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Your daughter will need to file partial year resident tax returns for any state where she received income and had taxes withheld during year. It looks to me like this would include both Pennsylvania and New Jersey.
The reciprocal agreement between New Jersey and Pennsylvania would likely not apply to her situation since she was a resident of New Jersey at the time income was earned there.
Thanks for your answer, but unless I can get a more definitive statement on the reciprocal agreement (and I understand if you don't know, given that you seem to be in Texas), I would prefer to throw this out to another expert who is familiar with the NJ/PA situation.
The reciprocal agreement states that residents of New Jersey do not have to pay income tax or have it withheld on income earned in Pennsylvania and vice-versa. If your daughter was a resident of New Jersey at the time she worked in that state, which she was if she lived in your home, then the reciprocal agreement does not apply to her situation.
In short, she owes the tax.
Let me clarify- she may not owe any tax if she didn't earn enough income, but she at least needs to file a New Jersey resident return to recover any amount withheld at the New Jersey job.