The main focus of this dissertation is to find ways to improve the power in cointegration tests. This dissertation consists of three essays. In the first essay, a modified testing procedure for the Engle and Granger (1987; EG) cointegration test is suggested. Specifically, we suggest augmenting the usual EG testing regression with the first difference of the integrated regressors. The limiting distribution of this modified EG test under the null hypothesis will depend on the nuisance parameter, which reflects the signal-to-noise ratio. This essay shows that the nuisance parameter issue can be resolved when we follow the asymptotic distribution of the modified EG test, and use the relevant new sets of critical values corresponding to the estimated value of the nuisance parameter. It is found that the size and power properties of the modified EG test are fairly good. The modified EG test gains improved power rather than losing power as the signal-to-noise ratio increases. In the second essay, we examine whether non-linear unit root tests is robust with non-normal errors, which provides a motivation for the third essay. Especially, the second essay demonstrates how popular nonlinear unit root tests perform in the presence of non-normal errors. Non-normal errors normally do not pose a problem in usual linear unit root tests since the least squares estimator will still be the most efficient under certain ideal conditions regardless of normal or non-normal errors. The asymptotic properties of the popular linear Dickey-Fuller tests, for example, will be unaffected by non-normal errors. As such, the literature has not paid much attention to this issue. Nevertheless, whether similar results will carry over to nonlinear unit root tests with non-normal errors is a question that merits examination. To our surprise, the extant literature on nonlinear unit root tests has not examined this important question. We find that, in general, nonlinear unit root tests will suffer a loss of power in the presence of non-normal errors. In this regard, this essay brings out the neglected point that the obvious analogies of linear processes do not necessarily hold for nonlinear models. The third essay suggests new cointegration tests that are more powerful in the presence of non-normal errors. We use a two-step procedure based on the "residual augmented least squares" (RALS) method to make use of nonlinear moment conditions driven by non-normal errors. By utilizing this neglected information, we can make the existing tests more powerful. The suggested testing procedure is easy to implement. The underlying idea is similar to adding stationary covariates to improve the power of the test, but the suggested procedure does not require any new covariates outside the system. Instead, we can exploit the information on the non-normal error distribution that is already available but ignored in the usual cointegration tests. Our simulation results show significant power gains over existing cointegration tests.