Is there any gain by adding correlated categorical variables (e.g.: city and neighborhood) in a model? It's a very straightforward question that I've never seen any discussion.
To keep it simple, let's say I have a Linear Regression and I want to predict housing prices. I have a dataset that contains both the city and the neighborhood of a city. Should I keep both? Does that matter? Another similar example could be: if I'm predicting the price of a car, and I have the car name and the brand.
Is my model getting any better when I add both features? Should I keep both? If not, how can I choose which feature should I keep?

Presumably the neighbourhoods are unique to their city, so once you know the neighbourhood you know the city.  Assuming this is the case, adding both variables will lead to an over-parameterised model; you should use the neighbourhood variable but not the city variable.  The problem is not merely that neighbourhood and city are correlated, but that the latter is a deterministic function of the former.
