Coughs:
1) originate from the chest, or,
2) from the sinus, as in post-nasal drip, or,
3) both
Or perhaps, a seasonal irritation, like pollens or dust or stuff in the air which you cannot even see.
For example: is there any road construction in the area, that is not the norm? This can cause lots of air pollution. Or, any trees/flowers flowering? Fruit trees and its pollen from flowering? Any other out of the norm type things, environmentally, that may cause irritation, for many people?
Or, is there anyone else at school, that is sick or has this coughing type thing? For example: last year, my son had that. So did lots of other kids. And per the Pediatrician, the "cough" was going around. It was JUST a cough. Coughing that dragged on awhile. Viral.
No other symptoms.
Per "seasonal" type irritations, I have that. Its not serious, but irritating. I've had it since I was a kid. Not a real "allergy" per say but just seasonal. I take Claritin and it helps. I HATE Zyrtec, because it makes me drowsy and grouchy like hell.