Consider the fixed effects, two-way error component panel data model

\[ y_{i t}=\alpha+\mathbf{x}_{i t}^{\prime} \boldsymbol{\beta}+\boldsymbol{\mu}_{i}+\lambda_{t}+\varepsilon_{i t} \]

where \(\alpha\) is a scalar, \(\mathbf{x}_{i t}\) is a \(k \times 1\) vector of exogenous regressors, \(\beta\) is a \(K \times 1\) vector, \(\mu\) and \(\lambda\) denote fixed individual and time effects, respectively, and \(\varepsilon_{i t} \sim\) iid \(\left[0, \sigma^{2}\right]\).

(a) Show that the within estimator of \(\beta\), which is best linear unbiased, can be obtained by applying two within (one-way) transformations on this model. The first is the within transformation ignoring the time effects followed by the within transformation ignoring the individual effects.

(b) Show that the order of these two within (one-way) transformations is unimportant. Give an intuitive explanation for this result.