Context
I have hierarchical data where individuals ($i$) are nested in groups ($j$) and am interested in examining the degree to which a continuous group-level variable ($X_j$) moderates the effect of a continuous individual-level variable ($W_{ij}$) on a binary individual-level outcome ($Y_{ij}$), i.e. $Y_{ij} = \alpha + \beta{X_j} + \beta{W_{ij}} + \beta{(X*W})_{ij}$
Assume $Z_j$ is a continuous, group-level instrument for $X_j$ and $W_{ij}$ is exogenous.
Questions
Is there anything inherently wrong with instrumenting a cross-level interaction that differs from instrumenting an interaction in OLS? I am not aware of any issues.
Is the first stage of instrumental variables regression with multilevel modeling (e.g. as implemented in xtivreg in Stata) simply OLS or should it be estimated via multilevel regression as well?
Given that the "second stage" of the multilevel IV is a binary logistic regression (insofar as the dependent variable is binary), am I erroneously trying to estimate Woolridge's "Forbidden Regression?" It is my understanding that if the first stage is either also binary logistic or OLS I have a consistent estimator.
Has anyone seen an implementation of something like this in statistical software packages? Stata's xtivreg does not implement binary dependent variables. I'd rather not try to implement this in Stan or BUGS.
More generally, is there anything terribly problematic about this? I have seen few multilevel IV estimates - although Gelman has a paper on this and discusses it in Gelman and Hill - and exactly zero using multilevel GLMs.
