Instrument selection is an extra modelling equation

I mentioned in a September post that when a growth model is prepared, a well-designed empirical estimation method can still find adequate estimation coefficients despite the incompleteness of the growth model. In a sense, the model is the equations plus the estimation method.

Here is another example. Instrumental variable estimation is a way of avoiding biases in estimation if the determinant variables are correlated with the error term. Ordinary Least Squares estimation gives biased estimates of a in the equation y=a.x+error if x is correlated with the error term.

OLS can be characterised in terms of the orthogonality condition E(x.error)=0. Then the estimate of a is x.y/x.x, and inserting the value of y and taking expectations shows this formula is unbiased if the orthogonality condition holds. The corresponding orthogonality condition for instrumental variables is that E(v.error)=0 where v is the instrumental variable, and the estimator is v.y/v.x.

The orthogonality approach is neat, but it is also helpful to consider the structure of the estimator as the projection of y on v divided by the projection of x on v. The derivation of instrumental variable estimators is given in many econometrics textbooks, often in terms of estimation when simultaneous equations give rise to the single observed relation. This happens when two variables can interact in more than one way, and the error term becomes correlated with the observed determinant variable.

Selection of an instrumental variable gets rid of the bias and selects one form of interaction between the variables, that which acts through the instrumented variable. So changing instruments can not only alter the biases on an estimated coefficient, but also change what is being modelled. It is like introducing a new equation into the model.