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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

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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.

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