Testing for the number of states in hidden Markov models

Scale mixtures of normal distributions are frequently used to model the heavy tails of asset returns. A simple specification is a three component scale mixture, where the states correspond to high, intermediate and low volatility phases of the market. Tests for the number of states in hidden Markov models are proposed and used to assess whether in view of recent financial turbulences, three volatility states are still sufficient. The tests extend tests for independent finite mixtures by using a quasi-likelihood which neglects the dependence structure of the regime. The main theoretical insight is the surprising fact that the asymptotic distribution of the proposed tests for HMMs is the same as for independent mixtures with corresponding weights. As application the number of volatility states for logarithmic returns of the S&P 500 index in two HMMs is determined, one with state-dependent normal distributions and switching mean and scale, and the other with state-dependent skew-normal distributions with switching scale and structural mean and skewness parameters. It turns out that in both models, four states are indeed required, and a maximum-a-posteriori analysis shows that the highest volatility state mainly corresponds to the recent financial crisis.