<blockquote>
  <p>Suppose you run a regression (over the entire sample) of hourly earnings <span class="math-container">$Y$</span> with an intercept and a binary variable <span class="math-container">$D$</span>, where <span class="math-container">$D$</span> takes on a value of <span class="math-container">$1$</span> if the individual has a college degree and <span class="math-container">$0$</span> otherwise:
  <span class="math-container">$$Y=\beta_0 + \beta_1D+U$$</span>
  with <span class="math-container">$E(U|D)=0$</span>. Does <span class="math-container">$\beta_1$</span> admit a causal interpretation? </p>
</blockquote>

<p>My instinct says yes since the zero conditional mean implies zero covariance, which then gives unbiasedness and consistency for <span class="math-container">$\hat{\beta_1}$</span>. But the solution says otherwise:</p>

<blockquote>
  <p>We should not expect <span class="math-container">$\beta_1$</span> to admit a causal interpretation here: If <span class="math-container">$U$</span> captures all unobserved factors which determine hourly earnings, it should not be the case that <span class="math-container">$E(U|D)=0$</span> as <span class="math-container">$U$</span> should include some measure of ability which we would expect it to be correlated with <span class="math-container">$D$</span>. <span class="math-container">$\beta_1$</span> simply captures the average difference in hourly earning between grads and non-grads. </p>
</blockquote>

<p>Can someone please explain to me how my intuition was wrong?</p>

CC BY-SA 4.0
2020-03-03T11:45:06.343Z
Either your quoted question or quotedsolution is mis-stated. They don't speak to each other. If E[U|D] = 0, then $\hat{\beta_1}$ admit causal interpretation. This model is akin to a difference-in-means between D = 0 and D = 1. E[U|D] = 0 means there's no selection bias between the two groups. Now, whether E[U|D] actually holds in your sample is a different question. But if it's assumed to hold, then beta hat has causal interpretation.
Michael
CC BY-SA 4.0
2020-03-03T20:37:50.533Z
@Michael I agree with you as this was my thought initially. I have edited the quoted solution so that it is verbatim. Does it still make no sense to you (it doesn't to me)?
CharlieCornell
CC BY-SA 4.0
2020-03-03T21:49:22.717Z
No, they are addressing different issues. Issue 1: if exogeneity holds, does beta admit causal interpretation? Issue 2: does exogeneity actually hold for the sample in question?
Michael
CC BY-SA 4.0
2020-03-03T22:15:43.103Z
What is $U$?  Is it the disturbance term in the population regression model?  Or is it the fitted residual?
Adam Bailey
CC BY-SA 4.0
2020-03-04T01:47:15.490Z
@AdamBailey Disturbance term, I believe
CharlieCornell
<econometrics><causality>
Causal Interpretation of $\beta_1$ with a Dummy explanatory variable
CharlieCornell
# Causal Interpretation of $\beta_1$ with a Dummy explanatory variable

(Asked by: username_0 on 2020-03-03)

> Suppose you run a regression (over the entire sample) of hourly earnings $Y$ with an intercept and a binary variable $D$, where $D$ takes on a value of $1$ if the individual has a college degree and $0$ otherwise:
> $$Y=\beta\_0 + \beta\_1D+U$$
> with $E(U|D)=0$. Does $\beta\_1$ admit a causal interpretation?

My instinct says yes since the zero conditional mean implies zero covariance, which then gives unbiasedness and consistency for $\hat{\beta\_1}$. But the solution says otherwise:

> We should not expect $\beta\_1$ to admit a causal interpretation here: If $U$ captures all unobserved factors which determine hourly earnings, it should not be the case that $E(U|D)=0$ as $U$ should include some measure of ability which we would expect it to be correlated with $D$. $\beta\_1$ simply captures the average difference in hourly earning between grads and non-grads.

Can someone please explain to me how my intuition was wrong?

**Tags:** econometrics, causality

> **Comments:**
> **username_1**: Either your quoted question or quotedsolution is mis-stated. They don't speak to each other. If E[U|D] = 0, then $\hat{\beta_1}$ admit causal interpretation. This model is akin to a difference-in-means between D = 0 and D = 1. E[U|D] = 0 means there's no selection bias between the two groups. Now, whether E[U|D] actually holds in your sample is a different question. But if it's assumed to hold, then beta hat has causal interpretation.  
> **username_0**: @username_1 I agree with you as this was my thought initially. I have edited the quoted solution so that it is verbatim. Does it still make no sense to you (it doesn't to me)?  
> **username_1**: No, they are addressing different issues. Issue 1: if exogeneity holds, does beta admit causal interpretation? Issue 2: does exogeneity actually hold for the sample in question?  
> **username_2**: What is $U$?  Is it the disturbance term in the population regression model?  Or is it the fitted residual?  
> **username_0**: @username_2 Disturbance term, I believe  

---

## 0 answers

