Dear users,

for my thesis I'm working with an IV regression, where I try to see what effect stock option compensation has on innovation of the company.
As an instrument for stock option compensation I use the predicted first year of the cycle. Which I've done below.



ivreghdfe xrd_w (option_value=predictedfirstyear) if hitech==1, a(fyear sic) cluster(gvkey)
(MWFE estimator converged in 4 iterations)

IV (2SLS) estimation
--------------------

Estimates efficient for homoskedasticity only
Statistics robust to heteroskedasticity and clustering on gvkey

Number of clusters (gvkey) = 343 Number of obs = 17037
F( 1, 342) = 0.10
Prob > F = 0.7491
Total (centered) SS = 5.37069e+10 Centered R2 = 0.0087
Total (uncentered) SS = 5.37069e+10 Uncentered R2 = 0.0087
Residual SS = 5.32398e+10 Root MSE = 1769

------------------------------------------------------------------------------
| Robust
xrd_w | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
option_value | .0297179 .0928436 0.32 0.749 -.1528984 .2123342
------------------------------------------------------------------------------
Underidentification test (Kleibergen-Paap rk LM statistic): 2.060
Chi-sq(1) P-val = 0.1512
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic): 238.727
(Kleibergen-Paap rk Wald F statistic): 2.191
Stock-Yogo weak ID test critical values: 10% maximal IV size 16.38
15% maximal IV size 8.96
20% maximal IV size 6.66
25% maximal IV size 5.53
Source: Stock-Yogo (2005). Reproduced by permission.
NB: Critical values are for Cragg-Donald F statistic and i.i.d. errors.
------------------------------------------------------------------------------
Hansen J statistic (overidentification test of all instruments): 0.000
(equation exactly identified)
------------------------------------------------------------------------------




However, now my question is that for my first stage I have found:



//first stage: instrument regressions
. reghdfe option_value predictedfirstyear, cl(gvkey) a(fyear)
(MWFE estimator converged in 1 iterations)

HDFE Linear regression Number of obs = 141,526
Absorbing 1 HDFE group F( 1, 2585) = 23.06
Statistics robust to heteroskedasticity Prob > F = 0.0000
R-squared = 0.0006
Adj R-squared = 0.0004
Within R-sq. = 0.0004
Number of clusters (gvkey) = 2,586 Root MSE = 6317.1100

(Std. Err. adjusted for 2,586 clusters in gvkey)
------------------------------------------------------------------------------------
| Robust
option_value | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------------+----------------------------------------------------------------
predictedfirstyear | 424.6596 88.43824 4.80 0.000 251.2426 598.0766
_cons | 421.2141 24.09054 17.48 0.000 373.9754 468.4528
------------------------------------------------------------------------------------


With an F-statistic of 23.06 I thought I could reject that my instrument is weak with Staiger and Stock's rule of thumb with F>10.
However, I now have trouble with interpreting the Kleibergen-Paap, Cragg-Donald and Stock-Yogo results in the second stage.
My questions are:
1. For the Kleibergen-Paap rk LM statistic is it true that when you are looking at underidentification you're testing if the instrument is irrelevant and therefore in this case with a p-value of 0.1512 I'm rejecting that the instrument is irrelevant?
2. For the Cragg-Donald Wald F-stat identification test, is it true that I'm again looking at whether the instrument is weak, the same way I did in the first stage regression and with an F-stat of 238.727 can reject that the instrument is weak?
3. How does the Kleibergen-Paap rk Wald F statistic differ from the Cragg-Donald Wald F statistic?
4. Is it true that SY’s tests can be used with multiple endogenous regressors and multiple instruments and therefore should not be used in this case, where I have only 1 instrument.

I would greatly appreciate all help I can get.