Selecting instruments in GMM

An instrument is a variable which is not correlated with the error term in an estimation model and is related in some way with the other variables in the model. This definition is mine and relates to generalised method of moments estimation, and is a bit broader than the classical definition described in a second. If one has enough instruments, it is possible to estimate the parameter values in an equation, such as b in y(t) = b.y(t-1) + error, without the estimation biases which would arise if they were estimated using the estimation variables correlated with the error term.

For example, if the error is positively correlated with y(t-1), then when y(t-1) is large, the error is large too and so b will tend to be estimated as being larger than it really is. It might not be y(t-1) which is increasing the size of the error (when one might say that the parameter estimate is sensible), but rather some missing variables which are correlated with y(t-1). So an incomplete model specification leads to a bias in b. Instruments overcome the problem since they only capture the effect of the y(t-1) which is not due to the missing variables.

Although the tendency in empirical economic growth analysis using GMM estimators is to select instruments from lagged variables within the estimation model, it is equally valid to choose variables which are not included among them. One important criterion in the case of GMM system and GMM difference estimators is that the instrumental variables should be at least moderately correlated with the variables in the estimation, to minimise the uncertainty of the resulting estimates of the estimation variables' effects on growth. In this respect, the two GMM estimators are the same as classical instrumental variable estimators, for whom instruments should preferably be highly correlated with the estimation variables.

The moment conditions which give rise to GMM estimation can also admit more complex conditions relating the instruments to the estimation variables. For example, an instrument may have to have its squared value highly correlated with the squared value of the estimation variables. The instrumental relationships required may also change with different variables. GMM instruments are a generalisation of classical instruments.