*Bounty: 100*

*Bounty: 100*

I am working on the intuition behind local instrumental variables (LIV), also known as the marginal treatment effect (MTE), developed by Heckman & Vytlacil. I have worked some time on this and would benefit from solving a simple example. 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 compared to others?

In revised form, the author states:

LATE and MTE are similar, except that LATE examines the

difference in outcomes for individuals with different average

treatment probability whereas MTE examines the derivative.

More specifically, MTE aims to answer what the is the average

effect for people who are justindifferentbetween receiving treatment

or not at a given value of the instrument.

The use of "marginally" and "indifferent" is key and what it specifically implies in this context eludes me. I can’t find an explanation for what these terms imply here.

Generally, I am used to thinking about the marginal effect as the change in outcome with a one unit change in the covariate of interest (discrete variable) or the instantenous change (continuous variable) and indifference in terms of indifference curves (consumer theory).

Aakvik et al. (2005) state:

MTE gives the average effect for persons who are indifferent between

participating or not for a given value of the instrument … [MTE] is

the average effect of participating in the program for people who are

on the margin of indifference between participation in the program

$D=1$ or not $D=0$ if the instrument is externally set … In brief,

MTE identifies the effect of an intervention on those induced to

change treatment states by the intervention

While Cornelissen et al. (2016) writes:

… MTE is identified by the derivate of the outcome with respect to

the change in the propensity score

From what I gather the MTE is, then, the change in outcome with the 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.

**Example**

To understand the mechanics and interpretation of MTE, I have set up a simple example that starts with the MTE estimator:

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

Where $X$ is covariates of interest, $U_{D}$ is the "unobserved distaste for treatment" (another term frequently used but not explained at length), $Y$ is the outcome, and $P(Z)$ is the probability of treatment (propensity score). I apply this to the effect of college on earnings.

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,

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

**Question**

My understanding of this partial derivate is that the current set up is invalid, and 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.

As for the interpretation, I would interpret the MTE as the change in earnings with a marginal increase in the probability of taking college education among the subset defined by $(X=5,P(Z)=.6)$.