# CUSUM test for regression model

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.

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

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

and

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

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.

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