According to Fama & MacBeth (1973) two-step regression, you start with estimating the beta factors. When applying the Fama-French 3-Factor model, you first run the linear regression
$$r_{i,t}=α_i+β_{i,MktRf}MktRf_t+β_{i,SMB}SMB_t+β_{i,HML}HML_t+ϵ_{i,t}$$
to estimate the corresponding factor loadings.
The second step is a cross-section regression for each t : $$r_{i,t}=λ_0+\hat{β}_iλ_t+α_{i,t}$$ with $\hat{β}_i≡[β_{i,MktRf},β_{i,SMB},β_{i,HML}]′$ as the estimated factor loadings from the first step.
The Wikipedia article describes the second step as follows:
Then regress all asset returns for a fixed time period against the estimated betas to determine the risk premium for each factor.
So in fact, the average value of the estimated $λ_t$ can be interpreted as the corresponding risk premium for each $β_{i,MktRf}$, $β_{i,SMB}$ and $β_{i,HML}$.
Question
I use data from Kenneth French`s website on the Fama-French portfolios for estimating the factor loadings in the first step of the regression. As far as i know, the data from Kenneth French are already the risk premium of the factors $MktRf$, $SMB$ and $HML$.
Can i just use the time-series data from Kenneth French, as they already are risk premiums on the corresponding portfolios, and interpret their average value as the estimated values of $λ_t$ following Fama & MacBeth regression?
Why should the results be different, if using Kenneth French data as input in the first step of Fama & MacBeth regression (when estimating the factor loadings following Fama & French 3 factor model) and then estimating the risk premiums or directly using Kenneth French data and calculate the average value of risk premiums?