# Fixed Effects vs. Random Effects vs. First Differences

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

$$$$y_{it} = x_{it}\beta + \alpha_i + \epsilon_{it}$$$$

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?

• Is the purpose of taking logs solely so that linear regression can be used rather than using non-linear regression? – James Phillips Dec 6 '18 at 13:38
• 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 – rbonac Dec 6 '18 at 13:39
• 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.. – James Phillips Dec 6 '18 at 13:42
• 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. – 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.

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