URL Shortener help for faster tracking of clicks and also can help with backlinks, by creating a short URL the service who created the link will create a backlink to the link desired to track.
If you are going to be hosting that service in your website your website score will go down since you will be creating backlinks to other, you become a directory of outside links and thats fucked up.
I am not sure if this can affect to a subdomain, currently for learning porpuses I am hosting a shortener service for my website links and use it for traffic reporting, for now it will be public but with limited access, such as the link tracker but feel free to get a backlink, hosted URLs will dissapear after some time.
If interested in learning how to do it check the following posts:
YOURLS Set Up
Sleeky Theme Set up
URL to Shortner Service