Instruments in estimation methods are variables which are correlated with the explanatory variables but not the error term. Whether or not a proposed instrument set is uncorrelated with the error term is tested using statistics like the Hausman statistic and Sargan statistic. Choosing sets of instruments can be difficult, and sometimes seems to be a brute force and ignorance procedure, where one tries a set, sees if it is accepted on the statistics, and if not tries another, sees if that is accepted, and so on. There are rules for rejecting instruments sets without testing them if they are very likely to be rejected anyway, for example if a variable is known to be influenced by changes in the explained variable, then it is probable that variable should not be used in the instrument set.
Here is a procedure which embeds some more rules for selecting reasonable instrument sets. The procedure is a work in progress - I am still trying to sort out an instrument selection process that is not frustratingly arbitrary - so it may be revised in future, and some of the steps might not work very well. I will explain some of the logic further in future posts, too.
1. Choose a base set.
2. Check the Hausman or Sargan statistics.
3. If the statistics have low probabilities, try removing the variables with low significance. If the probabilities rise, then the low significance may occur because the omitted variables are related to the instruments but should not be in the regressions.
4. If step 3 is not possible because there are no insignificant variables, then try using older lags of the variables in the instrument set. If the probabilities do not rise, the equation specification may contain omitted lags itself, or an incorrect functional form.
5. Try different instrument sets entirely.
6. Repeat from stage 2.
Every unsuccessful loop of the procedure increases the likelihood of misspecification.