# Random Regressors and Moment Based Estimation

In this chapter you will learn to use instrumental variables to model’s parameters when its independent variables are correlated

10.1 Basic Model

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.

Where is a certain sloppiness associated with the use of endogenous in this way, but it has become standard practice in econometrics.