What Causes Persistent Nagging Cough After Having Cold In A Child?

Hi! My son , who is 3 1/2 years old has a lingering cough for almost 2 months now. It started with a cold back on October. He got better for a week but was followed with another cold which he also recovered. However, his cough and a little bit of runny nose lingered specially when he is active. Some days are better. He specially coughs when he wakes up in the morning. His has a wet cough. And at his age he does not know how to cough out his phlegm. What could be the cause of his lingering cough? Thanks