Yes, you have to purchase replacements both to recover full missing elements or for missing hits in an element (the red part of an element)
It doesn't matter if the hits are because of battle, cold, hunger, desertion or whatever...
But with historical attrition, unless the affected unit is on a depot, it will only recover hits, never full missing elements.
One related thing i've noticed is that on F2 screen you will only see the missing hits you need to recover, but they don't include the missing full elements you may have around your units on map.
So on F2 you may see you don't need any replacement of a given type because you are not missing hits of this type on your units, but some of them may be in missing full elements of this type that won't never recover unless you buy some appropriate replacements.
This can be a little misleading...
Regards