Nonparametric Estimation of Additive Multivariate Diffusion - - PowerPoint PPT Presentation

Nonparametric Estimation of Additive Multivariate Diffusion Processes

Berthold R. Haag University of Mannheim European Young Statisticians Meeting 24.08.2005

Contents

• Statistical Model
• Smooth Backfitting Estimation
• Asymptotic Results
• Simulation
• Final Remarks

1

Statistical Model

Consider the d-dimensional time-homogenous diffusion process (Xt)t≥0 =

• (X1

t , . . . , Xd t )′ t≥0, satisfying the stochastic differential equation

dXt = µ(Xt) dt + Σ(Xt) dWt. Dynamics are described by µ(x) = (µ1(x), . . . , µd(x))′ the drift vector. Σ(x) = (σij(x))ij the dispersion matrix . A(x) = Σ(x)Σ(x)′ = (aij(x))ij diffusion matrix, defined for identification.

2

Assume that the process (Xt)t≥0 is strictly stationary, has compact support G and is strongly mixing with summable mixing coefficients. The stationary density is denoted by f(x). (Conditions see Veretennikov, 1997). The process is observed at nT + 1 equispaced time points in [0, T], denoted by Xi∆, i = 0, . . . , nT (∆ = n−1). For high frequency sampling n → ∞ the nonparametric estimation of the drift and diffusion function is based on lim

∆→0 E(∆−1(Xi (k+1)∆ − Xi k∆) | Xk∆ = x) = µi(x)

lim

∆→0 E(∆−1(Xi (k+1)∆ − Xi k∆)(Xj (k+1)∆ − Xj k∆) | Xk∆ = x) = aij(x) 3

Then the (multivariate) Nadraya-Watson estimator of the drift function is defined as the solution of ˆ µ1

h(x) = arg min ¯ µ1∈M

• 1

nT

nT

• k=1
• ∆−1(X1

k∆−X1 (k−1)∆)− ¯

µ1(x) 2

d

• i=1

Kh(xi, Xi

(k−1)∆) dx

and is given explicitly by ˆ µ1

h(x) = 1 nT

nT

k=1

d

i=1 Kh(xi, Xi (k−1)∆)∆−1(X1 k∆ − X1 (k−1)∆)

ˆ fh(x) with the usual kernel density estimator ˆ fh(x) = 1 nT

nT −1

• k=0

d

• i=1

Kh(xi, Xi

k∆)

Estimators of the diffusion matrix are defined analogously.

4

Smooth Backfitting Estimation (Mammen, Linton and Nielsen, 1999)

To circumvent the curse of dimensionality assume additivity for the drift function µ1(x) = µ1

0 + µ1(x1) + · · · + µ1(xd)

where for identifiability it is assumed that

• µ1(xi)f(xi) dxi = 0.

It is then natural to restrict the Nadraya-Watson minimization to the space of additive functions min

¯ µ1∈Madd

• 1

nT

nT

• k=1
• ∆−1(X1

k∆ − X1 (k−1)∆)

− ¯ µ1

0 − ¯

µ1(x1) − · · · − ¯ µ1(xd) 2

d

• i=1

Kh(xi, Xi

(k−1)∆) dx 5

This is equivalent to minimizing min

¯ µ1∈Madd

ˆ µ1

h(x) − ¯

µ1

0 − ¯

µ1(x1) − · · · − ¯ µ1(xd) 2 ˆ fh(x) dx where the minimization is restricted to functions with

• ¯

µ1(xj) ˆ fh(xj) dxj = 0. The solution of this minimization are called NW-SBF estimators.

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The solution (˜ µ1

0, ˜

µ1(x1), . . . , ˜ µ1(xd)) of the minimization is given by the set of equations ˜ µ1

h(xj) = ˆ

µ1

h(xj) −

• i=j
• ˜

µ1

h(xi)

ˆ fh(xj, xi) ˆ fh(xj) dxi − ˜ µ1 This can be solved iteratively, using the marginal Nadaraya-Watson estimators ˆ µ1

h(xj) as starting values.

For the applicability we have to use modified kernel estimators that fulfil

• ˆ

fh(xj, xi) dxi = ˆ fh(xj)

• e. g. by using modified kernels

Kh(u, v) = Kh(u − v) 1

0 Kh(w − v) dw

.

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Asymptotic Results

Theorem 1. Under regularity assumptions, h2 = O((Th)−1/2) and nh3 → ∞ we have that for the estimators ˜ µ1(xj) it holds that √ Th ˜ µ1

h(xj) − µ1(xj) − b1 h(xj) − h2 ˜

β1

µ(xj)

v1(xj)κ2(xj)/κ0(xj)

D

− → N(0, 1) for j = 1, . . . , d where b1

h(xj) = h ∂

∂xj (µ1(xj))κ1(xj) κ0(xj) −

• µ1(xj)f j(xj)κ0(xj) dxj

and v1(xj) = (f j(xj))−1 E(a11(X) | Xj = xj)

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The bias is not given in explicit form, it is only defined as the solution to (˜ βµ, ˜ βµ(x1), . . . , ˜ βµ(xd)) = arg min

β0

µ,...,βd µ

• (βµ(x)−β0

µ−β1 µ(x1)−· · ·−βd µ(xd))2f(x) dx

with βµ(x) = κ2

d

• j=1

∂ ∂xj (µ1(xj)f(x))(f(x))−1 + 1

2

∂2 ∂(xj)2 µ1(xj)

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• To judge the efficiency of the SBF estimator we compare it to the oracle

estimator ˇ µ1(xj), based on the unobservable data Y ⋆

(k−1)∆ = ∆−1(X1 k∆ − X1 (k−1)∆) −

• i=j

µ1(Xi

(k−1)∆)

NW-SBF achieves the same variance as the oracle estimator, but not the bias.

• Estimating elements of the diffusion matrix, the same efficiency is achieved,

the rate of convergence is given by √ Tnh.

10

Local Linear SBF

In analogy to the Nadraya-Watson estimator, local linear estimators are defined as minimizers of

• 1

nT

nT

• k=1
• ∆−1(X1

k∆−X1 (k−1)∆)−¯

µ1(x)−

d

• j=1

¯ µ1

j(x)

xj − Xj

(k−1)∆

h 2

d

• i=1

Kh(xi, Xi

(k−1)∆) dx

• r explicitly

ˆ µ1,LL

h

(x) = ˆ S−1(x)ˆ T(x) where ˆ S(x) = XT (x)K(x)X(x) and ˆ T(x) = XT (x)K(x)Y with Y = ∆−1 (X∆ − X0), . . . , (XT − XT −∆) T X(x) =      1

X1

0−x1

h

. . .

Xd

0 −xd

h

. . . ... . . . 1

X1

T −∆−x1

h

. . .

Xd

T −∆−xd

h

     K(x) = 1 nT

• diag
• d
• i=1

Kh(Xi

0, xi), . . . , d

• i=1

Kh(Xi

T −∆, xi)

• 11

Restricting the minimization to additive functions, this is equivalent to minimizing ˆ µ1,LL

h

(x) − ¯ µ1(x)ˆ

S =

• (ˆ

µ1,LL

h

(x) − ¯ µ1(x))T ˆ S(x)(ˆ µ1,LL

h

(x) − ¯ µ1(x)) dx with norming

• ¯

µ1(xj) ˆ fh(xj) dxj +

• ¯

µ1

j(xj) ˆ

f 1

h(xj) dxj = 0.

Here ¯ µ1(x) = (¯ µ(x), ¯ µ1

1(x1), . . . , ¯

µ1

d(xd))T is an element of Madd

The solution of this minimization can be derived iteratively. The algorithm converges with geometric rate.

12

• Asymptotic normality for the drift estimator with rate

√ Th.

• Asymptotic normality for the diffusion estimator with rate

√ Tnh.

• Local linear SBF leads to oracle variance and bias.

13

Simulation

For simulation a linear model is considered µ(x) =     0.75 − x1 + 0.25x2 + x3 0.75 + 0.5x1 − 2x2 + 0.25x3 1.5 + 0.25x1 + x2 − 3x3     Σ(x) =     √ x1 + x2 √ x2 √ x3     and a nonlinear model µ(x) =     0.75 + 0.5(sin(2x1) − x1) + 0.25x2 + x3 0.75 + 0.5x1 − 2x2 + 0.25x3 1.5 + 0.25x1 + x2 − 3x3     Σ(x) =    

• (x1)2 + (x2)2

√ x2 √ x3     . In both cases µ1(x) is estimated based on a sample of n = 30, T = 35

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1.0 1.5 2.0 −0.5 1.0 2.5

SBF estimator of µ1(x1)

0.2 0.4 0.6 0.8 1.0 1.2 1.4 −0.4 −0.1 0.2

SBF estimator of µ1(x2)

0.4 0.6 0.8 1.0 1.2 1.4 −0.5 0.5

SBF estimator of µ1(x3) 15

1.0 1.5 2.0 0.0 1.5

SBF estimator of µ1(x1)

1.0 1.5 2.0 0.2 0.4 0.6

Density of X1

16

3 4 5 6 7 −6 −2 2

SBF estimator of µ1(x1)

1.0 1.5 2.0 2.5 3.0 −1.0 0.5

SBF estimator of µ1(x2)

1.0 1.5 2.0 −2 1 2

SBF estimator of µ1(x3) 17

3 4 5 6 7 −6 −2 2

SBF estimator of µ1(x1)

3 4 5 6 7 0.10 0.20

Density of X1

18

Final Remarks

• Asymptotic distribution for SBF estimators of drift and diffusion of

multivariate diffusion processes.

• Extend to fixed time horizon (fixed T). The diffusion estimator should be

estimable with mixed aymptotic normality.

• Extension to low frequency data (fixed n). This would require extensions of

the estimators of Gobet, Hoffmann and Reiß (2004).

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