Monitoring Cointegration Breaks with Standardized Residuals
Summary
The document addresses how to monitor a previously estimated cointegrating relationship between two time series when repeatedly rerunning a cointegration test is too expensive. Given an existing regression of one series on the other, it proposes calculating the residual for each new paired observation using the fitted coefficients. The residual is then centered and scaled using the historical residual mean and standard deviation.
The answer suggests comparing standardized residuals with a critical range based on a t distribution and treating sustained excursions outside that range as a possible relationship break. It describes requiring the residual to remain outside for a researcher-chosen number of observations, making persistence part of the alert. This is a heuristic monitoring procedure, not a fresh cointegration test or definitive proof of a structural break. The threshold distribution, significance level, persistence length, and residual behavior need empirical judgment, and the document provides no validation study or code.
Key ideas
- New observations can be evaluated against an existing fitted cointegration regression.
- Centering and scaling the new residual makes its deviation comparable with historical residual variation.
- A t-distribution critical range is proposed as a threshold for unusually large deviations.
- Persistence outside the range is suggested as evidence of a possible relationship break.
- Threshold and persistence choices are empirical, and the procedure does not prove a break.
Tags
Full text
# when a co-integrated times series pair has broken the leash
# when a co-integrated times series pair has broken the leash
I have two times series, say $T_i$ and $S_i$ over a reasonably large time window, and I have calculated their cointegration (using python's OLS and Adfuller) . Say that the test has passed with high confidence.
I have just gotten two brand new values, $T_{new}$ and $S_{new}$, and I would like to have a gauge of how far apart they must be to decide that their cointegration is now "broken" (using the metaphor of the drunkard and his dog, I want to determine whether the dog's leash is ripped).
Intuitively, I use the information of the regression and check whether the new residual is beyond the range. Anyone has a better grasp and perhaps even some python code?
Thanks
PS IMPORTANT: the obvious answer would be to recalculate cointegration, but that is not an option: too computationally expensive.
## Answer by alexbougias (score 2, accepted)
https://quant.stackexchange.com/a/45419
You have estimated a cointegration relationship between $T_i, S_i$.
$$ T_i=\hat{\beta_1}+\hat{\beta_2} S_i + \hat{u_i}$$
For each new observation $(T_{new},S_{new})$, replace to the existing equation and find the residual $\hat{u}_{{new}}=T_{new}-\hat{\beta_1}+\hat{\beta_2} S_{new}$. Standardize this value with
$$\frac{\hat{u}_{{new}}-\bar{\hat{u}}}{\sigma_\hat{u}} \sim \text{for instance a }t_k \text{ (t-Dist with k degrees of freedom)} $$
Since residuals are mean-reverting, exceeding for a significant time the region $(-t_{(k,a)},t_{(k,a)})$, would indicate a possible break of the cointegration relationship between the two series. $(t_{(k,a)}$ is the critical value of the t-Distribution that corresponds to significance level $\alpha$: for instance $t_{(k,a)}=3$)
Denote, the region $(-t_{(k,a)},t_{(k,a)})$ with $\mathcal{D}$.
Cointegration is broken at:
$$\tau =inf\{t:\hat{u}_{new,t-k} \not \in \mathcal{D}, \forall k=1,2,..m\}$$
$m$ remains to researcher's dicretion (heavily depends on the data and is an empirical issue)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.