(EDIT: Reading the post over, I'm ashamed my motivating example was a vanilla college example and not something like what I usually do. So say we have data from an online dating site. Let X be whether or not you pay for premium, Y be the number of dates you go on, and Z be whether the site gives you a discount on the premium service. You'll get compliers obviously, but also never-takers (ultra attractive people who have 0 use for premium) and always-takers (super unattractive people who need every advantage they can get). Pretend that's the motivating example.)

Tl;dr Applied economists are running IV/TSLS with covariates and calling the treatment effect LATE when discussing potential heterogeneous treatment effects. That's not how it works.

Josh Angrist won the Nobel Prize last month in large part because of LATE. So researchers read Mostly Harmless Econometrics, use an IV in their specification with covariates, get asked about heterogeneous treatment effects, and simply say the IV gives you the LATE. You can see this in papers published in top journals, such as this JPE, this Econometrica, and this AER - they just say that Imbens & Angrist (1994) tells them it's the LATE, and everything's all good.

But with their covariate specifications, that's straight-up wrong. If you want a very short and sweet R1, I could just cite section 5 of Abadie (2003). But let's go the roundabout way.

Background for non-econ people:

A big question in economics is if some treatment, X, causes Y, and to what extent. Economists often use linear regression for this. Now, you probably know that you can't, say, look at the impact of college education (X) on future wages (Y), without first controlling for things, or else you get confounding bias. The things you control for are called covariates. However, we're often still not done. A lot of times, there are unobservable factors that affect decisions, and these factors can still cause bias, even after controlling for confounding. So economists often deal with this using instrumental variables, or IVs. For example, if I want to look at the impact of college education on future wages, an instrument (Z) one might consider is whether or not someone received a non-merit subsidy, say a somewhat random policy by some state governments to make college cheaper by some amount. Ideally, the instrument will cause "external variation" in X (the decision to go to college), so as to make things quasi-random and eliminate bias. (Regulars reading this, please don't go ham - just giving a quick and dirty explanation!)

However, instruments may still run into issues! The one we’ll discuss today is called heterogeneity in treatment effect. This means that the treatment may affect different people in different ways. Take the government's tuition subsidy. I would have gone to college no matter what, so a subsidy wouldn't have changed my choice. On the other hand, I have some friends from middle school who graduated high school and decided to start a landscaping business. Unless the tuition subsidy were huge, they probably would have always skipped college. There are also a lot of people in the middle, whose choices would be affected by a government tuition subsidy. With no subsidy, they'd skip college, and with a subsidy, they'd attend college.

People like me who would attend college no matter what are called always-takers. People like my friends who would skip college no matter what are called never-takers. People who would attend with a subsidy but skip without a subsidy are called compliers. There's also a wacky group, in theory, who would attend without a subsidy, and skip with a subsidy. They are called defiers. In any case, heterogeneous treatment effects can be a bit dangerous, because that means different people have different underlying incentives to take the treatment, and so the instrument's effectiveness may not be so great.

Enter what is known as the local average treatment effect (LATE). The beauty of this is that, if defiers don't exist, you'll only measure the effect of the treatment from compliers - people who are directly affected by your instrumental variable, i.e. those who choose the treatment if and only if they get the instrument. So it's "local" to that subset of people, and hence doesn't capture the effect of the treatment of people like myself or my friends, who are different in ways such that we choose to always go to college, or never go.

But it's not all fine and dandy with how it's used in practice.

Basic LATE Model

We have an outcome Y. We have one treatment D, and one instrument Z - both are binary, i.e. only take the values 0 and 1. Standard IV assumptions of relevance, exclusion, and exogeneity hold. In addition, we have the monotonicity condition: P[D(Z=1) >= D(Z=0)] = 1, so we don’t have defiers. The IV estimator is derived here.

So in this simple example, IV is LATE. Hooray!!!!!!!!

Including covariates

But people don't generally just have a treatment and an instrument. They also have covariates, which are often necessary to meet the exogeneity restriction for an instrumental variable. And this is where everything falls apart.

First, we have to strengthen the IV restrictions to account for covariates. Say W is a vector of covariates. Then for monotonicity, we must have P[D_1(1) >= D_i(0) | W_i = w] = 1.

Let's add some labels too. Call G your group. Then G = cp means you're a complier, G = at means you're an always-taker, and G = nt means you're a never-taker.

Now, note that if we condition everywhere then run an IV, we can follow the same steps as before to get this.

So simply conditioning on covariates then running an IV gives us a conditional LATE. But nobody does that, because conditioning on covariates first is a great way to take your dataset down from 500,000 observations to fewer than 10. So we control for W as we run our IV estimator. From the Frisch Waugh theorem, if we define L(Z|W) as the linear projection of Z on W, then the IV estimator is defined by this.

Note that Cov(Y_i, \tilde{Z}_i | W_i) = Cov(Y_i, Z_i | W_i) because L(Z_i|W_i) is constant, given W_i. Anyways, from our second photo, we see the first term can be written as a weighted average of LATEs.

From our monotonicity assumption, the weights are assumed to be positive.

The second term is where things get messy. Suppose that E[\tilde{Z}_i|W_i] =/= 0. Then the second term becomes a big, big, big mess. And I didn't even go into why the E[Yi(0)|W_i] term seemingly disappears in the second equality, which is here. But anyways, plugging this back into the main \beta{iv} equation and simplifying gets us a final expression for the IV estimator, in terms of conditional LATEs.

Look: the second term captures the treatment effect from always-takers, not compliers! And not only that, but we can get negative weights on the always-takers and compliers. If we choose W_i such that the fitted values from L(Z_i|W_i) > 1, then the complier groups with these covariates will be weighted negatively (since the weight on the complier treatment effects have the same sign as 1 - L(Z_i|W_i)). Including always takers and negative weights…that makes the LATE interpretation fall apart.

Now, it is possible to include covariates and still keep the LATE interpretation. But this requires a very precise selection of covariates, e.g. when covariates are discrete and W_i includes an indicator for each group of covariates. Then E[\tilde{Z}_i | W_i] = 0, and all the drama from before goes away. However, this is a sort of specification that nobody ever uses.

Point being? IV is not LATE for the specifications people generally use! You're capturing always-takers with negative weights, and also putting negative weights on compliers too! And when we extend the treatment and instruments to non-binary cases, or consider multiple of each, it's a total mess. But researchers keep doing this. Look at the three aforementioned papers:

Conditional on instrument validity, \alpha_{2,IV} captures the local average treatment effect (LATE) of electricity projects on community level employment growth. (Dinkelman, 2011)

If bands that had these unobservable ties were more likely to agree to shared reservations and if such ties also made it easier to cooperate on the reservation, then IV estimates the local average treatment effect (LATE) on only the bands that would not have agreed to shared reservations without the added pressure from mining. (Dippel, 2014)

However, if there are heterogeneous treatment effects, the IV estimate will be the local average treatment effect (LATE; Imbens and Angrist 1994), and a different pool of eligible women, or a different set of selectors, could lead to different IV estimates. (Dube and Harish, 2020)

None of their covariate specifications are saturated, and they offer no other defense as to why their IV strategies are the LATE. They're estimating some combined weighted effects of compliers and always-takers with both positive and negative weights. They're not estimating the LATE. Even Mostly Harmless say that covariates must be saturated:

2SLS with a fully saturated first stage and a saturated model for covariates in the second stage produces a weighted average of covariate-specific LATEs. (Section 4.5.2)

Angrist himself doesn't seem to invoke LATE all that often in his research. Well, he (and Imbens) out of anyone ought to know the limitations, right?

Conclusion

One could construct their own estimator, perhaps using marginal treatment effects, to estimate the parameter one cares about. But just taking the IV estimator and trying to interpret it as the LATE when your specification isn't as required is bad econometrics.