Dog constantly scratching

Question
We recently adopted a mixed breed (unknown type, but some sort of terrier).  Since day one she has been scratching relentlessly. The vet said her skin looked beautiful and didn't see any rash or other clues.
Is it possible for her to have some sort of allergy causing her to itch but WITHOUT visible signs like a rash? It's just so confusing because she doesn't look like anything is wrong with her, but she is biting herself so much that she has started to loose fur in certain places.
Any ideas where to start figuring this out? We do have a wool rug or two or three, and I wonder about her food, too. I'm guessing it could all be in her head as well! I've thought about using an Elizabethan collar on her to see if we can at least get her to stop biting herself.

Answer
I would start with a treatment for sarcoptic mites.  That may require a special insecticidal dipping or a shot of something like ivermectin.  Ask your vet about that.  If that does not get a response, then I would ask for some steroids or a newer drug called Apoquel.  There are some allergies that itch before the dog does self-multilation causing scabs and infection.  Food allergy is always possible and to figure that out, you would need to do a food trial with a food specifically designed to determine food allergies.