# How to calculate and interpret a marginal treatment effect (local instrumental variable)? (Intuition through simple example.)

I am working on the intuition behind local instrumental variables (LIV), also known as the marginal treatment effect (MTE). I have worked some time on this and would benefit from solving a simple. I hope I may get input on where my example goes awry.

As a starting point the standard local average treatment effect (LATE) is the treatment among individuals induced to take treatment by the instrument ("compliers"), while MTE is the limit form of LATE.

A helpful distinction between LATE and MTE is found between the questions:

• LATE: What is the difference in the treatment effect between those who are more likely to receive treatment compared to others?
• MTE: What is the difference in the treatment effect between those who are marginally more likely to receive treatment compare to others?

The term "marginally" is key and what it specifically implies in this context eludes me (in economics I am used to understanding it as the change in one variable given a one unit change in another). From what I gather, under MTE it is the change in outcome with a marginal change in the probability of receiving treatment, although I am not sure if this is correct. If it is correct I am not sure how to argue for policy or clinical relevance.

To understand the mechanics and interpretation of MTE I have set up a simple example that starts with the MTE estimator (from Cornelissen et al. 2016):

$$MTE(X=x, U_{D}=p) = \frac{\partial E(Y | X=x, P(Z)=p)}{\partial p}$$

Where $$X$$ is covariates of interest, $$U_{D}$$ is the "unobserved distaste for treatment" (not intuitively explained what implies), $$Y$$ is the outcome, and $$P(Z)$$ is the probability of treatment (propensity score).

Specifically, Shafrin (2007) state,

The LIV [MTE] estimates the derivative of the expected outcome conditional on observed characteristics and the probability of electing to be in the treatment group, $$E(Y|X=x, P(z,x))$$, with respect to the probability of treatment.

Example

We want to estimate the MTE of college ($$D=(0,1)$$) on earnings ($$Y>0$$), using the continous variable distance to college ($$Z$$) as the instrument. We start by obtaining the propensity score $$P(Z)$$, which I read as equal to the predicted value of treatment from the standard first stage in 2SLS:

$$D= \alpha + \beta Z + \epsilon$$

$$=\hat{D}=P(Z)$$

Now, to understand how to specifically estimate MTE, it would be helpful to think of the MTE for a specific set of observations defined by specific values of $$X$$ and $$P(Z)$$. Suppose there is only one covariate ($$X$$) necessary to condition on and that for the specific subset at hand we have $$X=5$$ and $$P(Z)=.6$$. Consequently, we have

$$MTE(5, .6) = \frac{\partial E(Y | X=5, P(Z)=.6)}{\partial .6}$$

Suppose further that $$Y$$ for the subset of observations defined by $$(X=5,P(Z)=.6)$$ is 15000,

$$MTE(5, .6) = \frac{\partial 15000}{\partial .6}$$

My understanding of this partial derivate is that the current set up is invalid, while substituting $$\partial .6$$ with $$\partial p$$ would simply result in 0 as it would be the derivate of a constant. I therefore wonder whether anyone has input on where I went wrong, and how I might arrive at MTE for this simple example.

Note that this post is moved from Cross Validated as this community may be more familiar with MTE.