You may have to run it in compatibility mode, but it should work.
Although it might well be worth upgrading Premiere Elements too! The latest versions are much faster, work with a much wider range of video formats AND are better able to take advantage of the advanced architecture of Windows 7 and 8.
Premiere Elements 4 works great in Windows 7 Professional SP1 64 bit computer. I have been there and am doing that along with other versions of the program on the same computer.
Do not go to the desktop icon for Premiere Elements 4 and set a compatibility setting in the Properties.
Just make sure that you are running the program from a User Account with Administrative Privileges and perhaps also with a Run As Administrator (right click desktop icon, select Run As Administrator). Latest version of QuickTime remains a must.
Does Premiere Elements 4 meet your video editing needs, feature-wise and otherwise? If so, stay with it. If not, explore the latest version 12 via a tryout from Adobe before any thoughts of purchase.
The above is my take on your inquiry. Please let us know if you need clarification on anything written.
Add On...If you need a computer environment with greater resources for your Premiere Elements workflows, then you need to look at specifically Premiere Elements 10, 11, or 12 specifically on Windows 7, 8, or 8.1 64 bit.