Let's say I have an explanatory variable and a response variable that represents counts. I want to see if the explanatory variable can predicts counts. I'm aware the response variable is overdispersed. What I probably should do is analyse the data with a generalised linear model using the negative binomial distribution. But let's say I ignore the overdispersion and analyse the data using poisson regression.
What would be consequences of analysing these data with a poisson distribution? Would consequences be the same as ignoring heteroscedasticity in linear regression therefore incorrect standard errors/P-values?