Quasi-maximum likelihood estimation for multivariate CARMA processes Quasi-maximum likelihood estimation for multivariate CARMA processes Eckhard Schlemm Institute for Advanced Study, Technische Universität München 2 nd Northern Triangular Seminar Stockholm, 17 March 2010 1/26
Quasi-maximum likelihood estimation for multivariate CARMA processes Outline Preliminaries Motivation Multivariate CARMA processes Main results Probabilistic properties of the sampled process Identifiability and quasi-maximum likelihood estimation Implementation and application Canonical parametrizations Simulation study Example from Economics Summary and future work 2/26
Quasi-maximum likelihood estimation for multivariate CARMA processes Preliminaries Motivation Introduction Versatile class of auto-regressive moving-average processes X n − ϕ 1 X n − 1 − . . . − ϕ p X n − p = ε n + θ 1 ε n − 1 + . . . + θ q ε n − q Extensions to ◮ multivariate models (Vector ARMA) ◮ continuous-time models (CARMA) Advantages: ◮ Modelling of dependent time series ◮ High-frequency and/or irregularly spaced observations Problem : Estimation 3/26
Quasi-maximum likelihood estimation for multivariate CARMA processes Preliminaries Multivariate CARMA processes Multivariate CARMA processes R m -valued Lévy process L satisfying E || L (1) || 2 < ∞ . An R d -valued second-order MCARMA(p,q) process solves D ≡ d P ( D ) Y ( t ) = Q ( D ) D L ( t ) , dt. Auto-regressive polynomial P ( z ) ≔ I d z p + A 1 z p − 1 . . . + A p ∈ M d ( R [ z ]) Moving-average polynomial Q ( z ) ≔ B 0 z q + B 1 z q − 1 . . . + B q ∈ M d,m ( R [ z ]) 4/26
Quasi-maximum likelihood estimation for multivariate CARMA processes Preliminaries Multivariate CARMA processes Multivariate CARMA processes Stationary solution to continuous-time state space model state equation d X ( t ) = A X ( t ) dt + B d L ( t ) observation equation Y ( t ) = [ I d , 0 d , . . . , 0 d ] X ( t ) , 0 I d 0 . . . 0 . ... . 0 0 I d . . A = ... ... , . 0 . 0 0 . . . . . . I d − A p − A p − 1 − A 1 . . . . . . � p − j − 1 � � β T � T , � β T B = · · · β p − j = − I [0: q ] ( j ) A i β p − j − i + B q − j 1 p i =1 4/26
Quasi-maximum likelihood estimation for multivariate CARMA processes Preliminaries Multivariate CARMA processes State space models I General N -dimensional continuous-time state space model: state equation d X ( t ) = A X ( t ) dt + Bd L ( t ) observation equation Y ( t ) = C X ( t ) , A ∈ M N ( R ) , B ∈ M N,m ( R ) , C ∈ M d,N ( R ) Sufficient condition for stationarity of the state process X : Re λ ν < 0 , λ ν , ν = 1 , . . . , N, eigenvalues of A. 5/26
Quasi-maximum likelihood estimation for multivariate CARMA processes Preliminaries Multivariate CARMA processes State space models II X satisfies � t X ( t ) = e A ( t − s ) X ( s ) + s e A ( t − u ) Bd L ( u ) The output process Y satisfies � t −∞ Ce A ( t − u ) Bd L ( u ) . Y ( t ) = Its spectral density is given by f Y ( ω ) = 1 2 πC ( iω − A ) − 1 B Σ L B T ( − iω − A T ) − 1 C T . 6/26
Quasi-maximum likelihood estimation for multivariate CARMA processes Preliminaries Multivariate CARMA processes Equivalence of MCARMA und multivariate state space models Theorem The stationary solution Y of the multivariate state space model ( A, B, C, L ) is an L -driven MCARMA process with auto-regressive polynomial P and moving-average polynomial Q if and only if C ( zI N − A ) − 1 B = P ( z ) − 1 Q ( z ) , ∀ z ∈ C . 7/26
Quasi-maximum likelihood estimation for multivariate CARMA processes Preliminaries Multivariate CARMA processes A useful decomposition Theorem Let Y be the output process of the SSM ( A, B, C, L ) , ◮ A ∈ M N ( R ) ◮ λ i ∈ σ ( A ) , λ i � λ j ∃ vectors s 1 , . . . , s N ∈ C m \{ 0 m } and b 1 , . . . , b N ∈ C d \{ 0 d } such that � t N −∞ e λ ν ( t − u ) d � s ν , L ( u ) � . � Y ( t ) = Y ν ( t ) , Y ν ( t ) = b ν ν =1 8/26
Quasi-maximum likelihood estimation for multivariate CARMA processes Main results Probabilistic properties of the sampled process Probabilistic properties of the sampled process We observe the process Y at discrete, equally spaced times Y ( h ) n ∈ Z , ≔ Y ( nh ) , h > 0 . n Linear innovations ε ( h ) = Y ( h ) n − P n − 1 Y ( h ) ( ε ( h ) n , n ) n ∈ Z ∼ white noise n ↑ � � Y ( h ) orthogonal projection onto span : −∞ < ν < n ν We define the polynomial N � � � 1 − e − λ ν h z ∈ C [ z ] . ϕ ( z ) = ν =1 9/26
Quasi-maximum likelihood estimation for multivariate CARMA processes Main results Probabilistic properties of the sampled process VARMA structure of Y ( h ) Theorem There exists a stable monic polynomial Θ ∈ M d ( C [ z ]) of degree at most N − 1 such that = Y ( h ) ϕ ( B ) Y ( h ) = Θ( B ) ε ( h ) B j Y ( h ) n , n − j , n n holds. ⇒ Y ( h ) is a weak VARMA( N, N − 1) process. 10/26
Quasi-maximum likelihood estimation for multivariate CARMA processes Main results Probabilistic properties of the sampled process The innovations process First r eigenvalues real: λ ν ∈ R for 1 ≤ ν ≤ r ; λ ν = λ ν +1 ∈ C \ R for ν = r + 1 , r + 3 . . . , N − 1 . � h 0 e ( h − u ) λ ν d L ( u ) M ν = � T � M T 1 · · · M T r , re M T r +1 , im M T r +1 · · · re M T N − 1 , im M T M = N − 1 Theorem If M has a non-trivial absolutely continuous component with respect to λ mN the innovations process ε ( h ) is strongly mixing with exponentially decaying coefficients. 11/26
Quasi-maximum likelihood estimation for multivariate CARMA processes Main results Identifiability and quasi-maximum likelihood estimation Parameter identifiability in MCARMA models I Quasi-maximum likelihood approach + discrete observations ⇒ distinction between models based only on second-order properties of the sampled process Definition (Identifiability) A collection of continuous-time stochastic processes ( Y ϑ , ϑ ∈ Θ) is identifiable if for any ϑ 1 � ϑ 2 the two processes Y ϑ 1 and Y ϑ 2 have different spectral densities. It is h -identifiable , h > 0 , if for any ϑ 1 � ϑ 2 the two processes Y ( h ) ϑ 1 and Y ( h ) ϑ 2 have different spectral densities. 12/26
Quasi-maximum likelihood estimation for multivariate CARMA processes Main results Identifiability and quasi-maximum likelihood estimation Parameter identifiability in MCARMA models II ψ : R q ⊃ Θ ∋ ϑ �→ ( A ϑ , B ϑ , C ϑ , L ϑ ) , Θ compact. Assumption (Minimality) For all ϑ ∈ Θ the triple ( A ϑ , B ϑ , C ϑ ) is minimal in the sense C ( zI m − A ) − 1 B = C ϑ ( zI N − A ϑ ) − 1 B ϑ ⇒ m ≥ N. Assumption (Eigenvalues) For all ϑ ∈ Θ the spectrum of A ϑ is contained in the strip { z ∈ C : − π/h < Im z < π/h } . 13/26
Quasi-maximum likelihood estimation for multivariate CARMA processes Main results Identifiability and quasi-maximum likelihood estimation Parameter identifiability in MCARMA models III Theorem Parametrization ψ : Θ ⊃ ϑ �→ ( A ϑ , B ϑ , C ϑ , L ϑ ) ◮ identifiable ◮ "Minimality" ◮ "Eigenvalues" Then the corresponding collection of output processes { Y ϑ , ϑ ∈ Θ } is h -identifiable. 14/26
Quasi-maximum likelihood estimation for multivariate CARMA processes Main results Identifiability and quasi-maximum likelihood estimation Gaussian maximum likelihood estimation I The QML estimator of ϑ based on y L = ( y 1 , . . . , y L ) is L ≔ argmax ϑ ∈ Θ L ϑ ˆ � y L � true parameter ϑ 0 ∈ int Θ , ϑ , where the Gaussian likelihood function is � L � − 1 / 2 L � � − 1 � y L � � � e T ϑ ,n V − 1 ∼ L ϑ det V ϑ ,n exp ϑ ,n e ϑ ,n 2 n =1 n =1 and e ϑ ,n = y n − P n − 1 Y ( h ) � � , � �� ϑ ,n Y ( h ) ϑ ,ν = y ν :1 ≤ ν<n � � � Y ( h ) � e ϑ ,n e T V ϑ ,n = E ϑ ,ν = y ν : 1 ≤ ν < n . � ϑ ,n 15/26
Quasi-maximum likelihood estimation for multivariate CARMA processes Main results Identifiability and quasi-maximum likelihood estimation Gaussian maximum likelihood estimation II The sampled process Y ( h ) satisfies the discrete-time state space model = e Ah X ( h ) X ( h ) n − 1 + Z n , n Y ( h ) = C X ( h ) n , n where the i.i.d. sequence ( Z n ) n ∈ Z is given by � nh ( n − 1) h e A ( nh − u ) Bd L ( u ) . Z n = ◮ Kalman Filter ◮ Numerical maximization 16/26
Quasi-maximum likelihood estimation for multivariate CARMA processes Main results Identifiability and quasi-maximum likelihood estimation QML estimation - Consistency Assumption: h-identifiable parametrization Theorem (Strong consistency) L is For every sampling interval h > 0 , the QML estimator ˆ ϑ strongly consistent, i.e. L → ϑ 0 ˆ a.s. as L → ∞ . ϑ 17/26
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