Economics 696F, Causal Inference and Program Evaluation Lecture Note 8: Instrumental Variables Part I

Suppose we are interested in estimating the effect of a treatment T on an outcome Y, and we believe the treatment does not satisfy the random assignment assumption, nor the unconfoundedness assumption. Then the approaches we have talked about up to now will not work. In some cases, we might have an additional variable, call it Z, that is known to be randomly assigned, and is thought to be related to the treatment. Example: suppose that Y is wages, the treatment T is whether or not the individual goes to college, and suppose that some individuals are randomly assigned a tuition subsidy. So Z would be the indicator for the tuition subsidy. We call such a variable Z an instrumental variable, or instrument for short, although perhaps a more descriptive word might be an encouragement. Later, we will define the term instrumental variable more precisely. It seems plausible that the instrument Z could affect the decision whether or not to go to college. Through the college decision, it might therefore also have an impact on wages. This suggests the following simple statistic, called the Wald estimator: b = ̂E(Y |Z = 1) − ̂E(Y |Z = 0) ̂E(T |Z = 1) − ̂E(T |Z = 0) where ̂E(Y |Z = 1) is the sample average of Y among observations with Z = 1, and so on. So we can also write b = YiZi