# Solving correlation between explanatory variables using instrumental variables

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!

• you need to estimate b using estimation equation y=xb + u where u = a + e, but x is correlated with u due to x being correlated with a (though not e). So in your estimation equation you do not have two explanatory variables correlated, only labor input correlated with error term. How to construct an instrument for I have no thoughts on. – Jesper Hybel Dec 6 '18 at 11:03
• I see thanks a lot for your input! Now the question is on how to implement an instrumental variable. – rbonac Dec 6 '18 at 12:02
• A clarifying question: You do not have any time variation in output and labor, so each firm is only observed once, hence data is cross section of firms? – Jesper Hybel Dec 6 '18 at 14:02
• Exactly! No time variation, therefore no panel data, just cross-sectional analysis of firms. – rbonac Dec 6 '18 at 14:15
• Then I would say you need instrumental variables or a proxy for soil-quality. P and W wont really help since they are constants (the way I read youre post). – Jesper Hybel Dec 6 '18 at 14:40