# Estimating the Local Average Treatment Effect (LATE) and always-takers

I am new to statistics and am particularly interested in RCTs so this is a very basic question.

If I have a program wherein 10% of my control group had access and used the treatment, how would that affect the Average Treatment Effect (ATE)? If the ATE can no longer be recovered, how would the LATE be computed? It seems to me that in this case, LATE is equal to ATE but I am not sure.

I understand how to address non-compliance in the treatment group using IV regression but I'm not sure if 2SLS could also be used if the only issue is that 10% of the control group received the treatment.

As for estimation, with a binary instrument (where the instrument $$Z$$ is 1 if in treatment group, 0 if in control group), we can just use a simple Wald estimator. Given outcome $$Y$$ and treatment $$D$$, let $$y_1$$ and $$d_1$$ be averages when $$Z=1$$, and define $$y_0,d_0$$ when $$Z=0$$. Then the LATE is given by $$\beta_{WALD} = \frac{y_1 - y_0}{d_1-d_0}$$