Random Regressors and Moment Based Estimation
Consider the linear regression model
Уі = ві + в2 Xi + ei i = 1,2,...,N (10.1)
Equation (10.1) suffers from a significant violation of the usual model assumptions when its explanatory variable is contemporaneously correlated with the random error, i. e., Cov(ei, xi) = E(eixi) = 0. When a regressor is correlated with the model’s errors, the regressor is often referred to as being endogenous.1 If a model includes an endogenous regressor, least squares is known to be both biased and inconsistent.
An instrument is a variable, z, that is correlated with x but not with the error, e. In addition, the instrument does not directly affect y and thus does not belong in the actual model as a separate regressor. It is common to have more than one instrument for x. All that is required is that these instruments, z1, z2,..., zs, be correlated with x, but not with e. Consistent estimation of (10.1) is possible if one uses the instrumental variables or two-stage least squares estimator, rather than the usual OLS estimator.