Mehmet Caner and Bruce E. Hansen
"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 non-standard 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 non-standard 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 non-linearity (thresholds) and non-stationarity (unit roots).
Our limit theory is based on a new set of tools which combine unit root asymptotics with empirical process methods. We work with a particular two-parameter empirical process 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 the threshold effect is in the short-run dynamics, rather than in the dominant root. While the conventional ADF test for a unit root is insignificant, out 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.
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Some of the above material is based upon work supported by the National Science Foundation under Grants No. SES-9022176, SES-9120576, SBR-9412339, and SBR-9807111.
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