Transforming panel data OLS into cross-sectional data model

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.

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