No, hobby income is not earned income. But, the services you are describing is not a hobby, it is self-employment. You do not have to have a specific location to be self-employed. This is no different than a tradesman working out of their van as their primary location traveling to other peoples houses or a walking counselor that walks in the park in people while counseling them.
The difference between hobby income and other payment performed for personal services is 1) how they are taxed and 2) your intent.
Hobby income is not taxed the same as income from performing services. You do not pay social security or Medicare taxes on it. You are not performing a service with the intent of earning an income. If you are, then it would be self-employment income instead of hobby income.
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You can't really. If you had no intention of making money doing this, this would be more of a volunteer activity instead of an activity where they paid you. Even if you are not setting the pay rate, this would be similar to getting a tip. You earned $5,000 providing a service to "gain experience".
A hobby is done for the love of the activity or for sport or recreation. A person with a workshop in their garage that likes to make different wood crafts, but doesn't need them all so he sells them to cover the cost of materials would be an example.
Doing something to gain work experience would fall into an apprenticeship type category which is not a hobby activity.
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Your question may be academic. $5000 of earned income is very unlikely to be more than half your support. Calling it earned income is not going to qualify you for the AOTC.
That said, this situation is simple. If you want it to be earned income, you must report it as self employment, on schedule C, and pay self employment (social security and Medicare) tax (SET) on the income. There is no way to report hobby income, as earned income, on the IRS forms.
That said, there is a workaround, in TurboTax to force the income onto line 1 of form 1040, as earned income, but that is a red flag to the IRS.
I used a bad word, earlier. Taxes are not "simple".
If you believe this is earned income (and you do have some mild support from that IRS link), here's the work around to enter it as earned income on line 1 of form 1040:
This is how to enter earned income without a W-2:
Select Federal on the left side menu.
Select Income and Expenses toward upper left.
Expand/ scroll down the list and find the section called, Less Common Income.
Scroll all the way down and select Miscellaneous Income, 1099-A, 1099-C.
Select Other income not already reported on a Form W-2 or Form 1099
Answer the first question Yes.
Click Continue until you get to the page titled, Any Other Earned Income?
Answer Yes and click Continue.
Select Other and click Continue.
Enter a description and amount on this page, click Continue.
I think you'll actually be waiving two red flags at the IRS; one for the earned income on line 1 and a bigger flag for a full time student, under 24, claiming the refundable portion of AOTC. For your situation, "get in trouble" only (usually) means a delay in your adjusted refund or more likely paying money back, sometime in the future, if the IRS disagrees.
Q. About the credit, I can still claim the nonrefundable part even if my earned income was less than one-half of my support (my W-2 wages), right?
A. Right. And in that case, the refundable part also becomes non-refundable. That is, if you have enough tax liability, you can still get the full $2500 AOTC.
Claiming the non-refundable credit is not a red flag item. Adding your hobby income under "activity not engaged in for-profit" always has some red flag potential, but is low risk unless somebody sent you a 1099.