Consider the panel model \(y_{i t}=\alpha+\beta x_{i t}+u_{i t}\), where \(\alpha\) and \(\beta\) are scalars.

(a) Show by appropriate subtraction that this model implies

\[ y_{i t}-\bar{y}=\beta\left(x_{i t}-\bar{x}_{i}\right)+\beta\left(\bar{x}_{i}-\bar{x}\right)+\left(u_{i t}-\bar{u}\right) \]

where \(\bar{y}=(N T)^{-1} \sum_{i, t} y_{i t}, \bar{x}=(N T)^{-1} \sum_{i, t} x_{i t}\) and \(\bar{x}_{i}=T^{-1} \sum_{t} x_{i t}\).

(b) For the corresponding unrestricted least-squares regression

\[ y_{i t}-\bar{y}=\beta_{1}\left(x_{i t}-\bar{x}_{i}\right)+\beta_{2}\left(\bar{x}_{i}-\bar{x}\right)+\left(u_{i t}-\bar{u}\right), \]

show that the least-squares estimator of \(\beta_{1}\) is the within estimator and that of \(\beta_{2}\) is the between estimator.

(c) Show that if \(u_{i t}=\mu_{i}+v_{i t}\), where \(\mu_{i} \sim i i d\left[0, \sigma_{\mu}^{2}\right]\) and \(v_{i t} \sim i i d\left[0, \sigma_{v}^{2}\right]\), and the two are mutually independent across both \(i\) and \(t\), the OLS and the GLS estimators are equivalent.