I'm running a regression model to test whether unionisation rates have an impact on wages.
I've introduced an instrumental variable: public support for unions. As far as I can tell, this instrument only affects wages through its impact on unionisation rates, but it seems likely that lower or higher wages affect support for unionisation.
Basically, z only affects y through x, but y likely affects z.
Is my instrument valid?
If it is the case that $Y$ affects $Z,$ then $Z$ cannot be an instrument. By definition, an instrumental variable has to be $d$-separated from $Y$ in $G_{\alpha}$ and $d$-connected to $X.$ Your instrument cannot be $d$-separated from $Y,$ so it's not really an instrument. Note: $G_\alpha$ is the causal graph obtained from $G$ by deleting the edge from $X$ to $Y.$
Now you essentially have this causal graph:
It is evident that $Z$ is a textbook confounding variable, since there is an unblocked backdoor path from $X$ to $Y.$ So you should be able to do any number of methods for mitigating the confounding effects of $Z.$ The backdoor adjustment might be the simplest, assuming all three variables are measured.
