Can you activate a planeswalker's loyalty ability in response to a spell targeting it?

Loyalty abilities have a strict timing restriction: they can only be activated any time you could cast a sorcery, and only on your own turn (CR 606.3). This means they cannot be activated in response to spells or abilities that are already on the stack, because that window occurs during an opponent's priority in the middle of the stack — not a sorcery-speed opportunity.

Additionally, CR 606.4 states that a player may activate a loyalty ability only once per turn per planeswalker. But the more fundamental barrier here is the sorcery-speed requirement: once your opponent has cast a spell targeting your planeswalker, priority passes and you are now in a response window — which is instant speed, not sorcery speed.

Some effects can grant the ability to activate loyalty abilities at instant speed (e.g., Teferi, Who Slows the Sunset's emblem or Carth the Lion's effect). In those specific cases, you could respond to a spell on the stack by activating a loyalty ability — but only if such an effect is in play.

Example: Your opponent casts Dreadbore targeting your Liliana of the Veil. You cannot respond by activating Liliana's −2 ability to sacrifice a creature, because activating that ability requires sorcery speed. The Dreadbore will resolve and destroy Liliana.