I guess my question is rather basic. Unfortunately, I still did not manage solve it, although searching for hours.

I have a linear regression model and need to do a CUSUM test for parameter stability.

I am fine to calculate the test statistic. However, I have not found a way to derive boundaries for the test statistic for a specific confidence interval.

I followed this guy here to calculated the test statistic: http://www2.econ.iastate.edu/classes/econ374/Falk/lecture_21_assessing_model_stability.doc

He refers that there should be a CUSUM(t-k) distribution, which I cannot find anywhere.

Additionally, I do not want to use a package to do CUSUM test, but I would prefer to calculate the test at least once on my own to make sure I got the mechanic.

Thanks for you help.


I am now trying to calculate the boundaries.

I found the following: http://www.uwyo.edu/aadland/classes/econ5350/slides.pdf

Here, the boundaries are said to be straight lines that go through the following points:

$$k \pm a \sqrt{T-k}$$


$$T \pm a \sqrt{T-k}$$

where $k$ is number of coefficients in the model and $T$ the number of recursive estimations.

BUT: how do I get the value of $a$? (it only says that $a$ depend on the chosen significance level) Does anyone know how to get $a$?

In another source, it says that the boundaries for $t$ are as follows:

$$\pm c \left( 1 + 2\frac{t-k}{T-k} \right)$$

where $c$ is the solution to the following equation:

$$\phi(3c) + e^{-4c^2} \big( 1 - \phi(c) \big) = 0.5 \alpha$$

I solved this equation for 95% significance level, but much, much lower values than on all the charts I found in the net. So I believe I must be doing something wrong.

Thanks again for any help.

  • $\begingroup$ I found/read the paper "Techniques for Testing the Constancy of Regression Relationships over Time" from Brown et. (1975). I got it now. :) $\endgroup$
    – Bombax
    Oct 13, 2017 at 15:57

1 Answer 1


You can go for Invidiual moving range chart(IMR chart) which is one of the type of control charts. This helps you to measure the variation within your process and also it gives you upper control limit and lower control limit.


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