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

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

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

As a special case I only observe one period so T = 1. Therefore my pooled OLS approach transforms into a cross-sectional analysis. How am I able to consistently estimate $\beta$, since I know that $\alpha_i$ is correlated with labor decisions $x_{it}$ and therefore a key assumption of the classical linear regression model is violated.

Glad for any help!

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  • $\begingroup$ You could try to find a suitable instrumental variable for $x$. $\endgroup$ – Christoph Hanck Dec 5 '18 at 15:21

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