<p>The CIA assumption is equivalent to <span class="math-container">$E\{ \nu_i D_i \vert X_i \} = 0$</span>. However, it is not the case that the unconditional expectation is equal to zero. The whole reason we need the CIA is that we don't have random assignment, but potentially we have random assignment <em>conditional</em> on the covariates we can observe. </p>

<p>I think it may help to think of <span class="math-container">$\nu_i$</span> as an error term that is composed of other variables <span class="math-container">$Z_i$</span> with effect <span class="math-container">$\gamma Z_i$</span>. In this case, <span class="math-container">$\nu_i \perp D_i \vert X_i$</span> means that <span class="math-container">$Z_i$</span> does not predict treatment after conditioning on the observed covariates <span class="math-container">$X_i$</span>. </p>

<p>To answer your IV question, this is indeed similar to the exclusion restriction in that you are making assumptions about the unobserved variables <span class="math-container">$Z_i$</span> in your data generating process and your treatment/instrumental variable. But I wouldn't try to connect the ideas too much or else you may come to wrong conclusions.</p>

CC BY-SA 4.0
2020-04-27T15:38:32.367Z
2020-04-27T15:38:32.367Z
Kyle Butts
<p>I have recently got myself confused about CIA and wondered if somebody could perhaps help me disentangle my thoughts. Consider the regression</p>

<p><span class="math-container">$Y_i = \alpha_0 + \Delta D_i + \beta X_i + \eta_i$</span></p>

<p>Where <span class="math-container">$X_i$</span> are controls/covariates and <span class="math-container">$D_i$</span> is the treatment dummy. The CIA says that <span class="math-container">$\eta_i \perp D_i |X_i$</span>. If this holds the causal effect of the treatment can be estimated with <span class="math-container">$E[Y_{1i} - Y_{0i}|X_i] = \Delta$</span>.</p>

<p>Is the CIA the same thing as saying <span class="math-container">$E[\eta_iD_i] = 0$</span>? (considering <span class="math-container">$\eta_i$</span> should have mean zero because a constant is included in the regression?) Which then is the same thing as <span class="math-container">$Cov(\eta_i, D_i) = 0$</span>? How is this related to <span class="math-container">$E[\eta_i|X_i] = 0$</span> Gauss-Markov assumption? Does this have anything to do with the <em>exclusion restriction</em> for IV which looks very similar?</p>

CC BY-SA 4.0
2020-04-27T21:29:59.670Z
What does CIA stand for?
Brennan
<econometrics><treatment-effect>
CIA and Exclusion restriction
Jhonny
# CIA and Exclusion restriction

(Asked by: username_0 on 2020-04-27)

I have recently got myself confused about CIA and wondered if somebody could perhaps help me disentangle my thoughts. Consider the regression

$Y\_i = \alpha\_0 + \Delta D\_i + \beta X\_i + \eta\_i$

Where $X\_i$ are controls/covariates and $D\_i$ is the treatment dummy. The CIA says that $\eta\_i \perp D\_i |X\_i$. If this holds the causal effect of the treatment can be estimated with $E[Y\_{1i} - Y\_{0i}|X\_i] = \Delta$.

Is the CIA the same thing as saying $E[\eta\_iD\_i] = 0$? (considering $\eta\_i$ should have mean zero because a constant is included in the regression?) Which then is the same thing as $Cov(\eta\_i, D\_i) = 0$? How is this related to $E[\eta\_i|X\_i] = 0$ Gauss-Markov assumption? Does this have anything to do with the *exclusion restriction* for IV which looks very similar?

**Tags:** econometrics, treatment-effect
> **Comments:**
> **username_1**: What does CIA stand for?  

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## 1 answers

### Answer by username_2 on 2020-04-27. Score: 1

The CIA assumption is equivalent to $E\{ \nu\_i D\_i \vert X\_i \} = 0$. However, it is not the case that the unconditional expectation is equal to zero. The whole reason we need the CIA is that we don't have random assignment, but potentially we have random assignment *conditional* on the covariates we can observe.

I think it may help to think of $\nu\_i$ as an error term that is composed of other variables $Z\_i$ with effect $\gamma Z\_i$. In this case, $\nu\_i \perp D\_i \vert X\_i$ means that $Z\_i$ does not predict treatment after conditioning on the observed covariates $X\_i$.

To answer your IV question, this is indeed similar to the exclusion restriction in that you are making assumptions about the unobserved variables $Z\_i$ in your data generating process and your treatment/instrumental variable. But I wouldn't try to connect the ideas too much or else you may come to wrong conclusions.


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