Threshold Autoregression with a Unit Root

This paper develops an asymptotic theory of inference 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 threshold are nonstandard and different
from the stationary case, and suggest basing inference on a bootstrap approximation. We
also study the asymptotic null distributions of tests for an autoregressive unit root, and
find that they are nonstandard and dependent on the presence of a threshold effect. We
propose both asymptotic and bootstrap-based tests. These tests and distribution theory
allow for the joint consideration of nonlinearity (thresholds) 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 weakly to a two-parameter Brownian motion. Our limit distributions
involve stochastic integrals with respect to this two-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 unemployment rate.
We find strong evidence of a threshold effect. The point estimates suggest that the
threshold effect is in the short-run dynamics, rather than in the dominate root. While the
conventional ADF test for a unit root is insignificant, our TAR unit root tests are arguably
significant. The evidence 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 process.
KEYWORDSB:o otstrap, nonlinear time series, identification, nonstationary, Brownian
motion, unemployment rate.