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Good day, dear Moderators.

i am estimating the model with instrumental variables using the GMM. I then run the regressor endogeneity test to test for the validity of instruments. The difference in J-stats I received is insignificant and hence, the null is rejected. Do I correctly understand that the null hypothesis under this test is that All instruments are valid and therefore can be used for estimation? Thank you in advance for clarifications. Cheers.

Sorry, I meant the null is accepted (not rejected), since the difference in J-stats is insignificant:

Endogenous variables to treat as exogenous: DEXPATS



Value df Probability
Difference in J-stats 0.005821 1 0.9392

J-statistic summary:
Value
Restricted J-statistic 42.76359
Unrestricted J-statistic 42.75777

Can someone please answer this simple question as to What is the null hypothesis under the Regressor endogenetiy test means? That all the instruments are valid, or the opposite? I have checked all possible manuals and do not see any interpretations examples. Thank you.

Null is no endogeneity.

Thanks Gareth! But does this suggest all the instruments are valid? Also, can J-stat with a related p-value which is available in the GMM output be used instead to make a validity conclusion about the instruments (rejection means instruments are invalid)? Isn't that a more direct way to assess the validity of instruments used to tackle the endogeneity issue? Thanks again!!

The Hausman-etc test does not say anything about whether the instruments are valid.

All right, so it is only used to assess whether or not the endogeneity is of concern in the model? But then why is the outcome of this test changing (accept versus reject the null) as we change the set of instruments? Doesn't the outcome then tell us something about the validity of the instruments used or how effectively they address the endogeneity issue? I have studied a lot of manuals and can't find anything related to this. Besides, this forum has no answer either. In fact, in one of the 2010 posts on of moderators say that the Null under this test is that instruments are valid.... Very confused.

The test assumes that the instruments are valid, it doesn't test that assumption.

As you add valid instruments, you may get a more powerful test so the result may change.

I don't understand. When the null is rejected upon using two instruments (endogeneity is present), but then is accepted when another instrument is added (not present anymore). What does this mean? That endogeneity has been effectively resolved upon introduction of the third instrument or something very different? If endogeneity is present in the original model, it is present, why would the outcome of the Hausman test change so much in opposite directions? Doesn't it have anything to do with the validity of instruments? Thanks again.

It is certainly possible that adding an invalid instrument would switch a rejection to a non-rejection. The way to test for validity of a set of over-identifying instruments, as you know, is to use the J-statistic.

Great, that's what I though, that adding a bad instrument essentially makes an OLS without this instrument preferred. Or at least this is how I interpret it. Thanks a lot again.

It does not mean that OLS is preferred. It means GMM or IV has a problem.

Got you!