As of ECMAScript-262 Edition 5.1, the constructor property is only set on function objects, see 13.2 Creating Function Objects (Step 17)
It is never read nor used by the language itself.
In particular,
obj instanceof C
does not use the constructor property, nor does
C.prototype.isPrototypeOf(obj)
So if you do not fashion it, and you do not use a framework which uses it, you may ignore the constructor property.
For further reading, see What’s up with the “constructor” property