The ‘difference’ and ‘system’ GMM estimator for dynamic panel models can be selected if there are endogeneity of multiple regressors. This estimator also avoids dynamic panel bias (Nickell, 1981). However, it is important to choose appropriate instruments to use in the model. It is an art to find the correct instruments to use. Problems can arise by over-fitting instruments, as well as by using weak instruments.
For example, if all available lagged instruments are used, then the number of instruments grows rapidly with the time dimension of the panel. If too many instruments overfit the endogenous variable, then the effect of the endogenous variable cannot be removed. It will fail to expunge their endogenous components and bias coefficient estimates. As shown in Alvarezand Arellano (Econometrica, 2003), over-fitting can lead first differenced GMM to converge on Within Groups.
On the other hand, it is possible that all of the instruments used are extremely weak, which can cause a weak instrument problem. In this case, the instruments do not contain much information about the endogenous variable. For example, water consumption is extremely persistent. Hence, the immediate lagged levels of consumption are weak instruments.
The Hansen test has a null hypothesis of ‘the instruments as a group are exogenous’. Generally is not easy to find suitable instruments, anything close to or above 0.05 of the p-value of the Hansen statistic is may be considered.