We have N companies with incomplete monthly data. For our research, we aim to run monthly cross$-$sectional regression where we take only companies each month where we have available data. In our regression, we regress excess stock returns on the Tax planning measure, and other control variables. Our tax planning measure is Cash effective tax rate $($CETR$),$ where lower the rate means aggressive tax planner the company is$.$ Then we extract coefficients from monthly regression for Tax planning measure $($Monthly betas for tax planning$)$ and regress them on Fama French five $-$factors in step $2$ as time series regression to analyse if Tax planning pays premia. We think if we get significant negative alpha it would mean that tax planning is associated with premia.

Question $1$: Question: Is the approach correct? If not, what would be the correct way to do this to find if the companies that do tax planning pays a premia to investor.

Question $2$: Given that a lower $\backslash$beta eta in the cross sectional regression indicates higher returns for lower CETR, a negative intercept in the time$-$series regression implies that the high returns associated with aggressive tax planning $($lower CETR$)$ are not fully explained by the Fama$-$French factors. Then how would we interpretate the other factor loading $?$