<p>I think you could proceed by regressing <span class="math-container">$Y$</span> on <span class="math-container">$X$</span>, <span class="math-container">$Z$</span> and <span class="math-container">$V = (V_1 + V_2)$</span>. Note that by substituting <span class="math-container">$U$</span> into the regression you get:</p>
<p><span class="math-container">$$
Y = \alpha X + \beta Z + \gamma \delta_4 V + (\varepsilon + \gamma \mu_3).
$$</span>
If <span class="math-container">$X, Z$</span> and <span class="math-container">$V$</span> are orthogonal to <span class="math-container">$\varepsilon$</span> and <span class="math-container">$\mu_3$</span>, we get the following moment conditions:
<span class="math-container">$$
\begin{align*}
&amp;cov(X,Y) = \alpha cov(X,X) + \beta cov(X,Z) + \gamma \delta_4 cov(X,V),\\
&amp;cov(Z,Y) = \alpha cov(Z,X) + \beta cov(Z,Z) + \gamma \delta_4 cov(Z,V),\\
&amp;cov(V,Y) = \alpha cov(V,X) + \beta cov(V,Z) + \gamma \delta_4 cov(V,V).
\end{align*}
$$</span>
This is a system of 3 equations in 3 unknowns <span class="math-container">$\alpha, \beta$</span> and <span class="math-container">$\gamma \delta_4$</span>. Note that <span class="math-container">$\gamma$</span> is not identified seperately from <span class="math-container">$\delta_4$</span>. Indeed multiplying <span class="math-container">$\gamma$</span> by a constant and dividing <span class="math-container">$\delta_4$</span> by the same constant gives the same data.</p>

CC BY-SA 4.0
2024-05-11T09:22:43.007Z
2024-05-11T09:22:43.007Z
tdm
<p>I am interested in estimating the relative effects of <span class="math-container">$X$</span> and <span class="math-container">$Z$</span> on <span class="math-container">$Y$</span> in the model <span class="math-container">$Y = \alpha X + \beta Z + \gamma U + \epsilon$</span>. Both <span class="math-container">$X$</span> and <span class="math-container">$Z$</span> are endogenous because <span class="math-container">$U$</span> is unobserved and potentially correlated with both <span class="math-container">$X$</span> and <span class="math-container">$Z$</span>.</p>
<p>I am trying to identify <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> using variation in another variable <span class="math-container">$V$</span>. This variable does not qualify as a valid instrument because it can also be correlated with <span class="math-container">$U$</span>, however, I observe two sources of variation in <span class="math-container">$V$</span> with a distinct impact on <span class="math-container">$X$</span>, <span class="math-container">$Z$</span>, and <span class="math-container">$U$</span>:</p>
<p><span class="math-container">$X = \delta_1  V1 +  \delta_2 V2 + \mu_1$</span></p>
<p><span class="math-container">$Z = \delta_3  V1 + \mu_2$</span></p>
<p><span class="math-container">$U = \delta_4 ( V1 +  V2) + \mu_3$</span></p>
<p>Here <span class="math-container">$V1$</span> and <span class="math-container">$V2$</span> are changes in <span class="math-container">$V$</span> that occur over different periods. <span class="math-container">$V2$</span> does not affect <span class="math-container">$Z$</span> and the impact on <span class="math-container">$X$</span> is exogenous (<span class="math-container">$\delta_2$</span> is identified).</p>
<p>What kind of empirical approaches can be used in this setting? Are there ways to use some kind of control function strategies? Any references that address this or similar problems would be very helpful. Thank you</p>

<econometrics><identification>
Two endogenous variables and two control functions
Peter
# Two endogenous variables and two control functions

(Asked by: username_0 on 2024-05-09)

I am interested in estimating the relative effects of $X$ and $Z$ on $Y$ in the model $Y = \alpha X + \beta Z + \gamma U + \epsilon$. Both $X$ and $Z$ are endogenous because $U$ is unobserved and potentially correlated with both $X$ and $Z$.

I am trying to identify $\alpha$ and $\beta$ using variation in another variable $V$. This variable does not qualify as a valid instrument because it can also be correlated with $U$, however, I observe two sources of variation in $V$ with a distinct impact on $X$, $Z$, and $U$:

$X = \delta\_1 V1 + \delta\_2 V2 + \mu\_1$

$Z = \delta\_3 V1 + \mu\_2$

$U = \delta\_4 ( V1 + V2) + \mu\_3$

Here $V1$ and $V2$ are changes in $V$ that occur over different periods. $V2$ does not affect $Z$ and the impact on $X$ is exogenous ($\delta\_2$ is identified).

What kind of empirical approaches can be used in this setting? Are there ways to use some kind of control function strategies? Any references that address this or similar problems would be very helpful. Thank you

**Tags:** econometrics, identification


---

## 1 answers

### Answer by username_1 on 2024-05-11. Score: 1

I think you could proceed by regressing $Y$ on $X$, $Z$ and $V = (V\_1 + V\_2)$. Note that by substituting $U$ into the regression you get:

$$
Y = \alpha X + \beta Z + \gamma \delta\_4 V + (\varepsilon + \gamma \mu\_3).
$$
If $X, Z$ and $V$ are orthogonal to $\varepsilon$ and $\mu\_3$, we get the following moment conditions:
$$
\begin{align\*}
&cov(X,Y) = \alpha cov(X,X) + \beta cov(X,Z) + \gamma \delta\_4 cov(X,V),\\
&cov(Z,Y) = \alpha cov(Z,X) + \beta cov(Z,Z) + \gamma \delta\_4 cov(Z,V),\\
&cov(V,Y) = \alpha cov(V,X) + \beta cov(V,Z) + \gamma \delta\_4 cov(V,V).
\end{align\*}
$$
This is a system of 3 equations in 3 unknowns $\alpha, \beta$ and $\gamma \delta\_4$. Note that $\gamma$ is not identified seperately from $\delta\_4$. Indeed multiplying $\gamma$ by a constant and dividing $\delta\_4$ by the same constant gives the same data.


---

