<p>I hope this meets your idea for intuition, but equation 1 comes from using the Law of Total Expectation with the independence condition (Condition 1). There are four possible values of <span class="math-container">$D_i(z)-D_i(w)$</span>. <span class="math-container">$$D_i(z)=D_i(w)=1$$</span> <span class="math-container">$$D_i(z)=D_i(w)=0$$</span> <span class="math-container">$$D_i(z)&gt;D_i(w)$$</span> <span class="math-container">$$D_i(z)&lt;D_i(w)$$</span>. Consider the LHS of (1). <span class="math-container">$$E[(D_i(z)-D_i(w))*(Y_i(1)-Y_i(0))]$$</span> From the four possibilities above it is clear that this expectation only takes on a value other than 0 when the last two are true, where the third option gives a value of 1 and the fourth a value of -1. So then using the Law of Total Expectation with the independence condition yields the equality from the paper.</p>

CC BY-SA 4.0
2019-02-05T14:51:07.837Z
Thanks for the answer. Let me study it a bit and I will get back to you.
yurnero
CC BY-SA 4.0
2019-02-05T15:56:08.740Z
Sounds good. If you have any questions let me know.
soccer_stats
CC BY-SA 4.0
2019-02-05T14:21:33.173Z
2019-02-05T14:21:33.173Z
soccer_stats
<p>I'm referring to the Econometrica (1994) <a href="https://pdfs.semanticscholar.org/8714/260129e51abb09fd89d6ff79065af17bb106.pdf" rel="nofollow noreferrer">paper</a> by Imbens and Angrist. In particular, I don't really understand the random variables <span class="math-container">$D_i(z)$</span> (defined on page 468). Could you please provide me with intuition about them and explain why, specifically, for <span class="math-container">$z\neq w$</span>, 
<span class="math-container">$$
D_i(z)-D_i(w)
$$</span>
can only be either <span class="math-container">$1$</span> and <span class="math-container">$-1$</span>? I understand that, in order to get <span class="math-container">$1$</span> or <span class="math-container">$-1$</span>, the authors are looking at 2 scenarios <span class="math-container">$0-1$</span> and <span class="math-container">$1-0$</span>. But why are the remaining scenarios <span class="math-container">$1-1$</span> and <span class="math-container">$0-0$</span> excluded? </p>

<econometrics><instrumental-variable><treatment-effect>
Imbens and Angrist (1994): the $D_i(z)$ variables
yurnero
# Imbens and Angrist (1994): the $D_i(z)$ variables

(Asked by: username_0 on 2019-02-04)

I'm referring to the Econometrica (1994) [paper](https://pdfs.semanticscholar.org/8714/260129e51abb09fd89d6ff79065af17bb106.pdf) by Imbens and Angrist. In particular, I don't really understand the random variables $D\_i(z)$ (defined on page 468). Could you please provide me with intuition about them and explain why, specifically, for $z\neq w$,
$$
D\_i(z)-D\_i(w)
$$
can only be either $1$ and $-1$? I understand that, in order to get $1$ or $-1$, the authors are looking at 2 scenarios $0-1$ and $1-0$. But why are the remaining scenarios $1-1$ and $0-0$ excluded?

**Tags:** econometrics, instrumental-variable, treatment-effect

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## 1 answers

### Answer by username_1 on 2019-02-05. Score: 2

I hope this meets your idea for intuition, but equation 1 comes from using the Law of Total Expectation with the independence condition (Condition 1). There are four possible values of $D\_i(z)-D\_i(w)$. $$D\_i(z)=D\_i(w)=1$$ $$D\_i(z)=D\_i(w)=0$$ $$D\_i(z)>D\_i(w)$$ $$D\_i(z). Consider the LHS of (1). $$E[(D\_i(z)-D\_i(w))\*(Y\_i(1)-Y\_i(0))]$$ From the four possibilities above it is clear that this expectation only takes on a value other than 0 when the last two are true, where the third option gives a value of 1 and the fourth a value of -1. So then using the Law of Total Expectation with the independence condition yields the equality from the paper.

> **Comments:**
> **username_0**: Thanks for the answer. Let me study it a bit and I will get back to you.  
> **username_1**: Sounds good. If you have any questions let me know.  

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