Issues about using interacted term as Instrumental Variable

[Shilong Zhuang]

If I want to explore y ~ x, where x is endogenous and IV is z1 * z2 (product), do I need to add z1 z2 as a separate variable for control or as a part of IV? Generally, such applied papers usually have fixed effects that absorb single terms in the first stage. However, here my z1 is a did term (city group dummy * post), z2 is an city characteristic, and z1 * z2 is strictly a triple term. Fixed effects do not absorb all double interaction terms (such as post * z2). So I am quite puzzled about the correct approach here. Currently, I am trying to find papers on econometric theory dealing with this issue, but I have not found suitable ones.

[Jeff Wooldridge]

You should control for the double interactions. For example, z2*post allows different trends by city in the untreated state.

[Shilong Zhuang]

You should control for the double interactions. For example, z2*post allows different trends by city in the untreated state.

Thanks prof Wooldridge! I think I understand your thoughts. However, what I would like to know more is in a more general context, shall we always control for single terms when we use interaction IV? If the single term is not included, will it undermine the exclusion restriction of IV?