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Unfortunately third party sick pay is not exempt from Michigan tax. As a part year resident, you would be responsible for paying Michigan tax on all income received during your period of MI residency. So if you received the third party sick pay while a resident of MI, even if the pay was earned in connection with work in TN, it would still be taxable to MI. MI has reciprocal state agreements, but not with TN. You would be eligible for a credit for taxes paid to another state if you paid TN tax, but since TN has no tax on wage income, this does not apply.
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