Normalizing Empirically Underidentified Linear State-Space Models

Normalizing latent-variable models in empirical work has sometimes more influence on statistical inference than commonly appreciated. In this paper, I show how relating non-identification to an invariance property of the likelihood function under certain groups of parameter transformations helps understanding the influence of normalization on inference and can guide the choice of identifying restrictions. In particular, I show that multimodal parameter estimator sampling and posterior distributions can result from poor normalization of countable groups of parameter transformations when a model is empirically underidentified. I also describe how group invariance affects parameter prior specification and how it can be exploited for improving the efficiency of optimization and posterior sampling algorithms.