A simple test to distinguish between deterministic and stochastic trends in non-stationary time series

For non-stationary times series, the main difficult is to identify the nature of non-stationarity. In macroeconomic and financial modeling, there are two kinds of non-stationarity. The first one is due to the presence of a deterministic trend and generally modeled by a d-order polynomial of time. The second is due to the presence of a stochastic trend which is modeled by integrating a stationary process up to a suitable order d. In this paper we propose a first-order (in mean) stationarity test to decide between these two kinds of trend. We assume that d∈N∗ is known to establish the limiting distribution under H0 and the consistency of the test, and propose a strategy to identify d if it is unknown. Simulations show that the test has a good empirical size and power. The test is finally applied to some real macroeconomic and financial datasets.