Interpreting Conflicting CUSUM Stability Tests in Finite Samples

Summary

The document addresses how to interpret differing stability-test results, including CUSUM and squared CUSUM tests applied after autoregressive distributed lag modeling. Its central point is that tests of the same parameter-stability hypothesis can disagree in finite samples. The tests may have different power depending on the form and source of instability, so a result that only slightly crosses a significance boundary does not by itself settle the question.

The answer contrasts finite-sample behavior with an asymptotic expectation: if parameters are truly stable and the dataset grows without bound, the tests should tend to reject instability. Small samples make conflicting findings more common. The discussion does not identify a particular cause of instability in the example, quantify test power, or provide a procedure for resolving the disagreement. It therefore supports cautious interpretation rather than a definitive stability verdict from the described plot alone.

Key ideas

• Different stability tests can disagree when the sample is finite.
• Test power depends on the type and source of possible parameter instability.
• Slightly crossing a significance boundary is not enough to identify the cause of instability.
• With stable parameters and increasingly large samples, the tests should tend to reject instability.

Tags

# serial correlation and CUSUM results


# serial correlation and CUSUM results












I have the following CUSUM test resulted from autoregressive distributed lag models (ARDL). Does the CUSUM results show that the model is stable? I am a bit confused because the red line in CUSUM square have only slightly crossed the 5% significance level boundaries .

## Answer by markowitz (score 0, accepted)

https://quant.stackexchange.com/a/45170

The answer is not trivial. The fact that two or more test about the same hypothesis, parameters stability in this case, give you different (also opposite) results is not uncommon. Basically the problem came from the fact that you do not know the source of potential instability. One test is more power than the other in some case but the opposite is true in others.

If your dataset go to infinity and the true parameters are stable, the tests should return the same result (refuse instability). However in finite sample your problem can appear. In small dataset is frequent.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.