I am frequently asked this question: "What to do when my child will not wear their coat and hat when it is cold outside?" First, before I answer that question, let me give some background.
If your child is not going to get hypothermia (a dangerously low temperature with being exposed to cold temperatures for a long time) or frost bite (skin damage due to cold exposure), how someone is dressed is really about their comfort. We all know people who, in the same house, are dressed very differently. One person is in shorts and a t-shirt. Their family member has a hoodie and jeans on that day. So comfortable temperature is certainly different for different people.
Despite what our grandparents were told, we get sick from germs we are exposed to and not from being too cold. It was for many years that parents were told "they will get pneumonia if they do not have a coat and hat on!". This is not true. It is the exposure to the illness that causes us to become ill. I had a job one Winter before going to med school where I worked at a stone quarry in East Liberty, Ohio. That Winter, I was around very few germs but worked outside in very cold conditions. One day, when it was 20 below (!!), we would work outside for 20 minutes then go in and warm up in for 30-40 minutes. We kept us that cycle most of the day. I did not get sick because I simply was not around many ill kids or adults.
Many children grumble around about wearing a coat and hat. I believe a lot of the "I do not need it!" from children is simply an issue of control. Children like to feel independence about many issues -- it gives them a sense of control and power in their life. And even if they ultimately do not get their way, they have frazzled you.
So what to do when your child will not wear a coat when it is cold outside? Nothing. And if they say "I was cold today"? Calmly say something non-judgmental like "Maybe wear an something warmer next time". And if someone asks you "Why do you allow your child to dress that way?", say "If they feel cold, they will dress warmer".