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

Question

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 just indifferent between 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

Cameron & Trivedi (2005, p. 886) reads as if MTE is just the effect estimate we obtain if we have a continous instrument compared to a binary instrument:

If we compare $TE_{WALD}$ with the LATE measure, we find that LATE is a measure of the effect of treatment on the subgroup of those at the margin of participating, denoted as compliers. In empirical economic applications the concept of a marginal impact caused by variation in a continuous variable, measured by a partial derivative, is well entrenched and is replaced by a discrete analogue when the variation in the causal variables is discrete.

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 $(X=5,P(Z)=.6)$.

Answer 1

I think this is a good way to explain the details. I got it from Counterfactuals and Causal Inference by Morgan and Winship, which is a wonderful book.

Let's say we are interested in the effect on wages from attending college ($D$). I am not a huge fan of distance, so imagine we had an instrumental variable $Z$ that is a lottery where winners get a voucher worth 25K. Let's assume that 10% of students win and everyone is auto-enrolled in the lottery to simplify things. The LATE estimated by the Wald estimator is the ATE for folks who go to school when they win 25K and don't go to school when they lose (the compliers). There's an intuitive derivation of this here, along with the familiar formula. So far this is pretty standard.

Now suppose we have a fancier lottery. Instead of 10% getting an identical 25K voucher, the winners get something random that is uniformly distributed between \$1 and tuition at Harvey Mudd College.$^*$ Now $Z$ is continuous, and let's assume it still satisfies (relevance, monotonicity, and random assignment).

An LIV is the limiting case of a component binary IV drawn from $Z$ in which $z′′$ approaches $z′$ for any two values of $Z$ such that $z′′ > z′$. Each LIV then defines a marginal treatment effect, which is the limiting form of a LATE, in which the IV is an LIV.

What does this mean? You could make some LIVs from $Z$ by stratifying the data by the values of $Z$ and then doing the Wald on adjacent strata (zero to one, one to two, etc). Assuming enough data, LIVs could be constructed for each dollar increase in the voucher. Each LIV could then be used to estimate its own LATE, and these LIV-identified LATEs are the MTEs.

LATEs and many other average treatment effects can be seen as weighted averages of the fundamental marginal treatment effects.

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$^*$I did this in dollar increments, but you could also imagine doing this in pennies or something even more infinitesimal instead. Harvey Mudd was the most expensive college in the US last year in terms of sticker price.