Stability results for nonlinear error correction models

This paper studies analogs of Granger's representation theorem in the context of a general nonlinear vector autoregressive error correction model. The model allows for nonlinear autoregressive conditional heteroskedasticity and the conditional distribution involved can be a mixture distribution of a rather general type. Mixture models of this kind can be thought of as generalizations of threshold models and they have attracted attention in the recent time series and econometrics literature. The paper develops a useful transformation which shows how the nonlinear error correction model can be transformed to a nonlinear vector autoregressive model so that available results on the stationarity or nonstationarity of the latter can be used for the former. The most satisfactory results are obtained in a model in which a specific structural relation between the nonlinearity and equilibrium correction prevails. Without this structural relation only a lower bound for the number of long-run equilibrium relations can explicitly be determined because the exact number depends on properties of the first and second moments of a nonlinear stationary component of the process.