# How to deal with weak instrument with GMM estimator?

I use the two-step system GMM estimator (panel data) and I get the following results:

GMM-type (missing=0, separate instruments for each period unless collapsed)
D.(Tobinsq pourc_femmes2)

Arellano-Bond test for AR(1) in first differences: z =  -2.53   Pr > z  =   0.011
Arellano-Bond test for AR(2) in first differences: z =  -2.05   Pr > z  =   0.041

Sargan test of overid. restrictions: chi2(34)   = 326.03  Prob  > chi2  =   0.000
(Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(34)   =  44.12  Prob  > chi2  =   0.115
(Robust, but weakened by many instruments.)


I think that I have a weak instrument (yt-1). How can I deal with this problem? Do I have to add other instruments?

The tests you're seeing are tests of over-identifying restrictions, in other words tests of $E[\Delta \epsilon_{i,t} y_{i,s}] = 0$. Rejection of the Hansen statistic (a standard $J$-test) suggests that there's some subset of instruments for which $E[\Delta \epsilon_{i,t} y_{i,s}] \neq 0$. In other words, your instruments appear to be invalid instead of weak. This means that instead of looking for more instruments, you should try to limit the instruments you're using. For instance, try just the difference estimator if you're using system, or cut down the number of lags. However, this rejection may mean the Arellano-Bond model has been rejected, so treat your results with some caution.

If you still suspect you're instruments are weak you can take a look at this.