The effect of “Time Wizard” is a Ignition Effect, and therefore can only be activated during your Main Phase 1 or 2.
Netrep Rulings:
The effect of “Time Wizard” is non-targeting.
Netrep Q&As:
Q: If I call “Time Wizard’s” Coin Flip wrong, can I use “Barrel Behind the Door” to give the damage to my opponent, even though my monsters are still destroyed?
A: In a nutshell, you must use “Barrel Behind the Door” in response to an effect that you KNOW will inflict damage to your Life Points (which is why you can’t use it in response to “Dice Jar”). Also, “Time Wizard” causes you to “lose” Life Points. Since “Barrel Behind the Door” cannot be used against such effects, you cannot use “Barrel Behind the Door” with “Time Wizard” anyway. Steve Okegawa, 01/08/2004