Solved – Dependent count variable with negative values

count-dataeconometricsnegative-binomial-distributionpoisson-regression

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

Best Answer

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.

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Generalized Linear Model – Count Explanatory Variable with Proportion Dependent Variable

You have a binary response. That is the important part of this. The count status of your explanatory variable doesn't matter. As a result, you should be doing some form of logistic regression. The part that makes this more difficult is that your data are clustered within just four participants. That means you need to either use a GLiMeM, or the GEE. This is a subtle decision, but I discuss it at some length here: Difference between generalized linear models & generalized linear mixed models in SPSS. Depending on the options that your software affords you, you may also have to un-group your data, so that you have a (very long) matrix where the response listed in each row is a 1 or a 0.

Solved – Quasi-poisson or negative binomial regression with continuous dependent variable

Rounding your response variable to an integer is NOT OK. For simplicity, lets assume you're conducting a Poisson regression. What you're modeling is the following:

$ \begin{align*} E(Y|x) &= \beta^{T}x + \beta_{0} \\[0.5em] \log \left( \frac{\mbox{No. of Deer}}{\mbox{Area}} \right) &= \beta^{T}x + \beta_{0} \\[0.5em] \log(\mbox{No. of Deer}) - \log(\mbox{Area}) &= \beta^{T}x + \beta_{0} \\[0.5em] \log(\mbox{No. of Deer}) &= \beta^{T}x + \beta_{0} + \log(\mbox{Area}) \end{align*} $

In R, this is done use the following command:

glm(No. of Deer ~ x + offset(log(Area)), family=poisson(link=log), data=data.frame)

This allows you to use Poisson (or Quasi-Poisson or Negative Binomial) regression for a continuous response, even though the No. of Deer is still a count. Your parameter estimates (i.e., $\beta_{0}$ and $\beta \mbox{s}$) will be on the log scale, so just exponentiate them to obtain estimates on the raw scale. Also, be careful of the parametrization used for the negative binomial distribution, if you decide to go with negative binomial regression.

Best Answer

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Generalized Linear Model – Count Explanatory Variable with Proportion Dependent Variable

Solved – Quasi-poisson or negative binomial regression with continuous dependent variable

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