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I am analysing purchasing behaviour using a negative binomial regression. I observe different products over several days, products differ in their popularity, there is no observable characteristic for that, therefore I would like to use a random or fixed model. I tried to find a solution by conducting hausman tests. Here I wonder whether to include all variables (controls) or only my time variant variables. If i include only my time variant variables hausman shows p>chi2 = 0,075. What would you recommend for this situation? thanks

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You could possibly model it using Linear Mixed Models

A commonly used test for such nested models is to determine if including a fixed effect or random effect is significant is the Likelihood Ratio Test which uses a Chi-square test too.

An example using the lme4 package in R and MixedLM in statsmodels is here