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I want to perform a Breusch-Pagan-Test. I did calculate some very small examples following the procedure described below, just to understand what the test does:

enter image description here

found here: (https://ipfs.io/ipfs/QmXoypizjW3WknFiJnKLwHCnL72vedxjQkDDP1mXWo6uco/wiki/Breusch%E2%80%93Pagan_test.html)

in Step 2: it is says, that "z could be partly replaced by independent variables x." Also in all hand-calculated examples I found z was just replaced by x. But can somone tell me in simple words what z really is? If it is x, why is it not named so, in the formulas?

Thanks for any help you can offer!

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Imagine you believe that your residuals are heteroskedastic. It might be the case, then, that they can be represented by some kind of function on some variables. When the authors assume that $\sigma^{2}_{t}=h(z'_{t}\alpha)$, they are simply suggesting this structure, where $h$ is the function and $z'_{t}\alpha$ is a linear combination of variables.

Nonetheless, in the case of linear regression models, it is common to observe that the residuals are somehow related to the independent variables when they're heteroskedastic. With this is mind, it might be a good shot to consider that, among all the possible variables that might constitute the $z$ vector, some of them might actually be the independent variables in your regression.

Both $h$ and $z$ could be totally different things, but since it might be that the simplest assumptions you could make actually correspond to reality, it is smart to try them out first.

I've skipped a bunch of mathematical formality in this explanation, and since they are not complicated to understand, I strongly suggest you try and read the second page of the original paper: A Simple Test for Heteroscedasticity and Random Coefficient Variation.

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