# 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?

## 1 Answer

This is maybe a late answer (I happened upon it while searching for a question of my own), but the answer to your question is that your 2SLS should include SES as a control variable in the first and second stages. The phrasing of your question doesn't make it clear if you want to know if you can include both together or if you want to actually compare their outcomes, but I am going with the former. To explain further:

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.

Both assumptions may not be true without proper conditioning on certain variables. The first may not hold if there are selection effects (families with higher SES may have parents that settle closer to college because they want to increase the odds that their children attend college), and the second assumption may not hold if the families who decided to reside closer to a college are different from those who decided to live far from a college (which is likely due to SES-related factors).

As such, to make those assumptions hold, you need to condition on the correct variables, in which case you can identify the LATE for the entire group. I'd recommend not partitioning the analysis (though you could argue dummying for SES and other effects is similar in effect) because the whole point of LATE is to estimate the treatment effect on the largest subpopulation, which are precisely the compliers.