st: ivreg2 and endogeneity

[Michele Mancini <mikmancinistata@gmail.com>]

Dear Statalisters,
I have some questions regarding my ivreg2 estimation:
ivreg2 exp a_pre tfp_pre llab_pre dsett* dwave* ( fd = X1 X2 X3 X4), robust
ffirst
and this is what I get:

Variable     Shea Partial R2    Partial R2      F(  4,  4192)    P-value
fd               0.8095           0.8095           2731.51       0.0000

NB: first-stage F-stat heteroskedasticity-robust

Underidentification tests
Ho: matrix of reduced form coefficients has rank=K1-1 (underidentified)
Ha: matrix has rank=K1 (identified)
Kleibergen-Paap rk LM statistic             Chi-sq(4)=1006.28  P-val=0.0000
Kleibergen-Paap rk Wald statistic           Chi-sq(4)=11004.22 P-val=0.0000

Weak identification test
Ho: equation is weakly identified
Kleibergen-Paap Wald rk F statistic              2731.51
See main output for Cragg-Donald weak id test critical values

Weak-instrument-robust inference
Tests of joint significance of endogenous regressors B1 in main equation
Ho: B1=0 and overidentifying restrictions are valid
Anderson-Rubin Wald test     F(4,4192)=9.37      P-val=0.0000
Anderson-Rubin Wald test     Chi-sq(4)=37.77     P-val=0.0000
Stock-Wright LM S statistic  Chi-sq(4)=35.45     P-val=0.0000

NB: Underidentification, weak identification and weak-identification-robust
test statistics heteroskedasticity-robust

Number of observations               N  =       4222
Number of regressors                 K  =         27
Number of instruments                L  =         30
Number of excluded instruments       L1 =          4


Underidentification test (Kleibergen-Paap rk LM statistic):
1006.284
Chi-sq(4) P-val =    0.0000

Weak identification test (Kleibergen-Paap rk Wald F statistic):
2731.507
Stock-Yogo weak ID test critical values:  5% maximal IV relative bias
16.85
10% maximal IV relative bias    10.27
20% maximal IV relative bias     6.71
30% maximal IV relative bias     5.34
10% maximal IV size             24.58
15% maximal IV size             13.96
20% maximal IV size             10.26
25% maximal IV size              8.31
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):
14.790
Chi-sq(3) P-val =    0.0020
-endog- option:
Endogeneity test of endogenous regressors:
0.810
Chi-sq(1) P-val =    0.3681
Regressors tested:    fd

My question is the following: as you can see the endogeneity test says that
fd in actually exogenous and the Hansen J statistic strongly reject the
null; should I treat fd as exogenous and use OLS instead of IV? The problem
is that if I remove X1 from the instruments
ivreg2 exp a_pre tfp_pre llab_pre dsett* dwave* (fd = X2 X3 X4), robust
first endog(fd)
I get this:
Partial R-squared of excluded instruments:   0.6199
Test of excluded instruments:
F(  3,  4193) =  2034.84
Prob > F      =   0.0000



Summary results for first-stage regressions


Variable     Shea Partial R2    Partial R2      F(  3,  4193)    P-value
fd               0.6199           0.6199           2034.84       0.0000

NB: first-stage F-stat heteroskedasticity-robust

Underidentification tests
Ho: matrix of reduced form coefficients has rank=K1-1 (underidentified)
Ha: matrix has rank=K1 (identified)
Kleibergen-Paap rk LM statistic             Chi-sq(3)=956.94   P-val=0.0000
Kleibergen-Paap rk Wald statistic           Chi-sq(3)=6146.74  P-val=0.0000

Weak identification test
Ho: equation is weakly identified
Kleibergen-Paap Wald rk F statistic              2034.84
See main output for Cragg-Donald weak id test critical values

Weak-instrument-robust inference
Tests of joint significance of endogenous regressors B1 in main equation
Ho: B1=0 and overidentifying restrictions are valid
Anderson-Rubin Wald test     F(3,4193)=2.59      P-val=0.0510
Anderson-Rubin Wald test     Chi-sq(3)=7.83      P-val=0.0497
Stock-Wright LM S statistic  Chi-sq(3)=7.72      P-val=0.0522

NB: Underidentification, weak identification and weak-identification-robust
test statistics heteroskedasticity-robust

Number of observations               N  =       4222
Number of regressors                 K  =         27
Number of instruments                L  =         29
Number of excluded instruments       L1 =          3


Underidentification test (Kleibergen-Paap rk LM statistic):
956.941
Chi-sq(3) P-val =    0.0000

Weak identification test (Kleibergen-Paap rk Wald F statistic):
2034.840
Stock-Yogo weak ID test critical values:  5% maximal IV relative bias
13.91
10% maximal IV relative bias     9.08
20% maximal IV relative bias     6.46
30% maximal IV relative bias     5.39
10% maximal IV size             22.30
15% maximal IV size             12.83
20% maximal IV size              9.54
25% maximal IV size              7.80
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):
0.189
Chi-sq(2) P-val =    0.9098
-endog- option:
Endogeneity test of endogenous regressors:
4.937
Chi-sq(1) P-val =    0.0263
Regressors tested:    fd

So now fd is endogenous and the null of the Hansen test is accepted.
Regarding X1, I can exclude it using orthog(X1):
ivreg2 exp a_pre tfp_pre llab_pre dsett* dwave* (fd = X1 X2 X3 X4), robust
first endog(fd) orthog(X1)
and the result is
-orthog- option:
Hansen J statistic (eqn. excluding suspect orthog. conditions):        0.189
Chi-sq(2) P-val    =    0.9096
C statistic (exogeneity/orthogonality of suspect instruments):        14.600
Chi-sq(1) P-val    =    0.0001

How can i interpret these results of the C statistics? Should I drop X1 and
treat fd as endogenous? Or using OLS instead of IV?
Thank you so much for your kind help.
Regards,

Michele Mancini

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