Threshold autoregression with a unit root

This paper develops an asymptotic theory of inferences for an unrestricted two-regime threshold autoregressive (TAR) model with an autoregressive unit root. We find that the asymptotic null distribution of Wald tests for a th reshold arc nonstandard and different from the stationary case, and suggest basing inference on a bootstrap approximation. We also study the asymptoti c null distributions of tests for an autoregressive unit root, and find tha t they are nonstandard and dependent on the presence of a threshold effect. We propose both asymptotic and bootstrap-based tests. These tests and dist ribution theory allow for the joint consideration of nonlinearity (threshol ds) and nonstationary (unit roots). Our limit theory is based on a new set of tools that combine unit root asymptotics with empirical process methods. We work with a particular two-parameter empirical process that converges w eakly to a two-parameter Brownian motion. Our limit distributions involve stochastic integrals with respect to this t wo-parameter process. This theory is entirely new and may find applications in other contexts. We illustrate the methods with an application to the U.S. monthly unemploym ent rate. We find strong evidence of a threshold effect. The point estimate s suggest that the threshold effect is in the short-run dynamics, rather th an in the dominate root. While the conventional ADF test for a unit root is insignificant, our TAR unit root tests are arguably significant. The evide nce is quite strong that the unemployment rate is not a unit root process, and there is considerable evidence that the series is a stationary TAR proc ess.