Selection of the Number of States by Birth-Death Processes

In this article we use spatial birth-death processes to estimate the number of states k of a switching model. Following Preston (1976) and Stephens (1998) matching the detailed balance condition for the underlying birth-death process results in an unique invariant probability measure with the corresponding stationary distribution of the number of states. This concept could be easily integrated to Bayesian sampling to derive the marginal posterior distribution of number of states within the sampling procedure. We apply this technique to simulated AR(1) data and to quarterly Austrian data on unemployment and real gross domestic product.