The very last step of the Fama McBeth procedure is to aggregate the estimated regression coefficients by taking their mean. The mean is then the estimate for the "overall" regression coefficient.
The significance of this mean is derived by looking at the variation of the regression coefficients. But by doing so, don't we ignore the fact that the different regression coefficients are not fix?
If I take the mean of three regression coefficients that are not significantly different than zero, but which's values are let's say 10, 11 12. It is obvious that treating them as fixed at 10, 11 and 12 would yield to the conclusion that the mean (11) is different from zero with a high significance. However, this stands in contrast to the fact that each of the single coefficients is not significantly different from zero.
Is my thinking correct? If not - how is the fact that the estimates are not fix considered in the Fama McBeth procedure