<p>The simple answer is <strong>no</strong>.</p>
<p>Randomness does not imply exogeneity.</p>
<p>In my opinion, the following example from <a href="https://www.princeton.edu/%7Edeaton/downloads/Instruments%20of%20development%20v1d_mar09_all.pdf" rel="nofollow noreferrer">Deaton: &quot;Randomization in the tropics, and the search for the elusive keys to economic development&quot;</a> is quite illuminating.</p>
<p>Assume that you want to estimate the effect of having a railway-station on poverty. We have data on the poverty rate <span class="math-container">$P_c$</span> and the presence of a station <span class="math-container">$R_c \in \{0,1\}$</span> for a large number of cities <span class="math-container">$c$</span>. Assume that the true data generating process is given by:
<span class="math-container">$$
P_c = \alpha + \beta_c R_c + \varepsilon_c,
$$</span>
Notice here that the slope <span class="math-container">$\beta_c$</span> is city specific, so the effect of a railroad is heterogeneous across cities.</p>
<p>The coefficient <span class="math-container">$\beta_c$</span> cannot be estimated, but you might wonder whether it is possible to estimate the mean of <span class="math-container">$\beta_c$</span> over all cities that decide to build a railroad.  <span class="math-container">$\overline{\beta} = \mathbb{E}(\beta_c|R_c = 1)$</span>. Then:
<span class="math-container">$$
P_c = \alpha + \overline{\beta} R_c + (\varepsilon_c + (\beta_c - \overline{\beta})R_c) = \alpha + \overline{\beta} R_c + \upsilon_c,
$$</span>
where <span class="math-container">$\upsilon_c = \varepsilon_c + (\beta_c - \overline{\beta})R_c$</span> is the new error. Although <span class="math-container">$\mathbb{E}(\upsilon_c) = 0$</span>, you might be worried that:
<span class="math-container">$$
\mathbb{E}(R_c \upsilon_c) \ne 0.
$$</span>
Indeed, we have:
<span class="math-container">$$
\mathbb{E}(R_c \upsilon_c) = \mathbb{E}(\varepsilon_c R_c = 1) + \mathbb{E}(\beta_c - \overline{\beta}|R_c = 1) \Pr(R_c = 1) = \mathbb{E}(\varepsilon_c R_c).
$$</span>
This will only be zero if the decision to build a station is uncorrelated to anything that might otherwise influence the poverty rate in the city. As this is probably not satisfied, we need an instrument.</p>
<h3>A random instrument</h3>
<p>Assume that the government randomly allocates cities to development areas, captured by the instrument <span class="math-container">$Z_c$</span> (equal to 1 if city <span class="math-container">$c$</span> is a development area and zero otherwise).</p>
<p>If designations to development areas motivate the building of railways, we have that there is positive correlation between <span class="math-container">$Z_c$</span> and <span class="math-container">$R_c$</span>. Also as <span class="math-container">$Z_c$</span> is random, we can assume that <span class="math-container">$\mathbb{E}(Z_c \varepsilon_c) = 0$</span>. However:
<span class="math-container">$$
\mathbb{E}(Z_c \upsilon_c) = \mathbb{E}(\varepsilon_c Z_c) + \mathbb{E}(\beta_c - \overline{\beta}|Z_c = 1, R_c = 1)\Pr(R_c = 1, Z_c = 1).
$$</span>
The first term on the right hand side is zero by the randomness of <span class="math-container">$Z_c$</span>. As such, <span class="math-container">$Z_c$</span> will be uncorrelated to <span class="math-container">$\upsilon_c$</span> iff:
<span class="math-container">$$
\mathbb{E}(\beta_c - \overline{\beta}|Z_c = 1, R_c = 1) = \mathbb{E}(\beta_c|Z_c = 1, R_c = 1) - \mathbb{E}(\beta_c|R_c = 1) = 0.
$$</span>
This will be the case if the mean effect of building a station over all cities that are a development area equals the mean effect over all cities that build the railroad (whether they are a development area or not).</p>
<p>The last condition can be rewritten as:
<span class="math-container">$$
\begin{align*}
&amp;\mathbb{E}(\beta_c|Z_c = 1, R_c = 1) = \mathbb{E}(\beta_c|R_c = 1),\\
\to &amp;\mathbb{E}(\beta_c|Z_c = 1, R_c = 1) = \mathbb{E}(\beta_c|Z_C = 1, R_c = 1) \Pr(Z_c = 1) + \mathbb{E}(\beta_c|Z_c = 0, R_c = 1) \Pr(Z_c = 0),\\
\to &amp;\mathbb{E}(\beta_c|Z_c = 1, R_c = 1)\Pr(Z_c = 0) = \mathbb{E}(\beta_c|Z_c = 0, R_c = 1) \Pr(Z_c = 0),\\
\to &amp;\mathbb{E}(\beta_c|Z_c = 1, R_c = 1)\Pr(Z_c = 0|R_c = 1) = \mathbb{E}(\beta_c|Z_c = 0, R_c = 1)\Pr(Z_c = 0|R_c = 1),\\
\to &amp;\mathbb{E}(\beta_c|Z_c = 1, R_c = 1)\Pr(Z_c = 0, R_c = 1) = \mathbb{E}(\beta_c|Z_c = 0, R_c = 1) \Pr(Z_c = 0, R_c = 1).
\end{align*}
$$</span>
This will be the case if either:</p>
<ol>
<li><span class="math-container">$\mathbb{E}(\beta_c|Z_c = 1, R_c = 1) = \mathbb{E}(\beta_c|Z_c = 0, R_c = 1)$</span> which means that the mean effect among all cities who are a development area and decided to build a railroad should be equal to the mean effect among those who build a railroad and are not a development area. So if the behaviour (i.e. the decision to build a station) changes due to being designated as a development area, this condition will not be satisfied.</li>
<li>Or <span class="math-container">$Pr(Z_c = 0, R_c = 1) = 0$</span> which means that no city that is not a development area builds a station.</li>
</ol>
<p>In short: randomness of the instrument makes sure that <span class="math-container">$\mathbb{E}(\varepsilon_c Z_c) = 0$</span> but does not guarantee that <span class="math-container">$\mathbb{E}(\upsilon_c Z_c) = 0$</span>.</p>

CC BY-SA 4.0
2021-05-11T09:49:50.820Z
2021-05-11T09:49:50.820Z
tdm
<p>If using <span class="math-container">$z$</span> as an instrument for <span class="math-container">$x$</span>, to study the effect <span class="math-container">$x$</span> has on <span class="math-container">$y$</span> and given that <span class="math-container">$z$</span> was indeed generated through a lottery, is it definite that exogeneity of <span class="math-container">$z$</span> will hold?</p>
<p>I have looked at this as two-stage OLS and I believe that the fact <span class="math-container">$z$</span> is lottery-generated can satisfy the exogeneity condition, i.e. <span class="math-container">$\text{corr}(z_i, \epsilon_i) = 0$</span>. Is that true?</p>

CC BY-SA 4.0
2021-05-10T19:06:20.810Z
do you mean by lottery simple random sampling or some real life program that used some sort of lottery? If some real life program then you should link to it/provide more details on how people were selected/drawn
1muflon1
CC BY-SA 4.0
2021-05-10T19:12:44.530Z
The instrument should  only affect y through x. Saying that z is generated by lottery does not in itself seem to garanty that the outcome of the lottery - that is z - do not somehow affect y (even if not through x).
Jesper Hybel
CC BY-SA 4.0
2021-05-10T23:21:17.457Z
@1muflon1 Yes, it was meant to be simple random sampling.
Bazinga
CC BY-SA 4.0
2021-05-10T23:21:47.550Z
@JesperHybel I can see the reasoning, but is it possible to formulate that more rigorously?
Bazinga
<econometrics><regression><instrumental-variable>
Is exogeneity guaranteed for a lottery-generated instrumental variable?
Bazinga
# Is exogeneity guaranteed for a lottery-generated instrumental variable?

(Asked by: username_0 on 2021-05-10)

If using $z$ as an instrument for $x$, to study the effect $x$ has on $y$ and given that $z$ was indeed generated through a lottery, is it definite that exogeneity of $z$ will hold?

I have looked at this as two-stage OLS and I believe that the fact $z$ is lottery-generated can satisfy the exogeneity condition, i.e. $\text{corr}(z\_i, \epsilon\_i) = 0$. Is that true?

**Tags:** econometrics, regression, instrumental-variable

> **Comments:**
> **username_1**: do you mean by lottery simple random sampling or some real life program that used some sort of lottery? If some real life program then you should link to it/provide more details on how people were selected/drawn  
> **username_2**: The instrument should  only affect y through x. Saying that z is generated by lottery does not in itself seem to garanty that the outcome of the lottery - that is z - do not somehow affect y (even if not through x).  
> **username_0**: @username_1 Yes, it was meant to be simple random sampling.  
> **username_0**: @username_2 I can see the reasoning, but is it possible to formulate that more rigorously?  

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## 1 answers

### Answer by username_3 on 2021-05-11. Score: 2

The simple answer is **no**.

Randomness does not imply exogeneity.

In my opinion, the following example from [Deaton: "Randomization in the tropics, and the search for the elusive keys to economic development"](https://www.princeton.edu/%7Edeaton/downloads/Instruments%20of%20development%20v1d_mar09_all.pdf) is quite illuminating.

Assume that you want to estimate the effect of having a railway-station on poverty. We have data on the poverty rate $P\_c$ and the presence of a station $R\_c \in \{0,1\}$ for a large number of cities $c$. Assume that the true data generating process is given by:
$$
P\_c = \alpha + \beta\_c R\_c + \varepsilon\_c,
$$
Notice here that the slope $\beta\_c$ is city specific, so the effect of a railroad is heterogeneous across cities.

The coefficient $\beta\_c$ cannot be estimated, but you might wonder whether it is possible to estimate the mean of $\beta\_c$ over all cities that decide to build a railroad. $\overline{\beta} = \mathbb{E}(\beta\_c|R\_c = 1)$. Then:
$$
P\_c = \alpha + \overline{\beta} R\_c + (\varepsilon\_c + (\beta\_c - \overline{\beta})R\_c) = \alpha + \overline{\beta} R\_c + \upsilon\_c,
$$
where $\upsilon\_c = \varepsilon\_c + (\beta\_c - \overline{\beta})R\_c$ is the new error. Although $\mathbb{E}(\upsilon\_c) = 0$, you might be worried that:
$$
\mathbb{E}(R\_c \upsilon\_c) \ne 0.
$$
Indeed, we have:
$$
\mathbb{E}(R\_c \upsilon\_c) = \mathbb{E}(\varepsilon\_c R\_c = 1) + \mathbb{E}(\beta\_c - \overline{\beta}|R\_c = 1) \Pr(R\_c = 1) = \mathbb{E}(\varepsilon\_c R\_c).
$$
This will only be zero if the decision to build a station is uncorrelated to anything that might otherwise influence the poverty rate in the city. As this is probably not satisfied, we need an instrument.

### A random instrument

Assume that the government randomly allocates cities to development areas, captured by the instrument $Z\_c$ (equal to 1 if city $c$ is a development area and zero otherwise).

If designations to development areas motivate the building of railways, we have that there is positive correlation between $Z\_c$ and $R\_c$. Also as $Z\_c$ is random, we can assume that $\mathbb{E}(Z\_c \varepsilon\_c) = 0$. However:
$$
\mathbb{E}(Z\_c \upsilon\_c) = \mathbb{E}(\varepsilon\_c Z\_c) + \mathbb{E}(\beta\_c - \overline{\beta}|Z\_c = 1, R\_c = 1)\Pr(R\_c = 1, Z\_c = 1).
$$
The first term on the right hand side is zero by the randomness of $Z\_c$. As such, $Z\_c$ will be uncorrelated to $\upsilon\_c$ iff:
$$
\mathbb{E}(\beta\_c - \overline{\beta}|Z\_c = 1, R\_c = 1) = \mathbb{E}(\beta\_c|Z\_c = 1, R\_c = 1) - \mathbb{E}(\beta\_c|R\_c = 1) = 0.
$$
This will be the case if the mean effect of building a station over all cities that are a development area equals the mean effect over all cities that build the railroad (whether they are a development area or not).

The last condition can be rewritten as:
$$
\begin{align\*}
&\mathbb{E}(\beta\_c|Z\_c = 1, R\_c = 1) = \mathbb{E}(\beta\_c|R\_c = 1),\\
\to &\mathbb{E}(\beta\_c|Z\_c = 1, R\_c = 1) = \mathbb{E}(\beta\_c|Z\_C = 1, R\_c = 1) \Pr(Z\_c = 1) + \mathbb{E}(\beta\_c|Z\_c = 0, R\_c = 1) \Pr(Z\_c = 0),\\
\to &\mathbb{E}(\beta\_c|Z\_c = 1, R\_c = 1)\Pr(Z\_c = 0) = \mathbb{E}(\beta\_c|Z\_c = 0, R\_c = 1) \Pr(Z\_c = 0),\\
\to &\mathbb{E}(\beta\_c|Z\_c = 1, R\_c = 1)\Pr(Z\_c = 0|R\_c = 1) = \mathbb{E}(\beta\_c|Z\_c = 0, R\_c = 1)\Pr(Z\_c = 0|R\_c = 1),\\
\to &\mathbb{E}(\beta\_c|Z\_c = 1, R\_c = 1)\Pr(Z\_c = 0, R\_c = 1) = \mathbb{E}(\beta\_c|Z\_c = 0, R\_c = 1) \Pr(Z\_c = 0, R\_c = 1).
\end{align\*}
$$
This will be the case if either:

1. $\mathbb{E}(\beta\_c|Z\_c = 1, R\_c = 1) = \mathbb{E}(\beta\_c|Z\_c = 0, R\_c = 1)$ which means that the mean effect among all cities who are a development area and decided to build a railroad should be equal to the mean effect among those who build a railroad and are not a development area. So if the behaviour (i.e. the decision to build a station) changes due to being designated as a development area, this condition will not be satisfied.
2. Or $Pr(Z\_c = 0, R\_c = 1) = 0$ which means that no city that is not a development area builds a station.

In short: randomness of the instrument makes sure that $\mathbb{E}(\varepsilon\_c Z\_c) = 0$ but does not guarantee that $\mathbb{E}(\upsilon\_c Z\_c) = 0$.


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