Instrumental variables methods are the backbone of causal inference because they can solve a wide variety of very thorny inference problems, including selection bias or measurement error in x, under some restrictive assumptions. The ideal setting for instrumental variables is an experiment, where a treatment is randomly assigned, but perhaps not every case assigned to treatment gets treated, and some cases not assigned to treatment do get treated. If we are interested in estimating the effect of treatment rather the effect of assignment to treatment, we can rely on instrumental variables.
To give an unbiased and consistent estimate of impact, a linear regression estimating y = Xb+e needs X to be uncorrelated with the error term, e. When a variable x in the set X is correlated with e, we say that x is endogenous and its associated coefficient is inconsistently estimated, which can arise from selection bias or measurement error in x . If we can find another variable, z, that is not correlated with e but is strongly correlated with x, we can use z as an instrument for x. This is trivially true in an experiment, where assignment to treatment z is random, so we know z cannot be correlated with e but will be strongly correlated with treatment x.
There are other situations where a “natural experiment” arises in which factors not correlated with individuals’ unobserved characteristics (all of which are absorbed into the error term e in a linear regression) can be used as in instrument for participation in treatment. For example, living near a hospital that has more psychiatric service intensity than an average hospital will tend to increase by a small amount the probability of getting any psychiatric treatment, and can be used as an instrument for receipt of treatment (Loprest and Nichols 2008). Living near a charter school in Washington, DC, can be used as an instrument for attending a charter (Nichols and Özek 2010).
For this method to work well, the excluded instrument z has to be entirely uncorrelated with the error e in the equation y = Xb+e, and it needs to be very strongly correlated with the endogenous treatment x. Even a small amount of positive or negative correlation between z and e can render instrumental variables methods worse than an uncorrected linear regression, and a weak correlation between z and x can produce higher levels of bias and size distortions (inference where the standard errors estimates are far smaller than they should be); see Nichols (2007) for further reading.