Econometrics Toolbox™ supports nonlinear models that describe the dynamic behavior of economic time series variables in the presence of structural breaks or regime changes. The models have two main components: a discrete state-space variable St representing the regime series, and a collection of dynamic regression (ARX or VARX) submodels that describe the dynamic behavior of the univariate or multivariate time series Yt within each regime. St is a fixed set of values or a random variable.
The threshold-switching dynamic regression model treats St as a fixed variable. The level of an observed threshold variable determines the regime at time t (the value of St), but threshold values that determine when regimes shift are unknown parameters. The threshold variable can be exogenous or endogenous, and transitions between states can be abrupt or smooth.
The Markov-switching dynamic regression model treats St as a latent, random discrete-time Markov chain, which is a state-space Markov process represented by a directed graph and described by a right-stochastic transition matrix P. The distribution of states at time t + 1 is the distribution of states at time t multiplied by P. The structure of P determines the evolutionary trajectory of the chain, including asymptotics.