Peter K. I owe you an apology. Apparently Dr. Hawking did make the argument much as you described it Here it is.. I also commend you for not going that far.

It is a profoundly stupid argument. It can be easily counted with "Where did the singularity and all of the physical laws unfurled by the Big Bang come from?"

His very definition of time is as a measurement of time is basically the measurement of change in this Universe. It has no meaning before the Big Bang. But obviously the singularity existed to Bang before the Universe we know existed and something made it go Bang. And by his own definition of singularity he has no possible source of information about what caused the bang or how long it took to cause it.