Covariance matrix estimation from noisy high-frequency ... · Covariance matrix estimation from noisy high-frequency observations: Local method of moments and efﬁciency Markus Bibinger

Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales

Covariance matrix estimation from noisy high-frequencyobservations: Local method of moments and efficiency

Markus BibingerHumboldt-Universitat zu Berlin

includes results from joint works with Markus Reiß, Randolf Altmeyer, NikolausHautsch and Peter Malec

Symposium on Financial Engineering and ERMHitotsubashi U. Tokyo

03/06/2014

Efficient covariance matrix estimation Markus Bibinger


Outline

1 Motivation & Contribution

2 Equivalence of non-synchronous and continuous-time model

3 The local method of moment approach

4 Asymptotic results and efficiency

5 Progress to approach more complex models



Outline








High-frequency data

Idealizedintra-day log-price model:Continuous martingale

dXt = Σ1/2

t dBt

discretely observed:Xti , i = 0, . . . ,n.

Volatility Matrix (Σs)s∈[0,1] reflects intrinsic risk.Main objective: Semi-parametric estimation of the quadraticcovariation matrix

∫ 10 Σs ds from discrete observations

Xti , i = 0, . . . ,n.



A well studied statistical experiment

Consider Ziiid∼ N(0,Σ),Zi ∈ Rd , and the empirical covariance matrix

Σn = n−1∑ni=1 ZiZ>i . It satisfies a multi-dimensional central limit

theorem (CLT) which can be written

√n vec

(Σn − Σ

) L−→ N(0, (Σ⊗ Σ)Z

)with the Kronecker product Σ⊗ Σ ∈ Rd2×d2

:

Σ⊗Σ =

Σ11 Σ · · · Σ1d Σ...

. . ....

Σ1d Σ · · · Σdd Σ

,

For (V1, . . . ,Vd ) Gaussian :E[VpVqVk Vl ] = E[VpVq]E[Vk Vl ]+E[VpVk ]E[VqVl ] + E[VpVl ]E[VqVk ]

“Isserlis formula”

,

and Z = COV(vec(UU>)), when U ∼ N(0,Ed ). Z satisfiesZ vec(A) = vec(A + A>) for all A ∈ Rd , Z/2 idempotent.



A well studied statistical experiment

Consider Ziiid∼ N(0,Σ),Zi ∈ Rd , and the empirical covariance matrix

Σn = n−1∑ni=1 ZiZ>i . It satisfies a multi-dimensional central limit

theorem (CLT) which can be written

√n vec

(Σn − Σ

) L−→ N(0, (Σ⊗ Σ)Z

)with the Kronecker product Σ⊗ Σ ∈ Rd2×d2

.

This classical problem has a close analogy to observing X on thegrid i/n, i = 0, . . . ,n, since

X(i+1)/n − Xi/n =

∫ (i+1)/n

i/nΣ

1/2

t dBt ≈ N(0,Σi/nn−1) .



Estimating quadratic covariation from discrete data

Consider a d-dimensional continuous martingale

Xt = X0 +

∫ t

0Σ

1/2s dBs , t ∈ [0,1],

discretely observed: Xi/n, i = 0, . . . ,n. The natural estimator

Realised volatility matrix RVn1 =

∑ni=1

(X i

n− X i−1

n

)(X i

n− X i−1

n

)>satisfies:

vec(√

n(

RVn1−∫ 1

0Σs ds

))L−→ N

(0,∫ 1

0

(Σs ⊗ Σs

)Zds

).

In the one-dimensional case, d = 1, we derive the CLT

√n

(n∑

i=1

(X i

n− X i−1

n

)2 −∫ 1

0σ2

s ds

)L−→ N

(0,2

∫ 1

0σ4

s ds).



Estimating quadratic covariation from discrete data

vec(√

n(

RVn1−∫ 1

0Σs ds

))L−→ N

(0,∫ 1

0

(Σs ⊗ Σs

)Zds

).

Two-dimensional setup, d = 2:

Σ =

(σ2

1 ρσ1σ2ρσ1σ2 σ2

2

), Z =

2 0 0 00 1 1 00 1 1 00 0 0 2

(Σ⊗ Σ

)Z =

2σ4

1 2ρσ31σ2 2ρσ3

1σ2 2ρ2σ21σ

22

2ρσ31σ2 (1 + ρ2)σ2

1σ22 (1 + ρ2)σ2

1σ22 2ρσ1σ

32

2ρσ31σ2 (1 + ρ2)σ2

1σ22 (1 + ρ2)σ2

1σ22 2ρσ1σ

32

2ρ2σ21σ

22 2ρσ1σ

32 2ρσ1σ

32 2σ4

2



Challenges & Groundwork

Account for market microstructure noise and non-synchronicity!

Non-synchronous observation model with noise:

Y (l)i = X (l)

t (l)i

+ ε(l)i ,1 ≤ l ≤ d , i = 0, . . . ,nl .

Some important contributions:In the one-dimensional parametric setup σs = σ, εi

iid∼ N(0, η2), ti = in ,

minimum asymptotic variance 8σ3η and optimal rate n1/4 obtained bylocal asymptotic normality of experiments (Gloter & Jacod (2001)).Observe that (εi )0≤i≤n and (Xi/n − X(i−1)/n)1≤i≤n uncorrelated with

E[(Xi/n − X(i−1)/n)2] = σ2/n ,

E[(εi − εi−1)2] = 2η2 not dependent on n .



Challenges & Groundwork

Account for market microstructure noise and non-synchronicity!

Non-synchronous observation model with noise:

Y (l)i = X (l)

t (l)i

+ ε(l)i ,1 ≤ l ≤ d , i = 0, . . . ,nl .

Some important contributions:In the one-dimensional parametric setup σs = σ, εi

iid∼ N(0, η2), ti = in ,

minimum asymptotic variance 8σ3η and optimal rate n1/4 obtained bylocal asymptotic normality of experiments (Gloter & Jacod (2001)).Barndorff-Nielsen, Hansen, Lunde, Shephard (realised kernels)Hayashi & Yoshida (for non-synchronous observations)Jacod, Li, Mykland, Podolskij, Vetter (pre-averaging)Reiß (one-dimensional non-parametric efficiency)Xiu (Quasi maximum likelihood)Zhang, Mykland, Aıt-Sahalia (two-scale, multi-scale)



Motivation & Contribution

Several methods for multivariate problem (lack of benchmark).

Natural questions in multi-dimensional framework:

(How) Does information content of experiment hinge onasynchronicity and different observation frequencies?

(How) Is it possible to construct an asymptotically efficientsemi-parametric estimator?

What is the Cramer-Rao efficiency bound in themulti-dimensional experiment?(Minimum variance of asymptotically unbiased estimators)

What is the Cramer-Rao efficiency bound in the non-parametricexperiment?

We shall establish the lower bound in the multi-dimensional andnon-parametric and non-synchronous setup.



Outline








Le Cam’s asymptotic equivalence of experiments

Compare statistical experiments E0 =(X0,F0, P0

θ |θ ∈ Θ)

andE1 =

(X1,F1, P1

θ |θ ∈ Θ)

with the same parameter set Θ.The Le Cam deficiency of E0 w.r.t. to E1 is

δ (E0,E1) = infK

supθ∈Θ‖KP0

θ − P1θ‖TV ,

where the infimum is taken over all Markov kernels or randomizationsK from (X0,F0) to (X1,F1).The Le Cam distance is

∆ (E0,E1) = max (δ (E1,E0) , δ (E0,E1)) .

(Strong) Asymptotic equivalence:If limn→∞∆ (En

0,En1) = 0, then the sequences (En

0)n and (En1)n are

called asymptotically equivalent.



Equivalence in a nutshell: Simple example

Statistical experiments with θ ∈ R unknown:E1: Observe i.i.d. sample X1, . . . ,Xn ∼ N(θ,1).E2: Observe X ∼ N(θ,n−1).

E1 is more informative than E2, since

X = n−1n∑

i=1

Xi ∼ N(θ,n−1) .

E2 is more informative than E1, since we obtain Xi := X + Yi ∼ N(θ,1)i.i.d. , i = 1, . . . ,n, with independent(Y1, . . . ,Yn) ∼ N(0,Σ),Σii = (n − 1)/n,Σij = −n−1(i 6= j).

E1 and E2 are equivalent.



Our statistical observation model

Consider a d-dimensional continuous martingale

Xt = X0 +

∫ t

0Σ

1/2s dBs , t ∈ [0,1],

non-synchronously observed with i.i.d. centered Gaussian noise

Y (l)i = X (l)

t (l)i

+ ε(l)i , i = 0, . . . ,nl ,1 ≤ l ≤ d ,(E0)

with Σ a function taking values in the class of symmetric positivesemi-definite matrices.Regularity assumptions:

Σ ∈ Hβ([0,1]), L2-Sobolev space of order β > 0, Σs ≥ ΣEd .

Sampling times t (l)i = F−1

l (i/nl ) with Fl ∈ Cα([0,1]),

nmax = O(n1+βmin ),nmax := max (nl ),nmin := min (nl ).



Equivalence to continuous-time model

THEOREMThe discrete non-synchronous observation model (E0) is forΣ ∈ Hβ(R),F ∈ Cβ(R),R > 0 and nmax = O

(n1+min(β,1/2)

min

)asymptotically equivalent to the continuous-time white noise model

dYt = Xt dt + diag(ηl (nlF ′l (t))

−1/2)

1≤l≤ddWt , t ∈ [0,1] ,(E1)

with W being a Brownian motion independent of B.

Conclusion: Asynchronicity at first order asymptotically negligible!



Outline








Covariance matrix estimation from noisy observations

Another excursion to a less well studied statistical problem:Zj

iid∼ N(0,Σ), observed with independent errors εj ∼ N(0, η2

j Ed):

Yj = Zj + εj , Yj ∈ Rd , j = 0, . . . ,n .

First estimator: Σ(1) =n∑

j=1

1n

YjY>j −n∑

j=1

η2j

nEd .

Second estimator: Σ(2) =n∑

j=1

wj(YjY>j − η2

j Ed).

In order to minimize the variance of Σ(2)11 , we choose

wj =(Σ11 + η2

j )−2∑ni=1(Σ11 + η2

i )−2, Var

((YjY>j − η2

j Ed )11)

= 2(Σ11 + η2j )2 = I−1

j

⇒ Var(

Σ(2)11

)=

n∑i=1

(∑u

Iu)−2

I2j I−1

j =(∑

u

Iu)−1

=: I−1 .



Covariance matrix estimation from noisy observations

Another excursion to a less well studied statistical problem:Zj

iid∼ N(0,Σ), observed with independent errors εj ∼ N(0, η2

j Ed):

Yj = Zj + εj , Yj ∈ Rd , j = 0, . . . ,n .

Third estimator: Σ(3) =n∑

j=1

Wj vec(YjY>j − η2

j Ed),

COV(Σ(3)

)=

n∑j=1

Wj vec(Cj ⊗ Cj

)ZW>

j , Cj = Σ + η2j Ed ,

becomes minimal for Wj =(∑

i C−⊗2i

)−1C−⊗2j where we write

C−⊗2j = C−1

j ⊗C−1j =

(Cj⊗Cj )

−1 ⇒ COV(Σ(3)

)= I−1, I =

∑j

Ij =∑

j

C−⊗2j .

It holds true that Var(

Σ(1)11

)≥ Var

(Σ

(2)11

)≥ Var

(Σ

(3)11

).



Local parametric approximation

Split [0,1] in bins [khn, (k + 1)hn), k = 1, . . . ,h−1n .

Approximate experiment (E0) by locally parametric one withΣs = Σkhn1[khn,(k+1)hn)(s).Efficient local parametric estimation: With the functions

Φjk (t) =√

2/hn sin(jπh−1

n (t − khn))1[khn,(k+1)hn)(t), j ≥ 1 ,

and the vector of spectral statistics (localized Fourier coefficients)

Sjk =( np∑

i=1

(Y (p)

i − Y (p)i−1

)Φjk

( t (p)i−1 + t (p)

i

2

))1≤p≤d

,

we have that Sjk ∼ N(0, I−1

jk

), I−1

jk =(Σkhn + π2j2h−2

n Hnkhn

)⊗2= C⊗2

jkindependent for all (j , k). Time-dependent noise level:

Hns = diag

(n−1

p η2p(F−1

p )′(s))

1≤p≤d .



Spectral Fourier approach

The spectral statistics,

Sjk =

∫ (k+1)hn

khn

Φ′jk (t) dYt in (E1)

de-correlate the observations and form their (bin-wise) principalcomponents.Moreover, we have a summation by parts property:

S(p)jk =

np∑i=1

(Y (p)

i − Y (p)i−1

)Φjk

( t (p)i−1 + t (p)

i

2

)

≈np∑

i=1

(X (p)

t (p)i

− X (p)

t (p)i−1

)Φjk

( t (p)i−1 + t (p)

i

2

)−

np−1∑i=1

ε(p)i Φ′jk

(t (p)i

) t (p)i+1 − t (p)

i−1

2.



LMM-estimation and optimal weightsThe oracle local method of moments (LMM) estimator is:

LMM(n)or =

h−1n∑

k=1

hn

Jn∑j=1

Wjk vec(

Sjk S>jk − π2j2h−2n Hn

khn

).

The oracle optimal weights are

Wjk = I−1k Ijk , Ik =

Jn∑j=1

Ijk =Jn∑

j=1

(Σkhn + π2j2h−2

n Hnkhn

)−⊗2.

A locally adaptive approach is obtained with a two-stage procedureplugging-in a pilot estimator:

Σpilots = (2Kn + 1)−1

bsh−1n c+Kn∑

k=bsh−1n c−Kn

J−1n,p

Jn,p∑j=1

(Sjk S>jk − π2j2h−2

n Hnkhn

).



Outline








Local method of moments: Central limit theorems

THEOREM (Feasible CLT) Let Σ ∈ Hβ ,F ∈ Cβ , β > 1/2,nmax = o(n2β

min). As nmin →∞ and h = h0n−1/2min with h0 →∞:

I1/2n

(LMM(n)

ad − vec( ∫ 1

0 Σs ds)) L−→ N (0,Z) ,

I−1n =

∑h−1n

k=1 h2nI−1

k . Tightness by uniform bound for ‖dWjk/dΣkhn‖.

COROLLARY Suppose nmin/np → νp ∈ (0,∞) and introduceH(t) = diag(ηpν

1/2p F ′p(t)−1/2)p and

(ΣH)1/2

:= H(H−1ΣH−1)1/2H.

n1/4min

(LMM(n)

ad − vec( ∫ 1

0 Σs ds)) L−→ N

(0, I−1Z

)with

I−1 = 2∫ 1

0 (Σt ⊗(ΣH

t)1/2

+(ΣH

t)1/2 ⊗ Σt ) dt .



Asymptotic efficiency

CLT for LMM in dimension d = 1 and equidistant observation times:

CLT: n1/4(

LMM(n)ad −

∫ 1

0σ2

s ds)

L−→ N(

0,8η∫ 1

0σ3

s ds).

Asymptotic efficiency concluded by Ibragimov & Khas’minskiı(1991),and an equivalence result by Reiß (2011).

Major result of multi-dimensional analysis:

THEOREM The asymptotic variance of the LMM-estimator coincideswith the semi-parametric Cramer-Rao bound, hence constitutes alower variance bound for all asymptotically unbiased estimators, andthus LMM is asymptotically efficient.



Overview: Lower bounds for (co-)volatility estimation

Lower Cramer-Rao bounds for asymptotic variance orvariance-covariance matrix of estimating quadratic (co-)variation:

Direct obs. Noisy obs.one-dim. p. 2σ4 8ησ3

one-dim. np. 2∫ 1

0 σ4s ds

∫ 10 8ησ3

s dsd-dim. p. (Σ⊗ Σ)Z 2(Σ⊗ (ΣH)1/2 + (ΣH)1/2 ⊗ Σ)Z

d-dim. np.∫ 1

0 (Σs ⊗ Σs)Zds 2∫ 1

0 (Σs ⊗ (ΣHs )1/2 + (ΣH

s )1/2 ⊗ Σs)Zds

Results by Gloter & Jacod (2001) for one-dim. p. noisy obs.;Renault, Sarisoy, Werker (2013) for one-dim. np. direct obs.;Reiß (2011) for one-dim. np. noisy obs.;BHMR (2013) for d-dim. p. and np. noisy obs;conjectured.



Overview: Lower bounds for (co-)volatility estimation

Lower Cramer-Rao bounds for asymptotic variance orvariance-covariance matrix of estimating quadratic (co-)variation:

Direct obs. Noisy obs.one-dim. p. 2σ4 8ησ3

one-dim. np. 2∫ 1

0 σ4s ds

∫ 10 8ησ3

s dsd-dim. p. (Σ⊗ Σ)Z 2(Σ⊗ (ΣH)1/2 + (ΣH)1/2 ⊗ Σ)Z

d-dim. np.∫ 1

0 (Σs ⊗ Σs)Zds 2∫ 1

0 (Σs ⊗ (ΣHs )1/2 + (ΣH

s )1/2 ⊗ Σs)Zds

Multi-dimensional lower bounds for indirect observations apply fornon-synchronous data as well. Not for direct observations! Efficient non-parametric estimator by Hayashi & Yoshida (2005) andparametric by Ogihara & Yoshida (2013).



Efficient asymptotic variance: Examples

Two-dimensional parametric with H(t) = E2, Σ11 = Σ22= 1,Σ12 = ρ.

Figure depicts asymptotic variances of Σ11 and Σ12 as function of ρ.




Two-dimensional non-parametric with equal volatilities:(Σ11)

t =(Σ22)

t = σ2t ,(Σ12)

t = ρtσ2t , noise levels H(t) = ηE2.

AVAR

(∫ 1

0(Σ12)t dt

)= 2η

∫ 1

0σ3

t((1 + ρt )

3/2 + (1− ρt )3/2)dt

AVAR(∫ 1

0σ2

t dt)

= 4η∫ 1

0σ3

t

(√1 + ρt +

√1− ρt

)dt , d = 2

Compare to one-dimensional asymptotic variance:

AVAR(∫ 1

0σ2

t dt)

= 8η∫ 1

0σ3

t dt , d = 1 .




Two-dimensional parametric Σ11 = Σ22 = 1, H(t) = diag (ηl )l=1,2,η1 = 1.

Variances of Σ11(left) and Σ12(right) dependent on correlation ρ and η2.



Outline








Semi-martingale model and stable convergence

Focus on a “usual setting” where X evolves as a continuous Itosemi-martingale:

Xt = X0 +

∫ t

0as ds +

∫ t

0σs dBs

on(Ω,F, (Ft ),P) with locally bounded drift process as and stochastic

volatility σs = F (Z (1)s ,Z (2)

s ), F differentiable in both coordinates andZ (1) itself a semi-martingale (leverage allowed), Z (2) ∈ Cα(R) forsome α > 1/2 and R > 0. Σs = σsσ

>s ≥ Σ.

Stable convergence: ZnF−st−→ Z means E[g(Zn)V ]→ E[g(Z )V ] for any

F-measurable bounded random variable V .



Semi-martingale model and stable convergence

Example: stable functional CLT for realised volatility:

√n

( bntc∑i=1

(Xi/n − X(i−1)/n

)2 −∫ t

0σ2

s ds

)F−st−→

√2∫ t

0σ2

s dWs,

with (Ws) a Brownian motion independent of F.

Limiting process on(Ω, F, P

)=(Ω,F,P

)⊗(Ω′,F′,P′

).

Feasible CLT

√n

(∑ni=1

(Xi/n − X(i−1)/n

)2 −∫ 1

0 σ2s ds

)23

∑ni=1

(Xi/n − X(i−1)/n

)4L−→ N(0,1) .



Stable CLTs for the spectral approach

Observation model Y = X + ε, F ′ ∈ Cα, α > 1/2 and m-dependentnoise η2

p = Var(εi ) + 2∑m

r=1 Cov(εi , εi+r ), E[εi ] = 0, independent of Xand eight moments exist.Stable functional CLT for the one-dimensional spectral estimator:

n1/4(

LMM(n)ad (t)−

∫ t0 σ

2s ds

)F−st−→ 2

√2∫ t

0 ((F−1)′(s))1/4√ησ3

s dWs

),

W Brownian motion independent of F. Feasible version implicitlyobtained. Stable CLT for LMM as n/np → νp,0 < νp <∞:

n1/4(

LMM(n)ad (t)− vec

( ∫ t0 Σs ds

))F−st−→ MN

(0, I−1Z

)with I of analogous form as above and MN means mixed normal. Afeasible version applies in a very general setup.



Estimation of the spot (co-)volatility

vec(Σs)

= (2Kn + 1)−1bsh−1

n c+Kn∑k=bsh−1

n c−Kn

Jn∑j=1

Wj vec(

Sjk S>jk − π2j2h−2n Hn

khn

).

If Σs is α-smooth, then MSE(Σs)

= OP(K−1

n)

+ OP(K 2α

n h2αn).

Kn ∝ nα/(2α+1) minimizes the MSE and facilitates an estimator with√Kn convergence rate. To obtain a CLT we have to undersmooth.

PROPOSITION

For hn = κ1 log (n)n−1/2, Kn = κ2nβ(log (n))−1 with0 < β < α(2α + 1)−1, for Jn →∞ and n/np → νp with 0 < νp <∞, asn→∞, we derive the CLT

nβ/2 vec(Σs − Σs

) F−st−→ MN(

0,2(Σ⊗

(ΣH)1/2

+(ΣH)1/2 ⊗ Σ

)s Z).

Spot correlation and betas with ∆-method.



So what about the jumps?In case of a general Ito semi-martingale, d = 1,

Xt = X0 +

∫ t

0bs ds +

∫ t

0σs dWs +

∫ t

0

∫Rκ(δ(s, z))(µ− ν)(ds,dz)

+

∫ t

0

∫Rκ′(δ(s, z))µ(ds,dz),

supω,s,x |δ(s, x)|/γ(x) locally bounded for non-negative γ satisfying∫R

(γr (x) ∧ 1)λ(dx) <∞ on the jump activity, the estimator

σ2s =

bsh−1n c+Kn∑

k=bsh−1n c−Kn

(2Kn + 1)−1ζnk1hn|ζn

k |≤un,

ζnk =

Jn∑j=1

wjk

(S2

jk − π2j2h−2n

η2

nF ′(khn)

),

satisfies for un ∝ hτn , τ ∈ (0,1), and 0 < β <(

α2α+1 ∧ τ

(1− r

2

))the

CLT:nβ/2

(σ2

s − σ2s) F−st−→ MN

(0,8σ3

s ((F−1)′(s))1/2η).



So what about the jumps?

In case of a general Ito semi-martingale, d = 1, supω,s,x |δ(s, x)|/γ(x)

locally bounded for non-negative γ satisfying∫R

(γr (x) ∧ 1)λ(dx) <∞on the jump activity, the estimator

σ2s =

bsh−1n c+Kn∑

k=bsh−1n c−Kn

(2Kn + 1)−1ζnk1hn|ζn

k |≤un,

ζnk =

Jn∑j=1

wjk

(S2

jk − π2j2h−2n

η2

nF ′(khn)

),

satisfies for un ∝ hτn , τ ∈ (0,1), and 0 < β <(

α2α+1 ∧ τ

(1− r

2

))the

CLT:nβ/2

(σ2

s − σ2s) F−st−→ MN

(0,8σ3

s ((F−1)′(s))1/2η).

On the restriction r < 3/2, for σ itself a semi-martingale the CLTabove is valid for the truncated spot volatility estimator.On the restriction r < 1, for σ itself a semi-martingale the CLT aboveis valid for the truncated integrated volatility estimator.



Monte Carlo: Parametric scenario

Two-dimensional setup, Σ11 = Σ22 = 1, η21 = η2

2 = 0.1, n = 30000synchronous observations, 20000 iterations, SPEC is the spectralestimator based on univariate weights.



Monte Carlo: Stochastic volatility model

Stochastic volatility model

dX (l)t = φl (t) σl (t) dZ (l)

t , dσ2l (t) = αl

(µl − σ2

l (t))

dt+ψl σl (t) dV (l)t ,

Random exogenous Poisson sampling En1 = 23400, En2 = 15600,40000 iterations.



Glimpse at data

Revisiting the introductory example (with one-dimensional method)

Bin-wise spectral estimates portray de-noised local quadraticvariation.



Glimpse at data

Estimated spot correlation of Apple and Google (NASDAQ) on2012/12/27.

09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time of Day

Correlation:AAPL

vs.GOOG

ρt CI(ρt,0.95) CI(ρt,0.95)



Glimpse at data

Averaged spot correlation vs. covariance of 30 NASDAQ100 stockswith highest market capitalization on 2012/12/27 and 2013/04/16.

10:00 11:00 12:00 13:00 14:00 15:00 16:000.2

0.4

0.6

0.8

1

Time of Day

Avg.Correlation

10:00 11:00 12:00 13:00 14:00 15:00 16:000.01

0.02

0.03

0.04

0.05

Avg.Covariance

ρt σt

10:00 11:00 12:00 13:00 14:00 15:00 16:000.2

0.4

0.6

Time of Day

Avg.Correlation

10:00 11:00 12:00 13:00 14:00 15:00 16:000

0.02

0.04

Avg.Covariance

ρt σt



Conclusion

Summary:

Covariation matrix estimation from HF-data is sensitive to both,noise and asynchronicity, but in combination noise prevails!

A locally parametric approach (LMM) provides an efficientsemi-parametric estimator.

Multivariate LMM-estimator of the integrated covolatility matrixprovides efficiency gains.

Estimators satisfy stable CLTs in the semi-martingale framework.



Outlook and remarks

Estimator LMM not guaranteed to be positive-definite. Yet,confidence relies on the positive definite estimated Fisherinformation matrices.

The model is still an idealization of real world financial data(endogeneities, oder book dynamics ...).

For further structural specification of the noise (bounded supportetc.) the efficiency theory can be (crucially) different.

The results on spot (co-)volatility estimation open up new waysfor statistical inference. Several generalizations of methods forthe direct observation model are possible.



Literature

Reiß, M., 2011.Asymptotic equivalence for inference on the volatility from noisy observations.Ann. Stat., 39(2),772-802.

Bibinger, M., and Reiß, M., 2014.Spectral estimation of covolatility using local weights.Scand. J . Stat., 41(1), 23-50.

Bibinger, M., Hautsch, N., Malec, P., Reiß, M., 2013.Estimating the quadratic covariation matrix from noisy observations: Local method ofmoments and efficiency. arXiv:1303.6146.

Altmeyer, R. and Bibinger, M. 2014.Functional stable limit theorems for efficient spectral covolatility estimators.arXiv:1401.2272.

Thank you for your attention!


Documents

Covariance matrix estimation from noisy high-frequency ... · Covariance matrix estimation from noisy high-frequency observations: Local method of moments and efﬁciency Markus Bibinger