On modeling of tall linear systems with multirate outputs

Motivated by problems of modeling high dimensional time series, this paper considers time-invariant, discrete-time linear systems which have a larger number of outputs than inputs, with the inputs being independent stationary white noise sequences. Moreover, different outputs are measured at different rates (in economic modeling, it is common that some variables are measured monthly and others quarterly). In particular, the paper focuses on the case where the number of measurements is extremely large compared to the number of inputs. In the current paper, our ultimate goal is to identify the parameter matrices of such systems from outputs covariance data. To achieve this main goal and avoid excessively high dimensionality in the model, we use the notion of static factor, which roughly is a special subvector of the latent vector i.e. those parts of output vector remaining after removal of contaminating additive noise in the measurement. Since the model associated with the static factor is periodic in the output parameters, we use the well-known technique of blocking to obtain a blocked linear time-invariant system associated with this model. It is illustrated that this blocked system is generically zero-free. Then we use the spectral factorization technique to obtain the parameter matrices associated with the blocked system. These parameter matrices can be obtained by a finite number of rational calculations from the spectral matrix due to the generic zero-freeness of tall spectral matrices. Finally, we use the parameter matrices associated with the blocked system to obtain the parameter matrices associated with the static factor and ultimately those of the original underlying unblocked system.