In this paper we consider a class of nonlinear autoregressive models in which a specific type of dependence structure between the error term and the lagged values of the state variable is assumed. We show that there exists an equivalent representation given by a p-th order state-dependent autoregressive (SDAR(p)) model where the error term is independent of the last p lagged values of the state variable (y_{t-1},…,y_{t-p}) and the autoregressive coefficients are specific functions of them. We discuss a quasi-maximum likelihood estimator of the model parameters and we prove its consistency and asymptotic normality. To test the forecasting ability of the SDAR(p) model, we propose an empirical application to the quarterly Japan GDP growth rate which is a time series characterized by a level-increment dependence. A comparative analyses is conducted taking into consideration some alternative and competitive models for nonlinear time series such as SETAR and AR-GARCH models.