Hidden Markov Models in Time Series, with Applications in Economics

Markov models introduce persistence in the mixture distribution. In time series analysis, the mixture components relate to different persistent states characterizing the state-specific time series process. Model specification is discussed in a general form. Emphasis is put on the functional form and the parametrization of timeinvariant and time-varying specifications of the state transition distribution. The concept of mean-square stability is introduced to discuss the condition under which Markov switching processes have finite first and second moments in the indefinite future. Not surprisingly, a time series process may be mean-square stable even if it switches between bounded and unbounded state-specific processes. Surprisingly, switching between stable state-specific processes is neither necessary nor sufficient to obtain a mean-square stable time series process. Model estimation proceeds by data augmentation. We derive the basic forward-filtering backward-smoothing/sampling algorithm to infer on the latent state indicator in maximum likelihood and Bayesian estimation procedures. Emphasis is again laid on the state transition distribution. We discuss the specification of state-invariant prior parameter distributions and posterior parameter inference under either a logit or probit functional form of the state transition distribution. With simulated data, we show that the estimation of parameters under a probit functional form is more efficient. However, a probit functional form renders estimation extremely slow if more than two states drive the time series process. Finally, various applications illustrate how to obtain informative switching in Markov switching models with time-invariant and time-varying transition distributions.