Consider the single factor model in time series form, e.g:
$$ r_t^i = \alpha_i + \beta_i f_t +\epsilon^i_t \tag{1} $$ Here $i$ is not an exponent but a superscript, e.g. it represents the return on security $i$.
Assuming we have a single factor model, which then states that: $$ \mathbb{E}[r^i] =\alpha_i + \lambda\beta_i \tag{2} $$ e.g. the average return for security $i$ is given by a constant times its exposure to the common factor $f$ (whatever it may be). As such, there seem to be two ways of estimating $\lambda$. The first is the rather simple/possibly too simple way - take equation $(1)$ and take empirical time-series averages so that: $$ \frac{1}{T}\sum_{t=1}^T r^i_t = \alpha_i + \beta_i \frac{1}{T}\sum_{t=1}^T f_t + \underbrace{\frac{1}{T}\sum_{t=1}^T \epsilon^i_{t}}_{\text{Assumed to be zero}} $$ so that the estimate of the risk premium is:$$ \hat{\lambda} = \frac{1}{T}\sum_{t=1}^T f_t $$ For example, in the case of CAPM - this would just be the average return of the benchmark/index.
The other method is the two stage, so-called Fama-Macbeth regression, which would estimate the $\beta_i$ for each stock through $N$ different time-series OLS regressions, and then for each estimated coefficient, run $T$ cross-sectional regressions to estimate $\lambda_t$ at each time $t$. It would the consider: $$ \hat{\lambda} = \frac{1}{T} \sum_{t=1}^T\lambda_t $$
My Question: What exactly is the difference between the results obtained in each of these procedures? I understand methodologically they are quite different, but I am wondering how their estimates of risk premia would differ in practice and why.
