st: ivreg2 questions (shea r2 and AP Statistic)

[Vidhya Soundararajan <vidhyahere@gmail.com>]

Dear statalist users,

I have a model with three endogenous variables. Three because I have
one endogenous variable (E), another that is an interaction of E with
an included instrument (A), which makes it endogenous (E * A), and
third one which is another interaction term (E*A*A).

My model is:

L = β0 + β1 * A+ β2 * A*A+ β3 * E + β4 * E*A+ β5 * E*A*A+ β6 * X + ε

I have three instruments for these ->  F, F*A, and F*A*A.

There are three first stage regressions and so the test for weak
instruments is not a simple test based on the F-statistic of each of
the first stage regressions. I realize that in the case of multiple
endogenous regressors, I can either look at the Angrist and Pischke
(2009) F-statistics on excluded instruments or the partial Shea R2.

In terms of results of ivreg2, I find that:

1. I find that the F-statistic from individual first stage regressions
for each of the three endogenous regressors are high, which is good!
2. The shea R2 is about 0.5 which is good as well.
3. But the AP F-statistic are very smal (in the order of 0.02)
4. Kleibergen-Paap Wald rk F statistic is 18.81

There is one statalist page which cleanly explains both shea and AP
statistics and what the difference is
(http://www.stata.com/statalist/archive/2010-02/msg01336.html). But it
is also mentioned here that the two are closely related and that AP is
preferred because of well defined distribution for testing purposes.
Even though, they are similar in spirit, I am not sure why my shea R2
is giving me different (better) results than the AP F-stat. Should I
now conclude now that my system is weakly identified? But I am
thinking may be that AP F-statistic is low because my three endogenous
regressors are probably correlated (that two endogenous regressors are
just interaction terms of E)? Does this interaction play a role?

Someone please help me with the interpretation.

Best,
Vidhya

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