Practical Issues With State Space Models with Mixed Stationary And Non-Stationary Dynamics

State-space models, and the state-space representation of data, are an important tool for econometric modeling and computation. However, when applied to observed (rather than detrended) data, many such models have a mixture of stationary and non-stationary roots. While Koopman (1997) and Durbin and Koopman (2002) provide "Exact" calculations for models with non-stationary roots, these have not yet been implemented in most software. This paper provides a unified framework for computing the finite and "infinite" components of the initial state variance matrix which is both flexible enough to handle mixed roots and also faster (even for purely stationary models) than standard methods. In addition, it examines some special problems that arise when the number of unit roots is unknown a priori.