In this paper we propose a Nonlinear Dickey-Fuller F test for unit root against first order Logistic Smooth Transition Autoregressive LISTAR (1) model with time as the transition variable. The Nonlinear Dickey-Fuller F test statistics is established under the null hypothesis of random walk without drift and the alternative model is a nonlinear LSTAR (1) model. The asymtotic distribution of the test is analytically derived while small sample distributiuons are investigated by Monte Carlo experiment. We next investigate the size and power properties of the test statistics. We use empirical example to compare our test with the traditional Dickey-Fuller F test.

Moreover, we find a serious size distortion for the Nonlinear Dickey-Fuller F test when GARCH errors appear in the Data Generating Process (DGP), which lead to an over-rejection of the unit root null hypothesis. To solve this problem, we use the wavelet method to count off the GARCH distortion and to improve the size property of the test under GARCH error. We also discuss the the asymptonic distributions of the test statistics in GARCH and wavelet environments. An empirical example is used to show the wavelet environments. An empirical example is used to show the wavelet improvement of the over-rejection problem.