We introduce nonnested hypotheses tests using indirect, simulation-based estimation procedures. We propose different test statistics and carefully examine the corresponding implicit mi hypotheses. A notion of indirect encompassing comes up naturally, which may be compared to the usual encompassing conditions. Then we show how these methods extend standard specification and overidentification tests.

Evaluating competing multifactor asset pricing models involves comparing the statistical significance of their mean pricing errors (alphas). Unfortunately, this comparison favors imprecisely estimated models because p-values tend to be higher in more noisy models. To avoid false impressions of relative success at tests for zero mean pricing errors, we develop a notion of comparative p-values and suggest comparing these instead of the raw p-values. This comparison gives more precisely estimated models a fairer chance or, equivalently, quantifies how much easier it is for imprecisely estimated models, by comparison, to pass the test.