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 � ( X 1 t , . . . , X d t ) ′ � ( X t ) t ≥ 0 = t ≥ 0 , satisfying the stochastic differential equation d X t = µ ( X t ) d t + Σ( X t ) d W t . Dynamics are described by µ ( x ) = ( µ 1 ( x ) , . . . , µ d ( x )) ′ the drift vector. Σ( x ) = ( σ ij ( x )) ij the dispersion matrix . A ( x ) = Σ( x )Σ( x ) ′ = ( a ij ( x )) ij diffusion matrix, defined for identification. 2

Assume that the process ( X t ) 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 X i ∆ , i = 0 , . . . , nT (∆ = n − 1 ). For high frequency sampling n → ∞ the nonparametric estimation of the drift and diffusion function is based on ∆ → 0 E (∆ − 1 ( X i ( k +1)∆ − X i k ∆ ) | X k ∆ = x ) = µ i ( x ) lim k ∆ )( X j ( k +1)∆ − X j ∆ → 0 E (∆ − 1 ( X i ( k +1)∆ − X i k ∆ ) | X k ∆ = x ) = a ij ( x ) lim 3

Then the (multivariate) Nadraya-Watson estimator of the drift function is defined as the solution of nT d 1 � � 2 � � µ 1 ∆ − 1 ( X 1 k ∆ − X 1 µ 1 ( x ) K h ( x i , X i � ˆ h ( x ) = arg min ( k − 1)∆ ) − ¯ ( k − 1)∆ ) d x nT µ 1 ∈M ¯ i =1 k =1 and is given explicitly by � nT � d 1 i =1 K h ( x i , X i ( k − 1)∆ )∆ − 1 ( X 1 k ∆ − X 1 ( k − 1)∆ ) k =1 nT µ 1 ˆ h ( x ) = ˆ f h ( x ) with the usual kernel density estimator nT − 1 d 1 ˆ � � K h ( x i , X i f h ( x ) = k ∆ ) nT i =1 k =0 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 µ 1 ( x ) = µ 1 0 + µ 1 ( x 1 ) + · · · + µ 1 ( x d ) µ 1 ( x i ) f ( x i ) d x i = 0. � where for identifiability it is assumed that It is then natural to restrict the Nadraya-Watson minimization to the space of additive functions nT � 1 � ∆ − 1 ( X 1 k ∆ − X 1 � min ( k − 1)∆ ) nT µ 1 ∈M add ¯ k =1 d � 2 � µ 1 µ 1 ( x 1 ) − · · · − ¯ µ 1 ( x d ) K h ( x i , X i − ¯ 0 − ¯ ( k − 1)∆ ) d x i =1 5

This is equivalent to minimizing � 2 ˆ � � µ 1 µ 1 µ 1 ( x 1 ) − · · · − ¯ µ 1 ( x d ) min ˆ h ( x ) − ¯ 0 − ¯ f h ( x ) d x µ 1 ∈M add ¯ where the minimization is restricted to functions with � f h ( x j ) d x j = 0 . µ 1 ( x j ) ˆ ¯ The solution of this minimization are called NW-SBF estimators . 6

µ 1 µ 1 ( x 1 ) , . . . , ˜ µ 1 ( x d )) of the minimization is given by the set of The solution (˜ 0 , ˜ equations ˆ f h ( x j , x i ) � d x i − ˜ � µ 1 h ( x j ) = ˆ µ 1 h ( x j ) − µ 1 h ( x i ) µ 1 ˜ ˜ 0 ˆ f h ( x j ) i � = j This can be solved iteratively, using the marginal Nadaraya-Watson estimators µ 1 h ( x j ) as starting values. ˆ For the applicability we have to use modified kernel estimators that fulfil � f h ( x j , x i ) d x i = ˆ ˆ f h ( x j ) e. g. by using modified kernels K h ( u − v ) K h ( u, v ) = . � 1 0 K h ( w − v ) d w 7

Asymptotic Results Theorem 1. Under regularity assumptions, h 2 = O (( Th ) − 1 / 2 ) and nh 3 → ∞ we µ 1 ( x j ) it holds that have that for the estimators ˜ h ( x j ) − h 2 ˜ √ µ 1 h ( x j ) − µ 1 ( x j ) − b 1 β 1 µ ( x j ) Th ˜ D − → N (0 , 1) v 1 ( x j ) κ 2 ( x j ) /κ 0 ( x j ) for j = 1 , . . . , d where ∂x j ( µ 1 ( x j )) κ 1 ( x j ) h ( x j ) = h ∂ � b 1 µ 1 ( x j ) f j ( x j ) κ 0 ( x j ) d x j κ 0 ( x j ) − and v 1 ( x j ) = ( f j ( x j )) − 1 E ( a 11 ( X ) | X j = x j ) 8

The bias is not given in explicit form, it is only defined as the solution to � (˜ β µ , ˜ β µ ( x 1 ) , . . . , ˜ β µ ( x d )) = arg ( β µ ( x ) − β 0 µ − β 1 µ ( x 1 ) −· · ·− β d µ ( x d )) 2 f ( x ) d x min β 0 µ ,...,β d µ with d ∂ 2 ∂ ∂x j ( µ 1 ( x j ) f ( x ))( f ( x )) − 1 + 1 � ∂ ( x j ) 2 µ 1 ( x j ) β µ ( x ) = κ 2 2 j =1 9

• To judge the efficiency of the SBF estimator we compare it to the oracle µ 1 ( x j ), based on the unobservable data estimator ˇ � Y ⋆ ( k − 1)∆ = ∆ − 1 ( X 1 k ∆ − X 1 µ 1 ( X i ( k − 1)∆ ) − ( k − 1)∆ ) i � = j 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 x j − X j nT d d � 1 � 2 ( k − 1)∆ � � � ∆ − 1 ( X 1 k ∆ − X 1 µ 1 ( x ) − µ 1 K h ( x i , X i � ( k − 1)∆ ) − ¯ ¯ j ( x ) ( k − 1)∆ ) d x nT h k =1 j =1 i =1 or explicitly µ 1 ,LL ( x ) = ˆ S − 1 ( x )ˆ T ( x ) ˆ h S ( x ) = X T ( x ) K ( x ) X ( x ) and ˆ T ( x ) = X T ( x ) K ( x ) Y with where ˆ � T Y = ∆ − 1 � ( X ∆ − X 0 ) , . . . , ( X T − X T − ∆ )  X 1 0 − x 1 X d 0 − x d  1 . . . h h . .  ...  . . X ( x ) =   . .     X 1 T − ∆ − x 1 X d T − ∆ − x d 1 . . . h h d d 1 � �� � � K h ( X i 0 , x i ) , . . . , K h ( X i T − ∆ , x i ) � K ( x ) = diag nT i =1 i =1 11

Restricting the minimization to additive functions, this is equivalent to minimizing � µ 1 ( x )) T ˆ µ 1 ,LL µ 1 ,LL µ 1 ,LL µ 1 ( x ) � ˆ µ 1 ( x )) d x � ˆ ( x ) − ¯ S = ( ˆ ( x ) − ¯ S ( x )( ˆ ( x ) − ¯ h h h with norming � � f h ( x j ) d x j + h ( x j ) d x j = 0 . µ 1 ( x j ) ˆ j ( x j ) ˆ µ 1 f 1 ¯ ¯ d ( x d )) T is an element of M add µ 1 ( x ) = (¯ µ 1 1 ( x 1 ) , . . . , ¯ µ 1 Here ¯ µ ( x ) , ¯ 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 √ 0 . 75 − x 1 + 0 . 25 x 2 + x 3     x 1 + x 2 0 0 √ 0 . 75 + 0 . 5 x 1 − 2 x 2 + 0 . 25 x 3     µ ( x ) = Σ( x ) = x 2 0 0     √     1 . 5 + 0 . 25 x 1 + x 2 − 3 x 3 x 3 0 0 and a nonlinear model 0 . 75 + 0 . 5(sin(2 x 1 ) − x 1 ) + 0 . 25 x 2 + x 3    ( x 1 ) 2 + ( x 2 ) 2  � 0 0 √ 0 . 75 + 0 . 5 x 1 − 2 x 2 + 0 . 25 x 3     µ ( x ) = Σ( x ) = x 2  . 0 0     √    1 . 5 + 0 . 25 x 1 + x 2 − 3 x 3 x 3 0 0 In both cases µ 1 ( x ) is estimated based on a sample of n = 30 , T = 35 14

SBF estimator of µ 1 ( x 1 ) 2.5 1.0 −0.5 1.0 1.5 2.0 SBF estimator of µ 1 ( x 2 ) 0.2 −0.1 −0.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 SBF estimator of µ 1 ( x 3 ) 0.5 −0.5 0.4 0.6 0.8 1.0 1.2 1.4 15

SBF estimator of µ 1 ( x 1 ) 1.5 0.0 1.0 1.5 2.0 Density of X 1 0.6 0.4 0.2 1.0 1.5 2.0 16

SBF estimator of µ 1 ( x 1 ) 2 −2 −6 3 4 5 6 7 SBF estimator of µ 1 ( x 2 ) 0.5 −1.0 1.0 1.5 2.0 2.5 3.0 SBF estimator of µ 1 ( x 3 ) 2 1 0 −2 1.0 1.5 2.0 17

SBF estimator of µ 1 ( x 1 ) 2 −2 −6 3 4 5 6 7 Density of X 1 0.20 0.10 3 4 5 6 7 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). 19