# pure factor return for factor model

I am reading a paper. The authors use the multivariate regression to calculate the pure factor return $\beta_F$ using the following equation: $$Return_{t+1}=\beta_F f_F + \beta_{RF_1} f_{RF_1} +⋯+ \beta_{RF_N} f_{RF_N} +\epsilon$$

where

• $\beta_F$ = pure factor return for the desired return factor,

• $\beta_{RF_i}$ = pure factor return for Risk factor $i$,

• $f_F$ = evaluated factors (ex: Dividend Yield)

• $f_{RF_i}$ = risk factor (ex: size)

After that, they produce a monthly factor return as the graph here. I just wonder from the pure factor return for the desired return factor, how they can get the return over months like the graph?

From that picture you took, it looks like the $\beta_F$ is a time series. It's computed by doing a cross-sectional regression at each point in time.