# Heterogeneous treatment effects with 2SLS local average treatment effect (LATE)

I am interested in a strategy to calculate heterogeneous treatment effects with an IV strategy for local average treatment effects (LATE).

I am estimating the effect of postsecondary educational selectivity on adult wages using an instrumental variables identification strategy. The instrument is linear distance to a selective institution and should identify the LATE, or the effect of the treatment on compliers.

In this case, the compliers would be those induced to attend a more selective institution due to closer proximity to a selective college or university. It is likely that the effect for students on the margin of attending a selective college (those who would not have otherwise attended a selective school) differs from the effect of those who would be an always-taker (not constrained by geography in choosing a college).

If the likelihood of being a complier is related with family socioeconomic background, what might be a valid strategy for estimating if the effect for low-SES students differs from high-SES students? Is it valid to run separate 2SLS models for students of low and high SES and then compare the LATE for each subgroup?

I understand that there are recent developments in strategies for marginal treatment effects. But, short of implementing a MTE strategy, would stratified models in a 2SLS framework be valid?

Instrumental variable estimation has 2 key assumptions regarding the instrument. For a simple scenario, consider an endogenous binary regressor $D$, a binary instrument $Z$, an outcome $Y$, and a vector of control variables $X$. The two key assumptions are:
$\textbf{unconfounded type}$ $$\forall x\in Supp(X), P(\tau_i=t\mid X_i=x,Z_i=0)=P(\tau_i=t\mid X_i=x,Z_i=1)$$ $\textbf{mean exclusion restriction}$ $$E[Y^{d}_{i,z}\mid X_i=x,Z_i=0,\tau_i=t]=E[Y^{d}_{i,z}\mid X_i=x,Z_i=1,\tau_i=t]$$ where we denote individual behaviors as $\tau_i \in \{a,n,c,d\}$ where a is always-taker, n is never-taker, c is complier, and d is defier.