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!

  • $\begingroup$ 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. $\endgroup$ – Jesper Hybel Dec 6 '18 at 11:03
  • $\begingroup$ I see thanks a lot for your input! Now the question is on how to implement an instrumental variable. $\endgroup$ – rbonac Dec 6 '18 at 12:02
  • $\begingroup$ 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? $\endgroup$ – Jesper Hybel Dec 6 '18 at 14:02
  • $\begingroup$ Exactly! No time variation, therefore no panel data, just cross-sectional analysis of firms. $\endgroup$ – rbonac Dec 6 '18 at 14:15
  • $\begingroup$ 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). $\endgroup$ – Jesper Hybel Dec 6 '18 at 14:40

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