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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
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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) .
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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).
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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 .
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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)
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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 ).
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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!
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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 .
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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
).
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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 .
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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
).
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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 .
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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 covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
Efficient asymptotic variance: Examples
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 .
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
Efficient asymptotic variance: Examples
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.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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 .
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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) .
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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).
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
Glimpse at data
Revisiting the introductory example (with one-dimensional method)
Bin-wise spectral estimates portray de-noised local quadraticvariation.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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)
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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.
Efficient covariance matrix estimation Markus Bibinger
Introduction Asymptotic Equivalence LMM Efficiency Semi-martingales
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!
Efficient covariance matrix estimation Markus Bibinger