The shape of drift function in continuous time interest rate models has been investigated by many authors during the past decade. The main concerns have been whether the drift function is linear or nonlinear, but no convincing conclusions have been seen. In this dissertation, we investigate the reason for this problem and test several models of the drift function using a nonparametric test. Furthermore, we study some related problems, including the empirical properties of the nonparametric test. First, we propose regression models for the estimation of the drift function in some continuous time models. The limiting distribution of the parameter estimator in the proposed regression model is derived under certain conditions. Based on our analyses, we conclude that the effect of drift function for some U.S. Treasury Bill yields data is negligible. Therefore, neither linear nor nonlinear modeling has a significant effect. Second, parametric linear and nonlinear proposed regression models are applied and the correctness of those models is examined using the consistent nonparametric model specification test introduced by Li (1994) and Zheng (1996), henceforth the Jn test. The test results indicate that there is no strong statistical evidence against the assumed drift models. Furthermore, the constant drift model is not rejected either. Third, we compare the Jn and generalized likelihood ratio (GLR) tests through Monte Carlo simulation studies concerning whether the sizes of tests are stable over a range of bandwidth values, which is an important indicator to measure the usefulness of nonparametric tests. The GLR test was applied to testing the linear drift function in continuous time models by Fan and Zhang (2003). Our simulation study shows that the GLR test does not provide stable sizes over a grid of bandwidth values in testing the drift function of some continuous time models, whereas the Jn test usually does.