Hansen, Bruce, and Mehmet Caner. “Threshold Autoregressions with a Unit Root”. Boston College Working Papers in Economics 381, 1997.
This paper develops an asymptotic theory of inference for a two-regime threshold autoregressive (TAR) model with an autoregressive root which is local-to-unity. We find that the asymptotic null distribution of the Wald test for a threshold is non-standard and mildly dependent on the local-to-unity coefficient. We also study the asymptotic null distribution of the Wald test for an autoregressive unit root, and find that it is non-standard and dependent on the presence of a threshold effect. These tests and distribution theory allow for the joint consideration of non-linearity (thresholds) and non-stationarity (unit roots). Our limit theory is based on a new set of tools which combines unit root asymptotics with empirical process methods. We work with a particular two-parameter empirical processes which 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 in about 80% of the observations, the regression function is close to a driftless I(1) process, and in the other 20% of the observations, the regression function is mean-reverting with an unconditional mean of 5%. While the conventional ADF test for a unit root is quite insignificant, our TAR unit root test is arguably significant, with an asymptotic p-value of 3.5%, suggesting that the unemployment rate follows a stationary TAR process.