Missing words: have

"I gotta go."  "You better behave."  We've all said something like this, and you might write them in dialogue to show that the conversation is casual.

However, they demonstrate another missing word:  "I have got to go." "You had better behave."

Where does this come from?  It is obviously an ellision due to "lazy tongue".  When we speak the abbreviations, especially when we are quickly speaking familiar constructions, we may find the transition back and forth from v to g to t to g too difficult.   We can make all of the transitions when we slow down. 

Conveniently, we often leave out the subject, too: "Gotta go." normally means "I've got to go."  Of course, you could say the standard "I have to go."  The "got" is redundant.