regression - Fama-Macbeth second step confusion

I am confused on how to run the second step of the Fama Macbeth (1973) two step procedure.

I have monthly stock returns and monthly Fama-French factors, for around 10,000 stocks. This creates an unbalanced panel, mainly because stocks start and stop trading within the period I examine (1991-2015, 25 years, 300 months).

In the first step I regress each stock's excess return on the Fama-French factors: $$ R_{i,t} = \alpha_i + \beta_{i, MktRf} MktRf_t + \beta_{i, SMB} SMB_t + \beta_{i, HML} HML_t + \epsilon_{i, t} $$

So, I get 10,000 "quadruplets" $\alpha_i, \text{ } \beta_{i, MktRf}, \text{ } \beta_{i, SMB}, \text{ } \beta_{i, HML}$ for each stock.

But how exactly do I proceed for the second step, which requires me to run 300 (number of months in sample) regressions??

What exactly are the dependent and the independent variables for each time period (month)?

Answer

Then for each month $t$, you run a cross-section regression:

$r_{i,t} = \lambda_0 + \hat{\beta}_i {\lambda}_t + \alpha_{i,t}$

Where: $\hat{\beta}_i \equiv [\beta_{i, MktRf}, \beta_{i, SMB}, \beta_{i, HML}]'$, is a vector of the coefficients estimated on the first step.

What you are looking for is to estimate the vector of $\hat{\lambda}_t \equiv [\lambda_{t, MktRf}, \lambda_{y, SMB}, \lambda_{t, HML}]$.

So after the second step you will have $T$ estimates for each $\lambda$ (price of risk).

Then you just need to average those $\lambda$'s:

$\hat{\lambda} = \frac{1}{T} \sum^{T}_{t=1} \hat{\lambda}_t$

And you can test their statistical significance using as a variance estimate the following:

$Est.Asy.Var(\hat{\lambda}) = \frac{1}{T^2} \sum^{T}_{t=1} (\hat{\lambda}_t - \hat{\lambda} )(\hat{\lambda}_t - \hat{\lambda} )'$