Dependent count variable with negative values In a regression model my dependent variable is a count variable that can have negative numbers. 
What econometric method can I use? Poisson regression and negative binomial only accept positive dependent variable
In particular, the dependent variable it's the difference in the number of managers of a company after and before an investment round. 
So a value of 2 indicates that there are 2 more managers, while a value of -2 indicates that there 2 less managers.
In my dataset, the dependent variable it's the subset of integer numbers from -6 to 12.
There is no literature on how far the dependent variable can go. 
In theory after the investment the new shareholder can fire all of the existing managers and replace them all. This is what I am trying to discover
 A: While the number of managers before and after are count variables, your dependent variable no longer is: Counts can't be negative, after all. So you don't need to use either poisson or negative binomial. 
A: For this kind of analysis I would suggest that you follow whuber's advice in his comment, and track the individual counts of the number of managers before and after each change.  It should not be too hard to create a GLM that uses the non-negative count variable managers as the response variable and uses a binary indicator investment to indicate whether or not an investment round occurred.  This kind of model would have a coefficient for the effect of the investment round on the count of managers, and this coefficient could be positive or negative.
Since your underlying count values for this kind of model would be non-negative integers, a good place to start would be a standard negative binomial GLM.  You could program this in R as shown below.  If you are able to add some of your data, and give some additional information on other covariates, we could have a look at what you get.
library(MASS)

MODEL <- glm.nb(managers ~ 1 + investment + covariates, data = DATA_FRAME);

A: Y is not actually a count - counts can only have positive values.  It's a difference, which is, well, different.
One option would be to treat Y as a continuous variable and use ordinary least squares regression. Another would be to add 5 to Y and use a count regression model.  You could try both and see if the results are similar. 
