## Bruce E. Hansen and Byeongseon Seo

"Testing for two-regime threshold cointegration in vector error correction models"

Journal of Econometrics (2002), 110, 293-318.

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Abstract:

This paper proposes a formal test for threshold cointegration and an
algorithm to estimate the model parameters. Our model is a vector error
correction model (VECM) with a single cointegrating vector, and a threshold
effect in the error-correction term.

We propose a relatively simple algorithm to obtain maximum likelihood
estimation (MLE) of the complete multivariate threshold cointegration model
for the bivariate case. However, we do not provide a proof of consistency,
nor a distribution theory for the MLE.

We propose testing for a threshold in this model with a Lagrange Multiplier
(LM) test. This is particularly convenient since the LM test can be computed
by an ordinary least squares regression involving the conventional maximum
likelihood estimate of the cointegrating vector. Since the threshold is not
identified under the null hypothesis, our tests take the Sup-LM form. We
derive the null asymptotic distribution of the test, and find that it takes
the form given in Hansen (1996). We show how to calculate asymptotic
critical values by simulation. We also present a bootstrap approximation to
the sampling distribution.

We investigate the performance of the test using Monte Carlo simulation, and
find that the test works quite well. The Type I error of the test is quite
close to the nominal level, even in a small sample with conditional
heteroskedasticity.

We apply our methods to the term structure model of interest rates. In
several bi-variate cointegrating VECMs, we find strong evidence for a
threshold error-correction model.

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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.
Any opinions, findings, and conclusions, or recommendations expressed in this material are those of the author(s), and do not necessarily reflect the views of the NSF.