st: AP F-statistic is 0 with multiple endogenous regressors that are correlated

[Vidhya Soundararajan <vidhyahere@gmail.com>]

Dear statalist users,

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

My model is:

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

I have three instruments for these say F, G, and H. Please note that
these instruments are "not" F, F*MW, and F*MW*MW. Rather, they are
three separate instruments F,G, and H.

There are three first stage regressions and so the test for weak
instruments is not a simple test based on the F-statistic of the first
stage regression but the one based on partialling out other endogenous
regressors from one, as suggested in Angrist and Pischke (2009) in
page 217-218. I am using the ivreg2 command in stata for obtaining
these AP F-statistic and a bunch of other test statistics. The problem
I encounter is that the F-statistic from individual first stage
regressions for each of the three endogenous regressors are high,
which is good! But the AP F-statistic are very small - literally 0.

So I guess I should conclude now that my system is weakly identified
(or even under-identified because the AP F-stat is 0). 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 in why
AP F-stat is 0?

Best,
Vidhya

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