I have 3 endogenous variables in my model and I have identified 3 (or max 4) instruments. I wanted to know appropriate test for identifying whether the instruments are weak or not given that I have heteroskedaticity (cluster option is being used) in my data. I read that one should employ the Kleibergen-Paap F statistic in this case and compare with Stock and Yogo Critical values but in the case of 3 endogenous variables the table in their paper starts with a min of 5 instruments. I am confused as to which test to employ in this case. Any suggestions would be useful in this regard.
Staiger and Stock (1997) formalized the definition of “weak instruments” and most researchers seem to have concluded (incorrectly) from that work (or hearsay) that if the F-statistic on the excluded instruments in the first stage is greater than 10, one need worry no further about weak instruments.Stock and Yogo (2005) go into more detail and provide useful rules of thumb regarding the weakness of instruments based on a statistic due to Cragg and Donald (1993). Stock, Wright, and Yogo (2002) provide a summary of this work.
Austin Nichols (2006)
While an F-statistic cutoff of 10 may be a crude rule of thumb and not sufficient to conclude that a set of instruments are not weak, Stock, Wright, and Yogo (2002) have a table about weak instrument cutoff values that goes down to 1 instrument. GMM, Weak Instruments, and Weak Identification