A. Colin Cameron and Pravin K. Trivedi
"Regression-Based Tests for Overdispersion in the Poisson Models"
Journal of Econometrics, 1990, Vol. 46(3), pp.347-364.
A property of the Poisson regression model is mean-variance equality, conditional on explanatory variables. "Regression-based" tests for this property are proposed in a very general setting. Unlike classical statistical tests, these tests require specification of only the mean-variance relationship under the alternative, rather than the complete distribution whose choice is usually arbitrary. The optimal regression-based test is easily computed as the t-test from an auxiliary regression. If a distribution under the alternative hypothesis is in fact specified and is in the Katz system of distributions or is Cox's local approximation to the Poisson, the score test for the Poisson distribution is equivalent to the optimal regression based test.