Some tests for unit roots in autoregressive-integrated-moving average models with deterministic trends

Abstract

SUMMARY For a nonstationary time series we consider a model which can accommodate the possibility of deterministic and stochastic trends. Assuming that we are primarily interested in testing for the stochastic trend, that is, for a unit root in the time series, we employ the Lagrange multiplier principle and obtain test statistics. The asymptotic distributions of the test statistics are characterized by functionals of stochastic integrals of a standard Brownian bridge. Unlike some of the existing test statistics for unit roots, the asymptotic distributions remain the same whether there is a deterministic trend or not under the null model, and therefore a priori knowledge about the deterministic trend is not needed for the test of a unit root. A numerical example is presented to illustrate the methods, and the powers of the proposed tests for finite samples are studied through a small Monte Carlo sampling experiment.