The Gaussian maximum likelihood estimate is investigated for time series models that have locally a stationary behaviour (e.g. for time varying autoregressive models). The asymptotic properties are studied in the case where the fitted model is either correct or misspecified. For example the behaviour of the maximum likelihood estimate is explained in the case where a stationary model is fitted to a nonstationary process. As a general model selection criterion the AIC is considered. It can for example automatically select between stationary models, nonstationary models and deterministic trends.