Interpret Hansen J statistic and other tests following ivreg2 command in stata

[Celine Tran]

Dear all,

Could you please help me to interpret some tests from ivreg2 command as follows:

Underidentification test (Kleibergen-Paap rk LM statistic):              5.555
                                                   Chi-sq(1) P-val =    0.0184
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Weak identification test (Cragg-Donald Wald F statistic):                6.019
                         (Kleibergen-Paap rk Wald F statistic):          5.656
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.

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Hansen J statistic (overidentification test of all instruments):         0.000
                                                 (equation exactly identified)
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Does it mean that my instrument is correlated with endogenous regressor but it is weak? And p-value of Hansen J statistic is 0.000 --> reject the null hypothesis that the validity of the overridentifying restriction is valid. However, I do not understand what it means? Could you please explain further?

Thank you very much in advance.