My dependent variable can take 3 distinct values so I am leaning towards using ordered-probit estimation. It is a cross-sectional analysis. I know that it is not advisable to use fixed effects regression in an ordered-probit estimation. But does the same reasoning apply for cross-sectional data? Also, I have almost 60,000 observations from around 250 regions, so can I add region dummies in the ordered-probit estimation because I have way more than 60 observation per cluster?

Thank you!