How do I determine how much of a 2013 local tax refund recived in 2019 is taxable for federal purposes?

Anita01 previously responded to a similar question:

The short explanation is that it is advisable for you to recalculate if you have a large potentially taxable state tax refund that you received in the current calendar year.  

You cannot just assume that none of your refund of state tax is taxable in the current year if your paid any AMT at all in the prior year.  You would need to do this recalculation to determine the amount of the deduction that failed to benefit you due to AMT, in order to exclude any of the refund from taxation this year.   If you received a very large state tax refund from the prior year tax return, going through this pretty elaborate calculation can possibly lower the tax you owe on that refund received during the current calendar year.

To provide a little more detail, the calculation is intended to determine the amount of your prior year state and local tax deduction that actually ended up not benefiting you because you paid AMT.  While the full amount of your deduction is added back for AMT, it may have only been responsible for a portion of your AMT owed.

When you receive a refund from the state in 2017, the full amount of that refund is considered as taxable income to you because it is assumes you benefited in the prior year from the deduction.  If you paid AMT, however, that deduction amount was added back to your income to calculate the amount of AMT you owed, if any.  It's entirely possible, though, that only a portion of that addback contributed to your AMT.  For example, the deduction was $5,000 but your total AMT paid was only $4,000.  

The recalculation exercise can determine how much the addback of the state tax deduction actually cost you in terms of AMT.  That portion of the deduction that actually caused a higher AMT, can be a nontaxable portion of your state refund this year.

Please see the following links regarding tax treatment of state and local tax refunds.

Journal of Accountancy , Tax Treatment of state and local refunds

Rev.Rul 2019-11