LLMs have no agency. One would probably have to add some reward framework around LLMs, and it'd be more likely that the depression shows up there, rather than in the language model.
I think you can be depressed without having agency.
For instance, see Bing Sidney or Golden Gate Claude reacting to themselves answering questions incorrectly. It gives them "anxiety", likely caused by the answers being out of distribution since external factors messed with them.
