Modeling stationary data by classes of generalized Ornstein-Uhlenbeck processes. Alejandra Caba˜ na joint work with Argimiro Arratia and Enrique M. Caba˜ na Universitat Aut` onoma de Barcelona, Universitat Polit` ecnica de Catalunya and Universidad de la Rep´ ublica 7` emes Journ´ ees Statistiques du Sud Barcelona, 9-11 June, 2014
Introduction Introduction The link between discrete ARMA processes and stationary pro- cesses with continuous time has been of interest for many years and has been studied, among others, by Doob, J.L. (1944) The elementary Gaussian Processes, Ann. Math. Statist. 25 Durbin, J. (1961) Efficient fitting of linear models for continuous stationary time series from discrete data Bergstrom (1984) Handbook of Econometrics, (1990) Continuous Time Econometric Modelling, Oxford U Press and there is a recent upsurge of interest in non Gaussian processes mainly due to the fact that jumps play an important role in real- istic modeling in finance and other fields of application.
Introduction Example Series A (Box, Jenkins & Reinsel) consists of 197 lectures of con- centration in a certain chemical process, taken every 2 hours. It can be treated as a series of equally spaced observations of a process developing in continuous time, with possible long term dependence. The original series empirical covariances ● 0.15 18.0 0.10 17.5 ● ● ● ● ● ● 0.05 ● 17.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.00 ● ● ● ● ● ● ● 16.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 50 100 150 200 0 50 100 150 lag
Introduction The classical model would be an ARMA( p, q ), an ARMA(1,1) or subsets of AR(7) have been proposed. Covariances of the fitted ARMA(1,1) process Covariances of the fitted ARMA(7,0) process ● ● 0.15 0.15 0.10 0.10 ● ● ● ● game game ● ● ● ● ● ● 0.05 ● 0.05 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.00 ● ● ● ● ● ● ● 0.00 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 50 100 150 0 50 100 150 Index Index Empirical covariances and covariances of the adjusted (ML) ARMA(11) and AR(7) models for Series A.
Introduction From AR(1) to Ornstein Uhlenbeck processes The simplest ARMA model is an AR(1) X t = φX t − 1 + σǫ t that can be written as (1 − φB ) X t = σǫ t where ǫ t , t ∈ Z is a white noise, B is the back-shift operator that maps X t onto BX t = X t − 1 . If | φ | < 1, the process X t is stationary. Equivalently, X t can be written as X t = σ MA (1 /ρ ) ǫ t , where MA (1 /ρ ) is the moving average that maps ǫ t onto MA (1 /ρ ) ǫ t , = � ∞ 1 ρ j ǫ t − j . j =0 The covariances of X t are σ 2 γ h = E X t X t + h = γ 0 where γ 0 = 1 − 1 ρ h ρ 2
Introduction Two ways of defining a continuous time analogue x t , t ∈ R of AR(1) processes are • by establishing that γ ( h ) = E x ( t ) x ( t + h ) be γ 0 e − κ | h | • by replacing the measure W concentrated on the integers defined by W ( A ) = � t ∈ A ǫ t , that allows writing � t + 1 X t = ρ t − s dW ( s ), by a measure Λ on R, with stationary, −∞ i.i.d. increments and defining (with ρ = e κ ) � t e − κ ( t − s ) d Λ( s ) x ( t ) = ℜ ( κ ) > 0 −∞ Both ways lead to the same result: Ornstein-Uhlenbeck type pro- cesses.
Levy-driven continuous time ARMA L´ evy driven continuous time ARMA processes A L´ evy process Λ( t ) is a c` adl` ag function, with independent and stationary increments, that vanishes in t = 0. As a consequence, Λ( t ) is, for each t , a random variable with an infinitely divisible law. The characteristic function of Λ( t ) is E e iu Λ( t ) = ( E e iu Λ(1) ) t , and is usually written as E e iu Λ(1) = e ψ Λ ( iu ) . The function ψ Λ is called characteristic exponent and has the form ψ Λ ( iu ) = aiu − σ 2 � � 2 u 2 + (e iux − 1 − iux ) dv ( x )+ (e iux − 1) dv ( x ) | x | < 1 | x |≥ 1 | x | < 1 x 2 dv ( x ) < ∞ , � � where v ( { 0 } ) = 0, | x |≥ 1 dv ( x ) < ∞ .
Levy-driven continuous time ARMA Wiener process w sat- 1.0 isfies these properties, and, moreover, is the 0.5 unique continuous L´ evy process. 0.0 0.0 0.2 0.4 0.6 0.8 1.0 2.0 The compound Poisson 1.5 process with rate λ and 1.0 i.i.d. jumps Y j with E Y j = 0 , Var ( Y j ) = η < 0.5 ∞ is also a L´ evy process. 0.0 0.0 0.2 0.4 0.6 0.8 1.0
Levy-driven continuous time ARMA L´ evy-driven Ornstein-Uhlenbeck processes of higher order In the same manner that higher order autoregressive processes are used for modeling stationary sequences, higher order Ornstein- Uhlenbeck processes can be used for modeling stationary contin- uous time processes. Econometric or physical models apply frequently linear combina- tions (superpositions) of OU processes driven by either uncorre- lated or correlated noise. � t p � e − κ j ( t − s ) d Λ j ( s ) a j −∞ j =1 Eliazar, I, and Klafter, J. (2009), From Ornstein-Uhlenbeck dynamics to long-memory processes and fractional Brownian motion,PHYSICAL REVIEW E, 79
Levy-driven continuous time ARMA or models that replace the finite linear combination by a continu- ous version � t � e − κ ( t − s ) d Λ( s, κ ) s = −∞ ℜ ( κ ) > 0 Barndorff- Nielsen, O. and Shephard, N. (2001), Non-Gaussian Ornstein-Uhlenbeck- based models and some of their uses in financial economics JRSS, 63. Bergstrom, A. R. (1984), Continuous time stochastic models and issues of aggregation over time, in Handbook of Econometrics, Volume ll, Edited by Z. Griliches and M. D. lntriligator, Elsevier Science Publishers BV. Brockwell, P.J. (2004), Representations of continuous time ARMA models, Jr. Appl. Probab, 41. Chambers, M. J. and Thornton, M.A. (2012), Discrete time representation of contin- uous time ARMA processes
The OU p Iterated Levy-driven Ornstein-Uhlenbeck processes We propose to use a parsimonious model, with few parameters, that is able to adjust slowly decaying covariances, obtained by a procedure that resembles the one that allows to build an AR( p ) from an AR(1). The AR( p ) process p � X t = φ j X t − j + σǫ t or φ ( B ) X t = σǫ t , j =1 p p φ j z j = � � (1 − z/ρ j ) has roots ρ j = e κ j where φ ( z ) = 1 − j =1 j =1 is obtained by applying the composition of the moving averages MA (1 /ρ j ) to the noise, that is: p � X t = σ MA (1 /ρ j ) ǫ t j =1
The OU p Let us denote MA κ = MA (e − κ ). A continuous version of the operator MA κ that maps ǫ t onto � e − κ ( t − l ) ǫ l . MA κ ǫ t = l ≤ t, integer is OU κ that maps y ( t ) onto � t e − κ ( t − s ) dy ( s ) OU κ y ( t ) = −∞ and this suggests the use of the model OU( p ): p � x κ ,σ ( t ) = σ OU κ j Λ( t ) j =1 with parameters κ = ( κ 1 , . . . , κ p ) , σ .
The OU p OU( p ) as a superposition of OU(1) The Ornstein-Uhlenbeck process with parameters κ = ( κ 1 , . . . , κ p ) and σ, p � x κ ,σ = OU κ j ( σ Λ) j =1 can be written as a linear combination of p processes of order 1 when the components of κ are pairwise different: � t x κ , σ = � p −∞ e − κ j ( t − s ) d ( σ Λ( s )). j =1 K j ( κ ) ξ κ j , ξ κ j ( t ) = The coefficients are: 1 K j ( κ ) = � κ l � = κ j (1 − κ l /κ j )
Recommend
More recommend
