the coefficient on the regressor of interest is given by . Substituting for gives where is what the estimated coefficient vector would be if x were not correlated with u. In this case, it can be shown that is an unbiased estimator of If in the underlying model that we believe, then OLS gives a coefficient which does not reflect the underlying causal effect of interest. IV helps to fix this problem by identifying the parameters not based on whether is uncorrelated with , but based on whether another variable is uncorrelated with . If theory suggests that is related to (the first stage) but uncorrelated with (the exclusion restriction), then IV may identify the causal parameter of interest where OLS fails. Because there are multiple specific ways of using and deriving IV estimators even in just the linear case (IV, 2SLS, GMM), we save further discussion for the Estimation section below. Example Informally, in attempting to estimate the causal effect of some variable X on another Y, an instrument is a third variable Z which affects Y only through its effect on X. For example, suppose a researcher wishes to estimate the causal effect of smoking on general health. Correlation between health and smoking does not imply that smoking causes poor health because other variables, such as depression, may affect both health and smoking, or because health may affect smoking. It is at best difficult and expensive to conduct controlled experiments on smoking status in the general population. The researcher may attempt to estimate the causal effect of smoking on health from observational data by using the tax rate for tobacco products as an instrument for smoking. The tax rate for tobacco products is a reasonable choice for an instrument because the researcher assumes that it can only be correlated with health through its effect on smoking. If the researcher then finds tobacco taxes and state of health to be correlated, this may be viewed as evidence that smoking causes changes in health. Angrist and Krueger (2001) present a survey of the history and uses of instrumental variable techniques. Graphical definition Of course, IV techniques have been developed among a much broader class of non-linear models. General definitions of instrumental variables, using counterfactual and graphical formalism, were given by Pearl (2000; p. 248). The graphical definition requires that Z satisfy the following conditions: where stands for d-separation and stands