Can we distinguish between common nonlinear time series models and long memory?

We show that specific nonlinear time series models such as SETAR, LSTAR, ESTAR and Markov switching which are common in econometric practice can hardly be distinguished from long memory by standard methods such as the GPH estimator for the memory parameter or linearity tests either general or against a specific nonlinear model. We show by Monte Carlo that under certain conditions, the nonlinear data generating process can have misleading either stationary or non-stationary long memory properties.