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Abstract
Tests of rational expectations ask whether the hypothesis can be rejected. This paper asks how much of a population could still be rationalized as satisfying it. In a mixture of a rational component and an unrestricted alternative, the mixing weight is bounded above but never below. The upper bound has a closed form for binary outcomes. In the Health and Retirement Study, full rational expectations is rejected decisively, yet at most 97.76 percent can be rationalized given the reported survival belief, and 93.50 percent once beliefs must also be efficient with respect to own sex, age, health and smoking. Testing whether that share is below one is exact in finite samples, and measuring how far below is not. The bound moves with the information asserted, the outcome forecast, and the population studied. On the same respondents, beliefs about working past sixty-two leave 11.38 percent non-rationalizable where survival beliefs leave 0.32.