This thesis consists of five essays on the application of wavelet methodology to different tests in time series analysis. Essay I proposes a nonlinear Dickey-Fuller F test for unit roots against the first order Logistic Smooth Transition Autoregressive LSTAR (1) model. The asymptotic distribution of the test statistic is analytically derived while the size and power properties for small samples of the tests are investigated using Monte Carlo experiments. The results show that there is a serious size distortion for the test when Autoregressive Conditional Heterskedasticity (GARCH) errors appear in the Data Generating Process (DGP). To solve this problem, we use the wavelet technique to filter out the GARCH distortion and to improve the size property of the test under GARCH error. We also discuss the asymptotic distributions of the test statistics in GARCH and wavelet environments. Essay II uses the wavelet technique to improve the over-rejection problem of the traditional linear Dickey-Fuller tests for unit root when the data is associated with volatility such as a GARCH (1, 1) effect. We prove that the asymptotic distribution of the test in the wavelet environment is still the same as the traditional Dickey-Fuller type of test. The finite sample property is improved when the data suffers from GARCH error. An empirical example is illustrated with data on the net immigration to Sweden during the period 1950 to 2000. Essay III applies the wavelet ratio statistic to the Im-Pesaran-Shin (IPS) type of test and compares it with the IPS test by using Dickey-Fuller t statistic. Simulation results show again in power by employing the wavelet ratio test instead of the Dickey-Fuller t statistic in the panel data case. As the IPS test is sensitive to the cross sectional dependence, we further compare the robustness of both test statistics to the cross sectional. Finally we apply a residual based waves trapping methodology to reduce the over biased size problem brought up by the cross correlation for both test statistics. Essay IV uses simulated data to investigate the power of different causality tests in a two dimensional vector autoregressive (VAR) model. The data are presented in a non-linearenvironment that is modelled using a logistic smooth transition autoregressive (LSTAR)function. We use both linear and non-linear causality tests to investigate the uni-direction causality relationship and compare the power of these tests. When implementing the nonlinear test, we use separately the original data, the linear VAR filtered residuals, and the wavelet decomposed series based on wavelet multi resolution analysis (MRA). The simulation results show that the non-parametric test based on the wavelet decomposition series (which is a model free approach) has the highest power for exploring causality relationships in nonlinear models. Essay V first presents a power controlled turning points detecting method based on the theory of likelihood ratio test in statistical surveillance. Next we show how the outlier will influence the performance of this methodology. Due to the sensitivity of the surveillance system to the outliers, we finally present a wavelet multi resolution (MRA) based outlier elimination approach, which can be combined with the on-line turning point detecting process and will also alleviate the false alarm problem introduced by the outliers.