Non-statistically significant effect of the instrument in the reduced form of the 2SLS I am using a 2SLS, with two endogenous variables and two instruments.
I conduct the Under- and the Weak-identification tests; results from these tests suggest that my instruments are not weak.
Moreover, in the two first stages (i.e. regression of the endogenous variable on the instrument), the instruments have always a statistically significant effect on the endogenous variable.
Finally, although I cannot conduct that over-identification test (i.e. I have as many endogenous variables as instruments), my arguments on the validity of the two instruments are solid (grounded on the literature).
In the second stage, the two predicted variables from the first stage have a statistically significant effect on the outcome.
It seems like things are working just fine. However, my question is this:
In the reduced form (i.e. regression of the outcome variable on the instruments), one of the two instruments does not have a statistically significant effect on the outcome variable. What does that imply?
It seems like displaying the results from the reduced form of the 2SLS might be needed to show that you are not "inventing" the results (my own interpretation of the third answer here) and thus just to reassure the referee (in the case you are working to revise a paper submitted to a journal). I cannot evaluate this answer, but might it hide some truth? (i.e. there might be no technical reason nowadays)
I have read somewhere that the reduced form might be used as an additional indirect way to test whether the instruments are weak; see section 11 of this document or see this short article. However, I am not sure I fully get the content of these two sources just now.
[Reduced form is used for the wald estimate when you have one endogenous variable and one instrument, but it is not my case...so I am wondering whether I might not even report the reduced form in the table with the 2sls output]
 A: "...first stages are all ok" is usually decided by some empirical rule of thumb that's more or less arbitrary.

In the reduced form (i.e. regression of the outcome variable on the
instruments), one of the two instruments does not have a statistically
significant effect on the outcome variable. What does that imply?

It need not imply the instrument is weak (which seems to be the conclusion you're driving at).
One possibility is that the first stage regression gives an $R^2$ that's not large (but a sufficiently large F-statistic to tell you "...ok"). This can easily happen when, for example, sample size is large. So there is enough variation in the instrument that is not explained by the endogenous regressor. As a result, in the reduced form regression, this unrelated variation washes out the significance you see in the second stage. (In the second stage, you're regressing on the part of the instrument that's fitted by the regressor. In the reduced form,  the part of the instrument that's not explained by the regressor also plays a role. )
In any case, same example shows that non-significance in reduced form regression should not be taken as a sufficient condition for weak instrument. Instead, it's probably more appropriate to invoke reduced form when it's known that the instrument is weak.
