Why are There So Many Dummies?

Lina from Essex writes:

When talking about grouped data and 2SLS (section 4.1.3) you mention
that expanding a continuous instrument is equivalent to have a set of
Wald estimators that consistent estimates the causal effect of
interest and in the Vietnam paper you mention that using the whole set
of dummies as instruments is more efficient.  I was wondering whether
using grouped data and instrumenting by the set of dummies for
different values of the continuous instrument differ from using the
continuous instrument (i.e. in your case using the continuous RSN). Is
there any gain of efficiency in the estimation? or is it just to
interpret the result under the set of Wald estimators? In other words.
If you have the continuous instrument why would you expand it? and
have over identification?. Thank you very much!!!, all the best.

Good question Lina.  One answer is the conceptual appeal of putting 
together Wald estimators.  Takes the mystery out of 2SLS!  But there 
is a more formal argument for dummying out intervals of a continuous 
instrument and then doing 2SLS with the dummies. As discussed in 
Section 4.1.3, in a homoskedastic constant-effects model with a 
continuous instrument, the efficient  method of moments estimator uses 
the (unknown) E[D|Z] as an instrument, where D is the variable to be 
instrumented and Z is the continuous instrument.  You
can think of a model with many dummies for intervals of Z as a 
nonparametric approximation to this efficient but infeasible procedure. 
Just using Z itself as an instrument would be a ** parametric ** approx
and therefore, perhaps, not as good. Of course, you could add polynomials 
in Z for a similar nonparametric flavor, but the first stage would be ugly, 
and as you conjecture, we would lose the conceptual appeal of combining 
Wald estimators.

My 1990 draft lottery paper shows this reasoning in action.  
See Newey (1990) for the theory.

JA