D.B.
If he's been sick that long, it's not a cold. If he's coughing up green phlegm, that's an infection. It could have started as a cold or allergies, but something else is going on. I'd push the doctors more. If that doesn't work, or if it goes away but comes back again, then you should look into a supplement made for kids. Something's "off" inside and his immune system needs to be strengthened.
I have several friends whose kids had food and pollen allergies, were on breathing treatments/nebulizers, and were sick all the time. They found a wonderful supplement, patented, research-based, by the same scientist who developed ProSoBee and Enfamil formulas. Liquid nutrition and a great immune system boost. My whole family is on it and no one has been sick since. Can't say enough about it! Contact me if you want more info.