I will be building two zero-inflated negative binomial (ZINB) regression models, where each model is aiming to predict different disease count outcomes based on the exact same independent variables and data. For example:
$$Y_1 =\beta_1 + \beta_2 + \beta_.... + \beta_n \tag*{Eq.(1)}$$ $$Y_2 = \beta_1 + \beta_2 + \beta_.... + \beta_n \tag*{Eq.(2)}$$
What would be the appropriate method to test if the $\beta$ coefficients are statistically significantly across models when the data from the independent variables are the same across the models? Such that:
$$H_0: \beta_1 ~\text{in Eq.(1)} = \beta_1 ~\text{in Eq.(2)} \\ H_A: \beta_1 ~\text{in Eq.(1)} \ne \beta_1 ~\text{in Eq.(2)}$$
I have investigated the use of Seemingly-Unrelated Regression (SUR) methods, but I am unsure if there is relevant literature to warrant their use in the form of a ZINB model. Is there another more simple means of testing if these $\beta$ coefficients are statistically significantly different from each other?