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

CC BY-SA 4.0
2020-02-18T11:10:32.617Z
2020-02-18T11:10:32.617Z
VARulle
<p>In Woodridge's textbook "Econometric Analysis of Cross Section and Panel Data", page 907, it says </p>

<blockquote>
  <p>...the treatment indicator <span class="math-container">$w$</span> is statistically independent of (<span class="math-container">$y_0$</span>, <span class="math-container">$y_1$</span>), as would occur when treatment is randomized across agents. </p>
</blockquote>

<p>In which <span class="math-container">$w$</span> would be 1 if the subject receive a treatment; <span class="math-container">$y_0$</span> is the outcome without treatment; <span class="math-container">$y_1$</span> is the outcome with treatment.</p>

<p>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).</p>

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

<p>Or is there something wrong with my interpretation?</p>

<paneldata><randomized-experiment>
How to interprete independence between treatment indicator and outcomes?
Aqqqq
# How to interprete independence between treatment indicator and outcomes?

(Asked by: username_0 on 2020-02-18)

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?

**Tags:** paneldata, randomized-experiment

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## 1 answers

### Answer by username_1 on 2020-02-18. Score: 2 | Accepted answer

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.


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