b
The latter inequality implies E(2)
⇒ 0 for all (u, v). It remains to show that E(1)
⇒ 0.
We remind some bounds under Assumption 2, see (B.83) in Aı̈t-Sahalia and Jacod (2014),
n
|E(ζ(1)n,gh
ζ(2)n,ab
|Fi−1
)| ≤ K∆n ,
i
i
(63)
n
|E(ζ(1)n,gh
ζ(1)n,ab
|Fi−1
)
i
i
(64)
n,ga n,hb
n,gb n,ha − Ci−1
Ci−1 + Ci−1
Ci−1 | ≤ K∆1/2
n ,
p
n,gh,ab
n
n
|Fi−1
ζ(2)n,ab
− C i−1 ∆n )| ≤ K∆3/2
|E(ζ(2)n,gh
n ( ∆n + ηi ).
i
i
(65)
Case (u, v) ∈ {(1, 2), (2, 1)}. By (63) we have
b
E(|E(1)|)
≤K
1
T
P
n
b
(∆1/4
eu,v
∆n ) ≤ K∆1/2
so E(1)
⇒ 0.
n γ
n
∆n ∆1/4
n
Case (u, v) ∈ {(1, 1), (2, 2)}. Set
b0
E (1) =
1
" [T /∆n ]
X
1/4
∆n
b 00 (1) =
E
b 000
E (1) =
i=2kn
" [T /∆n ]
X
1
1/4
∆n
1
#
(C),i,n n
Θ(u, v)0
Vi−2kn
#
(C),i,n
Θ(u, v)0
n
Vi−1
−
n
Vi−2k
n
i=2kn
" [T /∆n ]
X
1/4
∆n
(C),i,n
Θ(u, v)0
n
E(ζ(u)n,gh
ζ(v)n,ab
|Fi−1
)
i
i
−
n
Vi−1
#
i=2kn
where
n
Vi−1
=

n,ga n,hb
n,gb n,ha


Ci−1 Ci−1 + Ci−1 Ci−1
if (u, v) = (2, 2)


otherwise
n,gh,ab
C
∆n
 i−1
0
if (u, v) = (1, 1)
b
b 0 (1) + E
b 00 (1) + E
b 000 (1). Using (64) and (65), it can be shown that
Note that we have E(1)
=E
b 000 (1)|) ≤
E(|E

K
1/4 n
1/2
1
γ
eu,v )∆n
5/4 (∆n
∆n
1/4 n
3/2
1
K 5/4
eu,v )∆n
(∆n γ
∆n
if (u, v) = (1, 1)
if (u, v) = (2, 2)
≤ K∆1/2
in all cases.
n
P
b 0 (1) ⇒
Next, we prove E
0. To this end, write
b0
1
E (1) =
" [T /∆n ]−2kn +1
X
1/4
∆n
#
(C),i−1+2kn ,n
Θ(u, v)0
V(i−1)∆n
.
i=1
n
The fact that the summand in the last sum is Fi+2k
-measurable and lemma B.8 in Aı̈t-Sahalia and Jacod
n −2
(2014) imply that it is sufficient to show
1
1/4
∆n
" [T /∆n ]−2kn +1
X
#
(C),i−1+2kn ,n
n
|E(Θ(u, v)0
V(i−1)∆n |Fi−1
)|
i=1
55
P
⇒ 0 and
" [T /∆n ]−2kn +1
#
X
(C),i−1+2kn ,n
E |Θ(u, v)0
V(i−1)∆n )|2
⇒ 0.
2kn − 2
1/2
∆n
i=1
The first result readily follows from the inequality

K∆1/2
n
eu,v
n γ
(C),i−1+2kn ,n
n
|E(Θ(u, v)0
V(i−1)∆n |Fi−1 )| ≤
K∆1/2
e n ∆n
n γ
u,v
while the second is a direct consequence of

K∆1/2
n 2
γu,v
)
n (e
(C),i−1+2kn ,n
E(|Θ(u, v)0
V(i−1)∆n |2 ) ≤
1/2
K∆n (e
γ n ) 2 ∆2
u,v
n
if (u, v) = (1, 1)
if (u, v) = (2, 2)
if (u, v) = (1, 1)
if (u, v) = (2, 2)
≤ K∆3/2
in all cases
n
≤ K∆5/2
in all cases.
n
P
b 00 (1) =⇒
Finally, to prove that E
0, we use the fact that
(C),i,n
E(|Θ(u, v)0
(C),i,n 2 1/2
V(i−1)∆n − V(i−2kn )∆n |) ≤ E(|Θ(u, v)0
| ) E(|V(i−1)∆n − V(i−2kn )∆n |2 )1/2

K∆1/2
n
eu,v
if (u, v) = (1, 1)
n γ
≤
,
1/4
1/4
n
K∆n γ
e ∆n ∆n
if (u, v) = (2, 2)
u,v
which follows by the Cauchy-Schwartz inequality and earlier bounds. In particular, successive conditioning
1/2
together with Assumption 2 imply that for (u, v) = (1, 1) and (2, 2), E(|V(i−1)∆n − V(i−2kn )∆n |2 ) ≤ ∆n .
Proof of (61) for w = 2
Our aim here is to show that
b
E(2)
=
1
[T /∆n ]
X
1/4
∆n i=2kn
2k
n −1 X
m=1
3
2kn3
2knX
−m−1
(∂gh H∂ab G)(cni−j−m−1 ) − (∂gh H∂ab G)(cni−2kn ) ε(u)nj ε(v)nj+m ×
j=0
!
n,ab
ζ(u)n,gh
=⇒ 0.
i−m ζ(v)i
P
For this purpose, we introduce some new notation. For any 0 ≤ m ≤ 2kn − 1, set
Θ(u, v)(C),i,n
m
3
= 3
2kn
ρ(u, v)(C),i,n,gh =
2knX
−m−1
(∂gh H∂ab G)(cni−j−m−1 ) − (∂gh H∂ab G)(cni−2kn ) ε(u)nj ε(v)nj+m
j=0
2k
n −1
X
Θ(u, v)(C),i,n
ζ(u)n,gh
m
i−m .
m=1
(C),i,n
It is easy to see that Θ(u, v)m
n
is Fi−m−1
measurable and satisfies, by Hölder inequality,
n
(C),i,n q n
n q
|Θ(u, v)(C),i,n
|≤γ
eu,v
and E |Θ(u, v)m
| Fi−2kn ≤ Kq ∆q/4
γu,v
) .
m
n (e
56
Lemma 5 implies that for q ≥ 2,

K (∆1/4
n q q/2
eu,v
) kn
n γ
q
E(|ρ(u, v)(C),i,n,gh |q ) ≤
1/4
q/2
Kq (∆n γ
en )q /kn

K /k 2q
q
n
≤
q

if u = 2
Kq kn
if u = 1
u,v
if v = 1
.
(66)
if v = 2
Set
[T /∆n ]
1
b 0 (2) =
E
X
1/4
∆n i=2kn
b 00 (2) =
E
n
ρ(u, v)(C),i,n,gh E(ζ(v)n,ab
|Fi−1
),
i
[T /∆n ]
1
X
1/4
∆n i=2kn
n
ρ(u, v)(C),i,n,gh (ζ(v)n,ab
− E(ζ(v)n,ab
|Fi−1
)).
i
i
b 00 (2)|2 ) ≤ K∆1/2
The martingale increments property implies E(|E
in all the cases, which in turn implies
n
P
b 00 (2) =⇒ 0. Next, using the bounds on ρ(u, v)(C),i,n,gh and similarly to step 7 on page 549 of Aı̈t-Sahalia
E
P
b 0 (2) =⇒
and Jacod (2014), we obtain that E
0.
Return to the proof of Theorem 1
So far, we have proved that
1 1/4
∆n
(A1)
\
[H(C),
G(C)]T
d
X
−
2
X
A11(H, gh, u; G, ab, v)nT + A12(H, gh, u; G, ab, v)nT
g,h,a,b=1 u,v=1
+ A12(G, ab, v; H, gh, u)nT
P
−→ 0.
We next show that,
[T /∆n ]
1
X
1/4
∆n i=2kn
1 1/4
∆n
0
P
n
(∂gh H∂ab G)(Ci−2k
)ρgh (u, v)ni ζab (v)ni =⇒ 0, ∀ (u, v)
n
Z
T
A11(H, gh, u; G, ab, v) −
gh,ab
(∂gh H∂ab G)(Ct )C t
(67)
P
dt =⇒ 0 when (u, v) = (2, 2)
(68)
0
1 1/4
A11(H, gh, u; G, ab, v) −
∆n
3
θ2
Z
T
P
(∂gh H∂ab G)(Ct )(Ctga Cthb + Ctgb Ctha )dt =⇒ 0 when (u, v) = (1, 1)
0
(69)
1
1/4
∆n
P
A11(H, gh, u; G, ab, v) =⇒ 0 when (u, v) = (1, 2), (2, 1)
(70)
which will in turn imply
1 1/4
∆n
(A)
\
[H(C),
G(C)]T
d
2 [T /∆n ]
00
3 X X X h
n
− [H(C), G(C)]T − 3
(∂gh H∂ab G)(Ci−2k
)ρgh (u, v)ni ζab (v)ni
n
2kn
u,v=1
g,h,a,b
i=2kn
(71)
00
n
+ (∂ab H∂gh G)(Ci−2k
)ρab (v, u)ni ζgh (v)ni
n
57
i
P
=⇒ 0.
(72)
(67) can be proved easily following steps similar to step 7 on page 549 of Aı̈t-Sahalia and Jacod (2014) and
using the bounds of ρ(u, v)n,gh
in (66) . To show (68),(69) and (70), we set
i
[T /∆n ]
A11(H, gh, u; G, ab, v) = Γ(u, v)n0
X
.
ζ(v)n,ab
(∂gh H∂ab G)(Ci−1 )ζ(u)n,gh
i
i
i=2kn
Then it holds that,
1 1/4
∆n
P
A11(H, gh, u; G, ab, v) − A11(H, gh, u; G, ab, v) ⇒ 0.
(C),i,n
This result can be proved following similar steps as for (60) in case w = 1 by replacing Θ(u, v)0
by Γ(u, v)n0 ((∂gh H∂ab G)(Ci−1 ) − (∂gh H∂ab G)(Ci−2kn )), which has the same bounds as the former. Next,
decompose A11 as follows,
" [T /∆n ]
X
n
A11(H, gh, u; G, ab, v) = Γ(u, v)n0
(∂gh H∂ab G)(Ci−1 )Vi−1
i=2kn
[T /∆n ]
X
+
n,ab
n
n
(∂gh H∂ab G)(Ci−1 ) E(ζ(u)n,gh
ζ(v)
|F
)
−
V
i−1
i−1
i
i
i=2kn
[T /∆n ]
X
+
(∂gh H∂ab G)(Ci−1 )
ζ(u)n,gh
ζ(v)n,ab
i
i
−
n
E(ζ(u)n,gh
ζ(v)n,ab
|Fi−1
)
i
i
#
.
i=2kn
(C),i,n
We follow the proof of (61) for w = 1, and we replace Θ(u, v)0
by Γ(u, v)n0 (∂gh H∂ab G)(Ci−1 ), which
n
n
satisfies only the condition |Γ(u, v)0 (∂gh H∂ab G)(Ci−1 )| ≤ γ
eu,v . This calculation shows that that the last
1/4
two terms in the above decomposition of vanish at a rate slower that ∆n . Therefore,
1
1/4
A11(H, gh, u; G, ab, v) −
∆n
Γ(u, v)n0
/∆n ]
[TX
n
(∂gh H∂ab G)(Ci−1 )Vi−1
!
⇒ 0.
i=2kn
As a consequence, for (u, v) = (1, 2) and (2, 1),
1
1/4
A11(H, gh, u; G, ab, v) ⇒ 0.
∆n
The results follow from the following observation,
1
1/4
∆n
Γ(u, v)n0
d
X
[T /∆n ]
X
n
(∂gh H∂ab G)(Ci−1 )Vi−1
(u, v)
g,h,a,b=1 i=2kn
3
− 2
θ
Z
!
T
(∂gh H∂ab G)(Ct )(Ctga Cthb
+
Ctgb Ctha )dt
0
for (u, v) = (2, 2)
1
d
X
1/4
∆n
g,h,a,b=1
Γ(u, v)n0
/∆n ]
[TX
n
(∂gh H∂ab G)(Ci−1 )Vi−1
(u, v) − [H(C), G(C)]T
!
⇒ 0, for (u, v) = (1, 1).
i=2kn
Set
ξ(H, gh, u; G, ab, v)ni =
1
1/4
∆n
n
00
(∂gh H∂ab G)(Ci−2k
)ρgh (u, v)ni ζab
(v)ni ,
n
58
⇒ 0,
[t/∆n ]
Z(H, gh, u; G, ab, v)nt
=
X
∆1/4
n
ξ(H, gh, u; G, ab, v)ni .
i=2kn
Notice that (71) implies
1 1/4
∆n
(A)
\
[H(C),
G(C)]T −[H(C), G(C)]T
d
X
L
=
2
X
1 1/4
g,h,a,b=1 u,v=1
∆n
Z(H, gh, u; G, ab, v)nT +Z(H, ab, v; G, gh, u)nT .
(73)
(A)
(A) \
\
Next, observe that to derive the asymptotic distribution of [H1 (C),
G1 (C)]T , . . . , [Hκ (C),
Gκ (C)]T
, it
1
n
1/4 Z(H, gh, u; G, ab, v)T .
∆
suffices to study the joint asymptotic behavior of the family of processes
n
It is easy to see that ξ(H, gh, u; G, ab, v)ni are martingale increments, relative to the discrete filtration
(Fin ). Therefore, by Theorem 2.2.15 of Jacod and Protter (2012), to obtain the joint asymptotic distri1
bution of 1/4
Z(H, gh, u; G, ab, v)nT , it is enough to prove the following three properties, for all t > 0, all
∆n
(H, gh, u; G, ab, v), (H 0 , g 0 h0 , u0 ; G0 , a0 b0 , v 0 ) and all martingales N which are either bounded and orthogonal
to W , or equal to one component W j ,
[t/∆n ]
n
X
n
E(ξ(H, gh, u; G, ab, v)ni ξ(H 0 , g 0 h0 , u0 ; G0 , a0 b0 , v 0 )ni |Fi−1
)
A (H, gh, u; G, ab, v), (H 0 , g 0 h0 , u0 ; G0 , a0 b0 , v 0 ) :=
t
i=2kn
=⇒ A (H, gh, u; G, ab, v), (H 0 , g 0 h0 , u0 ; G0 , a0 b0 , v 0 )
P
t
[t/∆n ]
X
P
n
E(|ξ(H, gh, u; G, ab, v)ni |4 |Fi−1
) =⇒ 0
i=2kn
[t/∆n ]
B(N ; H, gh, u; G, ab, v)nt
:=
X
P
n
E(ξ(H, gh, u; G, ab, v)ni ∆ni N |Fi−1
) =⇒ 0.
i=2kn
Using the polynomial growth assumption on Hr and Gr , the second and the third results can be proved by
a natural extension to the multivariate case of (B.105) and (B.106) in Aı̈t-Sahalia and Jacod (2014).
Define

0
0
0
0

(Ctaa Ctbb + Ctab Ctba ) if (v, v 0 ) = (1, 1)


0
0
ab,a b
a0 b0
Vab
(v, v 0 )t = C t
if (v, v 0 ) = (2, 2)



0
otherwise,
and
0
0
g h
V gh (u, u0 )t =

gg 0 hh0
gh0 hg 0

(C
C
+
C
Ct )

t
t
t

if
(u, u0 ) = (1, 1)
Ct
if
(u, u0 ) = (2, 2)
0
otherwise.
0
0
gh,g h



Once again using the polynomial growth assumption on Hr and Gr and following steps similar to the proof
of (B.104) in Aı̈t-Sahalia and Jacod (2014), one can show that
A (H, gh, u; G, ab, v),(H 0 , g 0 h0 , u0 ; G0 , a0 b0 , v 0 ) =
t
Z t
g 0 h0
a0 b0
M (u, v; u0 , v 0 )
(∂gh H∂ab G∂g0 h0 H∂a0 b0 G)(Cs )Vab
(v, v 0 )s V gh (u, u0 )s ds,
0
59
with
M (u, v; u0 , v 0 ) =


3/θ3




3/4θ
if
(u, v; u0 , v 0 ) = (1, 1; 1, 1)
if
(u, v; u0 , v 0 ) = (1, 2; 1, 2), (2, 1; 2, 1)


151θ/280 if (u, v; u0 , v 0 ) = (2, 2; 2, 2)




0
otherwise.
Therefore,
we have
A (H, gh, u; G, ab, v), (H 0 , g 0 h0 , u0 ; G0 , a0 b0 , v 0 )
=
T
 R
0
0
0
0
0
0
T
gg 0 hh0
3
0 0 0 0

0 0
+ Ctgh Cthg )(Ctaa Ctbb + Ctab Ctba )dt

β 3 0 (∂gh H∂ab G∂g h H ∂a b G )(Ct )(Ct Ct


0
0

 3 R T (∂ H∂ G∂ 0 0 H 0 ∂ 0 0 G0 )(C )(C gg0 C hh0 + C gh0 C hg0 )C ab,a b dt

ab
t

t
t
t
t
t
 4β R0 gh ab g h
0
0
0
0
gh,g 0 h0
T
3
0
0
aa
bb
ab
ba
0 0
0 0
dt
4β 0 (∂gh H∂ab G∂g h H ∂a b G )(Ct )(Ct Ct + Ct Cs )ts


RT
ab,a0 b0 gh,g 0 h0

151β
0
0

0 0
0 0
Ct
dt


280 0 (∂gh H∂ab G∂g h H ∂a b G )(Ct )C s



0
(A)
\
Using (73), we deduce that the asymptotic covariance between [Hr (C),
Gr (C)]T
given by
d
X
d
X
2
X
A (Hr , gh, u; Gr , ab, v), (Hs , g 0 h0 , u0 ; Gs , a0 b0 , v 0 )
g,h,a,b=1 g 0 ,h0 ,a0 ,b0 =1 u,v,u0 ,v 0 =1
+ A (Hr , gh, u; Gr , ab, v), (Hs , a0 b0 , v 0 ; Gs , g 0 h0 , u0 )
if
(u, v; u0 , v 0 ) = (1, 1; 1, 1)
if
(u, v; u0 , v 0 ) = (1, 2; 1, 2)
if
(u, v; u0 , v 0 ) = (2, 1; 2, 1)
if
(u, v; u0 , v 0 ) = (2, 2; 2, 2)
otherwise.
(A)
\
and [Hs (C),
Gs (C)]T
is
T
+ A (Hr , ab, v; Gr , gh, u), (Hs , g 0 h0 , u0 ; Gs , a0 b0 , v 0 )
T
!
0 0 0
0 0
0
+ A (Hr , ab, v; Hr , gh, u), (Hs , a b , v ; Gs , g h , u )
.
T
T
After some simple calculations, the above expression can be rewritten as
d
X
d
X
g,h,a,b=1 j,k,l,m=1
6
θ3
Z
T
h
∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs (Ct ) (Ctgj Cthk + Ctgk Cthj )(Ctal Ctbm + Ctam Ctbl )
0
i
+ (Ctaj Ctbk + Ctak Ctbj )(Ctgl Cthm + Ctgm Cthl ) dt
Z
i
h gh,jk ab,lm
151θ t
ab,jk gh,lm
+
∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs (Ct ) C
C
+C
C
dt
140 0
Z t
h
3
ab,lm
gh,jk
+ (Ctal Ctbm + Ctam Ctbl )C t
+
∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs (Ct ) (Ctgj Cthk + Ctgk Cthj )C t
2θ 0
!
i
ab,jk
gh,lm
gl hm
gm hl
aj bk
ak bj
+ (Ct Cs + Ct Cs )C t
+ (Ct Ct + Ct Ct )C t
dt ,
which completes the proof.
B.2
Proof of Theorem 2
Using the polynomial growth assumption on Hr , Gr , Hs and Gs and Theorem 2.2 in Jacod and Rosenbaum
(2012), one can show that
6 b r,s,(1) P
r,s,(1)
−→ ΣT
.
Ω
θ3 T
60
Next, by equation (3.27) in Jacod and Rosenbaum (2012), we have
3 b r,s,(3) 6 b r,s,(1) P
r,s,(3)
[Ω
− ΩT
] −→ ΣT
.
2θ T
θ
Finally, to show that
151θ 9 b r,s,(2)
4 b r,s,(1) 4 b r,s,(3) P
r,s,(2)
[Ω
+ 2Ω
− ΩT
] −→ ΣT
,
140 4θ2 T
θ T
3
b n by C
b 0 n is negligible.
we first observe that as in Step 1, the approximation error induced by replacing C
i
i
For 1 ≤ g, h, a, b, j, k, l, m ≤ d and 1 ≤ r, s ≤ d, we define
[T /∆n ]−4kn +1
X
cn =
W
T
b n )γ n,gh γ n,jk γ n,ab γ n,lm
(∂gh Hr ∂ab Gr ∂gh Hs ∂lm Gs )(C
i
i
i
i+2kn i+2kn
i=1
w(1)
b ni
n,ab
= (∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs )(Cin )E(γin,gh γin,jk γi+2k
γ n,lm |Fin )
n i+2kn
n,ab
n,ab
w(2)
b ni = (∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs )(Cin )(γin,gh γin,jk γi+2k
γ n,lm − E(γin,gh γin,jk γi+2k
γ n,lm |Fin ))
n i+2kn
n i+2kn
bin ) − (∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs )(Cin ) γ n,gh γ n,jk γ n,ab γ n,lm
w(3)
b ni = (∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs )(C
i
i
i+2kn i+2kn
[T /∆n ]−4kn +1
X
c (u)n =
W
t
w
bi (u), u = 1, 2, 3.
i=1
c (3)nt . By Taylor expansion and using repeatedly the boundc (2)nt + W
c (1)nt + W
ctn = W
Note that we have W
edness of Ct , we have
n
|w(3)
b ni | ≤ (1 + kβin k4(p−1) )kβin kkγin k2 kγi+2k
k2 ,
n
P
5/4
c (3)nt −→
0. Using Cauchy-Schwartz inequality and the bound
which implies E(|w(3)
b ni |) ≤ K∆n and W
q/4
b ni is Fi+4kn −measurable,
b ni |2 ) ≤ K∆2n . Observing furthermore that w(2)
E(kγin kq |Fin ) ≤ K∆n , we have E(|w(2)
P
c (2)nt −→ 0. Also, define
we use Lemma B.8 in Aı̈t-Sahalia and Jacod (2014) to show that W
h
win = (∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs )(Cin )
4
kn2 ∆n
(Cin,ga Cin,hb + Cin,gb Cin,ha )(Cin,jl Cin,km + Cin,jm Cin,kl )
4
4(kn2 ∆n ) n,gh,ab n,jk,lm i
4
n,gh,ab
n,jk,lm
+ (Cin,ga Cin,hb + Cin,gb Cin,ha )C i
+
Ci
Ci
,
+ (Cin,jl Cin,km + Cin,jm Cin,kl )C i
3
3
9
[T /∆n ]−4kn +1
X
n
WT = ∆n
win .
i=1
√
P
The cadlag property of c and C, kn ∆n −→ θ, and the Riemann integral argument imply WTn −→ WT
where
Z
T
h4
4
gh,ab
(∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs )(Ct ) 2 (Ctga Cthb + Ctgb Ctha )(Ctjl Ctkm + Ctjm Ctkl ) + (Ctjl Ctkm + Ctjm Ctkl )C t
θ
3
0
4 ga hb
4θ2 gh,ab jk,lm i
jk,lm
+ (Ct Ci + Ctgb Ctha )C t
C
Ct
+
dt.
3
9 t
WT =
In addition, by Lemma 4, we have
[T /∆n ]−4kn +1
X
c (1)n − W n |) ≤ ∆n E
E(|W
T
T
i=1
61
(∆1/8
n
!
+ ηi,4kn ) .
P
c n −→
Hence, by the third result of Lemma 6 we have W
Wt , from which it can be deduced that
T
4
9 hc n
W (1)T + 2
4θ2
kn
−
2
kn
[T /∆n ]−4kn +1
X
bin )[Cin (jk, lm)Cin (gh, ab)]
(∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs )(C
i=1
[T /∆n ]−4kn +1
X
bin )Cin (gh, ab)γ n,jk γ n,lm
(∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs )(C
i
i
i=1
[T /∆n ]−4kn +1
2
kn
Z
P
−→
X
−
b n )C n (jk, lm)γ n,gh γ n,ab
(∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs )(C
i
i
i
i
i
i=1
T
gh,ab
(∂gh Hr ∂ab Gr ∂jk Hs ∂lm Gs )(Ct )C t
jk,lm
Ct
dt.
0
The result follows from the above convergence, a symmetry argument, and straightforward calculations.
62

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