Identification and spectral estimation of noisy multivariate autoregressive processes

Large sample identification spectral estimation problems of a noisy multivariate autoregressive process are solved independent of the probability law governing the observed data. Several different representations of a noisy multivariate autoregressive process are studied and linked to the properties of the block Toeplitz and Hankel matrices derived from the auto-correlation function of the process. Under a simple condition, the parameter estimators for the auto-regressive coefficients and noise statistics derived by solving block Toeplitz and Hankel matrix equations are shown to be strongly consistent. Asymptotic distributions of the parameter estimators are derived and used to compute the confidence bounds of the spectral estimators. For order selection, the Akaike Information Criterion (AIC) is modified into a form independent of the probability law of the observed data.