st: Weak instruments in models with more than one endogenous variable

[Daifeng He <dhe.statlist@gmail.com>]

Dear all,

I have a question about which statistics to look at for weak instrument in
models with more than one endogenous variable. In particular, I have the
following model:

y1=a0+a1*y2+ a2*X2 +e1
y2=b0+b1*Z  +b2*X2 +e2

Where y1 is the outcome variable of interest, y2 is an endogenous
explanatory variable, X2 is another set of exogenous controls, and Z
is an instrument for Y2.

Because I want to know whether the effect of y2 on y1 is similar, say,
across race (Black, White, and Hispanic), I interact the instrument Z with
the Hispanic and White indicators, and then use the three instruments (Z,
Z*Hispanic, Z*Black) to instrument for y2, y2*Hispanic, and y2*Black in the
full sample.  (I do not want to split the sample by race and estimate the
above model separately for each race because my instrument becomes very
weak in the Hispanic and Black subsamples given the much smaller sample
sizes).

My questions are:

(1)    What is the right statistics to look at for weak instruments?  The
A-F statistics which examines whether a particular instrument is weak (so I
should look at the A-F statistics separately for each of the two
instruments of interest, Z*Hispanic and Z*Black)?

Or I should use the F version of the Cragg-Donald Wald Statistics suggested
by the -ivreg2- help file (I am clustering standard errors), which is to
test whether the three instruments are jointly weak?

(2)    Currently my results show quite large A-F statistics (above 10) but
very small Cragg-Donald F-statistcs (in the magnitude of 0.0x).   Does
this inconsistency (or non-robustness) signal any underlying problems of my
model?

(3)    If I were to look at the A-F statics, what should I conclude about
the instrument strength if one of the A-F statistics (say for Z*black) is
large (above 15) but the other one (for say Z*Hispanic) is much smaller
(about 5)?  -- I run into this issue when I looked at the heterogeneous
effect across income tertiles instead across race.

(4)    Finally, is there any source that contains the critical values for
either of those statistics?

If you know the answer to any of my questions or can point me to some
source for further reading, I'd appreciate a great deal!

Thank you.

Daifeng
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