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

On 6/21/11 12:44 PM, 晖 曹 wrote:

On page 75:  [p(Xi=x|Di=1)(1-p(Xi=x|Di=1)] should be [p(Di=1|Xi=x)(1-P(Di=1|Xi=x)}

Yes indeed!

here’s a few more from Aron “eagle-eye” Tobias at Yale:

The book is really great but I need not tell you that since you
probably know that already. What I would like to share with you is
some possible typos (all are really minor, though):

 * pp. 123--125. There are multiple minor inconsistencies between
figures reported in table 4.1.1 and the text. On p. 123, it is claimed
that "[i]n both cases, the estimated return to schooling is around
.075[,]" whereas the figures reported in columns 1 and 2 in table
4.1.1 are .071 and .067, respectively. Similarly, the assertion that
"[t]hese estimates range from .10 to .11[,]" appears on the same page
but columns 3 and 4 in the table indicate .102 and .130, respectively.
In addition, on p. 125, "...the standard error declines from .019 to
.016 as we move from column 6 to column 7" but the table indicates
that it declines from .020 to .016, presumably because of the figures
in the table and the text being rounded differently.

* p. 130 "The Wald estimate of the effect of military service on 1981
earnings, reported in column 4[;]" in fact, they are reported in
column 5.

* p. 131. By "...defined using the 1952 lottery cutoff 95..." you
probably mean the 1972 lottery.

* p. 132. "...a multiple third birth increases this proportion to 1"
should be "...a multiple second birth [i.e. a second birth in which
the second and third babies are twin-born] increases this proportion
to 1."

* p. 351. The Hausman (2001) reference has the title of the journal
wrong (Journal of Econometric Perspectives should be The Journal of
Economic Perspectives).

I apologize in advance if any of my claims turns out to be erroneous.

I remain your faithful reader,
Aron Tobias
Ph.D. Student, Department of Economics, Yale University

thanks Aron
All I can say is, how come Steve didn't catch these last year?!


JA

Which isn’t to say there are no more mistakes, so keep those corrections coming.

JA

Steve must have been dozin’ on his galleys by this point.

How should you construct a first stage F stat to measure instrument strength when you have more than one endogenous variable?  Not by following the instructions we gave at the bottom of page 218.  Althought the theoretical expressions that motivate the p. 218 procedure are right, the computational algorithm we gave is not.

Specifically, where it says:

“regress the first-stage fitted values (for x_2) on the other first-stage fitted values and any exogenous covs . . .”

it should read:

“regress x_2 on the other first stage fitted values and any exogenous covs . . .”

If you do what we originally wrote, you’ll get an R2 of one, always a cause for concern.

Thanks guys for cleaning this up!

Thanks Raphael – this is indeed the wrong likelihood, so don’t try to maximize that at home, folks.  It works only if you don’t have an endogenous regressor in the first place.  For the correct biprobit likelihood, see, e.g., pp. 849-851 in Greene (2007) or better yet, just do it in stata using biprobit (if you must).

This of course raises the question of how we came to make such a mistake.  Is it because Angrist has such a strong aversion to latent-index models that he couldn’t stand the sight of the full likelihood?  Or is it just another silly mistake Steve missed in galleys?

Thanks to Peter Dizikes for pointing this out!