babies may be allergic to dust / dust mites too. so be sure to clean the house, mop/vaccuum floor (don't use bloom), and wash the bed sheets frequently. also suddenly change of weather can induce coughing too. good luck.
She has "nose water", but the color is clear and then, I suspect, just like what you said, the water went back to th throat and then has induced coughing. Just make sure the liquid is clear and not yellow. Then, I guess it should be allergic. Find out the cause..