Mastering 'Metrics: The Path from Cause to Effect

by Joshua D. Angrist

CONDITIONAL EXPECTATION The conditional expectation of a variable, Yi, given a dummy variable, Di = 1, is written E[Yi | Di = 1]. This is the average of Yi in the population that has Di equal to 1. Likewise, the conditional expectation of a variable, Yi, given Di = 0, written E[Yi|Di = 0], is the average of Yi in the population that has Di equal to 0. If Yi and Di are variables generated by a random process, such as throwing a die under different circumstances, E[Yi|Di = d] is the average of infinitely many repetitions of this process while holding the circumstances indicated by Di fixed at d. If Yi and Di come from a sample survey, E[Yi|Di = d] is the average computed when everyone in the population who has Di = d is sampled.