The main purpose of this dissertation is to compare the in-sample estimating and out-of-sample forecasting performance of a set of non-linear time series models, i.e., threshold autoregressive models, momentum threshold autoregressive models, exponential autoregressive models, generalized autoregressive models and bilinear models. First, a Monte Carlo simulation is used to study overfitting and forecasting. For AR processes, if the AIC and SBC criteria are used to select models, the possibility of overfitting is very high since the non-linear models and other linear models are very likely to have lower AIC and SBC. However, the MSPE for one-step ahead out-of-sample forecast can be used to identify the true AR processes. For TAR processes, the AIC can be used to identify the TAR-C models. The SBC and MSPE can identify the TAR process only if the difference of the persistence between the two regimes is large enough. Underfitting and misspecification are very likely to happen for a TAR process with small difference of the persistence between the two regimes. However, if we don't know the true AR or TAR process, the MSPE can't select the AR or TAR models in most cases. Thus, none of the AIC, SBC and MSPE can select the AR model for a given AR process with unknown order. For the TAR process, the AIC can consistently identify the TAR-C process and the SBC can identify the TAR-C process only if the difference of the persistency is large enough. Then, a set of linear and non-linear time series models are applied to the term structure of interest rates and the spread of wholesale and retail pork prices in U.S. It is shown that there are non-linear time series models can do better than the conventional ARMA models for both in-sample estimation and out-of-sample forecast. Also, it is very unlikely that the dominance of the non-linear time series models results from overfitting for both the term structure of interest rates and the spread of wholesale and retail pork prices in U.S. Thus, non-linear time series models are very useful for estimating and forecasting the non-linear time series.