for the graph in which all arrows entering X are cut off. The counterfactual definition requires that Z satisfies where Yx stands for the value that Y would attain had X been x and stands for independence. If there are additional covariates W then the above definitions are modified so that Z qualifies as an instrument if the given criteria hold conditional on W. The essence of Pearl's definition is: The equations of interest are "structural," not "regression". The error term U stands for all exogenous factors that affect Y when X is held constant. The instrument Z should be independent of U. The instrument Z should not affect Y when X is held constant (exclusion restriction). The instrument Z should not be independent of X. These conditions do not rely on specific functional form of the equations and are applicable therefore to nonlinear equations, where U can be non-additive (see Non-parametric analysis). They are also applicable to a system of multiple equations, in which X (and other factors) affect Y through several intermediate variables. An instrumental variable need not be a cause of X; a proxy of such cause may also be used, if it satisfies conditions 1–5. The exclusion restriction (condition 4) is redundant; it follows from conditions 2 and 3. Selecting suitable instruments Since U is unobserved, the requirement that Z be independent of U cannot be inferred from data and must instead be determined from the model structure, i.e., the data-generating process. Causal graphs are a representation of this structure, and the graphical definition given above can be used to quickly determine whether a variable Z qualifies as an instrumental variable given a set of covariates W. To see how, consider the following example. Suppose that we wish to estimate the effect of a university tutoring program on grade point average (GPA). The relationship between attending the tutoring program and GPA may be confounded by a number of factors. Students who attend the tutoring program may care more about their grades or may be struggling with their work. This confounding is depicted in the Figures 1–3 on the right through the bidirected arc between Tutoring Program and GPA. If students are assigned to dormitories at random, the proximity of the student's dorm to the tutoring program is a natural candidate for being an instrumental variable. However, what if the tutoring program is located in the college