Appropriate estimator for FDI "gravity" model I am trying to model Foreign Direct Investment (FDI) flows and am facing several issues;

There is the well-known condition that under heteroskedasticity log-linearized OLS models are biased and inconsistent. Santos-Silva and Tenreyro (2006) suggest using Poisson pseudo-maximum-likelihood (PPML) to overcome this, and subsequently PPML, negative binomial and Poisson modeling became commonplace in the literature, but...

... FDI flows are often negative (divestment), which cannot be accommodated by count data models or PPML. Negative flows are economically meaningful and therefore cannot just be dropped without loss of consistency. An alternative is to use stocks rather than flows, but this does not seem to fit very well with the theory of gravity equations which seem to me to inherently deal with flow concepts. 

Can anyone suggest an appropriate estimator? I have heard the suggestion of multinomial logit (where dep.var. is share of global FDI flows) but this would also seem to falter in the case of a negative flow.
An alternative is to use stocks rather than flows, but this does not
  seem to fit very well with the theory of gravity equations which seem
  to me to inherently deal with flow concepts.


Using stocks rather than flows and avoiding the disinvestment issue is a good alternative. Gravity equations have been shown to do a good job fitting stocks of FDI (see the Handbook of International Economics, Gravity Equations: Workhorse, Toolkit, and Cookbook, page 149). 

This paper considers a model in which FDI takes the form of acquisitions. Using the discrete choice framework in a way that resembles Eaton and Kortum (2002), the authors develop a gravity equation for FDI which fits the data well. This paper extends the model to greenfield investment by imagining that instead of bidding for assets, each corporation selects the best "investment project" across all host countries.