Unexplained factors and their effects on second pass R-squared's and t-tests

We construct the large sample distributions of the OLS and GLS R2’s of the second pass
regression of the Fama-MacBeth (1973) two pass procedure when the observed proxy factors are
minorly correlated with the true unobserved factors. The small correlation implies a sizeable
unexplained factor structure in the first pass residuals and, consequently, a sizeable estimation
error in the estimated beta’s which is spanned by the beta’s of the unexplained true factors. The
average portfolio returns and the estimation error of the estimated beta’s are then both linear in
the beta’s of the unobserved true factors which leads to possibly large values of the OLS R2 of the
second pass regression. These large values of the OLS R2 are not indicative of the strength of the
relationship between the expected portfolio returns and the (macro-) economic factors. We propose
an easy manner for diagnosing it using a statistic that reflects the unexplained factor structure in
the first pass residuals. Similar arguments apply to the second pass t-statistic which are resolved
using the identification robust factor statistics of Kleibergen (2009). Our results put into question
many of the empirical findings that concern the relationship between expected portfolio returns
and (macro-) economic factors. We discuss some prominent ones in passing.