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i did a regression with poisson model, and since the dependant variable is overdispersed (0.13) , i have also tried negative binomial. The problem is that my explanatory variable "repeated partner" is highly significant under poisson and not at all under NB. What should i do?

Dependent Variable: PATENT
Method: ML/QML - Poisson Count (Quadratic hill climbing) Date: 05/19/15 Time: 17:50
Sample (adjusted): 1996 2013
Included observations: 54 after adjustments Convergence achieved after 8 iterations Covariance matrix computed using second derivatives

Variable        Coefficient Std. Error  z-Statistic Prob.                   
LAG4_REPEATED   0.006209    0.000972    6.384682    0.0000

Dependent Variable: PATENT
Method: ML - Negative Binomial Count (Quadratic hill climbing) Date: 05/19/15 Time: 17:54
Sample (adjusted): 1996 2013
Included observations: 54 after adjustments Convergence achieved after 8 iterations Covariance matrix computed using second derivatives

Variable        Coefficient Std. Error  z-Statistic Prob.       
LAG4_REPEATED   0.005510    0.004541    1.213525    0.2249
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If the true distribution of the response is negative binomial (NB) and your regression equation is correctly specified, then the coefficient estimates from a Poisson fit are still consistent (albeit less efficient than the NB estimates). However, the corresponding standard errors are too small as they do not take the overdispersion into account.

This is what appears to happen for your data: The coefficient of LAG4_REPEATED is at least similar, roundabout 0.006. But the standard error in the Poisson model is much smaller (0.00097) than in the NB model (0.0045). Typically, this means that the significance in the Poisson model is spurious and only due to the misspecified dispersion.

The comments above are all under the assumption that your regression equation is otherwise correctly specified and that the true distribution is NB. Of course, there may be other misspecifications which is hard to tell from your description. However, assuming that the model fits well, the NB results are ok but the Poisson results are not.

Edit: Additionally, you may want to compare the usual standard errors from the Poisson model (from the maximum likelihood estimation) with the standard errors from a quasi-Poisson model (with estimated dispersion parameter) or the robust sandwich standard errors. I would expect these to be much closer to the NB standard errors than the Poisson standard errors, also leading to non-significant test results.

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