4SID linear regression

State-space subspace system identification (4SID) has been suggested as an alternative to more traditional prediction error system identification, such as ARX least squares estimation. The aim of this note is to analyse the connections between these two different approaches to system identification. The conclusion is that 4SID can be viewed as a linear regression multistep ahead prediction error method, with certain rank constraints. This allows us to analyse 4SID methods within the standard framework of system identification and linear regression estimation. For example, it is shown that ARX models have nice properties in terms of 4SID identification. From a linear regression model, estimates of the extended observability matrix are found. Results from an asymptotic analysis are presented, i.e. explicit formulas for the asymptotic variances of the pole estimation error are given. From these expressions, some difficulties in choosing user specified parameters are pointed out in an example