Resolving the ambiguity of random‐effects models with singular precision matrix

Random walks, intrinsic autoregression, state‐space models, smoothing splines, and so on have been widely used in various areas of statistics. However, practitioners wanting to fit these models using existing packages for random‐effects models are often faced with the difficulty that their covariance matrices are not uniquely determined. Unfortunately, different specifications of the model lead to different covariance structures, giving different analyses. Even if we make a decision on specification it is not immediately obvious how to make inferences from these models. There have been various suggestions on how to overcome such difficulties. However, they differ, implying that there is as yet no agreed remedy. In this article we provide a unified view on these alternatives and show how the analysis can be made invariant with respect to the choice of covariance by inclusion of a suitable set of covariates. Several examples are used to illustrate the approach.