How to interprete independence between treatment indicator and outcomes?

Question

In Woodridge's textbook "Econometric Analysis of Cross Section and Panel Data", page 907, it says

...the treatment indicator $w$ is statistically independent of ($y_0$, $y_1$), as would occur when treatment is randomized across agents.

In which $w$ would be 1 if the subject receive a treatment; $y_0$ is the outcome without treatment; $y_1$ is the outcome with treatment.

I found the statement a bit strange since this would mean that, e.g. in the case of double-blind experiment involving giving one group of patients placebo and the other group of patient real medication (i.e. randomized across agents), giving real medication or not is independent of the outcome of the patients taking the real medication and the outcome of the patients not taking the real medication (i.e. the real medication is assumed to have no effect if the agents is randomized, which does not make sense).

Furthermore, the book showed that when treatment is randomized, $E(y|w=1)=E(y_1|w=1) =E(y_1)$. However, it seems that randomizing treatment is not essential for this equation to be established as long as $y=(1-w)y_0+wy_1$ (showed on p. 906 at the buttom). What is the difference between $E(y|w=1)$ and $E(y_1)$? They seems to be the same (expected outcome with treatment). In what circumstance would $E(y|w=1)=E(y_1|w=1) =E(y_1)$ be not satisfied?

Or is there something wrong with my interpretation?

Answer 1

Note that this section in the book starts with the counterfactual that each agent has an outcome $y_0$ without treatment and an outcome $y_1$ with treatment. You observe only the $y_0$ of the untreated agents and only the $y_1$ of the treated agents, but both exist for each agent. If treatment is randomized across agents then $w$ is indeed independent of $(y_0,y_1)$, but of course not independent of the $y_i$ you actually observe.