Unit Root Tests Voor Ar(1) Processen (engelse Titel: Unit Root Testing for Ar(1) Processes) Bsc Verslag Technische Wiskunde " Unit Root Tests Voor Ar(1) Processen " (engelse Titel: " Unit Root Testing for Ar(1) Processes " )

The purpose of this study is to investigate the asymptotics of a first order auto regressive unit root process, AR(1). The goal is to determine which tests could be used to test for the presence of a unit root in a first order auto regressive process. A unit root is present when the root of the characteristic equation of this process equals unity. In order to test for the presence of a unit root, we developed an understanding of the characteristics of the AR(1) process, such that the difference between a trend stationary process and a unit root process is clear. The first test that will be examined is the Dickey-Fuller test. The estimator of this test is based on Ordinary Least Square Regression and a t-test statistic, which is why we have computed an ordinary least square estimator and the test statistic to test for the presence of unit root in the first order auto regressive process. Furthermore we examined the consistency of this estimator and its asymptotic properties. The limiting distribution of the test statistic is known as the Dickey-Fuller distribution. With a Monte Carlo approach, we implemented the Dickey-Fuller test statistic in Matlab and computed the (asymptotic) power of this test. Under the assumption of Gaussian innovations (or shocks) the limiting distribution of the unit root process is the same as without the normality assumption been made. When there is a reason to assume Gaussianity of the innovations, the Likelihood Ratio test can be used to test for a unit root. The asymptotic power envelope is obtained with help of the Likelihood Ratio test, since the Neyman-Pearson lemma states that the Likelihood Ratio test is the point optimal test for simple hypotheses. By calculating the likelihood functions the test statistic was obtained, such that an explicit formula for the power envelope was found. Since each fixed alternative results in a different critical value and thus in a different unit root test, there is no uniform most powerful test available. Instead we are interested in asymptotically point optimal tests and we will analyze which of these point optimal tests is the overall best performing test. By comparing the asymptotic powercurve to the asymptotic power envelope for each fixed alternative we could draw a conclusion on which fixed alternative results in the overall best performing test. On the basis of the results of this research, it can be concluded that there does not exist a uniform most powerful test, nonetheless we can define an overall best performing test.