ANOVA FOR DIFFUSIONS AND ITO PROCESSES By Per Aslak Mykland and Lan Zhang ∗ The ...

37 Pages·2012·0.26 MB·English

SubmittedtotheAnnalsofStatistics
ANOVA FOR DIFFUSIONS AND ITO PROCESSES
By Per Aslak Mykland and Lan Zhang (cid:3)
The University of Chicago, and
Carnegie Mellon University and University of Illinois at Chicago
Ito processes are the most common form of continuous semi-
martingales, andincludedi(cid:11)usionprocesses. Thepaperis concerned
withthenonparametricregression relationshipbetweentwosuchIto
processes. We are interested in the quadratic variation (integrated
volatility)of theresidualinthisregression, overaunitof time (such
asaday).Amainconceptual(cid:12)ndingisthatthisquadraticvariation
can be estimated almost as if the residual process were observed,
the di(cid:11)erence being that there is also a bias which is of the same
asymptotic order as the mixednormal error term.
The proposed methodology, \ANOVA for di(cid:11)usions and Ito pro-
cesses",canbeusedtomeasurethestatisticalqualityofaparametric
model,and,nonparametrically,theappropriatenessofaone-regressor
modelingeneral.Ontheotherhand,italsohelpsquantifyandchar-
acterize the trading (hedging) error in the case of (cid:12)nancial applica-
tions.
1. Introduction. We consider the regression relationship between two stochas-
tic processes (cid:4) and S ,
t t
d(cid:4) = (cid:26) dS +dZ ; 0 t T; (1.1)
t t t t
(cid:20) (cid:20)
whereZ isaresidualprocess.Wesupposethat theprocessesS and(cid:4) areobserved
t t t
at discrete samplingpoints0 = t <::: < t = T.With theadvent of highfrequency
0 k
(cid:12)nancial data, this type of regression has been a growing topic of interest in the
literature, cf. Section 2.4. Theprocesses (cid:4) andS will beIto processes,which is the
t t
mostcommonlyusedtypeofcontinuoussemimartingale.Di(cid:11)usionsareaspecialcase
(cid:3)This research was supported in part by National Science Foundation grants DMS 99-71738
(Mykland) and DMS-0204639 (Mykland and Zhang). The authors would like to thank The Bend-
heim Center for Finance at Princeton University,where part of the research was carried out.
Keywords and phrases:ANOVA,continuoussemimartingale, statisticaluncertainty,goodnessof
(cid:12)t,discretesampling,parametricandnonparametricestimation, smallintervalasymptotics,stable
convergence, option hedging
1
2 P.A. MYKLAND AND L. ZHANG
of Ito processes. De(cid:12)nitions are made precise in Section 2.1 below. The di(cid:11)erential
in (1.1) is that of an Ito stochastic integral, as de(cid:12)nedinChapter I.4.d of Jacod and
Shiryaev (2003) or Chapter 3.2 of Karatzas and Shreve (1991).
Our purpose is to assess nonparametrically what is the smallest possible residual
sum of squares in this regression. Speci(cid:12)cally, for two processes X and Y , denote
t t
the quadratic covariation between X and Y on the interval [0;T] by
t t
< X;Y > = lim (X X )(Y Y ); (1.2)
T maxti+1(cid:0)ti#0 ti+1 (cid:0) ti ti+1 (cid:0) ti
i
X
where 0 = t < ::: < t = T. (This object exists by De(cid:12)nition I.4.45 or Theorem
0 k
I.4.47 in Jacod and Shiryaev (2003) (pp. 51-52), and similar statements in Karatzas
and Shreve (1991) and Protter (2004)). In particular, < Z;Z > { the quadratic
T
variation of Z { is the sum of squares of the increments of the process Z under
t
the idealized condition of continuous observation. We wish to estimate, from time-
discrete data,
min< Z;Z > ; (1.3)
T
(cid:26)
where the minimum is over all adapted regression processes (cid:26).
Animportantmotivating application forthesystem(1.1) isthatofstatistical risk
management in (cid:12)nancial markets. We suppose that S and (cid:4) are the discounted
t t
values of two securities. At each time t, a (cid:12)nancial institution is short one unit of
the security represented by (cid:4), and at the same time seeks to o(cid:11)set as much risk as
possiblebyholding(cid:26) unitsofsecurity S.Z ,asgiven by(1.1), is thenthe gain/loss,
t t
up to time t, from following this \risk-neutral" procedure. In a complete (idealized)
(cid:12)nancial market, min < Z;Z > is zero; in a incomplete market, min < Z;Z >
(cid:26) (cid:26)
quanti(cid:12)es the unhedgeable part of the variation in asset (cid:4), when one adopts the
best possible strategy using only asset S. And this lower bound (1.3) is the target
that a risk management group wants to monitor and control.
The statistical importance of (1.3) is this. Once you know how to estimate (1.3),
you know how to assess the goodness of (cid:12)t of any given estimation method for (cid:26) .
t
You also know more about the appropriateness of a one regressor model of the form
(1.1). We return to the goodness of (cid:12)t questions in Section 4. A model example of
both statistical and (cid:12)nancial importance is given in Section 2.2.
To discuss the problem of estimating (1.3), consider (cid:12)rst how one would (cid:12)nd this
quantity if the processes (cid:4) and S were continuously observed. Note that from (1.1),
ANOVA FOR DIFFUSIONS 3
one can write
t t
< Z;Z > =<(cid:4);(cid:4) > + (cid:26)2d < S;S > 2 (cid:26) d < (cid:4);S > (1.4)
t t u u (cid:0) u u
Z0 Z0
(JacodandShiryaev(2003), I.4.54, p.55).Here,d < X;Y > is thedi(cid:11)erential ofthe
t
process < X;Y > with respect to time. We shall typically assume that < X;Y >
t t
is absolutely continuous as a function of time (for any realization). It is easy to see
that the solution in (cid:26) to the problem (1.3) is uniquely given by
t
d < (cid:4);S > < (cid:4);S >0
(cid:26) = t = t; (1.5)
t d < S;S > < S;S >0
t t
where < (cid:4);S >0 is the derivative of < (cid:4);S > with respect to time. Apart from its
t t
statistical signi(cid:12)cance, in (cid:12)nancial terms (cid:26) is the hedging strategy associated with
the minimal martingale measure (see, for example, Fo(cid:127)llmer and Sondermann(1986)
and Schweizer (1990)).
Theproblem(1.3) then connects to anANOVA, asfollows. Let Z betheresidual
t
in (1.1) for the optimal (cid:26) , so that the quantity in (1.3) can be written simply as
t
< Z;Z > . In analogy with regular regression, substituting (1.5) into (1.4), gives
T
rise to an ANOVA decomposition of the form
T
< (cid:4);(cid:4)> = (cid:26)2d < S;S > +< Z;Z > ; (1.6)
T u u T
Z0
total SS RSS
SS explained
| {z } | {z }
where\SS"istheabbreviationfo|r(contin{uzous)\su}mofsquares",and\RSS"stands
for \residual sum of squares". Under continuous observation, therefore, one solves
the problem (1.3) by using the (cid:26) and Z de(cid:12)ned above. Our target of inference,
< Z;Z > , would then be observable. Discreteness of observation, however, creates
t
the need for inference.
Themaintheoremsinthecurrentpaperareconcernedwiththeasymptoticbehav-
ior of the estimated RSS, as more discrete observations are available within a (cid:12)xed
\
time window. There will be some choice in how to select the estimator <Z;Z > .
t
\
We consideraclassofsuchestimators < Z;Z > .Nomatterwhichofourestimators
t
is used, we get the decomposition
\
< Z;Z > < Z;Z > bias + ([Z;Z] < Z;Z > ); (1.7)
t(cid:0) t (cid:25) t t(cid:0) t
to (cid:12)rst order asymptotically, where [Z;Z] is the sum of squares of the increments
of the (unseen) process Z at the sampling points, [Z;Z] = (Z Z )2 cf. the
t i ti+1 (cid:0) ti
de(cid:12)nition (2.4) below.
P
4 P.A. MYKLAND AND L. ZHANG
A primary conceptual (cid:12)nding in (1.7) is the clear cut e(cid:11)ect of the two sources
behind the asymptotics. The form of the bias depends only on the choice of the
estimator for the quadratic variation. On the other hand, the variation component
is common for all the estimators under study; it comes only from the discretization
error in time discrete sampling.
It is worthwhile to point out that our problem is only tangentially related to that
of estimating the regression coe(cid:14)cient (cid:26) . This is in the sense that the asymptotic
t
behavior of nonparametric estimators of (cid:26) do not directly imply anything about
t
the behavior of estimators of < Z;Z > . To illustrate this point, note that the
T
convergence rates are not the same for the two types of estimators (O (n(cid:0)1=4) vs.
p
O (n(cid:0)1=2)), and that the variance of the estimator we use for (cid:26) becomes the bias
p t
in one of our estimators of < Z;Z > (compare equation (2.9) in Section 2.4 with
T
Remark 1 in Section 3). For further discussion, see Section 2.4. Depending on the
goal ofinference, statistical estimates (cid:26)^ ofthe regressioncoe(cid:14)cient can beobtained
t
usingparametric methods,ornonparametriconesthat are eitherlocal inspaceor in
time, as discussed in Sections 2.4 and 4.1, and the references cited in these sections.
In addition, it is also common in (cid:12)nancial contexts to use calibration (\implied
quantities", see Chapters 11 and 17 in Hull (1999)).
The organization is as follows, in Section 2, we establish the framework for
ANOVA, and we introduce a class of estimators of the residual quadratic varia-
tion. Our main results, in Section 3, provide the distributional properties of the
estimation errors for RSS. See Theorem 1-Theorem 2. In Section 4, we discuss the
statistical application of the main theorems. Parametric and nonparametric esti-
mation are compared in the context of residual analysis. The goodness of (cid:12)t of a
model is addressed. Broad issues, including the analysis of variation versus analysis
ofvariance, the moderatelevel ofaggregation versuslongrun,theactual probability
distribution versus the risk neutral probability distribution in the derivative valua-
tion setting, are discussed in Section 4.4-4.5. After concluding in Section 5, we give
proofs in Sections 6-7.
2. ANOVA for Ito processes: framework.
2.1. Ito processes, quadratic variation, and di(cid:11)usions. The assumptions and def-
initions in the following two subsections are used throughout the paper, sometimes
without further reference. First of all, we shall be working with a (cid:12)xed (cid:12)ltered
probability space:
ANOVA FOR DIFFUSIONS 5
System Assumption I. We suppose that there is an underlying (cid:12)ltered proba-
bility space
((cid:10); ; ;P) , satisfying the \usual conditions" (see, for example, De(cid:12)nition
t 0(cid:20)t(cid:20)T
F F
1.3 (p. 2) of Jacod and Shiryaev (2003), also in Karatzas and Shreve (1991)).
We shall then be working with Ito processes adapted to this system, as follows.
Note that Markov di(cid:11)usions are a special case.
Definition 1. (Ito processes). By saying that X is an Ito process, we mean that
X is continuous (a.s.), ( )-adapted, and that it can be represented as a smooth
t
F
process plus a local martingale,
t t
X = X + X~ du+ (cid:27)XdWX; (2.1)
t 0 u u u
Z0 Z0
where W is a standard (( );P)-Brownian Motion, X is measurable, and the
t 0 0
F F
coe(cid:14)cients X~ and (cid:27)X are predictable, with T X~ du < + and T((cid:27)X)2du <
t t 0 j uj 1 0 u
+ . We also write
1 R R
X = X +XDR +XMG (2.2)
t 0 t t
as shorthand for the drift and local martingale parts of Doob-Meyer decomposition
in (2.1). 2
A more abstract way of putting this de(cid:12)nition is that X is an Ito process if
t
it is a continuous semimartingale (Jacod and Shiryaev (2003), De(cid:12)nition I.4.21, p.
42) whose (cid:12)nite variation and local martingale parts, given by (2.2), satisfy that
both XDR and the quadratic variation < XMG;XMG > are absolutely continuous.
t t
Obviously, an Ito process is a special semimartingale, also in the sense of the same
de(cid:12)nition of Jacod and Shiryaev (2003).
Di(cid:11)usions are normally taken to be a special case of Ito processes, where one can
write ((cid:27)X)2 = a(X ;t) and X~ = b(X ;t), and similarly in the multidimensional
t t t t
setting. For a descriptionof thelink, wereferto Chapter5.1 ofKaratzas andShreve
(1991).
The Ito process de(cid:12)nition extends to a two- or multidimensional process, say
(X ;Y ), by requiring each component X and Y individually to be an Ito process.
t t t t
Obviously, WX is typically di(cid:11)erent for di(cid:11)erent Ito processes X. For two processes
X and Y, the relationship between WX and WY can be arbitrary.
Thequadraticvariation< X;X > (formula(1.2)) cannowbeexpressedinterms
t
6 P.A. MYKLAND AND L. ZHANG
of the representation (2.1) by (Jacod and Shiryaev (2003), I.4.54, p.55)
t
< X;X > = ((cid:27)X)2du:
t u
Z0
We denote by < X;X >0 the derivative of < X;X > with respect to time t, then
t t
< X;X >0= ((cid:27)X)2,andthisquantity(oritssquareroot)isoftenknownasvolatility
t t
in the (cid:12)nance literature.
Both quadratic variation and covariation are absolutely continuous. This follows
from the Ito process assumptionand fromthe Kunita-Watanabe Inequality (see, for
example, p. 69 of Protter (2004)).
Definition 2. (Volatility as an Ito process). Denote by < X;Y >0 the derivative
t
of < X;Y > with respect to time. We shall often suppose that < X;Y >0 is itself
t t
an Ito process. For ease of notation, we then write its Doob-Meyer decomposition
as
d < X;Y >0= dDXY +dRXY = D~XYdt+dRXY:
t t t t t
2
Note that the quadratic variation of < X;Y >0 is the same as < RXY;RXY >,
andthat inournotation above, DXY = (< X;Y >0)DR andRXY = (< X;Y >0)MG
2.2. Model example: Adequacy of the one factor interest rate model. A common
model for the risk free short term interest rate is given by the di(cid:11)usion
dr = (cid:22)(r )dt+(cid:13)(r )dW ; (2.3)
t t t t
where W is a Brownian motion. For example, in Vasicek (1977), one takes (cid:22)(r) =
t
(cid:20)((cid:11) r), and (cid:13)(r) = (cid:13) = constant, while Cox et al. (1985) uses the same function
(cid:0)
(cid:22), but (cid:13)(r) = (cid:13)r1=2. For more such models, and a brief (cid:12)nancial introduction, see,
for example, Hull (1999). A discussion and review of estimation methods is given
by Fan (2005).
One of the implications of this so-called one factor model is the following. Sup-
pose S and (cid:4) are the values of two zero coupon government bonds with di(cid:11)erent
t t
maturities. Financial theory then predicts that until maturity, S = f(r ;t) and
t t
(cid:4) = g(r ;t) for two functions f and g (see Hull (1999), Chapter 21, for details
t t
and functional forms). Under this model, therefore, the relationship d(cid:4) = (cid:26) dS
t t t
holds from time zero until the maturity of the shorter term bond. It is easy to see
that d < S;S > = f0(r ;t)2(cid:13)(r )2dt and d < (cid:4);S > = f0(r ;t)g0(r ;t)(cid:13)(r )2dt, with
t r t t t r t r t t
ANOVA FOR DIFFUSIONS 7
(cid:26) = d < (cid:4);S > =d < S;S > = g0(r ;t)=f0(r ;t). Here, f0 is the derivative of f
t t t r t r t r
with respect to r, and similarly for g0.
r
The one factor model is only an approximation, and to assess the adequacy of
the model, one would now wish to estimate < Z;Z > over di(cid:11)erent time intervals.
T
This provides insight into whether it is worth while to use a one factor model at
all. If the conclusion is satisfactory, one can estimate the quantites (cid:22) and (cid:13) (and
hencef andg)withparametricornonparametricmethods(seeSection2.4 and4.1),
and again, use our methods to assess the (cid:12)t of the speci(cid:12)c model, for example as
discussed in Section 4.1.
2.3. Finitely many data. We nowsupposethatweobserveprocesses,inparticular
S and (cid:4) , on a (cid:12)nite set (partition, grid) = 0 = t < t < < t =T of time
t t 0 1 k
G f (cid:1)(cid:1)(cid:1) g
points in the interval [0;T]. We take the time points to be nonrandom, but possibly
irregularly spaced. Note that this also covers the case where the t are random but
i
independent of the processes we seek to observe, so that one can condition on to
G
get back to irregular but nonrandom spacing.
Definition 3. (Observed quadratic variation). For two Ito processes X and Y
observed on a grid ,
G
[X;Y]G = ((cid:1)X )((cid:1)Y ); (2.4)
t ti ti
tiX+1(cid:20)t
where (cid:1)X = X X . When there is no ambiguity, we use [X;Y] for [X;Y]G,
ti ti+1 (cid:0) ti t t
or [X;Y](n) in case of a sequence . 2
t Gn
Note that this is not the same as the usual de(cid:12)nition of [X;Y] for Ito processes.
t
We use < X;Y > to refer to a continuous process, while [X;Y] refers to a (ca(cid:18)dla(cid:18)g)
t t
process which only changes values at the partition points t .
i
Since the results of the paper rely on asymptotics, we shall take limits with the
(n) (n) (n)
help of a sequence of partitions = 0 = t < t < < t = T . As
Gn f 0 1 (cid:1)(cid:1)(cid:1) kn g
n + , we let become dense in [0;T], in the sense that the mesh
n
! 1 G
(cid:14)(n) = max (cid:1)t(n) 0: (2.5)
t j i j !
(n) (n) (n)
Here (cid:1)t = t t .In other words,the meshisthe maximumdistance between
i i+1(cid:0) i
(n)
the t ’s. On the other hand, T remains (cid:12)xed (except brie(cid:13)y in Section 4.5).
i
Inthiscase,[X;Y]= [X;Y]Gn convergesto< X;Y >uniformlyinprobability,see
Jacod and Shiryaev (2003), Theorem I.4.47 (p. 52), and Protter (2004), Theorem
8 P.A. MYKLAND AND L. ZHANG
II.23 (p. 68). More is true, see Section 5 of Jacod and Protter (1998), and (our)
Sections 2.8 and 6 below.
Note that, under (2.5), k = . It is often convenient to consider the
n n
jG j ! 1
average distance between successive observation points,
T
(n)
(cid:1)t = ; (2.6)
k
n
cf. Assumption A(i) below.
2.4. The regression problem, and the estimation of (cid:26) . The processes in (1.1) will
t
be taken to satisfy the following.
System Assumption II. We let (cid:4) and S be Ito processes. We assume that
inf < S;S >0 > 0 almost surely. (2.7)
t
t2[0;T]
This assumption assures that (cid:26) , given by (1.5), is well de(cid:12)ned under (2.7), by the
t
Kunita-Watanabe Inequality.
As noted in the Introduction, under continuous observation of (cid:4) and S , one
t t
can directly also observe the optimal (cid:26) and Z . Our target of inference, <Z;Z > ,
t t t
would then be observable. Discreteness of observation, however, creates the needfor
inference.
In a non-continuous world, where (cid:4) and S can only be observed over grid times,
the most straightforward estimator of (cid:26) is,
\
< (cid:4);S >0 [(cid:4);S] [(cid:4);S]
(cid:26)^ = t = t (cid:0) t(cid:0)hn: (2.8)
t <\S;S >0t [S;S]t (cid:0)[S;S]t(cid:0)hn
For simplicity, this estimator is the one we shall use in the following. The results
easily generalize when more general kernels are used. Note that we have to use a
(n)
smoothing bandwidth h . There will naturally be a tradeo(cid:11) between h and (cid:1)t .
n n
(n)1=2
As we now argue, this typically results in h = O((cid:1)t ).
n
\
Asymptoticsforestimatorsoftheform< X;Y >0 = ([X;Y] [X;Y] )=h and
t t(cid:0) t(cid:0)hn n
(n)
hence for (cid:26)^, are given by Foster and Nelson (1996) and Zhang (2001). Let (cid:1)t be
t
the average observation interval, assumedto converge to zero. If<X;Y >0 is an Ito
t
process with nonvanishing volatility, then it is optimal to take h = O(((cid:1)t(n))1=2),
n
and((cid:1)t(n))1=4(< X\;Y >0 < X;Y >0) converges inlaw (for each (cid:12)xedt) to a (con-
t(cid:0) t
ditionalonthedata)normaldistributionwithmeanzeroandrandomvariance.(The
ANOVA FOR DIFFUSIONS 9
mode of convergence is the same as in Proposition 1). The asymptotic distributions
are (conditionally) independent for di(cid:11)erent times t. If <X;Y >0 is smooth, on the
t
other hand, the rate becomes ((cid:1)t(n))1=3 rather than ((cid:1)t(n))1=4, and the asymptotic
distribution contains both bias and variance.
The same applies to the estimator (cid:26)^ . In the case when S and (cid:4) have a di(cid:11)usion
t
component, the estimator has (random) asymptotic variance
< (cid:26);(cid:26) >0 < (cid:4);(cid:4)>0
V (t) = t +cH0(t)( t (cid:26)2) (2.9)
(cid:26)^(cid:0)(cid:26) 3c < S;S >0 (cid:0) t
t
whenever h =((cid:1)t(n))1=2 c (0; ), see Zhang (2001). H(t) is de(cid:12)ned in Section
n
! 2 1
2.6 below.
The scheme given in (2.8) is only one of many for estimating < X;Y >0 by us-
t
ing methods that are local in time. In particular, Genon-Catalot et al. (1992) use
wavelets for this purpose, and determine rates of convergence and limit distribu-
tions under the assumption that < X;Y >0 is deterministic and has smoothness
t
properties.
Other important literature in this area seeks to estimate < X;Y >0 as a function
t
oftheunderlyingstatevariables,bymethodsthatarelocalinspace,seeinparticular
Florens-Zmirou(1993),Ho(cid:11)mann(1999)andJacod(2000). Thetypicalsetupisthat
U = (X;Y;:::) is a Markov process, so that < X;Y >0= f(U ) for some function f,
t t
andtheproblemistoestimatef.Ifallcoe(cid:14)cientsintheMarkovdi(cid:11)usionaresmooth
oforders,andsubjecttoregularityconditions,thefunctionf canbeestimatedwith
a rate of convergence of ((cid:1)t(n))s=(1+2s).
The convergence obtained for the estimator of f under Markov assumptions is
considerably faster than what can be obtained for (2.8). It does, however, rely on
stronger (Markov) assumptions than the ones (Ito processes) that we shall be work-
ing withinthis paper.Sincewe shall only beinterested in(cid:26) as a (random) function
t
of time, our development does not require a Markov speci(cid:12)cation, and in particular
does not require full knowledge of what potential state variables might be.
Ofcourse,thisisjustasubsetoftheliteratureforestimationofMarkovdi(cid:11)usions.
See Section 4.1 for further references.
We emphasize that the general ANOVA approach in this paper can be carried
out with other schemes for estimating (cid:26) than the one given in (2.8). We have seen
t
this as a matter of (cid:12)ne tuning and hence beyond the scope of this paper. This is
10 P.A. MYKLAND AND L. ZHANG
because Theorems 1 and 2 achieve the same rate of convergence as the one obtained
in Proposition 1.
2.5. Estimation schemes for the residual quadratic variation <Z;Z > . We now
t
return to the estimation of the quadratic variation < Z;Z > of residuals. Given the
discretedataof((cid:4);S),therearedi(cid:11)erentmethodstoestimatetheresidualvariation.
One scheme is to start with model (1.1). For a (cid:12)xed grid , one (cid:12)rst estimates
G
(cid:1)Z through the relation (cid:1)Z^ = (cid:1)(cid:4) (cid:26)^ ((cid:1)S ), where all increments are from
ti ti ti (cid:0) ti ti
time t to t , and then obtains the quadratic variation (q.v. hereafter) of Z^. This
i i+1
gives an estimator of < Z;Z > as
[Z^;Z^] = ((cid:1)Z^ )2 = [(cid:1)(cid:4) (cid:26)^ ((cid:1)S )]2 (2.10)
t ti ti (cid:0) ti ti
tiX+1(cid:20)t tiX+1(cid:20)t
where the notation of square brackets (discrete time-scale q.v.) is invoked, since
(cid:1)Z^ is the increment over discrete times.
ti
Alternatively, one can directly analyze the ANOVA version (1.6) of the model,
where d < Z;Z > = d < (cid:4);(cid:4) > (cid:26)2d < S;S > . This yields a second estimator of
t t (cid:0) t t
< Z;Z > ,
t
<\Z;Z >(1) = [((cid:1)(cid:4) )2 (cid:26)^2((cid:1)S )2]: (2.11)
t ti (cid:0) ti ti
tiX+1(cid:20)t
In general, any convex combination of these two,
<\Z;Z >((cid:11)) = (1 (cid:11))[Z^;Z^] +(cid:11)<\Z;Z >(1) (2.12)
t (cid:0) t t
would seem like a reasonable way to estimate < Z;Z > , and this is the class of
t
estimators that we shall consider. Particular properties will be seen to attach to
\ (1=2) ^
< Z;Z > , which we shall also denote by < Z;Z > . For a start, it is easy to see
t t
that
<^Z;Z > = [(cid:4);Z^] : (2.13)
t t
Note that (2.13) also has a direct motivation from the continuous model. Since
< S;Z > = 0, (1.1) yields that < (cid:4);Z > =<Z;Z > .
t t t
\ ((cid:11))
We establish the statistical properties of the estimator < Z;Z > , and in par-
t
ticular those of ([Z^;Z^], and <^Z;Z > ) in Section 3. Asymptotic properties are
t
naturally studied with the help of small interval asymptotics.
2.6. Paradigm for asymptotic operations. The asymptotic property of the esti-
mation error is considered under the following paradigm.