This paper studies some seemingly anomalous results that arise in possibly misspecified and unidentified linear asset-pricing models estimated by maximum likelihood and one-step generalized method of moments (GMM). Strikingly, when useless factors (that is, factors that are independent of the returns on the test assets) are present, the models exhibit perfect fit, as measured by the squared correlation between the model's fitted expected returns and the average realized returns, and the tests for correct model specification have asymptotic power that is equal to the nominal size. In other words, applied researchers will erroneously conclude that the model is correctly specified even when the degree of misspecification is arbitrarily large. We also derive the highly nonstandard limiting behavior of these invariant estimators and their t-tests in the presence of identification failure. These results reveal the spurious nature of inference as useless factors are selected with high probability, while useful factors are driven out from the model. The practical relevance of our findings is demonstrated using simulations and an empirical application.
asset pricing irrelevant risk factors unidentified models model misspecification continuously updated GMM maximum likelihood rank test test for overidentifying restrictions