# Threshold Cointegration Found in 6 of 9 Bond Rate Pairs

Cite as: Gorak, R. (2026). Threshold Cointegration Found in 6 of 9 Bond Rate Pairs. Tradicted. https://www.tradicted.com/research/hansen-threshold-2002/
Paper: Bruce E. Hansen and Byeongseon Seo — *Testing for two-regime threshold cointegration in vector error-correction models*
Published in: Journal of Econometrics (2002)
Original: https://doi.org/10.1016/s0304-4076(02)00097-0
DOI: 10.1016/s0304-4076(02)00097-0

Key finding: In six of the nine bivariate Treasury bond rate models, the SupLM0 statistic is significant at the 10% level when l = 1 and the cointegrating vector is fixed at unity.

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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.

## FAQ

### What is threshold cointegration?

Hansen and Seo (2002) estimated a threshold of -0.63 for the error-correction term in a two-regime model of U.S. Treasury bond rates. Threshold cointegration is a vector error-correction model whose adjustment term switches between regimes at an estimated threshold. Below the threshold, adjustment was minimal; beyond it, it became sharp and statistically significant.

### How much evidence is there for nonlinear adjustment in interest rate spreads?

6 of 9 bivariate Treasury bond pairs showed a significant SupLM0 threshold effect at the 10% level. That test used a one-lag VECM with the cointegrating vector fixed at unity. Raising the lag length to 2 raised the count to 7 of 9 pairs significant at 5%.

### Does estimating the cointegrating vector change the evidence for a threshold?

Only 4 of 9 Treasury bond pairs showed a significant SupLM statistic at the 5% level once the cointegrating vector was estimated. Fixing the vector at its theoretical value of 1 produced stronger evidence for threshold cointegration than estimating it freely.

### How do error-correction coefficients differ across regimes for 10-year versus 1-year Treasury bonds?

The short-rate error-correction coefficient was 1.41 in the "extreme" regime versus 0.04 in the "typical" regime for the 120-month/12-month bond pair. The long-rate coefficient showed the same pattern: 0.34 in the extreme regime against -0.02 in the typical regime. The extreme regime held only 8% of the sample's observations.

## Source

Hansen, B. E., & Seo, B. (2002). Testing for two-regime threshold cointegration in vector error-correction models. Journal of Econometrics, 110, 293-318.

[Read the full paper →](https://doi.org/10.1016/s0304-4076(02)00097-0)