Is Error Term Residual?

Abstract

Following Fama-French (1993), most researchers try to find new risk factors to complement the Fama-French three factors model. Most of them implement by ranking on the desirable risk factors or regression on the risk factors, and then check the beta or risk premium is significant from zero or not, and assess by the increment of the R2 or/and closer of alpha to zero. However, does adding new factors can really solve the puzzle? Unfortunately, this paper’s answers is no. By regression individual’s (or portfolio’s) excess return on the risk factors and rank the error term from the regression, then construct portfolios based on the ranking, you will get the result very similar to that you directly rank the average return of the portfolios. And by constructing portfolios with various kinds of strategies the result is unchanged and robust. That is similar error term have similar return, but this pattern isn’t persistent. When you rank on lag of (at least 1-12 is) error term or return this pattern disappears. This again, however, means no risk factors missing. This is because if there is missing factor, this pattern may survive when rank on lag due to the persistence of risk factor to some extent. Therefore, this pattern is like a puzzle.