Using Subspace Algorithms for the Estimation of Linear State Space Models for Over-Differenced Processes

Subspace algorithms like canonical variate analysis (CVA) are regression-based methods for the estimation of linear dynamic state space models. They have been shown to deliver accurate (consistent and asymptotically equivalent to quasi-maximum likelihood estimation using the Gaussian likelihood) estimators for stably invertible stationary autoregressive moving average (ARMA) processes. These results use the assumption that there are no zeros of the spectral density on the unit circle corresponding to the state space system. In this technical study, we consider vector processes made stationary by applying differencing to all variables, ignoring potential co-integrating relations. This leads to spectral zeros violating the above mentioned assumptions. We show consistency for the CVA estimators, closing a gap in the literature. However, a simulation exercise shows that over-differencing (while leading to consistent estimation of the transfer function) also complicates inference for CVA estimators, not just maximum likelihood-based estimators. This is also demonstrated in a real-world data example. The result also applies to seasonal differencing. The present paper hence suggests working with original data, not working in differences.