On the relationship between stochastic and deterministic polynomial trends with applications to the detection of the order of integration

We study the relationship between stochastic and deterministic polynomial trends over the long-run, as the sample size N→∞, and at a local level, by focusing on the last n observations of the sample, with n=o(N). First, we show that stochastic processes with integration order I(d), for integer d≥1, locally behave like deterministic polynomial trend models of degree d−1, scaled by asymptotically normal random variables that are constants at a local level. Second, we introduce statistical procedures to determine the order of polynomial trend models, thereby providing an indirect way to assess integration in stochastic processes. Using data on fourteen major U.S. macroeconomic variables, our method confirms that most are I(1), while Money Stock and Bond Yields exhibit I(2), highlighting the effectiveness of our approach in detecting higher-order integration.