We just acquired a new modem (old one died). It says to leave it on continuously for 10 days when you first activate it. Upon recollection, our old modem had the exact same instructions.
Why?
To give it time to collect any firmware updates.
If it’s an ADSL modem, it may be doing some calibration tests to determine the optimum connection speed (although usually that requires some action on the part of the ISP too).
Or could it just be that you’re doing the final QC burn-in test - i.e. if it’s going to fail, it’s most likely to happen within the first 240 hours.
If it’s cable, (or I’m guessing most types of DSL,) then the central office may be testing it at any time within that 10-day period to make sure that the activation went smoothly and you’re online with no detectable issues. If that test fails because you don’t have the thing plugged in, then there’s a modem test failure report generated and probably someone assigned to do troubleshooting on it, (which might involve calling you and asking you if it’s plugged in.) 
My company does contract work for various telcos, and these modem test failures for homes where we’ve done work count as part of what go into our quality of service scores. You probably didn’t have anyone inside your home hooking up the new modem if the old one was working fine before it died, but the series of modems from this particular supplier is probably being evaluated to see how reliable they are for self-installs.
It is DSL, if that makes any difference.
Thanks for the replies