Fixed-effects models allow you to control for time-constant unobserved heterogeneity. Therefore, if time-constant unobserved factors are correlated with covariates, the FE estimates are still consistent, but the RE estimates are not. If there is not this kind of problem (time-constant heterogeneity and covariates are not correlated), then both types of estimates are consistent but RE estimates are more efficiente (more precise, smaller standard errors).

Thank you very much. You are right. But this is an inference for general fixed and random effects models.There are categorical choices in ordered models. This will of course have an impact on the functioning. The logit model structure is at the forefront in the fixed effect model, while the probit structure is in the foreground in the random effect models. There is a incidental parameter problem in fixed effect logit model. However, there is a solution to this problem in the general model. The problem is solved with a minimum sufficient statistic. However, the problem persists, as there is no such statistics in sequential models. What I was wondering is: Is the random effect probit structure superior because there is no solution to this problem in the fixed effect model alone? What other advantages can they have when considering ordered discrete choice models for panel data?

The difference between the two models is due to assumptions on the error term of the underlying equation. Random effects or fixed effects refer to unobserved individual heterogeneity. With logit model, there is a way to difference out unobserved heterogeneity, the model (i.e. parameters) can be estimated. With the probit model, differencing does not work and unobserved heterogeneity has to be assumed to be random to make estimation feasible. A compromise is correlated random effects models where observed variables are allowed to be correlated with unobserved heterogeneity in a restrictive way. FE models no such a restriction is imposed (so it is superior), while in RE models observed variables are assumed to be independent of unobserved heterogeneity.

Thank you for your answer.

However, in the panel data model, the model operations are different in classical and ordered structures. In general, Robinson (1982) has proven that the estimates obtained from the panel probit model are more consistent. We can see from the numerical estimation that this is indeed the case. Standard errors from logit are larger. This weakens the sample's power to represent the population. This ensures probit is at the forefront.

I wonder about the differences in the ordered panel structure :)