I am currently stuck on a task where I am interested in estimating the production function for agricultural output as follows:

\begin{equation} y_{i} = x_{i}\beta + \alpha_i + \epsilon_{i} \end{equation}

where $y_{i}$ is log($output$), $x_{i}$ is log($labour$) - a variable input, $\alpha_i$ is log($soil quality$) - a fixed input, and $\epsilon_{i}$ is rainfall - a random input. Each farmer knows the price of output $P$, the wage rate $W$, and the soil quality of his farm $\alpha_i$. However, as the econometrician you only observe ($y_{i}$, $x_{i}$). Assume that $\epsilon_{i}$ is $iid$ and independent of everything in the model.

Since I know that $\alpha_i$ is correlated with labour decisions $x_{i}$, two explanatory variables are correlated and therefore violating key assumptions of the classical linear regression model. I know this can be solved by implementing instrumental variables, however I've only seen this when the error terms is correlated with an explanatory variable. How do I solve this in my problem where 2 explanatory variables are correlated? Could the variables $P$ and $W$ possibly help? Or do I need access to other variables?

Extremely glad for any help!

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  • Having two explanatory variables correlated does not violate the OLS assumptions unless they are perfectly colinear. Indeed, you say two explanatory variables are correlated. But the econometrician does not observe $\alpha_i$. Therefore, from the econometrician's point of view, $\alpha_i$ is subsumed into the error and we are left with the task of estimating $\beta$ in $y_i=\beta x_i+e_i$, where $e_i=\epsilon_i+\alpha_i$ is the residual. Thus, this problem is just like the cases you have seen where $x_i$ is correlated with the error. That's the real problem. – Ubiquitous Dec 6 at 16:38
  • I see, thanks a lot! – rbonac Dec 6 at 16:46

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