Threshold cointegration describes a vector error-correction model whose adjustment term switches between two regimes at an estimated threshold, rather than adjusting continuously. In the term structure of interest rates, a yield spread may sit flat within a band, then correct sharply once it crosses that threshold. In Testing for Two-Regime Threshold Cointegration in Vector Error-Correction Models, Hansen and Seo (2002) tested nine monthly U.S. Treasury bond pairs from the McCulloch and Kwon (1993) series, covering 1952 to 1991, and found a threshold effect significant in six of nine pairs. Their SupLM test uses a bootstrap procedure to compute p-values without assuming linear cointegration.
What the Study Found
The SupLM0 statistic (β fixed at unity, l=1) is significant at the 10% level in 6 of 9 bivariate Treasury bond pairs. Raising the lag length to 2 strengthens the evidence, with 7 of 9 pairs significant at the 5% level. Letting the cointegrating vector be estimated rather than fixed weakens the evidence, with only 4 of 9 pairs significant at the 5% level. The estimated threshold for the 120-month versus 12-month pair is -0.63, placing 8% of observations in the "extreme" regime and 92% in the "typical" regime. The estimated cointegrating relationship for that pair is wt = Rt - 0.984rt, close to the unit coefficient predicted by term structure theory.
"We have presented a quasi-MLE algorithm for constructing estimates of a two-regime threshold cointegration model and a SupLM statistic for the null hypothesis of no threshold."
Hansen and Seo (2002), Testing for two-regime threshold cointegration in vector error-correction models, p. 313.
Methodology
Hansen and Seo tested nine bivariate pairs drawn from the McCulloch and Kwon (1993) monthly U.S. Treasury bond term structure series covering 1952 to 1991. They estimated a two-regime vector error-correction model by quasi-maximum likelihood, using a grid search over the cointegrating vector and the threshold. The SupLM test was run with the cointegrating vector both fixed at unity and freely estimated, and with VAR lag lengths of 1 and 2. P-values came from a parametric residual bootstrap with 5,000 simulation replications.
Key Statistics
| Metric | Finding | Context |
|---|---|---|
| SupLM0 significance rate, l=1 | 6 of 9 pairs | 10% level, cointegrating vector fixed at unity |
| SupLM0 significance rate, l=2 | 7 of 9 pairs | 5% level, cointegrating vector fixed at unity |
| SupLM significance rate | 4 of 9 pairs | 5% level, cointegrating vector estimated |
| Estimated threshold, 120-mo/12-mo pair | θ = -0.63 | Percentage-point spread, wt = Rt - 0.984rt |
| Pointwise LM statistic | LM(β,θ) = vec(Â1-Â2)'(V̂1+V̂2)⁻¹vec(Â1-Â2) | Heteroskedasticity-robust LM statistic at fixed (β,θ) |
Typical vs Extreme Regime
Figures below are for the 120-month versus 12-month Treasury bond pair.
| Measure | Typical Regime | Extreme Regime |
|---|---|---|
| Share of observations | 92% | 8% |
| Error-correction coefficient, short-rate equation | 0.04 | 1.41 |
| Error-correction coefficient, long-rate equation | -0.02 | 0.34 |
Why This Matters
A linear cointegration model assumes bond yields correct toward equilibrium at a single, constant rate no matter how far a spread has drifted. The threshold result implies that correction is close to absent until the spread crosses the estimated threshold, then becomes forceful. Averaging across both regimes, as a linear model does, can mask this kind of episodic mean reversion entirely. For anyone testing term-structure or arbitrage relationships, treating an error-correction coefficient as constant risks understating how sharply prices snap back once a threshold is breached.