Hi, I am working on a paper to find the relationship between hiring a live-in domestic helper(DH) and  labour force participation(LFP) of mother. It is believed the DH will subsitute the child-care work of mother.
So the equation I want to estimate is LFP= a + b (DH) + controls
Both LFP and DH are binary varables.
DH is endgenous so that I have to use IV and estiamte by 2sls. The instrument I use is "the presence of the 4th room at home" to proxy the presence of DH. I restrict the sample to women with 3 & 4 rooms at home only.
The first-stage regression shows that the IV is valid.
Some concerns over the endogeneity of the IV have been noticed, for example women will move to lareger houses intentionally to hire a domestic helper.
Also the larger houses may imply more unobserved wealth.
May I ask if there are any more concerns about this IV? And what robust test should I use to test it in order to deplete the concern?