A child cannot move out without parental consent unless he/she has been legally emancipated. In the United States, a person is a minor (and therefore under the control of their parent(s)/guardian(s)) until they attain the Age of Majority (18 years in most states), at which point they're an adult. However, in special circumstances, a minor can be freed from control by their guardian before turning 18. In most states, the three circumstances in which a minor becomes emancipated are: (1) enlisting in the military (requires parent/guardian consent), (2) marrying (requires parent/guardian consent), (3) obtaining a court order from a judge (does not require parent/guardian consent).
The exact laws and protocols for obtaining emancipation vary from state to state. In most states, the minor must file a petition with the family court in his/her jurisdiction, formally requesting emancipation and citing reasons why it is in his/her best interest to be emancipated. He/she must prove that he/she can support himself/herself financially. Many states require that the minor have been living separate from his/her parent(s)/guardian(s) for a period of time; however, that requires the consent of the parent(s)/guardian(s) in order to not classify simply as "running away".
Emancipations are rarely granted, because of the subjectivity and narrowness of the definition of "best interest". On one end of the spectrum are minors who have been victims of abuse; in most cases, the state's department of child services is notified and the child is placed in foster care. On the other end of the spectrum are minors who are seeking emancipation for reasons such as not being pleased with their parents'/guardians' rules. In those cases, the emancipation will most likely be denied and the minor will be sent back home with the parent/guardian.
If the child has not done this, he/she cannot leave without your consent.