I'm working on a university task where I have to estimate the following using panel data:

\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. I have T = 10 periods.

Since I know that $\alpha_i$ is correlated with labour decisions $x_{it}$, the problem of endogeneity occurs. Now which practices is more suitable to have a consistent estimate of $\beta$ and why?

  • $\begingroup$ Is the purpose of taking logs solely so that linear regression can be used rather than using non-linear regression? $\endgroup$ – James Phillips Dec 6 '18 at 13:38
  • $\begingroup$ I think so, however you might also ignore the logs, since this is a theoretical exercise and I am able to make the assumption that our model is linear $\endgroup$ – rbonac Dec 6 '18 at 13:39
  • $\begingroup$ Would you please post a link to the data? I would like to run it through my zunzun.com "function finder" for an equation search and see what candidate equations it suggests.. $\endgroup$ – James Phillips Dec 6 '18 at 13:42
  • $\begingroup$ Unfortunately I do not have any data since this is only a theoretical exercise, however it is based on the seminal article in 1961 by Mundlak in the Journal of Farm Economics. $\endgroup$ – rbonac Dec 6 '18 at 13:53

You would want to estimate the regression

$$y_{it} = x_{it}\beta + u_{it},$$

where $u_{it} = \alpha_i + \epsilon_{it}$.

In random effects it is assumed that $$Cov(x_{it},u_{it}) = 0$$ which requires $$Cov(x_{it},\alpha_i) = 0$$ which ex hypotesi is not satisfied. So a random effects model will not work.

In the fixed effects model $$Cov(x_{it},\alpha_i) \not = 0$$ and it can be estimated using first differences.

Taking first differences to get

$$y_{it} - y_{i,t-1}= (x_{it}-x_{i,t-1})\beta + u_{it} - u_{i,t-1}$$

is the same as

$$y_{it} - y_{i,t-1}= (x_{it}-x_{i,t-1})\beta + \epsilon_{it} - \epsilon_{i,t-1}$$

so in estimating this equation your error term is $ \epsilon_{it} - \epsilon_{i,t-1}$ and you therefore need $Cov(x_{it}-x_{i,t-1},\epsilon_{it} - \epsilon_{i,t-1})=0$ which is satisfied because $\epsilon_{it}$ is independent "of everything in the model" as you put it.

  • 1
    $\begingroup$ Thanks a lot for the explanation, this made the aspects very clear! $\endgroup$ – rbonac Dec 6 '18 at 19:46

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