to the equation involving in the introduction (this is the matrix version of that equation). When X and e are uncorrelated, under certain regularity conditions the second term has an expected value conditional on X of zero and converges to zero in the limit, so the estimator is unbiased and consistent. When X and the other unmeasured, causal variables collapsed into the e term are correlated, however, the OLS estimator is generally biased and inconsistent for β. In this case, it is valid to use the estimates to predict values of y given values of X, but the estimate does not recover the causal effect of X on y. To recover the underlying parameter , we introduce a set of variables Z that is highly correlated with each endogenous component of X but (in our underlying model) is not correlated with e. For simplicity, one might consider X to be a T × 2 matrix composed of a column of constants and one endogenous variable, and Z to be a T × 2 consisting of a column of constants and one instrumental variable. However, this technique generalizes to X being a matrix of a constant and, say, 5 endogenous variables, with Z being a matrix composed of a constant and 5 instruments. In the discussion that follows, we will assume that X is a T × K matrix and leave this value K unspecified. An estimator in which X and Z are both T × K matrices is referred to as just-identified . Suppose that the relationship between each endogenous component xi and the instruments is given by The most common IV specification uses the following estimator: This specification approaches the true parameter as the sample gets large, so long as in the true model: As long as in the underlying process which generates the data, the appropriate use of the IV estimator will identify this parameter. This works because IV solves for the unique parameter that satisfies , and therefore hones in on the true underlying parameter as the sample size grows. Now an extension: suppose that there are more instruments than there are covariates in the equation of interest, so that Z is a T × M matrix with M > K. This is often called the over-identified case. In this case, the generalized method of moments (GMM) can be used. The GMM IV estimator is where refers