Markov-Switching Model Selection Using Kullback–Leibler Divergence. Smith, A., Naik, P. A, & Tsai, C. Journal of Econometrics, 134(2):553–577, North-Holland, 2006.
Markov-Switching Model Selection Using Kullback–Leibler Divergence [pdf]Paper  abstract   bibtex   
In Markov-switching regression models, we use Kullback–Leibler (KL) divergence between the true and candidate models to select the number of states and variables simultaneously. Specifically, we derive a new information criterion, Markov switching criterion (MSC), which is an estimate of KL divergence. MSC imposes an appropriate penalty to mitigate the over-retention of states in the Markov chain, and it performs well in Monte Carlo studies with single and multiple states, small and large samples, and low and high noise. We illustrate the usefulness of MSC via applications to the U.S. business cycle and to media advertising.

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