st: Computation of standard errors in an IV setting

[Sutirtha Bagchi <sutirthabagchi@gmail.com>]

To Stata users,

I have a question re: obtaining the right standard errors in an IV setting.

Let me first lay it out with generic variable names:

The original equation of interest is:

y1 = x1 + x2 + x3 + e, where x2 is defined as negative of the absolute
value of (0.5 - x1).

The independent variable, x1 is endogenous and hence, x2 is also endogenous.

I have an instrument for x1 - say z1.

Now I would like to estimate the original equation (y1 = x1 + x2 + x3
+ e) using the IV, z1. However, what makes it complicated relative to
a standard IV application is that x2 is a non-linear function of x1.
As a result, while z1 has a monotonic effect on x1, z1 has a
non-monotonic effect on x2.

As a result I am not sure how I can use Stata to obtain the correct
standard errors in this case.

Here is the specific application:

I have data on government deficits (y1) for all 50 U.S. states over a
20-year period. The hypothesis is that higher political competition
(x2) in the state leads to higher deficits. Political competition (x2)
is defined as the negative of the absolute deviation of the Democratic
vote share (x1) from 0.5. In other words, x2 = - abs(0.5 - x1)
(following Besley and co-authors paper in the Review of Economic
Studies, 2010). Thus, a state where Democrats get 80 percent of the
vote is as uncompetitive as a state where Democrats get 20% of the
vote.

I have two instruments that are good predictors of Democratic support:
percent of individuals that are Black and percent of individuals that
are members of a union. Democratic support goes up as the fraction of
individuals in the population who are Black and the fraction that are
union members go up.

Now my question is that how do I estimate the original equation
(Deficits = Political competition + Democratic vote share + other
controls + error) using these IVs (Percent of population Black &
Percent of labor force that are union members) and manage to get the
standard errors correct? Note that there is both an "i" and a "t"
dimension to the problem. If it is too hard with two dimensions, I
could drop the t-dimension and be content with just estimating it
using one year of data, thus exploiting the cross-sectional variation
alone.

Thanks for any and all suggestions,
Sutirtha

-- 
PhD Candidate, Business Economics,
Stephen M. Ross School of Business,
University of Michigan, Ann Arbor.
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