Replacing iPhone 4S logic board?

My iPhone 4S has become unresponsive to touch ton left and bottom part of my screen. At times I will only place my finger close to the screen and it will register as a touch. I took it to the iPhone Clinic and they replaced the digitizer and it did NOT correct the problem; they told me it was "the mother board" and they could not fix it.
Will replacing the mother board/logic board correct the problem or is there anything else that can be done?
P.S. I read that it was to do something with static. I do not have a screen protector and have never dropped the phone and never got it wet.

Replacing the motherboard should fix it but will cost more than the phone is worth.
sell it for parts and buy a new one.
if your in PA, email me i have a few for sale or trade.

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