# Conditional treatment effect and average treatment effect under no unmeasured confounders (ignorability)

The conditional treatment effect (CATE) is defined as:

$$\tau(x) = \mathbb{E} \left[ Y^1- Y^0 \mid X = x \right],$$

the average treatment effect (ATE) is defined as $$\tau_{ATE} = \mathbb{E}\left[ \tau(X) \right] = \mathbb{E} \left[ Y^1 - Y^0 \right].$$

If we assume no unmeasured confounders ($$A$$ = binary treatment indicator)

$$\left \lbrace Y^1, Y^0 \right \rbrace \perp \!\!\! \perp A \mid X$$

is then CATE=ATE ?