Instrumental Variables Introduction
Instrumental variables are powerful tools used (most often) by economists to infer causality in the absence of an experiment. Their use is worth the consideration of psychologists hoping to do the same thing.
Instrumental variables are appropriate for use when a covariate (your independent variable) is thought to be correlated with the error term in a regression equation (by definition, an endogenous variable). This is often a concern when an experiment, the “gold standard” for causal analysis, is lacking. An appropriate instrument will allow consistent estimation (estimation of a parameter that converges to the true parameter value as a sample increases to infinity) even without the “gold standard.” However, without such an instrument, an endogenous variable would be inconsistent and unable to approximate the true value of a parameter.
Of note, a researcher should have a theoretical explanation of why a covariate is related to the error term of a regression equation. For example, are there omitted X variables correlated with the error term? Is there possible measurement error in the X variables? Perhaps there is a problem of reverse causality—Y could be causing X instead of vice versa. A selection bias could also present reverse causality. Any or all of these problems could exist in a given regression equation, the point is, know which one(s) likely exist in order to make theoretical sense for the use of an instrumental variable.
What makes a good instrument? An appropriate instrument (or instruments) consists of a variable (or variables) that is (are) correlated with the aforementioned endogenous variable, but is (are) not correlated with the error term of the equation. In other words, an instrument(s) should not relate to the same possible omitted variables or have the same problem with reverse causation as the covariate, predictor variable in question.
For a good video introduction on the subject, see here. Alternatively, see the fourth chapter of Angrist and Pischke’s book, Mostly Harmless Econometrics (2009).