library? In that case, Proximity may also cause students to spend more time at the library, which in turn improves their GPA (see Figure 1). Using the causal graph depicted in the Figure 2, we see that Proximity does not qualify as an instrumental variable because it is connected to GPA through the path Proximity Library Hours GPA in . However, if we control for Library Hours by adding it as a covariate then Proximity becomes an instrumental variable, since Proximity is separated from GPA given Library Hours in . Now, suppose that we notice that a student's "natural ability" affects his or her number of hours in the library as well as his or her GPA, as in Figure 3. Using the causal graph, we see that Library Hours is a collider and conditioning on it opens the path Proximity Library Hours GPA. As a result, Proximity cannot be used as an instrumental variable. Finally, suppose that Library Hours does not actually affect GPA because students who do not study in the library simply study elsewhere, as in Figure 4. In this case, controlling for Library Hours still opens a spurious path from Proximity to GPA. However, if we do not control for Library Hours and remove it as a covariate then Proximity can again be used an instrumental variable. Estimation We now revisit and expand upon the mechanics of IV in greater detail. Suppose the data are generated by a process of the form where i indexes observations, is the i-th value of the dependent variable, is a vector of the i-th values of the independent variable(s) and a constant, is the i-th value of an unobserved error term representing all causes of other than , and is an unobserved parameter vector. The parameter vector is the causal effect on of a one unit change in each element of , holding all other causes of constant. The econometric goal is to estimate . For simplicity's sake assume the draws of e are uncorrelated and that they are drawn from distributions with the same variance (that is, that the errors are serially uncorrelated and homoskedastic). Suppose also that a regression model of nominally the same form is proposed. Given a random sample of T observations from this process, the ordinary least squares estimator is where X, y and e denote column vectors of length T. This equation is similar