Can a pet rat live outside, If not then what types of rodent can? **10 points**?

Hi, I have extremely severe asthma, there fore I absolutely can NOT have a pet that lives inside besides fish and reptiles. I really want a rodent because I used to have a pet mouse and he was the cutest little guy ever. I know rabbits can live outside (I know that they are not rodents), but what other types of rodents can live outside. I live in Illinois and as you probably know it becomes quite warm in the summer and very cold in the winter. In the summer I would just put ice packs so that the little guy could stay cool and in winter I would use an electric heater. Thanks for helping in advance!