Please, see below.
> A) On how to compute the std. Errors of the APEs manually.
Look at the documentation of -margeff- kindly posted by its author at
http://www.bke.hu/bartus/stata/margeff.htm. In particular go through the
paper published in the Stata journal. There are some misprints however so be
careful. I would suggest you derive everything you need. Greene (1993) also
has some insights that could be helpful.
> B) On why the difference between the standard errors of the APEs and the
> marginal effects is so large, giving completely different results
> regarding the significance of the estimates.
This is possibly due to the errors in the approximations used to evaluate
the derivatives of the densities involved in those calculations. Some
routines are better than others but their performance usually increases with
the size of the dataset. In addition, you have to be careful what you are
comparing: -mfx- provides marginal effects evaluated at the means, whereas
-margeff- could give you either marginal effects evaluated at the means or
APEs. You should expect those two to be same (or very close to each other)
in the first case, but not in the second.
I would however once again advise you to derive everything yourself.
Simo Goshev
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