Smooth Transition Autoregressive (STAR) Model

Both are STAR models that differ only in the transition function. LSTAR uses a logistic function that gives an asymmetric S-shaped switch between two regimes, suited to asymmetric business cycles. ESTAR uses an exponential function that is symmetric around the location c, so the dynamics return to the same outer regime on both sides — well suited to mean-reverting deviations such as PPP gaps.

A threshold model switches abruptly between regimes once the transition variable crosses a threshold. A STAR model replaces that hard switch with a smooth transition function valued between 0 and 1, so the dynamics blend continuously across regimes instead of jumping.

Yes. Because STAR nests the linear autoregression as a special case, you should first confirm nonlinearity with an LM-type test (Luukkonen, Saikkonen & Teräsvirta, 1988). If the test does not reject linearity, a plain AR model is preferable.

The slope γ and location c enter the model nonlinearly and the parameter count is high, so the model is fitted by nonlinear least squares. Good starting values usually require a grid search, and convergence can still be difficult, which is why a long series (T greater than 100) is recommended.