Is it possible to jailbreak my Iphone 5 to a lower version?

My Iphone 5 is at the most recent version 6.1.4 and I already know that there's not a jailbreak available for this ios version. But can I jailbreak my phone use the 6.1.3 firmware during the jailbreak process?

No, because you have to use the correct firmware, if you use a different one you might jam your iphone.

This was possible a few years ago, when apple did not sign their firmware and there was no such thing as SHSH blobs. Now however, if you had not jailbroke in 6.1.3 and obtained it's SHSH blob, it will not be possible for you to go back to IOS 6.1.3