We consider a time series switching between different dynamics or phases, e.g. a Generalized Mixture of first order Nonlinear and Nonparametric AR-ARCH models with two dynamics driven by a hidden Markov process.

We first introduce some conditions implying the asymptotic stability of the process and define a version of the likelihood function that takes into account the hidden process. Further, based on the likelihood function we investigate the behavior of feed-forward neural networks for estimating
the autoregressive and volatility functions and for identifying the change-points between different phases.

Since the hidden process is not observable we construct a version of the Expectation Maximization (EM) algorithm that accounts for solving the problem numerically.

We illustrate our results with some applications. For example, we construct a trading strategy that we apply to real data and compare the performance with that of a classical Buy and Hold Strategy.