<p>Endogeneity can arise from several reasons and in each case the explanation will be slightly different. I won't show full review of all possible reasons but just two important examples:</p>
<p><strong>Simultaneity</strong></p>
<p>For example, following Verbeek a guide to modern econometrics pp 146, suppose that true model is given by system of equations:</p>
<p><span class="math-container">$$y = \beta_1 + \beta_2 x_{2t}  + \epsilon_t \tag{1} $$</span></p>
<p><span class="math-container">$$x_{2t} = y_t + z_{2t} \tag{2}$$</span></p>
<p>where <span class="math-container">$z$</span> will be exogenous <span class="math-container">$(cov(z,\epsilon)  = 0)$</span>  but clearly <span class="math-container">$x$</span> is endogenous. We can see from the equations above that if <span class="math-container">$x$</span> increases not only does <span class="math-container">$y$</span> increase but <span class="math-container">$x$</span> increases again through effect of <span class="math-container">$y$</span> on <span class="math-container">$x$</span> given by <span class="math-container">$2$</span> which then again increases <span class="math-container">$x$</span>.</p>
<p>Solving the 1 and 2 for <span class="math-container">$x$</span> and <span class="math-container">$y$</span> respectively we get:</p>
<p><span class="math-container">$$ x_{2t} = \frac{\beta_1}{1-\beta_2} + \frac{1}{1-\beta_2} z_{2t} +  \frac{1}{1-\beta_2} \epsilon_t \tag{3} $$</span></p>
<p><span class="math-container">$$ y_t = \frac{\beta_1}{1-\beta_2} + \frac{\beta_2}{1-\beta_2} z_{2t} + \frac{1}{1-\beta_2} \epsilon_t \tag{4}$$</span></p>
<p>It follows from 3:</p>
<p><span class="math-container">$$ cov(x_{2t} \epsilon_t ) = \frac{1}{1-\beta_2} cov(z_{2t} \epsilon_t) + \frac{1}{1-\beta_2} V(\epsilon) = \frac{\sigma^2}{1-\beta_2} \tag{5} $$</span></p>
<p>The 5 just shows that <span class="math-container">$x$</span> will be correlated with <span class="math-container">$\epsilon$</span>, but a direct consequence of this correlation is that:</p>
<p><span class="math-container">$$ \text{plim } b_2 = \beta_2 + \frac{cov(x_{2t}, \epsilon_t}{V(x_{2t})}$$</span></p>
<p>So what you are estimating is not just your beta coefficient you want <span class="math-container">$\beta_2$</span> but you are estimating sum of the true <span class="math-container">$\beta$</span> coefficient together with ratio of covariance between <span class="math-container">$x$</span> and <span class="math-container">$z$</span> to variance in <span class="math-container">$x$</span> or  <span class="math-container">$\frac{cov(x_{2t}, \epsilon_t}{V(x_{2t})}$</span>.  This indeed can even force coefficients to flip. For example, suppose true effect <span class="math-container">$\beta_2=5$</span> but <span class="math-container">$cov(x,z)=-20$</span> and <span class="math-container">$V(x)=1$</span> then if you just naively run model  <span class="math-container">$y = b_1 + b_2 x_{2t}  + \epsilon_t $</span> your estimated <span class="math-container">$\hat{b_2}$</span> will be <span class="math-container">$\hat{b_2} = 2 -20 = -18$</span> which has opposite sign and the magnitude of <span class="math-container">$b$</span> will be distorted. So you will be getting unreliable results.</p>
<p><strong>Ommited Variable Bias</strong></p>
<p>This would be example, where we do not observe (or fail to include) variable that should be in the model. Again following Verbeek pp 145 consider individual wage equation:</p>
<p><span class="math-container">$$ y_ = x_{1i}' \beta_1 + x_{2i} \beta_2 +  u_i \gamma + v_i$$</span></p>
<p>here <span class="math-container">$y$</span> would be wage, <span class="math-container">$x_{1i}$</span> is vector of individual characteristics (e.g. gender, age, location etc), <span class="math-container">$x_{2i}$</span> years of schooling and <span class="math-container">$u_i$</span> some unobserved innate ability. We expect that <span class="math-container">$cov(x_{2i},u_i)&gt;0$</span> because it makes sense that if you have higher innate ability you will have higher education as people with high innate ability should have easier time at the school/university.</p>
<p>Now let us suppose that because innate ability is not observable you are forced to estimate:</p>
<p><span class="math-container">$$y x_i'\beta + \epsilon_i$$</span></p>
<p>with <span class="math-container">$x_i'=(x_{1i}', x_{2i}) $</span> and <span class="math-container">$\beta'= (\beta_1', \beta_2), $</span> and naturally since we omitted ability  <span class="math-container">$\epsilon_i = u_i\gamma + v_i$</span>.</p>
<p>Now the <span class="math-container">$b$</span> estimates of <span class="math-container">$\beta$</span> are given by:</p>
<p><span class="math-container">$$b= \beta +  \left( \sum_{i=1}^N x_i x_i' \right)^{-1} \sum_{i=1}^N x_i u_i \gamma + \left( \sum_{i=1}^N x_i x_i' \right)^{-1} \sum_{i=1}^N x_i v_i$$</span></p>
<p>Even assuming <span class="math-container">$E(x_i, v) = 0$</span> (i.e. there is no additional simultaneity or other endogeneity issues in the true model), we get:</p>
<p><span class="math-container">$$\text{plim } b = \beta + \sum_{xx}^{-1}E[x_i u_i] \gamma$$</span></p>
<p>So as long either <span class="math-container">$\gamma \neq 0 $</span> (in which case it should not be in the first model to begin with as that would mean ability does not affect wage), or <span class="math-container">$E[x_i u_i] \neq 0$</span> which in our case wont hold because we assumed ability does affect education, your estimated beta is <strong>not</strong> going to be actual effect of the independent variable on the dependent variable but the actual effect plus term determined by correlation between omitted and included independent variable and <span class="math-container">$\gamma$</span>. Again this can change both magnitude and sign of <span class="math-container">$\beta$</span> so you simply can't trust it at all.</p>
<p>Lastly, there could be other ways in which endogeneity arises in econometric setting, but exploring all of them is beyond scope of SE, and the two above mentioned ones are probably the most common ones and relevant.</p>

CC BY-SA 4.0
2021-05-08T11:26:45.087Z
2021-05-08T11:34:29.910Z
2021-05-08T11:34:29.910Z
1muflon1
<p>When learning about endogenous variable, I found an answer <a href="https://economics.stackexchange.com/a/18136/33142">in a topic on StackExchange</a> very direct:</p>
<blockquote>
<p>An Exogenous Variable is defined as a variable which is unaffected by
other variables within an model.</p>
<p>take a multivariable regression model as an example:</p>
<p><span class="math-container">$y=\beta_0+\beta_1x_1+\beta_2x_2+u$</span></p>
<p><span class="math-container">$x_1$</span> is called an exogenous variable when its determination is
unaffected by <span class="math-container">$x_2$</span> and error term <span class="math-container">$u$</span>.</p>
<p>An Endogenous Variable is defined when <span class="math-container">$x_1$</span> is influneced by <span class="math-container">$x_2$</span> or
<span class="math-container">$u$</span>.</p>
<p>This is important because when we run a regression we are producing a
function which assumes a dependent and independent variable. If we
find endogeneity we wont get accurate estimates for the effect of
<span class="math-container">$x_1$</span> on <span class="math-container">$y$</span>.</p>
</blockquote>
<p>Why in endogeneity case, we wont get accurate estimates for the effect of <span class="math-container">$x_1$</span> on <span class="math-container">$y$</span> . Is there any sample for that statement? In reality, <a href="https://stats.stackexchange.com/questions/521821/what-is-the-solution-of-simpson-paradox">I faced this problem that when I add some regressors to a regression equation, one of variable coefficient changed the magnitude or even flip the sign</a>. But I want to know the reason behind.<br />
One theoretical explanation is: <span class="math-container">$x_1$</span> is correlated with <span class="math-container">$u$</span> or <span class="math-container">$x_2$</span>, producing a bias in <span class="math-container">$beta_1$</span> that depends on the direction and magnitude of the correlation</p>

<endogeneity>
What is the example of inaccurate impact due to endogeneity?
Phil Nguyen
# What is the example of inaccurate impact due to endogeneity?

(Asked by: username_0 on 2021-05-08)

When learning about endogenous variable, I found an answer [in a topic on StackExchange](https://economics.stackexchange.com/a/18136/33142) very direct:

> An Exogenous Variable is defined as a variable which is unaffected by
> other variables within an model.
>
> take a multivariable regression model as an example:
>
> $y=\beta\_0+\beta\_1x\_1+\beta\_2x\_2+u$
>
> $x\_1$ is called an exogenous variable when its determination is
> unaffected by $x\_2$ and error term $u$.
>
> An Endogenous Variable is defined when $x\_1$ is influneced by $x\_2$ or
> $u$.
>
> This is important because when we run a regression we are producing a
> function which assumes a dependent and independent variable. If we
> find endogeneity we wont get accurate estimates for the effect of
> $x\_1$ on $y$.

Why in endogeneity case, we wont get accurate estimates for the effect of $x\_1$ on $y$ . Is there any sample for that statement? In reality, [I faced this problem that when I add some regressors to a regression equation, one of variable coefficient changed the magnitude or even flip the sign](https://stats.stackexchange.com/questions/521821/what-is-the-solution-of-simpson-paradox). But I want to know the reason behind.  
One theoretical explanation is: $x\_1$ is correlated with $u$ or $x\_2$, producing a bias in $beta\_1$ that depends on the direction and magnitude of the correlation

**Tags:** endogeneity

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## 1 answers

### Answer by username_1 on 2021-05-08. Score: 3 | Accepted answer

Endogeneity can arise from several reasons and in each case the explanation will be slightly different. I won't show full review of all possible reasons but just two important examples:

**Simultaneity**

For example, following Verbeek a guide to modern econometrics pp 146, suppose that true model is given by system of equations:

$$y = \beta\_1 + \beta\_2 x\_{2t} + \epsilon\_t \tag{1} $$

$$x\_{2t} = y\_t + z\_{2t} \tag{2}$$

where $z$ will be exogenous $(cov(z,\epsilon) = 0)$ but clearly $x$ is endogenous. We can see from the equations above that if $x$ increases not only does $y$ increase but $x$ increases again through effect of $y$ on $x$ given by $2$ which then again increases $x$.

Solving the 1 and 2 for $x$ and $y$ respectively we get:

$$ x\_{2t} = \frac{\beta\_1}{1-\beta\_2} + \frac{1}{1-\beta\_2} z\_{2t} + \frac{1}{1-\beta\_2} \epsilon\_t \tag{3} $$

$$ y\_t = \frac{\beta\_1}{1-\beta\_2} + \frac{\beta\_2}{1-\beta\_2} z\_{2t} + \frac{1}{1-\beta\_2} \epsilon\_t \tag{4}$$

It follows from 3:

$$ cov(x\_{2t} \epsilon\_t ) = \frac{1}{1-\beta\_2} cov(z\_{2t} \epsilon\_t) + \frac{1}{1-\beta\_2} V(\epsilon) = \frac{\sigma^2}{1-\beta\_2} \tag{5} $$

The 5 just shows that $x$ will be correlated with $\epsilon$, but a direct consequence of this correlation is that:

$$ \text{plim } b\_2 = \beta\_2 + \frac{cov(x\_{2t}, \epsilon\_t}{V(x\_{2t})}$$

So what you are estimating is not just your beta coefficient you want $\beta\_2$ but you are estimating sum of the true $\beta$ coefficient together with ratio of covariance between $x$ and $z$ to variance in $x$ or $\frac{cov(x\_{2t}, \epsilon\_t}{V(x\_{2t})}$. This indeed can even force coefficients to flip. For example, suppose true effect $\beta\_2=5$ but $cov(x,z)=-20$ and $V(x)=1$ then if you just naively run model $y = b\_1 + b\_2 x\_{2t} + \epsilon\_t $ your estimated $\hat{b\_2}$ will be $\hat{b\_2} = 2 -20 = -18$ which has opposite sign and the magnitude of $b$ will be distorted. So you will be getting unreliable results.

**Ommited Variable Bias**

This would be example, where we do not observe (or fail to include) variable that should be in the model. Again following Verbeek pp 145 consider individual wage equation:

$$ y\_ = x\_{1i}' \beta\_1 + x\_{2i} \beta\_2 + u\_i \gamma + v\_i$$

here $y$ would be wage, $x\_{1i}$ is vector of individual characteristics (e.g. gender, age, location etc), $x\_{2i}$ years of schooling and $u\_i$ some unobserved innate ability. We expect that $cov(x\_{2i},u\_i)>0$ because it makes sense that if you have higher innate ability you will have higher education as people with high innate ability should have easier time at the school/university.

Now let us suppose that because innate ability is not observable you are forced to estimate:

$$y x\_i'\beta + \epsilon\_i$$

with $x\_i'=(x\_{1i}', x\_{2i}) $ and $\beta'= (\beta\_1', \beta\_2), $ and naturally since we omitted ability $\epsilon\_i = u\_i\gamma + v\_i$.

Now the $b$ estimates of $\beta$ are given by:

$$b= \beta + \left( \sum\_{i=1}^N x\_i x\_i' \right)^{-1} \sum\_{i=1}^N x\_i u\_i \gamma + \left( \sum\_{i=1}^N x\_i x\_i' \right)^{-1} \sum\_{i=1}^N x\_i v\_i$$

Even assuming $E(x\_i, v) = 0$ (i.e. there is no additional simultaneity or other endogeneity issues in the true model), we get:

$$\text{plim } b = \beta + \sum\_{xx}^{-1}E[x\_i u\_i] \gamma$$

So as long either $\gamma \neq 0 $ (in which case it should not be in the first model to begin with as that would mean ability does not affect wage), or $E[x\_i u\_i] \neq 0$ which in our case wont hold because we assumed ability does affect education, your estimated beta is **not** going to be actual effect of the independent variable on the dependent variable but the actual effect plus term determined by correlation between omitted and included independent variable and $\gamma$. Again this can change both magnitude and sign of $\beta$ so you simply can't trust it at all.

Lastly, there could be other ways in which endogeneity arises in econometric setting, but exploring all of them is beyond scope of SE, and the two above mentioned ones are probably the most common ones and relevant.


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