# Solving correlation between explanatory variables using instrumental variables in economic production function

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

$$$$y_{i} = x_{i}\beta + \alpha_i + \epsilon_{i}$$$$

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?

• 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