The analysis of time series has long been the subject of interest in different fields. For decades time series were analysed with linear models, which have many advantages. Nevertheless, an issue which has been raised is whether there exist other models that can explain and forecast real data better than linear ones. In this thesis, new nonlinear time series models are suggested, which consist of a nonlinear conditional mean model, such as an ExpAR or an Extended ExpAR, and a nonlinear conditional variance model, such as an ARCH or a GARCH. Since new models are introduced, simulated series of the new models are presented, as it is important in order to see what characteristics real data which could be explained by them should have. In addition, the models are applied to various stationary and nonstationary economic and financial time series and are compared to the classic AR-ARCH and AR-GARCH models, in terms of fitting and forecasting. It is shown that, although it is difficult to beat the AR-ARCH and AR-GARCH models, the ExpAR and Extended ExpAR models and their special cases, combined with conditional heteroscedastic errors, can be useful tools in fitting, describing and forecasting nonlinear behaviour in financial and economic time series, and can provide some improvement in terms of both fitting and forecasting compared to the AR-ARCH and AR-GARCH models.

In this study, we present two new frequency domain tests for testing the Gaussianity and linearity of a sixth-order stationary univariate time series. Both are two-stage tests. The first stage is a test for the Gaussianity of the series. Under Gaussianity, the estimated normalized bispectrum has an asymptotic chi-square distribution with two degrees of freedom. If Gaussianity is rejected, the test proceeds to the second stage, which tests for linearity. Under linearity, with non-Gaussian errors, the estimated normalized bispectrum has an asymptotic non-central chi-square distribution with two degrees of freedom and constant noncentrality parameter. If the process is nonlinear, the noncentrality parameter is nonconstant. At each stage, empirical distribution function (EDF) goodness-ofit (GOF) techniques are applied to the estimated normalized bispectrum by comparing the empirical CDF with the appropriate null asymptotic distribution. The two specific methods investigated are the Anderson-Darling and Cramer-von Mises tests. Under Gaussianity, the distribution is completely specified, and application is straight forward. However, if Gaussianity is rejected, the proposed application of the EDF tests involves a transformation to normality. The performance of the tests and a comparison of the EDF tests to existing time and frequency domain tests are investigated under a variety of circumstances through simulation. For illustration, the tests are applied to a number of data sets popular in the time series literature.