How do I apply for the Missouri state income tax exemption for living outside the state for more than 6 months in 2017?

You need to claim Part-Year residency for Missouri.  Once you do this, Missouri will want to know the dates of residency so that they are only taxing your income on what is earned in Missouri and not in other states.

The following information from Missouri may be helpful:

http://dor.mo.gov/personal/nonresident/