Joao Santos Silva is correct, my blog post is directly related to Santos Silva and Tenreyro's paper,
Santos Silva, J.M.C., S. Tenreyro 2006. "The log of gravity", The Review of Economics and Statistics 88(4):641-658.
I should have cited it. The use of poisson regression is related to a strand of research in econometrics anad statistics that is huge and goes under the name quasi-likelihood theory. In addition to Santos Silva and Tenreyro's paper, I recommend that interested readers see
Jeffery Wooldridge, Econometric Analysis of Cross Section and Panel Data, 2nd ed.
which I did cite in by blog entry, and two more that I did not,
A. Colin Cameron and Pravin K. Trivedi, Microeconometrics: Theory and Applications.
McCullagh P. and J. A. Nelder, Generalized Linear Models, 2nd ed.
Joao mentioned that they provided Stata code that checks identification conditions and we are looking into adding those features to Stata's poisson command. These checks are discussed in
Santos Silva, J.M.C., S. Tenreyro 2010. "On the existence of the maximum likelihood estimates in Poisson regression", Economics Letters 107(2):310-312.
Santos Silva, J.M.C., S. Tenreyro 2011. "poisson: Some convergence issues", The Stata Journal 11(2): 207-212.