Covariate Contradiction?

Thoughtful reader Nikhil from UBC asks:

I had a question regarding LATE. In your book you say in a model with
covariates, 2SLS leads to a sort of "covariate averaged LATE" even
when one does not have a saturated model. Does this mean that as one
introduces covariates the 2SLS estimator is most likely to change and
that change in the 2SLS estimate is not a comment on the validity of
the instrument?However in your empirical examples you seem to suggest
that invariance of 2SLS estimates to introduction of covariates is a
desirable thing.For example in the first paragraph on pg-152 of
Chapter 4, below Table 4.6.1, you state, "The invariance to covariates
seems desirable: since the same-sex instrument is essentially
independent of the covariates, control for covariates is unnecessary
to eliminate bias and should primarily affect precision." Essentially
my question is: should I start worrying if I see my 2SLS estimates
change as I introduce more covariates in my model? Thanks

Wow, awesome question!  MHE is indeed a little fast and loose on this.
Let me take a stab at clarification.

In Section 4.6.2, we talk about how models with covariates can be
understood as generating a weighted average of cov-specific LATEs across
covariate cells.  True enough ... if the the instrument is
discrete and the first stage saturates (includes a full set of covariate
interactions).  So far so good.  Of course, in practice, you might not
want to saturate.  OK, so do Abadie kappa weighting and get the
best-in-class linear approx to the fully saturated model.
Too lazy to do Abadie?  Just do plain old 2SLS, and that will likely be
close enough to a more rigorously justified approx or weighted average.

Later, however, as Nikhil notes - below table 4.6.1 and on the following page -
we express relief (or satisfaction at least) when IV estimates come out
insensitive to covariates (using samesex) on the grounds that samesex is
independent of covs.  

Contradiction? 

Marginal LATE, that is, LATE with no covs, is also a weighted average
of covariate-specific LATE.  The weight here is the histogram of X 
(convince yourself of this using the law of iterated expectations).
Now, sticking the covariates in and saturating (where we start in 4.5.2)
produces a weighted average with different, more complex, weighting scheme
(instead of the histogram of X as for marginal LATE, it's the histogram
times the variance of conditional-on-covs first-stage, as in Thm 4.5.1).
In practice, tho, w/o too much heterogeneity, we don't expect weighting
this way or that to be a big deal.  On the other hand, even under constant
effects, covs may matter big time when there's substantial omitted variables.
bias. Seeing that randomly assigned instrument generates IV estimates
invariant to covs makes me happy - as always, its the OVB I worry about first!  

So to be specific - Nikhil asks if he should worry when IV ests are
sensitive to covs - I'd say, yes, worry a little.
Try to figure out if what you thought was a good instrument is in
fact highly confounded with covariates. If so, its maybe not such a
great experiment after all.   If not, then perhaps the senitivity you're
seeing is just a difference in weighting schemes at work

JA