In this paper we investigate the properties of the standard two-pass method
ology of testing beta pricing models with misspecified factors. In a settin
g where a factor is useless, defined as being independent of all the asset
returns, we provide theoretical results and simulation evidence that the se
cond-pass cross-sectional regression tends to find the beta risk of the use
less factor priced more often than it should. More surprisingly, this missp
ecification bias exacerbates when the number of time series observations in
creases. Possible ways of detecting useless factors are also examined.