Nov 20.-22., 2008, G ottingen Estimation of Different Scales in - - PowerPoint PPT Presentation

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Nov 20.-22., 2008, G¨

• ttingen

DFG-SNF Research Group Opening Conference ’Statistical Regularisation’

Tony Cai, University of Pennsylvania Rama Cont, Columbia University Jos Manuel Corcuera, Universitat de Barcelona Jean-Pierre Florens, Universit Toulouse I Peter Gottschalk, Boston College Janine Illian, University of St Andrews Karl Kunisch, Universitt Graz Guillaume Lecue, CNRS, LATP, Marseille Jens Perch Nielsen, Cass Business School, London Nicolai Meinshausen, University of Oxford Ya’acov Ritov, The Hebrew University of Jerusalem Naftali Tishby, The Hebrew University of Jerusalem Alexandre Tsybakov, Universit Paris VI

http: //www.stochastik.math.uni-goettingen.de/forschergruppe/

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

The Estimation of Different Scales in Microstructure Noise Models from a Nonparametric Regression Perspective

Axel Munk Joint work with T. Cai and J. Schmidt-Hieber

Institut f¨ ur Mathematische Stochastik, G¨

• ttingen

www.stochastik.math.uni-goettingen.de/munk munk@math.uni-goettingen.de

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Introduction

1 Introduction

Motivating example Data transformation

2 Estimation of τ 2 3 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

4 Non-constant σ and τ 5 Summary/ Outlook

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

The model

Suppose we observe (Stein ’87, Gloter and Jacod ’01) Yi,n = σWi/n + τǫi,n, i = 1, . . . , n, where (i) Wt is a (Standard) Brownian Motion, (ii) ǫi,n

i.i.d

∼ N (0, 1), (iii) σ, τ > 0 are unknown scale parameters. (iv) ǫi,n and Wt are assumed to be independent for all i and all t ∈ [0, 1].

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

The model

Suppose we observe (Stein ’87, Gloter and Jacod ’01) Yi,n = σWi/n + τǫi,n, i = 1, . . . , n, where (i) Wt is a (Standard) Brownian Motion, (ii) ǫi,n

i.i.d

∼ N (0, 1), (iii) σ, τ > 0 are unknown scale parameters. (iv) ǫi,n and Wt are assumed to be independent for all i and all t ∈ [0, 1]. Statistical Problem Estimation of the parameters σ2 and τ 2.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Four data sets

• 0.0

0.2 0.4 0.6 0.8 1.0 −2 −1 1 2

• 0.0

0.2 0.4 0.6 0.8 1.0 −2 −1 1 2

• 0.0

0.2 0.4 0.6 0.8 1.0 −2 −1 1 2

• 0.0

0.2 0.4 0.6 0.8 1.0 −2 −1 1 2

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Data with underlying path of Brownian motion

• 0.0

0.2 0.4 0.6 0.8 1.0 −2 −1 1 2 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 1 2

• 0.0

0.2 0.4 0.6 0.8 1.0 −2 −1 1 2 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 1 2

• 0.0

0.2 0.4 0.6 0.8 1.0 −2 −1 1 2 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 1 2

• 0.0

0.2 0.4 0.6 0.8 1.0 −2 −1 1 2 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 1 2

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Var (Yi,n) = σ2 i

n + τ 2,

σ2/2 + τ 2 ≈ 1

• 0.0

0.2 0.4 0.6 0.8 1.0 −2 −1 1 2 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 1 2

sigma^2 = 0.01, tau^2=1

• 0.0

0.2 0.4 0.6 0.8 1.0 −2 −1 1 2 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 1 2

sigma^2 = 1/2, tau^2=3/4

• 0.0

0.2 0.4 0.6 0.8 1.0 −2 −1 1 2 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 1 2

sigma^2 = 2/3, tau^2=2/3

• 0.0

0.2 0.4 0.6 0.8 1.0 −2 −1 1 2 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 1 2

sigma^2 = 2, tau^2=0.01

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Estimation of σ2 (and τ 2)

Difficulties: Observations are not independent. The path σWt is hard to identify by inspection of the data. Idea: Make the observations independent by an orthogonal transformation and look at them from the viewpoint of statistical inverse problems.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Covariance diagonalization

(Cov [Y ])i,j = σ2

i n ∧ j n

• + τ 2δi,j,

i, j = 1, . . . , n, δi,j Kronecker delta, Y = (Y1,n, . . . , Yn,n)t . = ⇒ Find an orthogonal transformation Dn such that Z := DnY has independent components.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

The observations in Fourier space

Eigenvalues and eigenvectors of (i/n ∧ j/n)i,j=1,...,n are solutions of the second order difference equation (Durbin, Knott ’72) 1 λin∆2vi,l−1 + vi,l = 0, vi,0 := 0, vi,n+1 := vi,n, where ∆2 denotes the second finite difference operator, i.e. ∆2vi,l−1 = vi,l+1 − 2vi,l + vi,l−1, i = 1, . . . , n. Matrix of eigenvectors is a discrete sine transformation Dn :=

• 4

2n + 1

• sin

(2j − 1) iπ 2n + 1

• i,j=1,...,n

.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

The observations in Fourier space

Hence Z := DnY has independent components. It holds Zi ∼ N

• 0, σ2λi + τ 2

, i = 1, . . . , n, where λi are eigenvalues of (i/n ∧ j/n)i,j=1,...,n , λi =

• 4n sin2

2i − 1 4n + 2π −1 = 1 n Dir2

n

(2i − 1) π 2n + 1

• ,

Dirn(x) = 1/2 + n

i=1 cos(iπx) (Dirichlet kernel).

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Eigenvalues and approximation

It holds for all n ≥ 1 and i = 1, . . . , n that π−2 n

i2 ≤ λi ≤ 4 n i2 .

2 4 6 8 10 1 2 3 4

Eigenvalues (black) and approximation 2n/

• i2π2

(blue).

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

How to construct estimators in the transformed model?

Sufficient observations: Z 2

i ≈

• σ2n/
• π2i2

+ τ 2 Ui, i = 1, . . . , n, Ui

iid

∼ χ2

1.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

How to construct estimators in the transformed model?

Sufficient observations: Z 2

i ≈

• σ2n/
• π2i2

+ τ 2 Ui, i = 1, . . . , n, Ui

iid

∼ χ2

1.

Which observations are suitable for estimating τ 2?

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

How to construct estimators in the transformed model?

Sufficient observations: Z 2

i ≈

• σ2n/
• π2i2

+ τ 2 Ui, i = 1, . . . , n, Ui

iid

∼ χ2

1.

Which observations are suitable for estimating τ 2? The last mn, . . . , n observations, such that mn/√n → ∞ and mn/n → 0.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

How to construct estimators in the transformed model?

Sufficient observations: Z 2

i ≈

• σ2n/
• π2i2

+ τ 2 Ui, i = 1, . . . , n, Ui

iid

∼ χ2

1.

Which observations are suitable for estimating τ 2? The last mn, . . . , n observations, such that mn/√n → ∞ and mn/n → 0. Linear Estimator: (m = mn) ˆ τ 2

m = (n − m)−1 n

• i=m+1

Z 2

i .

These are n − m ≍ n observations.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

MSE analysis

For any ǫ > 0, sup

σ,τ>ǫ (στ)−4

n MSE

• ˆ

τ 2

m

• − 2τ 4

= o (1) . In particular for fixed σ, τ > 0 it holds MSE

• ˆ

τ 2

m

• = 2τ 4n−1 (1 + o (1)) .

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Lower bound

Gloter, Jacod ’01 and C.M.S. ’08 (i) For any estimator ˆ τ 2, and any σ ≥ 0 lim

n→∞

inf

ˆ τ 2 sup τ>0

1 2τ 4 E

• n
• ˆ

τ 2 − τ 22 ≥ 1, (ii) and for any 0 < ǫ < c < ∞, lim

n inf ˆ τ 2

sup

σ,τ>ǫ, σ<c

1 2τ 4 E

• n
• ˆ

τ 2 − τ 22 = 1.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Estimation of σ2: a first approach

New observations Z 2

i =

• σ2λi + τ 2

Ui ≈

• σ2n/
• π2i2

+ τ 2 Ui, i = 1, . . . , n, Ui

iid

∼ χ2

1.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Estimation of σ2: a first approach

New observations Z 2

i =

• σ2λi + τ 2

Ui ≈

• σ2n/
• π2i2

+ τ 2 Ui, i = 1, . . . , n, Ui

iid

∼ χ2

1.

Which observations can be used for estimating σ2?

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Estimation of σ2: a first approach

New observations Z 2

i =

• σ2λi + τ 2

Ui ≈

• σ2n/
• π2i2

+ τ 2 Ui, i = 1, . . . , n, Ui

iid

∼ χ2

1.

Which observations can be used for estimating σ2? Essentially, the first √n

• bservations!

ˆ σ2

n =

1 √n

• [

√n]

• i=1

Z 2

i − ˆ

τ 2 λi .

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Estimation of σ2: a first approach

The estimator for σ2 is n−1/4 consistent and we will see that this is the optimal rate of convergence. Heuristics: Z 2

i remains bounded for i = O

√n

• .

These are √n observations! This estimator is not asymptotically sharp, i.e. does not attain the optimal asymptotic constant.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Sharp estimator for σ2: optimal weights

Find wi, i = 1, . . . , n, such that the mean squared error of σ2 = n

i=1 wi

• Z 2

i − τ 2

is minimized.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Sharp estimator for σ2: optimal weights

Find wi, i = 1, . . . , n, such that the mean squared error of σ2 = n

i=1 wi

• Z 2

i − τ 2

is minimized. w∗

i =

λi (σ2λi + τ 2)2 /  

n

• j=1

λ2

j

(σ2λj + τ 2)2   , i = 1, . . . , n. Depend on unknown σ2 and τ 2 → oracle.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Splitting technique

Let k =

• n1/2−b

and m =

• n1/2+b

, 0 < b < ǫ. Split the spectrum:

1 k m n

1 Use observations m + 1, ..., . . . , n to get an estimator of

τ 2, ˆ τ 2

m := (n − m)−1 n

• i=m+1

Z 2

i . 2 Use observations 1, . . . , k to get a first estimate of σ2, i.e.

˜ σ2

k,m := 1

k

k

• i=1

λ−1

i

• Z 2

i − ˆ

τ 2

m

• .

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Splitting technique

Let k =

• n1/2−b

and m =

• n1/2+b

, 0 < b < ǫ. Split the spectrum:

1 k m n

3 Use the remaining indep. observations to mimic the oracle

ˆ σ2 :=

m

• i=k+1

ˆ wi

• ˜

σ2

k,m, ˆ

τ 2

m

Z 2

i − ˆ

τ 2

m

• .

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Asymptotic properties of the estimator

Upper Bound, Asymptotic Normality. C.M.S. ’08 Let k = [n1/2−b] and m = [n1/2+b] with 0 < b < 1/20. Then (i) for any ǫ > 0 sup

σ,τ>ǫ (στ)−8

• MSE
• ˆ

σ2 − 8τσ3n−1/2

• = o
• n−1/2

, (ii) n1/4(ˆ σ2 − σ2)

L

− → N(0, 8τσ3).

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Asymptotic properties of the estimator

Lower Bound. Gloter, Jacod ’01 and C.M.S. ’08 (i) For any estimator ˆ σ2, we have lim

n→∞ sup τ,σ>ǫ

1 8τσ3 E

• n1/2(ˆ

σ2 − σ2)2 ≥ 1 and equality holds if in addition σ, τ ≤ K < ∞. (ii) Furthermore, lim

n→∞ inf ˆ σ2 sup τ,σ>ǫ (στ)−8

E

• n1/2(ˆ

σ2 − σ2)2 − 8τσ3 = 0.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Sketch of proof for (i). Part 1/3

Use information inequality method (Lehmann (1983), pp. 266). Assume (i) does not hold. Then there exists an estimator ˆ σ2 = ˆ σ2

n and a subsequence {nk}, such that for fixed

τ = τ0 > ǫ MSE

• ˆ

σ2 ≤ (1 − 3δ) 8τσ3n−1/2

k

, for all σ2 > ǫ For the Fisher information (and all sufficiently large nk) Ink

• σ2

≤ (1 + δ) 1 8τσ3 n1/2

k

, for all ǫ < σ2 ≤ 3ǫ2.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Sketch of proof for (i). Part 2/3

Let b(σ2) denote the bias of ˆ σ2 if σ2 is the true parameter. Information inequality b2 σ2 +

• 1 + b′

σ22 Ink (σ2) ≤ MSE

• ˆ

σ2 .

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Sketch of proof for (i). Part 3/3

Some calculations show that this implies for all ǫ < σ2 ≤ 3ǫ2 b

• σ2

≤ −δ

• σ2 − 2ǫ
• + b (2ǫ) .

In addition: b2 σ2 ≤ MSE

• ˆ

σ2 ≤ 8τσ3n−1/2

k

, If σ2 = 3ǫ then lim

k→∞ b (3ǫ) = 0,

lim

k→∞ b (3ǫ) ≤ −δǫ.

This is a contradiction!

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Other approaches

MLE in MA (1) processes (A¨ ıt-Sahalia et al. ’05). Contrast estimator (Gloter, Jacod ’01).

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Numerical results

The proposed estimator ˆ σ2 can be implemented explicitly, no optimization step necessary. Computational cost is O (n log n). Might be a good starting value for MLE. Robust against model misspecifications.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Extensions of the model

Asset price dynamics in financial models, σ non-constant (Fan et al. ’03). In fact more realistic models (Barndorff-Nielsen et al. ’06) are of the type Yi,n = Xi,n + Ui,n, i = 1, . . . , n, where

Xt is a semimartingale and dXt = btdt + σtdWt. σt can be stochastic itself. Ui,n is so called market microstructure noise, i.e. Ui,n are centered iid with finite fourth moment. Ui,n and Xt are independent for all i = 1, . . . , n.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Extensions of the model

We consider two ”intermediate” models. For i = 1, . . . , n Yi,n = σ i n

• Wi/n + τ

i n

• ǫi,n,

(1) ˜ Yi,n = i/n σ (s) dWs + τ i n

• ǫi,n.

(2) Model (2) will not be discussed further. σ, τ are now deterministic, unknown, positive functions. ǫi,n

i.i.d

∼ N (0, 1). ǫi,n and Wt are considered to be independent for i = 1, . . . , n.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Estimation of σ2 and τ 2

Non-parametric approach necessary. Transformation that diagonalizes the process depends on the unkown quantities σ(t) and τ(t) and can not be computed explicitly in general.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Mimicking the constant case

Idea: Apply similar estimators as in the constant case. However, for technical reasons, to the increments ∆iY := Yi+1,n − Yi,n.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Mimicking the constant case

Idea: Apply similar estimators as in the constant case. However, for technical reasons, to the increments ∆iY := Yi+1,n − Yi,n. ∆Y := (∆1Y , . . . , ∆n−1Y )t . Therefore, we use another DST, i.e. Dn :=

• 2

n sin ijπ n

• i,j=1,...,n−1

.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Behaviour of ˆ τ 2 and ˆ σ2 in the non-constant case

Let Z := Dn (∆Y ) and define the estimator ˆ τ 2 := 1 n − m

n−1

• i=m+1

˜ λ−1

i

Z 2

i ,

where ˜ λi = 4 sin2 (iπ/ (2n)) and m = mn, s.t. m/√n → ∞ and m/n → 0. Under smoothness assumptions on σ, τ, ˆ τ 2 estimates 1

0 τ 2(s)ds at a convergence rate n−1/4.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Behaviour of ˆ τ 2 and ˆ σ2 in the non-constant case

Similar for ˆ σ2 ˆ σ2 := √n

2[n1/2]

• i=[n1/2]+1

Z 2

i −

˜ λi ˆ τ 2

• bias correction

. Under smoothness assumptions on σ, τ, ˆ σ2 estimates 1

0 σ2(s)ds at a convergence rate n−1/4.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Mimicking the constant case

We will focus on estimation of σ2(t). Use this to construct a Fourier series estimator.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Mimicking the constant case

We will focus on estimation of σ2(t). Use this to construct a Fourier series estimator. Consider the functions fk : [0, 1] → R, fk(x) := √ 2 cos (kπx/2) , k = 0, 1, . . . Note f 2

k (x) = 1 + cos(kπx) =: ψ0(x) + 2−1/2ψk(x)

{ψk} =

• 1,

√ 2 cos (kπt) , k = 1, . . .

• is an ONS of L2[0, 1].

ψiψj = 2−1/2(ψi−j + ψi+j) sin( 2i−1

2 π) sin( 2j−1 2 π) = 2−3/2(ψi−j + ψi+j+1)

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Mimicking the constant case

Transform model (1) into fk i n

• Yi,n = fk

i n

• σ

i n

• Wi/n + fk

i n

• τ

i n

• ǫi,n

Take these as new observations and apply the estimators above gives estimators ˆ sk for k = 0, 1, . . . of 1 σ2(s)f 2

k (s)ds =

1 σ2(s)ds + 1 σ2(s) cos (kπs) ds.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

The pointwise estimator

The estimator of σ2(t) is then given by ˆ σ2

N (t) = 1

2ˆ s0 +

N

• i=1

(2ˆ si − ˆ s0) cos (iπt) , where N is some threshold parameter.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Assumptions

Function space: Sobolev s-ellipsoid Θs = Θs(α, C) =

• f ∈ L2[0, 1] : ∃ (θn)n ,
• s. t. f (x) = θ0 + 2

∞

• i=1

θi cos (iπx) ,

∞

• i=1

i2αθ2

i ≤ C

• Characterisation:

For any l odd, l < α ∈ N, f (l)(0) = f (l)(1) = 0 and 1 (f (α))2(x)dx ≤ ˜ C

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Theoretical results of the estimator

Upper bound on the risk, M., Schmidt-Hieber ’08 Assume τ 2 ∈ Θs (β, Qτ 2) and σ2 ∈ Θs (α, Qσ2), where Qτ 2, Qσ2 > 0 are fixed constants. Assume model (1) and α > 3/2, β > 5/4. Then it holds for N∗ = n1/(4α+2) MISE

• ˆ

σ2

N∗

• = O
• n−α/(2α+1)

. Note that this is ”half” of the minimax rate in nonparametric

• regression. Recall: λi bounded as long as i = O

√n

• .

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Lower bound

Many results known about lower bounds, mainly for independent observations and regression. Here:

Estimation of the scale of a Brownian motion. Dependent observations.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Idea of proof

Similar as in nonparametric regression: We use a multiple testing argument. Main problem: Bound Kullback-Leibler divergence between two multivariate centered normal random variables. Results by Golubev ’08 and Reiß ’08.: Bounds for Hellinger distance of multivariate centered normal r. v.s under the restriction that eigenvalues of covariance matrix are uniformly bounded or as in Reiß ’08 that one covariance matrix is the identity. However for our purpose one has to allow that eigenvalues tend to 0 and infinity.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

KL distance bound

M., Schmidt-Hieber ’08 Let X ∼ N (µ, Σ0) and Y ∼ N (µ, Σ1) and denote by PX and PY the corresponding probability measures. Assume 0 < CΣ0 ≤ Σ1 for some constant 0 < C ≤ 1. Then dK(PY , PX) ≤ 1 4C 2

• Σ−1

0 Σ1 − In

• 2

F .

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Lower bound

M., Schmidt-Hieber ’08 Assume model (1) or model (2), α ∈ N∗ and τ > 0. Then there exists a C > 0, such that lim

n→∞

inf

ˆ σ2

n

sup

σ2∈Θc(α,Q)

E

• n

α 2α+1

ˆ σ2 − σ2 2

2

• ≥ C.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Summary

Microstructure noise models with deterministic volatility provide some insight into volatility estimation from a minimax point of view. Viewing these models as a statistical inverse problem problems reveals some similarity to deconvolution. However, scale estmation makes it different. Degree of ill posedness corresponds to 1/2. Our approach relies heavily on Fourier methods and hence

• n a Gaussian process. Nevertheless, it seems quite robust.

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

Summary/Outlook

σ, τ constant.

Spectral methods allow explicit, sharp estimators. Fast computable O (n log n). Robust against model misspecifications.

σ, τ non-constant.

Fourier type estimator achieves optimal rates of convergence. Open issues: Adaptation, SVMs, . . .

Estimation of Different Scales in Microstructure Noise Models Munk Introduction

Motivating example Data transformation

Estimation of τ2 Estimation of σ2

Construction of sharp estimator for σ2 Numerics

Non-constant σ and τ Summary/ Outlook

References

Ait-Sahalia, Y., Mykland, P. and Zhang, L. (2005). How often to sample a continuous-time process in the presence of market microstructure noise. Review of Financial Studies, Vol. 18, 351-416. Barndorff-Nielsen, O., Hansen, R., Lunde, A. and Stephard, N. (2006). Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise. Working Paper. Cai, T., Munk, A. and Schmidt-Hieber, J. (2008) Sharp minimax estimation of the variance of Brownian motion corrupted by Gaussian noise. Stat. Sinica, Under Revision. Durbin, J. and Knott, M. (1972). Components of the Cramer-von Mises statistics I. Journal of the Royal Statistical Society Series B 34, 290-307. Fan, J., Jiang, J., Zhang, C. and Zhou, Z. (2003). Time-dependent diffusion models for term structure dynamics. Stat. Sinica 13, 965-992. Gloter, A. and Jacod, J. (2001). Diffusions with measurement errors. I. Local Asymptotic Normality. ESAIM: Probability and Statistics, 5, 225-242. Gloter, A. and Jacod, J. (2001). Diffusions with measurement errors. II. Optimal estimators. ESAIM: Probability ans Statistics, 5, 243-260. Golubev, G., Nussbaum, M. and Zhou, H. (2008). Asymptotic equivalence of spectral density estimation and Gaussian white noise. Submitted. Hoffmann, M. (2002). Rate of convergence for parametric estimation in a stochastic volatility model.

• Stoch. Proc. and their Appl., 97, 147-170.

Malliavin, P. and Mancino, M. E. (2005) Harmonic analysis methods for nonparametric estimation of

• volatility. Preprint.

Reiß, M. (2008). Asymptotic equivalence for nonparametric regression with multivariate and random

• design. Annals of Statistics, to appear.

Stein, M. (1987). Minimum norm quadratic estimation of spatial variograms. Journal of the American Statistical Association, 82, 765-772. www.stochastik.math.uni-goettingen.de/munk munk@math.uni-goettingen.de