Abstract
In this paper, we investigate quasi-maximum likelihood (QML) estimation for the parameters of a cointegrated solution of a continuous-time linear state space model observed at discrete time points. The class of cointegrated solutions of continuous-time linear state space models is equivalent to the class of cointegrated continuous-time ARMA (MCARMA) processes. As a start, some pseudo-innovations are constructed to be able to define a QML-function. Moreover, the parameter vector is divided appropriately in long-run and short-run parameters using a representation for cointegrated solutions of continuous-time linear state space models as a sum of a Levy process plus a stationary solution of a linear state space model. Then, we establish the consistency of our estimator in three steps. First, we show the consistency for the QML estimator of the long-run parameters. In the next step, we calculate its consistency rate. Finally, we use these results to prove the consistency for the QML estimator of the short-run parameters. After all, we derive the limiting distributions of the estimators. The long-run parameters are asymptotically mixed normally distributed, whereas the short-run parameters are asymptotically normally distributed. The performance of the QML estimator is demonstrated by a simulation study.
Highlights
The state vector process X = (X(t))t≥0 is an RN -valued process and the output process Y = (Y (t))t≥0 is an Rd-valued process
The advantages of continuous-time modelling over discrete-time modelling in economics and finance are described in detail, i.a., in the distinguished papers of Bergstrom [7], Phillips [43], Chambers, McCrorie and Thornton [15] and in signal processing, systems and control they are described in Sinha and Rao [54]
In the context of non-stationary multivariate continuous-time ARMA (MCARMA) processes most attention is paid to Gaussian MCAR(p) processes: An algorithm to estimate the structural parameters in a Gaussian MCAR(p) model by maximum-likelihood started already by Harvey and Stock [26, 27, 28] and were further explored in the well-known paper of Bergstrom [8]
Summary
For two matrices A ∈ Rd×s and B ∈ Rr×n, we denote by A ⊗ B the Kronecker product which is an element of Rdr×sn, by vec(A) the operator which converts the matrix A into a column vector and by vech(A) the operator which converts a symmetric matrix A into a column vector by vectorizing only the lower triangular part of A. We write ∂i for the partial derivative operator with respect to the ith coordinate and ∂i,j for the second partial derivative operator with respect to the ith and jth coordinate. For a matrix function f (θ) in Rd×m with θ ∈ Rs the gradient with respect to the parameter vector θ is denoted by. Let ξ (ξk )k∈N and η (ηk )k∈N be d-dimensional stochastic processes Γξ,η(l) = Cov(ξ1, η1+l) and Γξ(l) =. Cov(ξ1, ξ1+l), l ∈ N0, are the covariance functions. In general C denotes a constant which may change from line to line
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