Multivariate Autoregressive and Ornstein-uhlenbeck Processes: Estimates for Order, Parameters, Spectral Information, and Conndence Regions

Fast methods are presented for identifying a multivariate autoregressive model that is appropriate to represent large, potentially high-dimensional time series data as they occur, e.g., in geophysical applications. The algorithms are based on the concept of least-squares estimation, which is known to yield consistent and asymptotically unbiased coeecient matrix estimates that also perform well on small samples. For order selection, a new and useful modiication of Schwarz' Bayesian Criterion is introduced. As the interpretation of a model is often facilitated by an analysis of its eigenmodes, the spectral decomposition of autoregressive processes of arbitrary order is considered. The discussion includes the computation of conndence intervals for the estimated eigenmodes, eigenvalues, and other spectral information. Numerical experiments demonstrate the eeciency of the proposed estimation techniques. Further, it is shown that a discrete sample of an Ornstein-Uhlenbeck process forms a rst order autoregressive (AR(1)) process. This fact reduces both the simulation and estimation of Ornstein-Uhlenbeck processes to the simulation and estimation of a corresponding AR(1) process.