3
$\begingroup$

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?

$\endgroup$
4
  • $\begingroup$ Is the purpose of taking logs solely so that linear regression can be used rather than using non-linear regression? $\endgroup$ Commented Dec 6, 2018 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
    Commented Dec 6, 2018 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.. $\endgroup$ Commented Dec 6, 2018 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
    Commented Dec 6, 2018 at 13:53

1 Answer 1

7
$\begingroup$

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.

$\endgroup$
1
  • 1
    $\begingroup$ Thanks a lot for the explanation, this made the aspects very clear! $\endgroup$
    – rbonac
    Commented Dec 6, 2018 at 19:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.