I would like to know whether the following reasoning regarding the instrumental variable approach is acceptable. I understand there are case-by-case factors that affect the applicability of instruments that I am not discussing below. But I just want to know if the general logic is correct or if I am missing something

Let's say that we are studying the effect of some state-level policy on some state-level outcome. We could run a regression with the policy variable as a covariate, but it is possible that there is reverse causality. Let's say that one of the factors driving the reverse causality is that there are special interests that would benefit from the policy, and states with higher values for the outcome variable have stronger special interests that lobby policymakers to implement it. If campaign contributions only have an indirect causal influence on the dependent variable through the policy, then it would satisfy the exclusion restriction for instruments.

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However, if there are multiple policies that benefit the special interests, it is likely that the special interests lobby for the other policies as well. If these policies were exogenous, then we would just need to include these other policies in the second-stage regression. But if the special interests are lobbying for the second policy, the policy would not be exogenous; there is reverse causality, as with the first policy. Thus, we need to treat the second policy as an endogenous variable as well.

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Therefore, we should try to find all the policies that could influence the outcome variable and treat them as endogenous variables. To do so, we would need as many instruments as there are endogenous variables. Assuming that the policies are the only endogenous variables, the instruments have no direct causal impact on the dependent variable, and the instruments are not correlated with the error in the second-stage regression, we should be able to adequately control for endogeneity in the estimation of the second-stage regression.

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