The MacBook Air would have had some version of iMovie pre-installed when new.
iMovie version 10 was released in Oct 2013 , so it is possible that your mid 2013 Air shipped with
iMovie version 9.If it was purchased after Oct 2013 then it may have been version 10.
Have the University given you the Apple ID that is associated with the original applications?
If iMovie ver 10 has ever been pre- installed at purchase or subsequently installed by the university then you will need the Apple ID that is associated with it, so that you can re-download it from the Purchased page in the App Store.
You may have to go back to the university to see if they can re-download iMovie for you from the App Store Purchased page.
Do you know any body that has iMovie ver 10 on their Mac?