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No.  Actually household employee income is reported on a W-2, but it is not that simple.   A babysitter can either be a W-2 household employee, or a self-employed (schedule C) person.  It depends on the circumstances.

Pub 17:
Babysitting.   If you are paid to babysit, even for relatives or neighborhood children, whether on a regular basis or only periodically, the rules for childcare providers apply to you.

Childcare providers.    If you provide childcare, either in the child's home or in your home or other place of business, the pay you receive must be included in your income. If you are not an employee, you are probably self-employed and must include payments for your services on Schedule C (Form 1040), Profit or Loss From Business, or Schedule C-EZ (Form 1040), Net Profit From Business. You generally are not an employee unless you are subject to the will and control of the person who employs you as to what you are to do and how you are to do it.
**Disclaimer: This post is for discussion purposes only and is NOT tax advice. The author takes no responsibility for the accuracy of any information in this post.**