Shift permutation invariance in linear random factor models

The objective of this paper is to consider shift invariance, a specific type of exchangeability, of random factors in linearmodels. The randomfactors are described via their covariance matrices and it is shown that shift invariance implies circular Toeplitz covariancematrices and marginally shift invariance implies block circular Toeplitz covariance matrices. In order to get interpretable linear models reparametrization is performed. It is shown that by putting restrictions on the spectrum of the shift invariant covariance matrices natural reparametrization conditions for the corresponding factors are obtained which then, among others, can be used to obtain unique parametrizations under shift invariance.