We study asymptotic inferences of the OLS estimator in the first order autoregressive model with an explosive root and a nonzero drift. Recent literatures focus on driftless model in dealing with explosive parameter in relation with financial bubbles detection. We consider an extension by allowing a non-zero drift, where the process behaves as a linear time trend during the non- bubble period, and it exhibits an exponential trend during the explosive era. Consistency of the least squares estimator and of the right-tailed coefficient-based Dickey-Fuller unit root test are shown in case of the presence of drift term.