Absolving beta of volatility's effects

The beta anomaly, negative (positive) alpha on stocks with high (low) beta, arises from beta's positive correlation with idiosyncratic volatility (IVOL). The relation between IVOL and alpha is positive among underpriced stocks but negative and stronger among overpriced stocks (Stambaugh, Yu, and Yuan, 2015). That stronger negative relation combines with the positive IVOL-beta correlation to produce the beta anomaly. The anomaly is significant only within overpriced stocks and only in periods when the beta-IVOL correlation and the likelihood of overpricing are simultaneously high. Either controlling for IVOL or simply excluding overpriced stocks with high IVOL renders the beta anomaly insignificant.