Consider the model

\[y_{i t}=\beta_{1 i}+\beta_{2} x_{i t}+e_{i t}\]

a. Show that the fixed effects estimator for \(\beta_{2}\) can be written as

\[\hat{\beta}_{2, F E}=\frac{\sum_{i=1}^{N} \sum_{t=1}^{T}\left(x_{i t}-\bar{x}_{i}\right)\left(y_{i t}-\bar{y}_{i}\right)}{\sum_{i=1}^{N} \sum_{t=1}^{T}\left(x_{i t}-\bar{x}_{i}\right)^{2}}\]

b. Show that the random effects estimator for \(\beta_{2}\) can be written as

\[\hat{\beta}_{2, R E}=\frac{\sum_{i=1}^{N} \sum_{t=1}^{T}\left[x_{i t}-\hat{\alpha}\left(\bar{x}_{i}-\overline{\bar{x}}\right)-\overline{\bar{x}}\right]\left[y_{i t}-\hat{\alpha}\left(\bar{y}_{i}-\overline{\bar{y}}\right)-\overline{\bar{y}}\right]}{\sum_{i=1}^{N} \sum_{t=1}^{T}\left[x_{i t}-\hat{\alpha}\left(\bar{x}_{i}-\overline{\bar{x}}\right)-\overline{\bar{x}}\right]^{2}}\]

where \(\overline{\bar{y}}\) and \(\overline{\bar{x}}\) are the overall means.

c. Write down an expression for the pooled least squares estimator of \(\beta_{2}\). Discuss the differences between the three estimators.