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I am conducting an instrumental variable approach (IV) in my paper, and I am puzzled how to correctly test for the relevance / strength of the instrument assumption. I recently read in a paper by Bastardoz et al. (2023) (page 12, 22) that it is a common mistake, that researcher include controls in the first stage when testing for strength of the instrument (obviously, for the "real" first stage regression controls have to be included, but as they say, not if you test for the weakness of the instrument).

In the lecture slides by Benjamin Elsner (Download Slides, page 59) it says that if controls are included in IV, then the first stage assumption changes to Cov(Z, D | X)≠0, which I understand like controls must be included when testing instrument strength.

Has anybody practical experience with this? Are controls included for relevance tetsing or not?

When I run my first stage without controls, then my F-statistic is sufficiently high (ca. F=150). When I include controls, it ranges between F=2 and F=50, which would make it weak and I might have to discard the whole paper.

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"In instrumental variables (IV) regression, the instruments are called weak if their correlation with the endogenous regressors, conditional on any controls, is close to zero." [1, emphasis mine]

However, this does not mean that you should throw away your whole paper but just that you should acknowledge your instruments are weak and use weak-identification-robust inference. See e.g. [2]

[1] https://scholar.harvard.edu/files/stock/files/wirev_080218-_posted.pdf

[2] https://econ.lse.ac.uk/staff/spischke/ec533/Weak%20IV.pdf

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Are controls included for relevance tetsing or not?

You should include them when checking for relevance.

The whole point of checking for relevance is to know how well can the instrument explain the endogenous regressor. However, the auxiliary first stage regression will only be valid if you include relevant controls. For example, if you instrument$x$ with $z$, but $z$ only explains $x$ in unconditional model, but stop explaining $x$ once you add some vector of controls $\mathbf{v}$ then I believe it’s reasonable to say $x$ is not relevant instrument.

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