Dog at Columbia?

What about Brooklyn? Much more affordable and spacious.

By myself, although I'd love to find roommates if I went.

Since you'll be by yourself, you'll need a dog walker or something if you're planning to live that far away. I'm sure you'll be able to find something, but I don't know what affordable means to you... If you like the school, apply and figure it out later. I'm sure there are other dog owners at CBS. I know when I voiced concern about having time for my pup at cornell they gave me contact info of a few people with dogs there. Living in major cities with a dog is tough, living in a city with a big dog is tougher, and living in one with one of those blacklisted "aggressive" breeds is almost impossible. But, if you try hard enough, you'll find something.

You can move to Morningside Heights or Bronx given they are not the best area but they tend to be cheaper and a quick commute to Columbia by the metro/subway

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There are parks in Morningside Heights but you aren't going to get anything with a yard. You're still looking at a lot of walk-ups and fairly small apartments. They're more affordable than UWS but I don't think they will work for the original poster. Also I don't know what dogs are considered bully-breeds in NYC but dobermans are usually on the list which makes renting an apartment with one very difficult. You are probably going to have to have an hour+ commute if you want quiet and yard for your dog.

Riverdale. It's commutable to the UWS (it's the last stop on the 1 line) and it's almost the 'burbs compared to neighborhoods in Manhattan. It's possible to find a flat with a yard. You don't want to be any further away from school than that - nobody going to Columbia should be living in Brooklyn, the commute is ridiculous (you could do NYU from Brooklyn, but uptown is quite a bit different). You'd still likely need a dog walker since you'll be on campus for really long hours. If your dog is that skittish then none of these options are going to be so good though. NYC is NYC, it's a tough place to be if you're high-strung, whether you've got two legs or four. And, echo to the comment re: bully breeds - even if a Dobie isn't literally classified as one, you're going to be faced with some prejudice with some landlords. Expect to pay a finders fee for the apartment PLUS a big pet deposit.

All that being said, there's more dogs per capita in Manhattan than like anywhere else in the country. It's not easy to find the right setup but it can be done.