<p>Short answer: No.</p>
<p>Your model is <span class="math-container">$Y=\alpha + \beta X + \varepsilon$</span>. Even when <span class="math-container">$X$</span> is <em>exogenous</em>, if you regress <span class="math-container">$Y$</span> on <span class="math-container">$X$</span>, <span class="math-container">$W_1$</span> and <span class="math-container">$W_2$</span>, then the OLS estimator is inconsistent (for <span class="math-container">$\beta$</span>) unless <span class="math-container">$W_1$</span> and <span class="math-container">$W_2$</span> do not affect <span class="math-container">$Y$</span> on average (after controlling for <span class="math-container">$X$</span>) or <span class="math-container">$X$</span> is uncorrelated with <span class="math-container">$W_1$</span> and <span class="math-container">$W_2$</span>. When <span class="math-container">$X$</span> is <em>endogenous</em>, there is no reason why your strategy should work.</p>
<p>Now suppose that your model is <span class="math-container">$Y=\alpha + \delta X + \gamma_1 W_1 + \gamma_2 W_2 + \varepsilon$</span>, instead. (This is not your model; <span class="math-container">$\delta \ne \beta$</span>.) You are saying <span class="math-container">$X$</span>, <span class="math-container">$W_1$</span> and <span class="math-container">$W_2$</span> are endogenous, while <span class="math-container">$Z$</span> is exogenous. You need instruments for <span class="math-container">$X$</span>, <span class="math-container">$W_1$</span> and <span class="math-container">$W_2$</span>, but you don't. The parameters are not identified.</p>

CC BY-SA 4.0
2021-02-21T23:05:24.913Z
I already accepted this answer, but just realized that I actually do not completely follow. Would $\eta_1$ not be interpretable as causal under the conditional mean independence assumption? And why would regressing Y on an exogenous X (and endogenous Ws) yield an inconsistent estimator for $\beta$? From my understanding it would only be a problem if the Ws where not included and are correlated with X and Y.
ArOk
CC BY-SA 4.0
2021-02-22T01:23:44.550Z
I think it's a matter of what the model is. Let us accept it that the relationship of interest is $Y = \alpha + \beta X + \varepsilon$. If X is exogenous but W is included in the RHS, the estimator is inconsistent (in the sense that $\eta_1$ is not equal to $\beta$) in general (unless X and W are uncorrelated) - the opposite case of omitted important variables.
chan1142
CC BY-SA 4.0
2021-02-21T04:47:51.060Z
2021-02-21T16:42:34.113Z
2021-02-21T16:42:34.113Z
chan1142
<p>If I want to run a 2 stage least squares (2SLS) regression with:</p>
<p>Relationship of interest: <span class="math-container">$Y = \alpha + \beta X + \varepsilon $</span>, where <span class="math-container">$X$</span> is the endogenous explanatory variable of interest.</p>
<p>If I have an instrument <span class="math-container">$Z$</span> where I can safely assume that it is both relevant and fulfils the exclusion restriction, can I interact it with other variables in the first stage, that do not fulfil the exclusion restriction, as long as I control for these other variables in the second stage?</p>
<p>So I am thinking</p>
<p>Estimating the first stage: <span class="math-container">$X = \gamma + \delta_1 Z + \delta_2 W_1 + \delta_3 W_2 + \delta_4 Z*W_1 + \delta_5 Z* W_2 + \epsilon$</span>, where the exclusion restriction only holds for <span class="math-container">$Z$</span> but not for the <span class="math-container">$W$</span>s, to get <span class="math-container">$\hat X$</span></p>
<p>Second stage: <span class="math-container">$Y = \omega + \eta_1 \hat X + \eta_2 W_1 + \eta_3 W_2 + e $</span></p>
<p>So does this procedure allow me to kind of side-step the exclusion restriction? If so, is there any paper / book / article that talks about that?</p>

<econometrics><applied-econometrics><instrumental-variable><causality>
Interacting covariates with the instrument in the first stage
ArOk
# Interacting covariates with the instrument in the first stage

(Asked by: username_0 on 2021-02-19)

If I want to run a 2 stage least squares (2SLS) regression with:

Relationship of interest: $Y = \alpha + \beta X + \varepsilon $, where $X$ is the endogenous explanatory variable of interest.

If I have an instrument $Z$ where I can safely assume that it is both relevant and fulfils the exclusion restriction, can I interact it with other variables in the first stage, that do not fulfil the exclusion restriction, as long as I control for these other variables in the second stage?

So I am thinking

Estimating the first stage: $X = \gamma + \delta\_1 Z + \delta\_2 W\_1 + \delta\_3 W\_2 + \delta\_4 Z\*W\_1 + \delta\_5 Z\* W\_2 + \epsilon$, where the exclusion restriction only holds for $Z$ but not for the $W$s, to get $\hat X$

Second stage: $Y = \omega + \eta\_1 \hat X + \eta\_2 W\_1 + \eta\_3 W\_2 + e $

So does this procedure allow me to kind of side-step the exclusion restriction? If so, is there any paper / book / article that talks about that?

**Tags:** econometrics, applied-econometrics, instrumental-variable, causality

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## 1 answers

### Answer by username_1 on 2021-02-21. Score: 2 | Accepted answer

Short answer: No.

Your model is $Y=\alpha + \beta X + \varepsilon$. Even when $X$ is *exogenous*, if you regress $Y$ on $X$, $W\_1$ and $W\_2$, then the OLS estimator is inconsistent (for $\beta$) unless $W\_1$ and $W\_2$ do not affect $Y$ on average (after controlling for $X$) or $X$ is uncorrelated with $W\_1$ and $W\_2$. When $X$ is *endogenous*, there is no reason why your strategy should work.

Now suppose that your model is $Y=\alpha + \delta X + \gamma\_1 W\_1 + \gamma\_2 W\_2 + \varepsilon$, instead. (This is not your model; $\delta \ne \beta$.) You are saying $X$, $W\_1$ and $W\_2$ are endogenous, while $Z$ is exogenous. You need instruments for $X$, $W\_1$ and $W\_2$, but you don't. The parameters are not identified.

> **Comments:**
> **username_0**: I already accepted this answer, but just realized that I actually do not completely follow. Would $\eta_1$ not be interpretable as causal under the conditional mean independence assumption? And why would regressing Y on an exogenous X (and endogenous Ws) yield an inconsistent estimator for $\beta$? From my understanding it would only be a problem if the Ws where not included and are correlated with X and Y.  
> **username_1**: I think it's a matter of what the model is. Let us accept it that the relationship of interest is $Y = \alpha + \beta X + \varepsilon$. If X is exogenous but W is included in the RHS, the estimator is inconsistent (in the sense that $\eta_1$ is not equal to $\beta$) in general (unless X and W are uncorrelated) - the opposite case of omitted important variables.  

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