Unexplained Factors and Their Effects on Second Pass Rsquared'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. This implies an unexplained factor structure in the first pass residuals and, consequently, a large 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. To diagnose this, we propose a statistic that reflects the unexplained factor structure in the first pass residuals. The same argument holds for the second pass t-statistic which are resolved using identification robust factor statistics. Our results question many empirical findings that concern the relationship between expected portfolio returns and (macro-) economic factors.

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