Comparing the CCA subspace method to pseudo maximum likelihood methods in the case of no exogenous inputs

This paper deals with the CCA subspace algorithm proposed in Larimore [Proceeding of 1983 American Control Conference (1983) pp. 445-451], which constitutes an alternative to the classical criteria optimization based approach to the identification of linear dynamic models for a stationary process. Subspace algorithms for the estimation of linear models have been advocated mainly due to their numerical properties. A large variety of different subspace algorithms is known to provide strongly consistent and asymptotically normal estimates of the system under mild assumptions on the noise and the underlying true system. This paper shows that for certain versions of CCA described in the paper the estimates are asymptotically equivalent to pseudo maximum-likelihood estimates in the sense that the difference in the estimators multiplied by the square root of the sample size converges to zero (in probability). Therefore these versions of CCA are asymptotically efficient for Gaussian innovations.