Nested Reduced-Rank Autoregressive Models for Multiple Time Series

The nested reduced-rank autoregressive (AR) model is considered in order to simplify and provide a more detailed description of the structure of the multivariate time series and to reduce the number of parameters in the time series modeling. The multivariate AR model is Yt = Σ p j=1 Φ j Y t-j + εt , where Yt is m × 1, and the structure of the model considered is such that the rj = rank(Φ j ) are nonincreasing as the lag j increases, so the Φ j have the factorization Φ j = AjBj and range(Aj ) ⊃ range(A j+1). Specification of the coefficient rank structures for such models through the use of canonical correlation analysis between Yk,t = [Y′t, …, Y′t-k ]′ and Y k,t-1 is discussed. A canonical variable transformation that produces simpler structure in the model and explicitly illustrates how different components of the vector series depend on past lagged values to differing degrees is also examined. A Gaussian parameter-estimation procedure is described and asymptotic properties of the Gaussian estimator are studied for a more general class of multivariate AR models in which the coefficient matrices Φ j are assumed to be restricted to be functions of an unknown parameter vector β. A numerical example involving grain price data in the U.S. is considered to illustrate the methods, and this example clearly exhibits the simplification in structure of the nested reduced-rank AR model.